Behaviour of Quantum Chromodynamics Near an Infrared Fixed Point

Galib O. M. Souadi Special Research Centre for the Subatomic Structure of Matter School of Chemistry and Physics

A Thesis Submitted for the Degree of Doctor of Philosophy April 2014 Abstract

In perturbation theory, the QCD running coupling depends on the renormalization scheme or is parameterised by a physical process. The problem is that artefacts of this ambiguity may upset physical conclusions outside the asymptotically free region, in particular near an infrared fixed point. Thus, a non-perturbative definition for the QCD running coupling is required that should be a monotonic and analytic function of the space-like energy scale Q2. The most physical coupling is Grunberg’s definition for the running 2 coupling as an effective charge αG(Q ). However, we find that it works only for sufficiently high energy scales. At some finite values of the energy scale, near the top of the resonance region, the 2 β-function associated with αG(Q ) has a false zero below which 2 αG(Q ) decreases. We test this conclusion further by applying chiral perturbation theory to the running coupling based on the method of effective charges. We consider the Drell-Yan ratio R(q2) and Adler- function D(Q2) in the time-like and space-like domains, respectively. These quantities tend to a finite value in the both infrared and ultraviolet limits: R(0) = D(0) = 0.5 for π±,K± and R(∞) = D(∞) = 2 for the light quarks. This means that the running coupling becomes negative in the infrared limit. There- fore, neither the time-like nor the space-like effective charges is a monotonic function over the whole energy scale. We also try to cancel the space-like pole of the proposed exact β-function, using the property of renormalization scheme dependence. Our modified exact β-function is non-singular for all finite values of the running coupling α and has an infrared fixed point even for a small number of quark flavours. Statement of Originality

I certify that this work contains no material which has been accepted for the award of any other degree or diploma in my name, in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission in my name, for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Ade- laide and where applicable, any partner institution responsible for the joint-award of this degree. I give consent to this copy of my thesis, when deposited in the Uni- versity Library, being made available for loan and photocopying, subject to the provisions of the Copyright Act 1968. I also give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library Search and also through web search en- gines, unless permission has been granted by the University to restrict access for a period of time.

Galib O. SOUADI List of workshop proceeding and future publication based on this thesis:

• G. O. Souadi, When is an Infrared Fixed Point of QCD Phys- ical?, poster presented at workshop ”Determination of the Fundamental Parameters of QCD”, Nanyang Technological University, Singapore, March 18-21, 2013. • R. J. Crewther and G. O. Souadi, When is a QCD Infrared Fixed Point Physical?, in preparation. Acknowledgements

The completion of any work is too hard without the contribution of other people. During my PhD, I had a support from many people, whose should have a great acknowledgement from me. First and foremost, I have to thank my supervisors for their support, time, great ideas to complete my PhD. In particular, I would like give a great thank to my principal supervisor Dr. Rod Crewther for his effort, ideas and guidance, wisdom and helpful comments throughout my PhD. The support from him stems from the beginning of my studying at the University of Adelaide for the degree of Master and then PhD. Also, thanks to him for the great comments and feedback during writing this thesis. Furthermore, I would like to thank my second supervisor Dr. Ross Young for useful comments. I thank all staff of CSSM and School of Chemistry and Physics for administering and providing technical support. I owe deep gratitude to the government of Saudi Arabia for the financial support to complete my PhD, in particular King Khalid University. Thanks to Saudi Arabia Culture Mission (SACM) in Australia for the excellent communication with my sponsor in Saudi Arabia. To Department of Physics at King Khalid Uni- versity for choosing me to complete my postgraduate education, Master and then PhD at the University of Adelaide. In particular, I am grateful to Profs. Abdulaziz AlShahrani , Ali AlHajry and Ali AlKamli for giving me this opportunity. I also would like to thank my friends at Adelaide University who are help me during this work. In particular, I thank Lewis Tun- stall for his helpful discussions and comments on my PhD project. Also, thanks to him for pointing me to many important papers related to my work. Further, I would like to thank my friend Ali Alkathiri for providing a technical help at some stages of writing my thesis. Also, thanks to other friends outside of physics commu- nity for their moral support. I also have to thank my nephew A. AlFraji and Elite Editing Centre for proofreading at some stages of my thesis. Last but not least, I would like to send a special thank to all members of my family for their encouragement to complete my postgraduate education, in particular my mother for her love and prayer. The biggest thank is to my wife for her love and support to complete my PhD. She shared me all problems that I had met during my work and always encouraged me to pass these problems. Thanks to her very much. To my children, Yazad, Ziyad, Ayan and Ellen, who are made my world so beautiful. Contents

Contents vi

List of Figures ix

List of Tables xiii

1 Introduction1

2 Quantum Chromodynamics7 2.1 QCD Lagrangian ...... 7 2.1.1 Gauge Invariance ...... 7 2.1.2 QCD Quantisation ...... 10 2.2 Renormalization Procedure ...... 12 2.3 Perturbative QCD ...... 15 2.3.1 The QCD β-Function ...... 15 2.3.2 Asymptotic Freedom ...... 16 2.3.3 Annihilation of e+e− into Hadrons ...... 20 2.4 Non-Perturbative QCD ...... 23 2.4.1 Eﬀective Field Theories ...... 23 2.4.2 Proposed Behaviour for the QCD β-function at low- energy scales ...... 25

3 Perturbative Running Coupling and Renormalization Scheme 28 3.1 Perturbative Running Coupling ...... 28 3.2 The Problem of Renormalization Scheme ...... 31 3.3 Proposed Solutions for the Problem ...... 33 3.3.1 The ’t Hooft Scheme ...... 34

vi CONTENTS

3.3.2 Eﬀective Charges Method ...... 37 3.3.2.1 The Gell-Mann-Low ψ-Function ...... 37 3.3.2.2 The Generalisation of the GML ψ-Function into QCD ...... 39 3.4 Physical Running Coupling ...... 41

4 Analysis of Perturbative and Non-Perturbative Infrared Fixed Point 43 4.1 Perturbative Infrared Fixed Point ...... 43 4.1.1 The Banks-Zaks Infrared Fixed Point ...... 44 4.1.2 The Eﬀect of Higher-Loop Orders ...... 46

4.1.3 Large Limits: Nf ,Nc → ∞ ...... 50 4.2 Non-Perturbative Infrared Fixed Point ...... 51 4.2.1 Schr¨odinger Functional Analysis ...... 52 4.2.2 Dyson-Schwinger Equations Analysis ...... 54 4.2.3 Anti-de Sitter Analysis ...... 58

5 The Infrared Behaviour of Proposed Exact β-Functions 62 5.1 Supersymmetric β-Function ...... 63 5.2 Ryttov-Sannino (RS) β-Function ...... 65 5.2.1 The Singularity of the βRS-Function ...... 65 5.2.2 Non-trivial Fixed Point ...... 67 5.2.3 Comparison of the Banks-Zaks and Ryttov-Sannino Fixed Points ...... 69 5.3 Modiﬁed βRS-Function ...... 72 5.3.1 Renormalization Gauge Transformation ...... 72 5.3.2 Infrared Fixed point ...... 73

6 Chiral Analysis of QCD at Low-Energy Scales 79 6.1 Chiral Perturbation Theory ...... 80 6.1.1 Chiral Symmetry ...... 80 6.1.2 Eﬀective Lagrangians for Strong Interactions ...... 82 6.1.2.1 Transformation Properties of the Goldstone Bosons ...... 82 6.1.2.2 The Lowest-Order Eﬀective Lagrangian . . . 84

vii CONTENTS

6.2 Electromagnetic Interactions ...... 85 6.2.1 External Fields ...... 85 6.2.2 Eﬀective Lagrangian for Electromagnetic Interactions . 87 6.3 The Physical Quantity R(q2)...... 90 6.4 QCD Eﬀective Charges ...... 93

7 Conclusion 96 7.1 Summary of Results ...... 96 7.2 What Next? ...... 98

A Feynman Diagrams 100

B Reference Formulaes 102 B.1 Gell-Mann Matrices ...... 102 B.2 Gamma Matrices ...... 103 B.3 Loop Integrals ...... 103 B.4 Dimensional Regularization ...... 104

C The β-Function and Anomalous Dimension 107 C.1 Perturbative Coeﬃcients ...... 107

C.2 The SU(Nc) Coloured Gauge Group ...... 110

D Proposed Exact β-Function at a Small α 112 D.1 Proof 1 ...... 112 D.2 Proof 2 ...... 113 D.3 Proof 3 ...... 115

E Loop Integrals 116 E.1 Meson-Loop Integral ...... 116 E.2 Quark-Loop Integral ...... 119

References 121

viii List of Figures

2.1 Contributions of fermions and all gauge boson interaction terms of order α to gluon propagator in QCD...... 15 2.2 In QCD, the point at the origin is an UV ﬁxed point, where β < 0. In this case, QCD may have an IR ﬁxed point (solid

line). For Nc → ∞, QCD does not have an IR ﬁxed point

(dashed line). As Nf → ∞, asymptotic freedom is lost and the ﬁxed point at the origin becomes an IR ﬁxed point (dash-

dotted line), it can be seen when Nf > 17 for Nc = 3...... 17 2.3 In QED, the β-function is a positive function near the origin, and hence the fixed point at the origin is an IR fixed point. When high-order corrections are added to the β-function, it may develop another fixed point (an UV fixed point) at a non- vanishing value for α...... 18 2.4 Diagrams contribute to annihilation of e+e−into hadrons in QCD. The first diagram gives the leading-order contribution and the others give the correction of order α...... 21 2.5 The possibilities of the QCD β-function in the IR region. . . . 26

3.1 Beyond the perturbative region, the running coupling α grows faster than the ’t Hooft running couplingα ˜ based on the ﬁrst three terms in Eq. (3.22)...... 36 3.2 In the ’t Hooft scheme, the original running coupling α becomes negative for some positive values ofα ˜ in the SU(3) gauge group, but beyond the perturbative region...... 36

ix LIST OF FIGURES

4.1 In the SU(3) gauge group, the BZ IR ﬁxed point is a decreasing min max function of Nf with Nf < Nf < Nf . It becomes more re-

liable in the framework of perturbation theory as Nf increases to 16...... 46 4.2 The shape of the AdS β-function in the scheme of the Bjorken sum rule is completely diﬀerent compared to the perturbative

QCD β-function with Nf = Nc = 3. Clearly, the AdS β- function is not compatible with perturbative QCD...... 60

RS 5.1 The behaviour of the β -function’s pole as a function of Nf

for ﬁnite values of Nc...... 67 5.2 Comparison between the value of the pole (solid line) and non-trivial ﬁxed point (dashed line) of the βRS-function at the

leading-order of the anomalous dimension as a function of Nf

for Nc =3...... 69 RS 5.3 For a small Nf , the β -function has a non-trivial UV fixed UV point αEx. (unphysical), while the perturbative β-function does not have any fixed point at two-loop order...... 70 IR 5.4 Comparison of the Banks-Zaks IR fixed point αBZ and the IR IR RS min max fixed point αEx. of the β -function for Nf < Nf < Nf , where both are renormalization scheme independent...... 71 IR IR 5.5 Comparison between the non-trivial fixed pointsα ˜Ex., αEx. of the modified (k = 1) and unmodified βRS-functions in the RS SU(3) gauge group with Nf = 16. Both β -functions have an IR fixed point at the same value, which is larger than the IR Banks-Zaks IR fixed point αBZ ...... 74 5.6 Comparison between the modified (k = 1) and βRS-functions ˜RS for the SU(3) gauge group with Nf = 6. In this case, the β - function is well defined, while the βRS-function is undefined for all 0 < α < αFP...... 75

x LIST OF FIGURES

5.7 At k = d, the β˜RS-function develops two non-trivial fixed points that could be an IR or UV fixed point. These fixed points

merge with each other as Nf increases from 17Nc/5 to 11Nc/2.

The special ﬁxed point is an UV ﬁxed point as Nf → 17Nc/5

(Figure A) and becomes an IR ﬁxed point as Nf → 11Nc/2 (Figure B)...... 77 ˜RS 5.8 At a particular value of Nf , the ﬁxed points of the special β -

functions merge with each other. This can be seen for Nf = 14 in the SU(3) gauge group to leading-order...... 78

6.1 The contribution of pseudoscalar mesons (π±,K±) to the photon vacuum polarisation in the IR limit...... 88 6.2 The contribution of the light quark fields (u, d, s) to the photon vacuum polarisation in the UV limit...... 89 6.3 Feynman diagram whose imaginary part gives the total cross section for e+e− annihilation to hadrons. In the IR limit, the imaginary part yields the mesons loop of π±,K± and the light quarks loop in the perturbative domain...... 91 6.4 As q2 decreases, the Drell-Yan ratio R(q2) associated with the time-like effective charge loses its monotonic property at the maximum of the first resonance...... 94 6.5 The Adler function D(Q2) associated with the space-like effective charge loses its monotonic property as well, but at an unknown value...... 94 6.6 The ψ-function has a false zero A and an IR limit at a negative value for the associated effective charge...... 95

A.1 The Feynman rules for gqq¯ vertex and quark propagator in QCD.100 A.2 The Feynman rules for γφ−φ+, γγφ−φ+ and γqq¯ vertices in χPT, including the rule of scalar propagator...... 101

B.1 Leading-order contribution of the quark loop to the gluon propagator...... 105

xi LIST OF FIGURES

E.1 Leading-order diagrams for mesons loop (π±,K±) in the IR limit, including momentum labels...... 117 E.2 Leading-order diagram for the light quark loop in the UV limit, including momentum labels...... 119

xii List of Tables

4.1 The values of the perturbative IR fixed point at two-, three-, and four-loop order for the SU(3) gauge group...... 48 4.2 The values of the perturbative IR fixed point at different loop

orders in large limits Nf ,Nc → ∞ with Nf /Nc → a ﬁnite value. 51

xiii Chapter 1

Introduction

1 In the Standard Model, fermions have spin- 2 and are divided into two groups: quarks and leptons. They interact with each other via electromagnetic, weak, and strong interactions. The strong interactions arise from interactions between quarks and intermediate gauge bosons (gluons) in a manner governed by the theory of Quantum Chromodynamics (QCD) [1]. QCD is based on the SU(3) coloured gauge group, which is a sub-group of the Standard Model SU(3) × SU(2) × U(1) gauge group. Excellent agreement between QCD and experiment, mostly at high energies, has led to QCD being generally accepted as the theory of strong interactions. The key property of QCD which allows this extensive comparison with experiment is asymptotic freedom [2,3], which relates phenomena at high- energy scales Q 1 GeV to calculations based on perturbation theory in the gluon constant g. The value of the renormalized gluon coupling α = g2/4π depends on the energy scale at which it is defined. Its variation with respect to that scale defines a ψ- or β-function [4,5,6] of the running coupling. In lowest-order perturbation theory, the QCD β-function is negative. As a result, the gluon coupling α “runs” to zero as the energy scale becomes large, i.e. there is an ultraviolet (UV) fixed point at the origin. The problem then is to understand QCD at other energy scales, where strong interactions are obviously not explained by a theory of weakly coupled gluons. Some general assumptions must be made about non-perturbative QCD. First, it is not known how to define QCD non-perturbatively without

1 INTRODUCTION introducing a regulator, such as a lattice spacing. Quarks and gluons are not seen as real particles, so it must be supposed that they are “confined”, i.e. gluonic forces bind them with sufficient strength to prevent them appearing in scattering states. More generally, no explicit remnants of colour SU(3) symmetry are seen experimentally: the absence of colour multiplet structure or Higgs-like mechanisms suggests that all physical states are manifest colour singlets, which is consistent with Fermi statistics for 3-colour quarks in baryons. It is also believed that the light-quark sector of QCD exhibits approximate chiral symmetry realised in the Nambu-Goldstone (NG) mode. These non-perturbative phenomena are presumably a consequence of the infrared (IR) properties of QCD [7,8]. As the energy scale is lowered, the running coupling α moves from zero to finite values at which perturbation theory is no longer valid. In the IR limit, there are two main possibilities of interest:

1. The β-function has no zeros for α > 0, and (to avoid space-like singularities in α) is bounded below by a linear function of α for large α. Then the running coupling α tends to inﬁnity, and the IR properties of amplitudes are controlled by their asymptotic behaviour for α ∼ ∞.

2. There is an IR ﬁxed point αIR, i.e. the running coupling tends to a ﬁnite value αIR at which the β-function vanishes. At αIR (if it exists), QCD becomes a theory of broken scale invariance realised in one of two ways in the limit of massless quarks:

(a) There is a “conformal window”. The vacuum is scale invariant, and Green’s functions are manifestly conformal covariant for all momenta, scaling with (in general) anomalous dimensions. Con- sequently, there is no mass gap. (b) Scale and conformal symmetry are hidden by the NG mechanism. Strong gluonic interactions produce scale condensates which require that there is a massless 0++ NG boson or “dilaton”. Non-NG particles can be all massive, with a spectrum of non-NG particles similar to what we observe for 0 < α < αIR (presumably). Green’s functions exhibit analytic conformal covariant behaviour only in

2 INTRODUCTION

an asymptotic sense: when all 4-momenta are large and space-like.

For case 2(a), it makes no sense to talk about confinement. Scale covariant Green’s functions for interpolating operators would be evidence against having particles in the normal sense and possibly for the existence of an “unparticle” spectrum [9]. In “walking” gauge theories [10], the running coupling comes close to a fixed point of this type. For either case 1 or case 2(b), where a particle spectrum seems to be allowed, it is reasonable to suppose that the confinement hypothesis is valid. Sometimes, it is argued that confinement implies case 1 alone. The argu- ment supposes that confinement is due to the exchange of a single dressed gluon with an IR running coupling. We see no reason to accept single gluon exchange, an implicitly perturbative idea, in a non-perturbative context. The ultimate aim of our work is to find ways of determining whether an IR fixed point is physical or not. Many definitions of the running coupling are effectively perturbative, so their extension to the non-perturbative regime may introduce artefacts, which are responsible for an IR fixed point seeming to be present or absent. If that is the case, conclusions about the IR limit cannot be trusted. Even if the running coupling is related to a physically observable non-perturbative process, as in the effective charge method of Grunberg [11], the problem remains: how do we know that the relation we set up is free of unphysical artefacts, and to what extent is this procedure independent of the process we choose? Can we find a way of dealing with all energy scales, from the UV perturbative region where we feel secure, through intermediate scales, and then on to the far IR region where the possibility of an IR fixed point can be investigated? The arbitrariness in definition of the running coupling is already obvious in perturbation theory. The running of the QCD coupling is caused by its dependence on the renormalization scale, which in turn is determined by the renormalization scheme used to define α itself. The physical consequences of having an IR fixed point have to be projected out of this arbitrariness. In other words, physical quantities should not depend on the renormalization scheme (RS). In the perturbative sector, the most interesting scheme is that due to ’t

3 INTRODUCTION

Hooft [12]. He made two key observations, which can be stated generally as follows:

(I) If the RS is mass independent, then the one- and two-loop terms of the QCD β-function do not depend on the scheme.

(II) All other orders of the β-function are entirely scheme dependent, i.e. all β-function coeﬃcients at three or more loops can be independently adjusted to any set of values by changing the renormalization prescription. ’t Hooft considered the simplest possibility where they are all set to zero, in which case the prescription is known as the ’t Hooft scheme.

Sometimes, it is said that this scheme is renormalization group invariant, but that is a misconception which arises from the first observation and ignores the second. Generally, setting an infinite set of coefficients to particular values, such as {0, 0, 0,...}, represents an extremely arbitrary choice of scheme. The exceptional case is where it can be argued that the region of interest is sufficiently perturbative that contributions beyond two loops do not matter, so dependence on the RS can be ignored. Scale condensates will not be present, and so this is an example of case 2(a) above. It was observed [13, 14] that the two-loop β-function can have an IR fixed point if the (integer) number of flavours Nf lies in the range 9 6 Nf 6 16 for the SU(3) gauge group. Such an IR fixed point could be physical if αIR is small enough for it to lie in the region where asymptotic freedom works. This is more likely to happen for

Nf close to the upper limit Nf = 16 allowed for asymptotic freedom. Case 2(a) is being studied on the lattice [10] to see if the conformal window can be extended to Nf values of 9 or smaller. In this thesis, our main interest is to find ways of distinguishing cases 1 and 2(b). In the literature, many discussions of this entirely non-perturbative problem are based on lattice techniques or the Dyson-Schwinger equations (DSEs). A pioneering study of pure Yang-Mills theory on the lattice [15] concluded that there is no IR fixed point and that the lattice running coupling (as defined in that work) tends to infinity in the low-energy limit. A similar conclusion was reported in a study of the DSE for the gluon propagator in Landau gauge with ghosts neglected [16], but when ghosts were

4 INTRODUCTION included in the DSEs analysis, the result was reversed [17]. Support for this last conclusion of DSEs came from a lattice study [18] of the gluon and ghost propagators in the Landau gauge. The situation becomes even more contra- dictory when fermions are included [19, 20] and quark condensation becomes another factor to be considered. It is not clear if the differing conclusions are the result of some approximations not working or of some definitions of the running coupling involving unphysical defects in the non-perturbative region. The literature also contains proposals to extrapolate an exact expression for β-functions to finite values of α on the basis of simple ansaetze which avoid obvious problems, such as Landau-Pomeranchuk poles, at space-like momenta. The most interesting proposal [21], which depends on an analogy with an explicit formula for the β-function in supersymmetric gauge theory, gives rise to a non-trivial fixed point αFP. A first step to deciding whether this proposal is physical or not is to check that it is consistent with analyticity for space-like momenta. We observe that the proposed β-function has a singularity at a point αpole. This does not matter if αpole lies outside the range 0 < α < αFP where physics is supposed to occur. However, there are cases where we find 0 < αpole < αFP and the prescription clearly breaks down. We show that the singularity can be removed by a change of renormalization prescription (necessarily singular) such that the new β-function and running coupling appear consistent with analyticity. A key objective of this thesis is to find one or more physical prescriptions for the QCD running coupling. Such prescriptions should depend on essential non-perturbative input, bear an analytic relation to each other, and be monotonic in the energy scale. The non-perturbative aspects of our approach are as follows:

• We consider Grunberg’s method of effective charges [11], where the running coupling α is defined in terms of a measurable amplitude or Green’s function. The definition is made such that, in the limit of asymptotic freedom, the leading gluonic correction is reproduced. The result is clearly “process dependent”, i.e. it depends on the choice of measurable amplitude.

• We decouple heavy quarks and use chiral perturbation theory to deter-

5 INTRODUCTION

mine the IR behaviour of the measurable amplitude for massless light quarks.

Since a physical amplitude is involved, the requirement of analyticity at space- like momenta is automatically satisfied for each process. The main difficulty is to show that the definition for a given process is monotonic in the energy scale. Only the subclass of monotonic analytic definitions can be considered sufficiently physical to permit a reliable conclusion about the existence of αIR to be drawn. We apply this approach to the effective charge associated with the Drell- Yan ratio R(q2) for the time-like momentum transfer q [11, 22] and extend it to the corresponding charge derived from the Adler function [23] D(Q2) for q → space-like Q. In the IR limit, for massless light quarks, chiral perturbation theory is dominated by the channels containing two charged NG bosons, i.e. π+π− and K+K−. A comparison of the results for the UV and IR limits shows that these effective charges cannot be monotonic. This thesis is structured as follows. The following chapter (Chapter2) introduces the basic background of QCD, including the main concepts of the perturbative QCD. It also introduces the principles of effective field theories, decoupling heavy quarks. In Chapter3, the problem of renormalization scheme dependence is discussed with some proposed solutions. In this chapter, we propose some conditions under which a definition for the running coupling can be regarded a physical definition. In Chapter4, we analyse the most known perturbative and non-perturbative IR fixed points. The major works of this thesis are presented in Chapters5 and6. In Chapter5, we modify the proposed exact β-function for non-supersymmetric QCD. In Chapter6, we use chiral perturbation theory to discuss the behaviour of QCD at low- energy scales. Finally, we draw our conclusion, including the main results of this thesis and directions for future work, in Chapter7.

6 Chapter 2

Quantum Chromodynamics

This chapter reviews the main features of QCD, particularly its behaviour at high-energy scales. The intent is to discuss the perturbative behaviour of the QCD β-function and proposals for its behaviour beyond perturbation theory. This chapter is organised as follows. In the first section (Section 2.1), the principles of QCD Lagrangian are introduced, beginning with the properties of QCD symmetry under both global and local gauge transformations. Section 2.2 reviews how the renormalization procedure is used to remove divergences of loop integrals, leading to the problem of renormalization scheme dependence. The main concepts of perturbative QCD are reviewed in Sec- tion 2.3. This section discusses the behaviour of the perturbative QCD β- function, including the asymptotic freedom property of QCD. Beyond perturbation theory, non-perturbative methods are required to understand QCD. Thus, Section 2.4 introduces the essential concepts of effective field theories, as useful tools to study QCD at high- and low-energy scales, including the possibilities of the QCD β-function in the IR domain.

2.1 QCD Lagrangian

2.1.1 Gauge Invariance

In general, QCD is gauge invariant under both global and local gauge transformations. In order to understand the properties of QCD symmetry, let us

7 2.1. QCD LAGRANGIAN

ﬁrst consider the free quark Lagrangian density

¯ µ Lquark = Ψ(iγ ∂µ − m)Ψ, (2.1) where γµ and Ψ are the gamma matrices and Dirac ﬁelds, respectively. The Lagrangian in Eq. (2.1) is gauge invariant under arbitrary global gauge transformations U(1) at which Dirac ﬁelds transform as

Ψ(x) → UΨ(x), (2.2)

where U is a Nc × Nc unitary matrix (U ∈ SU(Nc)) and its determinant equals one (det U = 1). This matrix is usually expressed as an exponential function λa U = exp −iφa , (2.3) 2 where φa are arbitrary parameters that are independent of the space-time coordinates. The λa matrices are the generators1 of the fundamental representation. They are traceless and satisfy the following relations

tr[λaλb] = 2δab, (2.4)

λa λb λc , = if abc , (2.5) 2 2 2 where f abc are the antisymmetric structure constants of the gauge group.2 We want a QCD Lagrangian which is invariant under local gauge transformation. Under local gauge transformations, the arbitrary parameter φa becomes a function of the space-time coordinates, and hence the unitary matrix in Eq. (2.3) becomes a function of x. Thus, the free quark Lagrangian transforms locally as

Lquark → Lquark + extra terms, (2.6)

1For the SU(3) gauge group, these generators are given by the Gell-Mann matrices, which are listed in AppendixB. 2These structure constants vanish in Abelian gauge ﬁeld theories, such as Quantum Electrodynamics (QED).

8 2.1. QCD LAGRANGIAN which is not gauge invariant under the local gauge transformation. This is because of the partial derivative ∂µ that acts on the unitary matrix U(x), producing extra terms. This problem is ﬁxed by introducing a new gauge ﬁeld 8 X λa A = Aa , (2.7) µ µ 2 a=1 which should transform under the local gauge transformation as

1 A (x) → U(x) A (x) − ∂ U †(x). (2.8) µ µ ig µ

Moreover, the partial derivative must be replaced by a covariant derivative

(∂µ → Dµ) with

Dµ = ∂µ − igAµ, (2.9) where it transforms locally as

DµΨ(x) → U(x)DµΨ(x). (2.10)

¯ µ Thus, quarks interact with gluons because the term Ψiγ ∂µΨ in Eq. (2.1) ¯ µ becomes Ψiγ DµΨ.

There must also be a term which depends on the spin-1 gauge ﬁeld Aµ alone, so that gluons have kinetic energy. Let Fµν be the ﬁeld strength tensor for Aµ

Fµν = ∂µAν − ∂νAµ − ig[Aµ,Aν] (2.11) with gauge covariant transformation law

† Fµν(x) → U(x)Fµν(x)U (x). (2.12)

Then the term 1 Lgluon = − Tr[F F µν] (2.13) 2 µν contains a kinetic term

1 h i Lgluon = − Tr (∂ A − ∂ A )(∂µAν − ∂νAµ) (2.14) kinetic 2 µ ν ν µ

9 2.1. QCD LAGRANGIAN

and also cubic and quartic interaction terms for the gauge ﬁeld Aµ

ig h i Lgluon = Tr (∂ A − ∂ A )[Aµ,Aν] + [A ,A ](∂µAν − ∂νAµ) (2.15) cubic 2 µ ν ν µ µ ν

g2 Lgluon = Tr[A ,A ][Aµ,Aν]. (2.16) quartic 2 µ ν Consequently, the locally gauge invariant Lagrangian of QCD takes the following form

1 Linv. = Ψ(¯ iγµD − m)Ψ − Tr [F µνF ] . (2.17) QCD µ 2 µν

This Lagrangian is gauge invariant under both global and local gauge transformations. It supports all known conservation symmetries of strong interactions, such as parity and charge conjugation. Moreover, it has all ﬂavour symmetries of a free quark model, since gluons are ﬂavours independent.

2.1.2 QCD Quantisation

a When QCD is quantised, unphysical degrees of freedom of Aµ become a problem because of the self-coupling of the gauge bosons. This problem is solved by adding new terms to the local QCD Lagrangian, the gauge ﬁxing term

LG.F and its corresponding ghost ﬁeld Lagrangian Lghost,

1 Lfull = Ψ(¯ iγµD − m)Ψ − TrF µνF + L + L . (2.18) QCD µ 2 µν G.F ghost

The general forms for the gauge ﬁxing and ghost terms are

ξ L = − [G(A )]2 , (2.19) G.F 2 µ and δG(A ) L = C¯b µ Cb, (2.20) ghost δρ where ξ, Cb and C¯b are the gauge parameter, ghost and anti-ghost ﬁelds, respectively. There are several choices for the gauge ﬁxing term; however, the physical results are independent of this choice. In a covariant gauge, the

10 2.1. QCD LAGRANGIAN well-known example of the gauge ﬁxing term is

1 L = − ∂µAa 2 , (2.21) G.F 2ξ µ where the corresponding ghost term is given by Faddeev-Popov [24]

¯a µ abc b c ¯a µ ac c Lghost = −C ∂ ∂µ + gf Aµ C = −C ∂ Dµ C . (2.22)

The gauge fixing term allows us to quantise QCD, but breaks the local gauge symmetry of the QCD Lagrangian in Eq. (2.17). However, the full Lagrangian has a remnant symmetry found by Becchi, Rouet, Stora and Tyutin (BRST symmetry) [25, 26]. It is formed by intro- a 1 µ a ducing a new commuting scalar field B = − ξ ∂ Aµ and then rewriting the full QCD Lagrangian in terms of this field as

1 ξ Lfull = Ψ(¯ iγµD − m)Ψ − TrF µνF + (Ba)2 + Ba∂µAa − C¯a∂µDacCc. QCD µ 2 µν 2 µ µ (2.23) The BRST symmetry requires that all ﬁelds, including in the full QCD La- grangian, should transform under a local inﬁnitesimal gauge transformation as

δΨ = igCaT aΨ a ac c δAµ = Dµ C 1 δCa = − gf abcCbCc 2 δC¯a = Ba δBa = 0. (2.24)

Under such a transformation, the ﬁrst three terms in the full QCD La- grangian are invariant since they are independent of the ghost ﬁeld. The fourth term in Eq. (2.23) is trivially invariant under the BRST symmetry because δBa = 0. The remaining two terms in Eq. (2.23) are invariant as follows

11 2.2. RENORMALIZATION PROCEDURE

a µ a ¯a µ ac c a µ a a µ a δ(B ∂ Aµ − C ∂ Dµ C ) = δB ∂ Aµ + B ∂ δAµ ¯a µ ac c ¯a µ ac c −δC ∂ Dµ C − C δ(∂ Dµ C ). (2.25)

Under the BRST symmetry Eq. (2.24), the ﬁrst term in Eq. (2.25) vanishes and the second two terms cancel each other . Finally, the last term in Eq. (2.25) also vanishes under the BRST symmetry as:

ac c c abc b c δ(Dµ C ) = δ ∂µC + gf AµC 1 1 = − gf cde∂ (CdCe) + gf abc(DbdCd)Cc − g2f abcf cdeAb CdCe 2 µ µ 2 µ 1 = − gf cde∂ (CdCe) + gf abc∂ (Cb)Cc 2 µ µ 1 − g2f abcf cdeAb CdCe + g2f abcf bdeAd CcCe. (2.26) 2 µ µ

The terms of order g cancel each other, and the terms of order g2 vanish by the property of Jacobi identity.1 It is now clear that the BRST symmetry is an invariance of the full QCD Lagrangian.

2.2 Renormalization Procedure

In perturbation theory, quantities are usually expressed as power expansions in the gauge coupling (α = g2/4π). Some coeﬃcients of these expansions are singular due to ultraviolet divergences of loop integrals. These singularities come from the integration of the quark, gluon and ghost loops over large values of momenta. Thus, it is important to cancel these divergences in order to end with a physical result. The standard method of canceling these divergences is to introduce a renormalization prescription. This prescription contains two important steps: regularization and renormalization. The ﬁrst step removes ultraviolet diver-

1In order for the Jacobi identity to be used, the terms of order g2 should be rewritten 1 2 abc cde b d e d e b e b d as − 2 g f f AµC C + AµC C + AµC C using the anti-commuting relation of the ghost ﬁeld. Now, we can apply the Jacobi identity through the ﬁnite structure constant as f adef bcd + f bdef cad + f cdef abd = 0.

12 2.2. RENORMALIZATION PROCEDURE gences, while the second step is used to specify finite parts of divergent terms and remove any dependence on the intermediate steps of regularization. The second step is usually done by redefining unnormalized quantities in terms of renormalized quantities, such as gauge coupling constant, mass and wave- functions. The most common technique of regularization is dimensional regularization [27, 28]. It is the only type of regularization technique that can be readily used to remove the loop divergences in non-Abelian gauge field theories.1 Moreover, dimensional regularization is the simplest technique that can be used to perform multi-loop calculations. In order to understand this technique, let us consider the gluon self-energy loop. Using the Feynman rules, one can write the contribution of the fermion loop in Fig. 2.1 to the gluon self-energy as

Z 4 " # µν 2 2 d k µ a i(6k + mf ) ν b i(6k+ 6q + mf ) Π (q ) = (−1)(ig) 4 Tr γ t 2 2 γ t 2 2 . (2π) (k − mf ) ((k + q) − mf ) (2.27) For large k, the loop integral is proportional to 1/k2, and hence it is diver- R d4k gent when one performs the integration over all momenta ( k2 ∼ ∞). The problem of this loop divergence can be solved as follows.2 Dimensional regularization respects gauge invariance, so the contribution of the fermion loop to the gluon self-energy takes the following form

Πµν(q2) = (gµνq2 − qµqν)iΠ(q2), (2.28) where Π(q2) is the scalar part (function) of the loop integral

Z 1 2−d/2 2 1 Γ(2 − d/2) 1 Π(q ) = −2πα dx(1 − x)x d/2 , (2.29) 0 (4π) Γ(2) ∆

1There are other types of regularization technique such as cut-off, Pauli-Villars [29] and analytical regularization techniques (see [30, 31] for more details) . However, they are difficult to apply to QCD because they break the gauge invariance of non-Abelian gauge field theories. 2Detailed calculations of how one can use dimensional regularization to remove the loop divergence can be viewed in textbooks, such as [32, 33, 34, 35]. We also summarise these calculations in AppendixB.

13 2.2. RENORMALIZATION PROCEDURE

2 2 where ∆ = mf − x(1 − x)q and Γ is a function of the dimension d. For a physical value of the dimension (d → 4), the ﬁnal result of the integration is divergent due to the pole of Γ at d = 4. This divergence can be easily seen, Γ(2−d/2) 1 2−d/2 using the Laurent series to expand the factor (4π)d/2 ( ∆ ) as

α Z 1 1 Π(q2) = − dx(1 − x)x − Log(∆) + Log(4π) − γ + O() , (2.30) π 0 where = 2 − d/2 and γ is Euler-Mascheroni constant. The scalar function Π(q2) has a dimension due to the ∆ term, while it should be dimensionless. In this case, it is important to introduce a dimensional parameter µ2, in order to have a dimensionless expression for the scalar function Π(q2)

α Z 1 1 ∆ 2 −2 O Π(q ) = − Nf µ dx(1 − x)x − Log 2 + Log(4π) − γ + () , π 0 µ (2.31) where Nf comes from the summation over all fermion fields Nf . The final result of the loop integral is still singular at d = 4 due to the 1/ term in Eq. (2.31). The simplest way to remove the remaining divergence is to subtract the 1/ term. This type of the renormalization scheme is called the minimal subtraction (MS) scheme [28]. Since the divergent term 1/ appears with other finite terms (log(4π) − γ), it is often convenient to eliminate all of these terms, using the modified minimal subtraction (MS) scheme [36]. Within the MS scheme, the divergent and finite terms can be combined to form the counterterm of fermion fields

α 1 δf = − N µ−2 + Log(4π) − γ + O() . (2.32) 3 6π f

In the same manner, the counterterm of the other gauge bosons in Fig. 2.1 reads 5α 1 δother = N µ−2 + Log(4π) − γ + O() , (2.33) 3 12π c where Nc is the number of colours. By Combining these two counterterm, we end with one counterterm, which can be used to cancel all divergences of

14 2.3. PERTURBATIVE QCD

Figure 2.1: Contributions of fermions and all gauge boson interaction terms of order α to gluon propagator in QCD. loop integrals in Fig. 2.1,

α 5 1 δ = Z − 1 = µ−2 N − N + Log(4π) − γ + O() , (2.34) 3 3 6π 2 c f where Z3 is the renormalization constant. These counterterms depend explicitly on the renormalization scale µ2 due to the factor µ−2 in Eq. (2.32). The dependence on µ2 seems to disappear in the limit of d → 4, but it is not really true due to the singular term 1/ at this limit. Thus, the dependence on the renormalization scale µ2 cannot be cancelled. This dependence gives rises to the problem of the renormalization scheme dependence in the perturbative QCD.

2.3 Perturbative QCD

The behaviour of QCD is well deﬁned at large-energy scales (Q 1 GeV), using a perturbative expansion for the running coupling. The success of the perturbative method is due to asymptotic freedom, which is the main feature of perturbative QCD, playing important roles in understanding QCD phenomenology at high-energy scales.

2.3.1 The QCD β-Function

In quantum field theory, QCD is a normalizable field theory, thus its coupling constant α is defined as a function of the renormalization scale µ2. In perturbation theory, the QCD β-function had been calculated up to the fourth-loop

15 2.3. PERTURBATIVE QCD order given by

dα(µ) β(α) = µ2 = −β α2 − β α3 − β α4 − β α5 + O(α6), (2.35) dµ2 1 2 3 4

1 where β1, β2, β3 and β4 are the coefficients of the one-, two-, three- and four-loop orders, respectively. It is important to note that only the first two coefficients of the QCD β-function are renormalization scheme independent.

Historical Overview

The first calculation of the QCD β-function was undertaken for the one- loop order by Gross and Wilczek [2] and separately by Politzer [3]. These calculations led to the discovery of asymptotic freedom. It was also discovered independently by t Hooft (not published) [37]. The two-loop contribution was calculated first for non-Abelian theories without fermions [38] and then with fermions [13]. The first calculation of the three-loop β-function was extracted [39] by calculating renormalization constants for the ghost-ghost-gluon vertex and corresponding inverse gluon and ghost propagators. This calculation was confirmed by calculating renormalization constants for the quark-quark-gluon vertex and corresponding inverse quark and gluon propagators [40]. The four- loop coefficient was obtained [41] by calculating renormalization constants for the ghost-ghost-gluon vertex and the inverse ghost and gluon propagators. It was also confirmed [42] by calculating different renormalization constants for the quark-gluon vertex and the inverse quark and gluon propagators. All these were calculated using the dimensional regularization within the MS scheme.

2.3.2 Asymptotic Freedom

In QCD, asymptotic freedom refers to a small running coupling at high-energy scales in which the running coupling approaches zero as Q → ∞. A theory is considered asymptotically free if its β-function satisﬁes two important conditions: 1The coeﬃcients of the QCD β-function are listed in AppendixC and will be recalled during this thesis when they are required.

16 2.3. PERTURBATIVE QCD

Figure 2.2: In QCD, the point at the origin is an UV fixed point, where β < 0. In this case, QCD may have an IR fixed point (solid line). For Nc → ∞, QCD does not have an IR fixed point (dashed line). As Nf → ∞, asymptotic freedom is lost and the fixed point at the origin becomes an IR fixed point (dash-dotted line), it can be seen when Nf > 17 for Nc = 3.

1. The β-function has an UV stable ﬁxed point at the origin with

β(α) < 0 for α & 0. (2.36)

2. The physical region corresponds to 0 < α < αmax, where αmax may be finite or infinite. In the case of finite value, the β-function has another zero (an IR fixed point).

Fixed Point

In gauge field theory, a fixed point refers to a point on the abscissa, where the β-function approaches zero for a finite value of the running coupling. We have to distinguish between two types of fixed points: an UV and IR stable fixed points. The difference between these points can be summarised as follows.

• If the β-function is a negative function and the running coupling decreases when increasing the energy scale, then the point at the origin

17 2.3. PERTURBATIVE QCD

Figure 2.3: In QED, the β-function is a positive function near the origin, and hence the fixed point at the origin is an IR fixed point. When high-order corrections are added to the β-function, it may develop another fixed point (an UV fixed point) at a non-vanishing value for α.

is an UV stable fixed point i.e. the theory is asymptotically free. How- ever, the β-function may develop an extra zero at some finite value of α, which in this case is an IR fixed point, as shown in Fig. 2.2.

• If the β-function has a positive value and the running coupling increases as the momenta scale decreases, then the point at the origin is an IR stable fixed point, as in QED, Fig. 2.3. In this case, the β-function may develop another fixed point (an UV fixed point), but current opinion [43] is that QED does not have an UV fixed point.

Flavour Dependence

The coeﬃcients of the QCD β-function are functions of the numbers of quark

flavours Nf and colours Nc. The value of the β-function, and hence the asymptotic freedom property, depends on these numbers. For a finite Nc, the QCD β-function should follow one of three different behaviours in the perturbative region, depending on the sign of these coefficients:

18 2.3. PERTURBATIVE QCD

1. All coeﬃcients are positive, then the QCD β-function is negative everywhere and theory is asymptotically free for α ∼ 0.

2. One or more coeﬃcients of the β-function changes its sign, but not the ﬁrst one. In this case, QCD is asymptotically free for 0 < α < αIR, see the solid line in Fig. 2.2.

3. When Nf → ∞, the first coefficient of the perturbative β-function becomes positive. In this situation, the β-function is positive and fixed point at the origin is an IR fixed point; hence, asymptotic freedom does not hold any more, as shown by the dash-dotted line in Fig. 2.2.

In the large Nc limit [44, 45], the dependence of asymptotic freedom on

Nc can be discussed by rewriting the QCD β-function as

2 3 4 β(αc) = −β1αc − β2αc + O(αc ) (2.37) with 1 Nf 11 β1 = 11 − 2 ≈ , (2.38) 12π Nc 12π 1 Nf Nf 1 β2 = 2 34 + Nf − 10 2 − ≈ 2 (34 + Nf ) , (2.39) 12(2π) Nc Nc 12(2π) where αc = Ncα keeps fixed as Nc → ∞. In this limit, all terms proportional to O(1/Nc) approach zero and the coefficients βn of the β-function are still positive for all Nf . Thus, the β-function is a negative function everywhere, whatever the number of quark flavours, and theory is asymptotically free for α ∼ 0. In this limit, the two-loop order of the QCD β-function cannot develop an IR fixed point, as shown by the dashed line in Fig. 2.2. It has been shown that only non-Abelian gauge field theories with no U(1) product groups can be asymptotically free [46, 47, 48]. Sometimes adding matter to a non-Abelian gauge field theory destroys asymptotic freedom, as has been shown for a large number of quark flavours, e.g. Nf > 17 for Nc = 3 (see Fig. 2.2). In contrast, all pure Yang-Mills theories involving just non- Abelian gauge fields are asymptotically free [33].

19 2.3. PERTURBATIVE QCD

2.3.3 Annihilation of e+e− into Hadrons

The standard example of QCD is the measurement of the total cross section for e+e− annihilation into hadrons. The simplest process of producing hadrons from e+e− interaction is the annihilation of e+e− into a quark and anti-quark pair, through a virtual photon,

e+e− → γ → qq.¯ (2.40)

The quark and anti-quark pair interact with each other, via strong interactions, to produce more quark and anti-quark pairs that combine to form hadrons. The cross section for quark and anti-quark annihilation to a lepton pair via an intermediate massive photon is easily obtained from the fundamental e+e− → µ+µ− cross section with some major changes [32]:

• The charge of µ must be replaced by Qf e.

• The eﬀect of the strong interactions should be included.

+ − • The ﬁnal result of µ µ production must be multiplied by Nc corresponding to the number of colours in QCD.

In perturbation theory, the eﬀect of strong interactions appears in the ﬁnal state of quarks due to the asymptotic freedom property of QCD. The total cross section for e+e− annihilation into hadrons is usually expressed in terms of the total cross section for e+e− → µ+µ− through the ratio R(s)

σ(e+e− → hadrons,s) R(s) = + − + − , (2.41) σ0(e e → µ µ , s)

+ − + − where σ0 is the QED total cross section for e e → µ µ

4πα σ (e+e− → µ+µ−, s) = QED (2.42) 0 3s with s = q2 (time-like scale) is the square of the centre of mass energy. The expression of the total cross section of e+e− → hadrons is the sum of two processes: e+e− → qq¯ and e+e− → gqq¯, as shown in Fig. 2.4[32]. Each process is divergent by itself (in the limit of massless gluon ﬁeld), but this

20 2.3. PERTURBATIVE QCD

Figure 2.4: Diagrams contribute to annihilation of e+e−into hadrons in QCD. The ﬁrst diagram gives the leading-order contribution and the others give the correction of order α. divergence is cancelled when both processes combine. Moreover, the total cross section of e+e− → hadrons depends on the coupling constant of QCD, which is usually deﬁned at some renormalization scales µ2. Hence, it can be 2 written as a function of s, µ and αs as follows

+ − 2 σ(e e → hadrons,s) = σ(s, µ , αs). (2.43)

In fact, the total cross section σ is an observable quantity that should be independent of any conventions. It implies the Callan-Symanzik equation [5,6] 2 ∂ ∂ 2 µ 2 + β(αs) σ(s, µ , αs) = 0 (2.44) ∂µ ∂αs with no anomalous dimension (γ = 0). In the strong interaction process, there is a leading-order contribution to the cross section, which depends on the renormalized coupling. Therefore, the ﬁxed coupling must be replaced by the running coupling in order to satisfy the Callan-Symanzik equation.

The Adler Function

The total cross section of e+e− annihilation into hadrons is related to the renormalized hadronic vacuum polarisation Π(q2),

iΠµν(q) = (qµqν − q2gµν)Π(q2), (2.45)

21 2.3. PERTURBATIVE QCD through the imaginary part of the vacuum polarisation as

4πα σ(e+e− → hadrons, s) = QED ImΠ(s), (2.46) s where s > 0. The ﬁrst derivative of the renormalized hadronic vacuum polarisation can be directly related to its imaginary part via the dispersion relation

dΠ(q2) Z ∞ ImΠ(s) π 2 = ds 2 2 . (2.47) dq 0 (s − q )

This dispersion relation is also expressed in terms of the Drell-Yan ratio R(s), introducing the Adler function [23]

Z ∞ 2 2 R(s) D(q ) = −q ds 2 2 . (2.48) 0 (s − q )

The Adler function plays an important role in describing strong interaction processes, such as the running coupling [49] and hadronic τ-decay [50, 51, 52, 53] at high- and low-energy scales, respectively. In the space-like region (Q2 = −q2), the Adler function reads

Z ∞ 2 2 R(s) D(Q ) = Q ds 2 2 , (2.49) 0 (s + Q ) where Q2 is the space-like energy scale. For large-energy scales, the Adler function can be expressed as a power series in the space-like running coupling 2 αs(Q ). In the MS scheme, this expression is given by

" 2 n # X αs(Q ) X D(Q2) ∼ 3 Q2 × 1 + + c α1+i(Q2) , (2.50) f π i s f i=1

1 where ci are constants, depending on the number of active quark ﬂavours

Nf . In addition, the Drell-Yan ratio R(s) is also expressed perturbatively as

1More details about these coeﬃcients can be found in [54, 55, 56, 57, 58, 59].

22 2.4. NON-PERTURBATIVE QCD

a power series, but in the time-like running coupling αs(s), as

" n # X αs(s) X R(s) ∼ 3 Q2 × 1 + + d α1+i(s) , (2.51) f π i s f i=1 where c1 = d1 and other coefficients are different compared to the coefficients of the Adler function.1 The difference starts at the fourth order, which is due to the analytic continuation from the Euclidean to the Minkowski regions. Finally, it is important to note that the asymptotic series in Eq. (2.50) and (2.51) refer to the UV limit and so cannot be applied to the low-energy limit.

2.4 Non-Perturbative QCD

Below a few GeV, non-perturbative methods are required to study QCD. Lat- tice QCD2 [66] and Dyson-Schwinger equations [67, 68, 69] are the standard methods used to find a non-perturbative definition for the strong coupling (see Chapter4). In this thesis, we propose using the approximate chiral symmetry, as a non-perturbative method to investigate the behaviour of QCD at low-energy scales (see Chapter6). In order to do that, we must first decouple the heavy quarks (t, b, c) through a decoupling theorem, as a type of effective field theories.

2.4.1 Eﬀective Field Theories

Effective field theories are theoretical approaches to develop a theory with multiple energy scales Qi. They can be used to describe theory at both high- and low-energy scales. There are different types of effective field theories depending on the relevant degrees of freedom. Based on the structure of the transition from high- to low-energy scales, there are two types of effective field theories [70]: decoupling and non-decoupling theory. At high-energy scales, the relevant degrees of freedom are quarks and gluons. According to the decoupling theorem [71], the only relevant degrees of

1The relationships between the coeﬃcients of the Adler function D(Q2) and ratio R(s) can be found in [59, 60] and references therein. 2Detailed discussions of lattice QCD can be viewed in [61, 62, 63, 64, 65].

23 2.4. NON-PERTURBATIVE QCD freedom are the light quarks and gluons, where the heavy quarks are inte- grated out. In this case, the eﬀective QCD Lagrangian reads

1 Leff =q ¯(iγµD − m )q − TrF µνF , (2.52) QCD µ q 2 µν where q refers to the light quarks field (u, d, s). The effective QCD Lagrangian is still invariant under parity and time reversal transformations. It contains four physical parameters: the coupling constant (α) of the strong interaction and masses of the light quarks (mu, md, ms). However, the masses of the light quarks can be neglected within the chiral limit,

mq = mu = md = ms = 0, (2.53) ending with one dimensionless parameter, the strong running coupling. At low-energy scales, the relevant degrees of freedom are hadrons. The transition from the high-energy to low-energy region occurs through the chiral symmetry breaking. The related effective Lagrangian can be non- renormalizable with physical consequences, independent of the way in which fields are introduced. An example of this type of effective field theory is chiral perturbation theory, which is our method to study QCD at low-energy scales in this thesis. In chiral perturbation theory, the relevant degrees of freedom are pseudoscalar mesons. The lowest-order effective chiral Lagrangian for meson sector will be introduced in Chapter6, including the spontaneous breaking of chiral symmetry. In order to build an effective field theory, some principles should be considered [72, 73, 74]:

• Only a ﬁnite set of parameter k is necessary to describe the physics of a theory at an energy scale Q to an accuracy scale [72, 73]:

Q k ln(1/) ≈ ⇒ k ≈ , (2.54) M ln(M/Q)

where M is some mass with Q < M.

• The physics of a theory at low-energy scales bears little resemblance to the physics of the high-energy region. The eﬀect of the physics at high-

24 2.4. NON-PERTURBATIVE QCD

energy scales is replaced by a tower of non-renormalizable interactions. The eﬀects of the high- and intermediate-energy physics appear in the values of the low-energy parameters that are either computable or can be estimated in an experiment.

• If there is a large gap between an intermediate energy scale Q and a quark mass scale, the mass scale should go to zero for the light quarks (chiral perturbation theory) and to inﬁnity for the heavy quarks (decoupling theorem)

Q mq ⇒ mq → 0, q = u, d, s

Q Mq ⇒ Mq → ∞ q = t, b, c. (2.55)

• Only hadrons are incorporated as the degrees of freedom when one obtains an eﬀective ﬁeld theory for QCD at low-energy scales. Here, low-energy scales refer to a small scale compared to a typical interaction

scale Λ. In QCD, low-energy scales refer to energies below scale Λχ ∼ 1 GeV set by the quark condensate.1

2.4.2 Proposed Behaviour for the QCD β-function at low-energy scales

As we have noticed above, non-perturbative methods are required to investigate the behaviour of QCD at low-energy scales. Within the framework of these methods, the QCD β-function should follow one of two main possibilities in the IR limit. These possibilities can be summarised as follows:

1. The QCD β-function approaches a zero away from the origin (an IR ﬁxed point), as shown by the dash-dotted line in Fig. 2.5. In this case, the running coupling tends to a ﬁnite value as the renormalization scale decreases to zero,

α → αIR, β(α) → 0 as µ2 → 0. (2.56)

1 Distinguish the “chiral scale” Λχ for low-energy expansions from the “QCD scale” ΛQCD ∼ 200 MeV for high-energy expansions. Both are scheme independent.

25 2.4. NON-PERTURBATIVE QCD

Figure 2.5: The possibilities of the QCD β-function in the IR region.

Note that the existence of an IR fixed point should not be affected by the renormalization scheme that is used to renormalize theory. How- ever, the value of this point may change under a change of renormalization prescription since we deal with a renormalizable field theory. In this possibility, the QCD β-function is negative between its zeros and positive after the non-trivial zero,

β(α) < 0 for 0 < α < αIR β(α) > 0 for α > αIR, (2.57)

where the integral

Z α dx ∼ −∞, α + αIR (2.58) β(x)

diverges at this IR ﬁxed point because β(αIR) = 0.

2. There is no IR ﬁxed point at which the running coupling increases as µ2 decreases. In this case, we have to distinguish between two diﬀerent possibilities:

26 2.4. NON-PERTURBATIVE QCD

(a) The running coupling grows to inﬁnity as the renormalization scale approaches zero. Then, the QCD β-function is bounded below by a linear negative function,

2 α → ∞, β(α) & −(constant) α as µ → 0, (2.59)

as shown by the dashed line in Fig. 2.5. In this possibility, the integral Z α dx → −∞, α → ∞. (2.60) β(x) (b) The running coupling approaches an inﬁnite value for a ﬁnite scale µ2, then the QCD β-function runs faster to −∞,

2 2 α → ∞, α β(α) → −∞ as µ → ﬁnite µ0, (2.61)

as shown by the dotted line in Fig. 2.5. It is important to note that no physical conclusion can be reached from this behaviour since it corresponds to unphysical poles1 in the space-like region. In this case, the integral

µ2 Z α dx ln 2 = (2.62) µ 2 β(x) 0 α(µ0)

2 2 is converges at µ → µ0.

1Some examples of these poles are discussed in [75].

27 Chapter 3

Perturbative Running Coupling and Renormalization Scheme

In general, the QCD β-function is renormalization scheme (RS) dependent to all loop orders. This dependence leads to introduce the problem of RS [76, 11, 77, 78, 79]. The aim of this chapter is to review the most known solutions to this problem. The discussion begins by introducing the main concepts of the perturbative running coupling in Section 3.1. In this section, we discuss the problem of a Landau singularity at low-energy scales and then introduce the time-like running coupling. In Section 3.2, the dependence of the β-function on the RS is shown by considering various examples of RS. Section 3.3 shows a selection of solutions for the problem of RS, in particular the ’t Hooft scheme [12] and eﬀective charges method [11, 22, 80]. At the end of this chapter, we propose some conditions for the physical running coupling in Section 3.4.

3.1 Perturbative Running Coupling

In QCD, there is a single coupling constant that depends on the renormalization scale µ2. The variation of this coupling as a function of µ2 is known as the running (eﬀective) coupling α(µ2). It can be evaluated by integrating

28 3.1. PERTURBATIVE RUNNING COUPLING the diﬀerential equation Eq. (2.35)

2 2 Z µ dµ2 Z α(µ ) dα 2 = . (3.1) 2 µ 2 β(α) µ0 α(µ0)

An approximated solution for this integrable equation is often proposed by substituting the perturbative deﬁnition of the β-function. It takes the form

2 (1) 2 α(µ0) α (µ ) ≈ 2 2 2 (3.2) 1 + β1α(µ0) ln(µ /µ0) for the one-loop order. Eq. (3.2) shows explicitly the dependence of the running coupling on the renormalization scale µ2. The dependence on the 2 initial renormalization scale µ0 can be removed by rewriting the one-loop running coupling in terms of a scale, such as ΛQCD, as follows

(1) 2 1 α (µ ) ≈ 2 2 , (3.3) β1 ln(µ /ΛQCD) where 2 2 1 µ0 ≈ ΛQCD exp 2 . (3.4) β1α(µ0) These approximations for the perturbative running coupling are accurate only for high-energy scales Q ΛQCD. Contradictions become evident if 2 2 this rule is not observed. For example, at µ = ΛQCD, the right-hand side of Eq. (3.3) has a space-like pole at a ﬁnite value of the renormalization scale,

2 2 2 α(µ ) → ∞ as µ → ΛQCD. (3.5)

This problem is known as a Landau singularity, which leads to unphysical 2 2 properties for the QCD running coupling. For µ < ΛQCD, the value of the running coupling becomes negative, which is even less acceptable.

Higher-Loops Eﬀect

In general, the Landau singularity is still present when higher-loop orders are included. These orders can aﬀect the structure of the Landau singularity. For

29 3.1. PERTURBATIVE RUNNING COUPLING example, the approximated two-loop deﬁnition for the running coupling1 is given by the asymptotic formula

" 2 2 # 1 β2 ln ln(µ /Λ ) α(2)(µ2) ≈ 1 − QCD . (3.6) µ2 β2 ln µ2/Λ2 β1 ln 2 1 QCD ΛQCD

Again, the rule is that this works only for large scales compared to ΛQCD. Breaking this rule produces even worse results. For example, the right-hand 2 2 side of Eq. (3.6) has a double-pole Landau singularity at µ = ΛQCD. This 2 2 double pole can be seen by expanding the ln µ /ΛQCD factor in Eq. (3.6):

2 2 β2 ln(µ /Λ ) α(2)(µ2) ≈ − QCD . (3.7) 2 2 2 2 β1 µ /ΛQCD − 1

Time-like Running Coupling

For the time-like region, a perturbative deﬁnition for the running coupling was proposed in [81, 82]. The key point of getting such a deﬁnition is to start with the inverse of the Adler function Eq. (2.49)

Z s+i 2 i dQ 2 R(s) = 2 D(Q ). (3.8) 2π s−i Q and then approximate D(Q2) and R(s) by the two-loop part of Eq. (2.50) and (2.51), respectively. In this approximation, the one-loop time-like running coupling reads2

2 (1) 1 1 1 ln(s/Λ ) αtime(s) = − arctan . (3.9) β1 2 π π

1Detailed calculations of the two- and higher-loop corrections to the QCD running coupling can be found in recent reviews, see e.g. [60]. 2It is derived by applying the one-loop space-like (perturbative) running coupling in Eq. (3.3) into the inverse of the Adler function Eq. (3.8), and then form the integration 2 2 over Q [81, 82]. The integration goes below the real axis for all s − i 6 Q 6 0 and above 2 the real axis for all 0 < Q 6 s + i (see [83] for more details).

30 3.2. THE PROBLEM OF RENORMALIZATION SCHEME

It has been suggested [82] that the time-like running coupling is free from IR singularities, which appear in the space-like region for low-energy scales,

( 2 (1) 1/2β1 s → Λ not singular ? αtime(s) = (3.10) 1/β1 s → 0 IR ﬁxed point ? but this is not signiﬁcant because Eq. (3.9) is valid only in the UV limit s ∼ ∞. Moreover, the time-like running coupling Eq. (3.9) fails when it is applied to the space-like region (s → −Q2 < 0) because it does not vanish on the negative real axis as it showed [60]. Clearly, neither two-loop expressions for R(s) and D(Q2) nor the one-loop expression for α(s) or α(Q2) work outside the asymptotically free region. In order to get reasonable results at low-energy scales, the Drell-Yan ratio R(s) and Adler function D(Q2) should be treated non-perturbatively (see Chapter6).

3.2 The Problem of Renormalization Scheme

As stated above, the running coupling depends on the RS that is used to cancel divergences of theory. This dependence implies that an arbitrary change in the RS leads to a redefinition of the running coupling. The redefinition does not affect the first two orders of the β-function, where the effect occurs at third- and higher-loop orders. To get a clear view, let us assume that α transforms under a change of RS to a new running couplingα ˜ where α begins a function ofα ˜, α = F (˜α) as (α, µ2) → (˜α, µ˜2). (3.11)

Since the QCD β-function is a function of the running coupling, it should be redeﬁned as follows ∂α˜ β (F (˜α)) β˜(˜α) =µ ˜ = (3.12) ∂µ˜ F 0(˜α) with ∂α F 0(˜α) = . (3.13) ∂α˜ It is important to note that any scheme transformation must satisfy some conditions in order to be physically acceptable. These conditions can be

31 3.2. THE PROBLEM OF RENORMALIZATION SCHEME summarised as follows [84]:

1. It should not transform a small value of α to a large value ofα ˜, in order to be acceptable in perturbation theory. In other words, α andα ˜ should be equal α ≈ α˜ in the deep UV region.

2. It must not create any singularity beyond the perturbative region; both α andα ˜ must stay finite, unlessα ˜ tends to a finite valueα ˜IR in the IR limit. In this case, the old running coupling α may become only finite IR for 0 6 α˜ < α˜ and singular at the IR fixed point,

α → ∞ asα ˜ → α˜IR. (3.14)

3. It must transform a real positive value of α to a real positive value of α˜ in both perturbative and non-perturbative regions.

4. The function in Eq. (3.13) must be a non-vanishing and non-singular function for all values ofα ˜ in order to avoid an unphysical value for the β˜-function

F 0(˜α) 6= 0, ∞ for allα ˜ ∈ (0, α˜IR or ∞). (3.15)

In order to examine the eﬀect of RS on the coeﬃcients of the QCD β- function, let us consider some examples of scheme transformations,1 starting with the standard example

n X a+1 α = F (˜α) =α ˜ + caα˜ , (3.16) a=1 where ca are constants and a is an integer number. Under this transformation, the β˜(˜α)-function reads

˜ ˜ 2 ˜ 3 ˜ 4 ˜ 5 6 β(˜α) = −β1α˜ − β2α˜ − β3α˜ − β4α˜ + O(˜α ), (3.17)

1In this thesis, we only consider two examples of scheme transformations, Eq. (3.16) and (3.19). More examples can be viewed in [84, 85, 86].

32 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM with

˜ β1 = β1, ˜ β2 = β2, ˜ 2 β3 = β3 + c1β2 + (c1 − c2)β1, ˜ 2 3 β4 = β4 + 2c1β3 + c1β2 + (−2c1 + 4c1c2 − 2c3)β1. (3.18)

It is also possible to deﬁne a scheme that can keep some higher coeﬃcients invariant. For instance, let us assume the following transformation

n X i α = F (˜α) = diα˜ , (3.19) i=1

˜ with i = 1, 3, 5, ... and d1 = 1. In this case, the coeﬃcients of β-function read

˜ ˜ ˜ ˜ β1 = β1, β2 = β2, β3 = β3 − d3β1, β4 = β4. (3.20)

It is clear from both transformations in Eq. (3.16) and (3.19) that only the one- and two-loop orders of the β˜-function are universal and the dependence on the RS enters at the third-loop order. For the transformation in Eq. (3.19), the fourth-loop order is universal; however, this does not matter because the final result is still scheme dependent due the correction of the third-loop order. This problem is known as the problem of RS dependence. The problem of RS dependence affects conclusions about the physical behaviour of QCD when the correction of higher-loops is taken into an account. The next section presents some interesting methods that have been proposed to extract a physical definition for the running coupling, and then obtain a RS independent QCD β-function.

3.3 Proposed Solutions for the Problem

Several methods have been considered to solve the problem of RS dependence, and obtain reliable results (physical parameters). One of these methods is the ’t Hooft scheme [12] in which the QCD β-function includes only one- and

33 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM two-loop orders. Grunberg [11, 22, 80] suggested that the running coupling should be defined as the gluonic correction to a physical process, so that any quantity extracted via this method is physical. This is known as the method of effective charges; it produces a QCD generalisation of the Gell- Mann-Low (GML) ψ-function for QED [4]. So instead of a scheme dependent β-function, we have a process dependent ψ-function. Subsequently, Brodsky, Lepage and Mackenzie (BLM) [78, 87, 88] suggested that the renormalization scale be adjusted iteratively to maximise convergence of the perturbative series. Also, Stevenson proposed the method of the principle of minimum sensitivity (PMS) [89, 90, 91, 92]. This method1 states that if there is an approximation depending on some unphysical parameters, then the value of these parameters should be chosen to be much less sensitive to a small change in the RS. The purpose of this section is to discuss the two methods which have attracted the most interest in the literature: the ’t Hooft scheme and effective charge method.

3.3.1 The ’t Hooft Scheme

The ’t Hooft scheme [12] uses the fact that it is possible to deﬁne a scheme making the third- and higher-loop orders of the QCD β-function vanish, re- sulting with just the scheme independent one- and two-loop terms. The ’t Hooft scheme can be established by setting the coeﬃcients of the transformation in Eq. (3.16) into

β3 β4 c1 = 0, c2 = , c3 = . (3.21) β1 2β1

In this case, the old running coupling α is related to the ’t Hooft running couplingα ˜ as follows:

β β α =α ˜ + 3 α˜3 + 4 α˜4 + O(˜α5). (3.22) β1 2β1

1This method will not be discussed in this thesis because it still gives a small correction that depends on the RS, while we are looking for a totally RS independent result. However, it has been used to investigate the QCD coupling constant at low-energy scales in many studies, such as [93, 94, 95, 96].

34 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM

It is now clear how the third and fourth terms in Eq. (3.17) vanish under this scheme. In the same manner, the higher order terms of the β˜-function vanish, leaving us with an explicit β˜-function,

˜ ˜ 2 ˜ 3 β(˜α) = −β1α˜ − β2α˜ . (3.23)

Note that the ’t Hooft scheme is a very special scheme where all higher ˜ ˜ coefficients β3, β4 and so on have been set equal to zero. So, the fact that ˜ ˜ only the scheme independent coefficients β1 and β2 appear does not imply that ’t Hooft’s scheme is less dependent on the renormalization prescription or more physical than other schemes. The transformation in Eq. (3.22) must satisfy the conditions of a physical RS transformation to be physically acceptable. These conditions are listed in Section 3.2 and can be tested on the ’t Hooft running couplingα ˜, assuming the old running coupling α is defined within the MS scheme, as follows:

• The ﬁrst and second conditions require that ’t Hooft’s running coupling must be non-singular in all regions of interest. In the perturbative region, there is no problem with these conditions since the running coupling is small, such that higher-order terms in Eq. (3.22) can be ignored: α ≈ α˜. The problem appears beyond the perturbative region since α grows faster thanα ˜ (α α˜), and hence may be singular for a ﬁnite value ofα ˜ [97], as shown in Fig. 3.1. In this case, we have to examine the last two conditions of the physical RS transformation in the framework of perturbation theory.

• In perturbation theory, let us set the higher terms O(˜α5) in Eq. (3.22) to zero and consider just the ﬁrst three terms. The third condition of the physical RS transformation requires that each real positiveα ˜ should map into a real positive α. Assumingα ˜ is real and positive, then the sign of α depends on the coeﬃcients of the relationship betweenα ˜ and α

Eq. (3.22), which are determined by β1, β3 and β4. Asymptotic freedom max 1 requires that β1 is positive for all Nf < Nf , where

1 max Nf refers to a value at which the ﬁrst coeﬃcient of the β-function vanishes β1 = 0 max and then becomes negative for Nf > Nf .

35 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM

Figure 3.1: Beyond the perturbative region, the running coupling α grows faster than the ’t Hooft running couplingα ˜ based on the ﬁrst three terms in Eq. (3.22).

Figure 3.2: In the ’t Hooft scheme, the original running coupling α becomes negative for some positive values ofα ˜ in the SU(3) gauge group, but beyond the perturbative region.

36 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM

11 N max = N (3.24) f 2 c

max for the SU(Nc) gauge group. In the MS scheme, β4 > 0 for Nf < Nf , max thus the sign of α is only controlled by the sign of β3. For Nf < Nf ,

β3 could be positive or negative, depending on the number of quark 1 2 ﬂavours. For a small Nf , β3 > 0 and hence each positiveα ˜ maps to

a real and positive α in the perturbative region. In contrast, when Nf max increases to its maximum value (Nf → Nf ), β3 becomes negative and then a real positiveα ˜ will map to a real negative α, but beyond the perturbative region, as shown in Fig. 3.2.

• The remaining fourth condition is satisfied by the ’t Hooft scheme, since the coefficient of the first term in Eq. (3.22) is fixed to one, and the other coefficients should be positive in the perturbative region for max all Nf < Nf , ∂α = 1 + positive terms. (3.25) ∂α˜ Thus, we conclude that the ’t Hooft scheme is valid only in the perturbative region.

3.3.2 Eﬀective Charges Method

It is clear from the last section that the ’t Hooft method is limited to the perturbative region, and hence new solutions are required. Over many studies, it was found that the best way to solve the problem of RS to all orders is the method of effective charges [11, 22, 80]. The idea is to define the QCD running coupling in terms of a physical quantity. This approach was suggested by the GML analysis of QED [4] in which the QED effective charge was related to the physical quantity hJµ,Jνi, where Jµ is the electromagnetic current density.

3.3.2.1 The Gell-Mann-Low ψ-Function

The GML ψ-function describes the variation of the QED eﬀective charge

αeﬀ as a function of the space-like momentum. First, we must discuss the

1 We have assumed that the number of colours Nc is ﬁxed to a ﬁnite value. 2 In the SU(3) gauge group, β3 > 0 for all Nf < 6.

37 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM deﬁnition and properties of this eﬀective charge. In QED, the sum over one- particle-irreducible (1PI) amplitudes can be written as

2 2 Πµν(q, α) = i(gµνq − qµqν)Π(q , α), (3.26) where Π(q2) is the scalar part of the photon self-energy that depends on a 2 scale given by the momentum squared q . The QED eﬀective charge αQED was deﬁned by relating it to the scalar part of the photon self-energy as follows

α α (q2) = , (3.27) QED 1 − Π(q2, α) where α is the renormalized coupling constant for QED. This effective charge can be determined in experiments by measuring the effective potential between two charges. The definition of the effective charge in Eq. (3.27) has some important properties that can be summarised in two points [4]:

• The QED eﬀective charge is RS independent to all orders since the

photon self-energy contribution is governed by a physical operator Jµ, unlike the QCD running coupling.

• The on-shell renormalization condition Π(0, α) = 0 deﬁnes the ﬁne structure constant of QED in the IR limit q2 → 0,

1 α (0) = α = constant ≈ . (3.28) QED 137

In QED, Gell Mann and Low [4] used the definition of the effective charge in Eq. (3.27) to obtain a function, describing a change in the QED running coupling as a function of the momentum scale q2. The GML ψ-function is given by the derivative of the QED effective charge with respect to q2,

∂α (q2) ψ(α ) = QED QED ∂lnq2 2 3 4 5 = ψ1αQED + ψ2αQED + ψ3αQED + O(αQED), (3.29) where ψi are positive coeﬃcients. Based on the above properties for the QED

38 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM effective charge, the GML ψ-function is RS independent to all orders (unlike the QCD β-function). The first two terms of the GML ψ-function are the only process independent terms. In fact, if the idea of the QED effective charge can be applied to QCD, then one can extract a RS independent β/ψ-function for QCD to all orders. Then if this β/ψ-function has an IR fixed point, it would be more acceptable as a physical IR fixed point.

3.3.2.2 The Generalisation of the GML ψ-Function into QCD

In order to apply the eﬀective charge method to QCD, it is necessary to identify a suitable physical quantity. The best known example is the ratio of the total cross section of e+e− annihilation into hadrons,

+ − 2 σ(e e → hadrons) R(Q ) = + − + − . (3.30) σ0(e e → µ µ )

In general, let us consider any physical quantity σ(Q2) which depends only on a single scale Q2. The quantity σ(Q2) is renormalization group invariant, but it can be expanded in terms of a renormalized coupling constant [22] as

2 2 d 2 2 2 2 2 2 2 σ(Q ) = a + b[α(µ )] [1 + σ1(Q /µ )α(µ ) + σ2(Q /µ )α (µ ) + ...], (3.31) where a, b and d (usually d = 1) are constants. It is clear that when Eq. (3.31) truncated to a ﬁnite order, the prescription for α depends on the renormalization scale µ. This dependence can be removed if the renormalized coupling is related to a low-energy scale Λ2. Using the idea of the dimensional transmu- tation [98], the quantity σ(Q2) can be written as a function of Q2/Λ2, with σ(Q2) ≡ F (Q2/Λ2). By expanding it in powers of 1/ln(Q2/Λ2) and using the renormalization group, one ﬁnds [22]

2 2 2 2 b σ1(1) + cβ1 − β2/β1 ln ln(Q /Λ ) σ(Q ) = a + 2 2 1 + 2 2 , (3.32) β1 ln(Q /Λ ) β1 ln(Q /Λ ) where c is a constant that is determined through the deﬁnition of Λ2. Here,

β1 and β2 are the coeﬃcients of one- and two-loop orders of the perturbative QCD β-function. The quantity σ(Q2) is independent of the renormalized coupling α(µ2). The only dependence in Eq. (3.32) can be found by redeﬁning

39 3.3. PROPOSED SOLUTIONS FOR THE PROBLEM the scale parameter Λ2 or Q2 as

1 Λ2 → Λ¯ 2 = λ2Λ2 OR Q2 → Q¯2 = Q2. (3.33) λ2

The exact deﬁnition for the physical quantity in Eq. (3.32) is not invariant under these transformations, but becomes dependent on a single parameter λ. Thus, it is better to deal with the inverse function Q2/Λ2 = F −1(σ) in order to eliminate the dependence on Q2 and Λ2. Now, it is clear that the inverse function only depends on the physical quantity; hence, it is RS independent even if the expression for the physical quantity in Eq. (3.32) is truncated to a particular order, where the observable quantity itself cannot be truncated. An exact expression for the inverse function can be found by introducing a 2 new running coupling αeﬀ (Q ) as follows

2 2 σ(Q ) = a + bαeﬀ (Q ), (3.34)

2 1 where αeff (Q ) defines the QCD effective charge that should include all higher-order corrections to the physical quantity σ(Q2). In the same manner of the GML ψ-function, the QCD βeff -function associated with the effective 2 charge αeff (Q ) can be defined by

∂α (Q2) Q2 eff = β (α ) = −β¯ α2 (Q2) − β¯ α3 (Q2) − β¯ α4 (Q2) + O(α5 ). ∂Q2 eff eff 1 eff 2 eff 3 eff eff (3.35)

The first two coefficients of the βeff (αeff )-function are equal to the first two ¯ ¯ coefficients of the perturbative β(α)-function (β1 = β1 and β2 = β2) and the higher coefficients become process dependent.2

The exact solution for the associated βeﬀ -function is given by [22]

2 Z αeff Q 1 β2 1 β2 β1 β1ln 2 = + ln(β1αeff ) + k + dx 2 − + , (3.36) Λ αeff β1 0 x β1 βeff (x)

1Eq. (3.34) looks like the result of leading-order and the effective charge coincides with the strong running coupling extracted from experimental data for the physical quantity σ(Q2). 2Measuring the QCD effective charge directly from data is presented in [99]. It is found that the first two terms of the QCD β/ψ-function are universal.

40 3.4. PHYSICAL RUNNING COUPLING where k is the coefficient of the correction of next-to-leading order 1 (NLO) to the effective charge at Q2 = µ2. Comparing the exact solution2 with the definition of the physical quantity in Eq. (3.34), one can find the exact form for the inverse function, F −1(σ). From the exact solution in Eq. (3.36), the definition of the QCD effective charge takes the following form

1 β ln(ln(Q2/Λ¯ 2)) α (Q2) = 1 − 2 (3.37) eﬀ 2 ¯ 2 2 2 ¯ 2 β1ln(Q /Λ ) β1 ln(Q /Λ ) with Q2 Λ¯ 2. (3.38)

This expression for the QCD running coupling is RS independent and does not have a pole, but cannot be applied to low-energy scales due to the condition Q2 Λ¯ 2. The result of the effective charge method depends on the choice of the physical quantity σ(Q2)[22]. For example, if there are two different physical quantities σ1 and σ2, which can be applied to the effective charge method, the result of their sum, S = σ1 + σ2, may depend on whether the method is applied directly to the sum or applied to σ1 and σ2, separately. This dependence raises the question: which is the best physical quantity that can give the correct definition for the QCD running coupling?

3.4 Physical Running Coupling

As shown above, the problem of RS dependence has been eliminated to all orders through the method of the effective charge at the expense of introducing process dependence. This says nothing about the QCD running coupling at low-energy scales. Thus, the purpose of this section is to introduce some conditions under which a definition for the QCD running coupling can be considered a physical definition. These conditions can be summarised as fol-

1The value of this correction can be obtained by calculating the physical quantity σ at NLO. 2Since the solution of the associated β¯-function is exact, any approximation to the solution can be applied to it.

41 3.4. PHYSICAL RUNNING COUPLING lows.

• It must be defined non-perturbatively. This condition can be satisfied by relating the QCD running coupling to a non-perturbative physical quantity through the effective charge method.

• It must provide a real and and positive (physical) value for the running coupling at all energy scales,

IR 2 αeﬀ ∈ (0, αeﬀ or ∞) for all Q ∈ (0, ∞) (3.39)

• It should be a monotonic and analytic function of the space-like energy scale Q2 everywhere, e.g. when moving from non-perturbative to perturbative regions. In the UV region, it is clear from the eﬀective charge method’s deﬁnition for the QCD running coupling in Eq. (3.37) that it is a monotonic and decreasing function of the space-like energy scale in the perturbative region. The remaining question is that does it have a monotonic property in the non-perturbative region?

The central purpose of this thesis is to find such a physical definition for the strong running coupling within the framework of chiral perturbation theory and the effective charge method. If such a definition is found, then we will discuss its IR behaviour. It may be singular or tend to a finite value (an IR fixed point),

2 2 αeff (Q ) → ∞ as Q → 0 2 IR 2 αeff (Q ) → αeff as Q → 0. (3.40)

42 Chapter 4

Analysis of Perturbative and Non-Perturbative Infrared Fixed Point

In previous chapters, it has been indicated that there is a large number of definitions for the running coupling, some of which lead to an IR fixed point. The purpose of this chapter is to analyse the most known IR fixed point, perturbative and non-perturbative fixed point. This chapter contains two main sections. Section 4.1 reviews the IR fixed point of the two-loop β-function, the Banks-Zaks IR fixed point, Section 4.1.1. The contribution of higher-loop orders to this IR fixed point is discussed in Section 4.1.2. In the latter case

(Section 4.1.3), we show the effect of large Nf and Nc limits on this IR fixed point. Section 4.2 reviews three of the known non-perturbative definitions for the strong running coupling, Schrödingerfunctional (Section 4.2.1), Dyson- Schwinger equations (Section 4.2.2) and anti-de Sitter (Section 4.2.3) running couplings, including their conclusions about an IR fixed point.

4.1 Perturbative Infrared Fixed Point

If it exists, an IR fixed point should appear at low-energy scales, where the running coupling relatively becomes large. It is difficult to find an acceptable perturbative IR fixed point because the perturbative running coupling must

43 4.1. PERTURBATIVE INFRARED FIXED POINT be small, according to asymptotic freedom. The analysis of a perturbative IR ﬁxed point is presented in [13, 14], which is known as the Banks-Zaks IR ﬁxed point [14].1

4.1.1 The Banks-Zaks Infrared Fixed Point

The Banks-Zaks (BZ) IR ﬁxed point is based on the idea that the perturbative QCD β-function is given by the two-loop order formula, with higher-loop orders neglected. There are two ways to form this order: through the ’t Hooft scheme, or by truncating the β-function at the two-loop order. First, let us consider the ’t Hooft scheme in which the β-function of the SU(Nc) gauge group reads ˜ 2 3 β(˜α) = −β1α˜ − β2α˜ , (4.1) then the BZ IR ﬁxed point is given by

IR 2 β1 α˜BZ(µ ) = − (4.2) β2 with 1 11 2 β = N − N , (4.3) 1 (4π) 3 c 3 f 2 1 34 2 13Nc − 3 β2 = 2 Nc − Nf . (4.4) (4π) 3 3Nc The BZ IR ﬁxed point is physically acceptable (positive) when one co- eﬃcient of the β˜-function changes its sign. Since we are dealing with an asymptotically free theory, β1 must be positive, which implies that

11 β > 0 ⇒ N < N = N max. (4.5) 1 f 2 c f 1There is other works (see for example [93, 100, 101, 102, 103, 104, 102, 105]) based on perturbative analysis, which suggest that the QCD running coupling freezes in the IR limit.

44 4.1. PERTURBATIVE INFRARED FIXED POINT

1 In this case, β2 must have the opposite sign, leading to

3 34Nc min β2 < 0 ⇒ Nf > 2 = Nf . (4.6) 13Nc − 3

In other words, the BZ IR fixed point exists, Eq. (4.2) has a positive value, if the number of quark flavours Nf satisfies the following range

min max Nf < Nf < Nf . (4.7)

The value of the BZ IR ﬁxed point must be suﬃciently small in order to be acceptable in the framework of perturbation theory. The truth is that it min becomes larger as Nf goes to Nf . For example, let us consider the SU(3) gauge group; then the condition in Eq. (4.7) reads

9 6 Nf 6 16. (4.8)

If the number of quark flavours is close to nine (Nf ' 9), then the value of the IR BZ IR fixed point isα ˜BZ ≈ 5, which is presumably too large for perturbation theory to be reliable. In contrast, if Nf becomes larger (Nf → 16), then the BZ IR fixed point becomes small, which is more consistent with perturbation theory, as shown in Fig. 4.1. One can conclude that the BZ IR fixed point is perturbatively acceptable as long as the number of quark flavours is close max max to Nf . This condition (Nf ≈ Nf ) can be considered to be a physical condition for a BZ IR fixed point. The second way to extract the BZ IR fixed point is to truncate the β- function at the two-loop order. However, it is impossible to truncate the β-function to a particular order when one runs an experiment. In this case, the BZ IR fixed point has a correction from the higher-loop levels. Thus, it is the purpose of the next section to discuss the effect of higher-loop contribution to the BZ IR fixed point.

1 min Nf refers to a value in which the second coeﬃcient of the β-function vanishes β2 = 0 min and then becomes negative for Nf > Nf .

45 4.1. PERTURBATIVE INFRARED FIXED POINT

Figure 4.1: In the SU(3) gauge group, the BZ IR ﬁxed point is a decreasing min max function of Nf with Nf < Nf < Nf . It becomes more reliable in the framework of perturbation theory as Nf increases to 16.

4.1.2 The Eﬀect of Higher-Loop Orders

Let us assume that higher-loop orders preserve the BZ IR fixed point. Then these orders will certainly change the location of the BZ IR fixed point. In order to study this change, let us first recall the four-loop MS β-function1

2 3 4 5 6 β(α) = −β1α − β2α − β3α − β4α + O(α ) (4.9) with

1 2857 5033 325 β = − N + N 2 , 3 (4π)3 2 18 f 54 f 1 149753 1078361 6508 β = + 3564ζ − + ζ N 4 (4π)4 6 3 162 27 3 f 50065 6472 1093 + + ζ N 2 + N 3 , (4.10) 162 81 3 f 729 f

1 For simplicity, we have fixed Nc to three through this section by considering the SU(3) gauge group. The third coefficient is a quadratic function of Nf and vanishes at two different values of Nf , while the fourth one is a cubic function in Nf . A detailed discussion is presented in [106, 107].

46 4.1. PERTURBATIVE INFRARED FIXED POINT

max where β4 is positive for all Nf < Nf compared to the coeﬃcients of two- and three-loop orders.

Contribution of Three-Loop Order

The three-loop order of the β-function has an extra zero for α > 0, i.e. at two diﬀerent values of the running coupling,

p 2 (3) −β2 ± β2 − 4β3β1 α± = (4.11) 2β3

2 with β2 − 4β3β1 > 0 in order to avoid the complex (unphysical) value for the running coupling. One of these values is negative, and hence is not a part of the discussion as it is unphysical. Therefore, there is only one physical value for the IR ﬁxed point at the three-loop order that is given by

p 2 IR −β2 − β2 − 4β3β1 α(3) = . (4.12) 2β3

2 max Since β2 and β1 are positive for all Nf < Nf , then β3 must be negative 2 in order to satisfy β2 > 4β3β1, which is satisfied for all Nf in Eq. (4.7), see Table 4.1. Thus, the three-loop β-function has an IR fixed point for all values of Nf that satisfy Eq. (4.7). For the SU(3) gauge group, it is found that the IR fixed point of the three-loop order is smaller than the BZ IR fixed point,

IR IR α(2) > α(3), (4.13) which is more acceptable in perturbation theory. It is easy to prove the last equation by considering the diﬀerence between these points, which gives a positive quantity [106, 107],

IR IR 2 2 α(2) − α(3) > 0 ⇒ β3 β1 > 0. (4.14)

min For Nf < Nf , the three-loop β-function also has an IR ﬁxed point min,3 1 [108, 109]. In this case, the minimum value of Nf changes to Nf , where

1 min,3 Nf refers to a value in which the third coeﬃcient of the β-function vanishes β3 = 0

47 4.1. PERTURBATIVE INFRARED FIXED POINT

2 3 4 IR IR IR Nf (4π)β1 (4π) β2 (4π) β3 (4π) β4 αBZ α(3) α(4) 1 10.33 89.33 1154.91 22703.1 - - - 2 9.67 76.67 893.35 22703.1 - - - 3 9.00 64.00 643.83 12090.2 - - - 4 8.33 51.33 406.35 8034.97 - - - 5 7.67 38.67 180.91 4825.91 - - - 6 7.00 26.00 -32.50 2472.00 - 12.719 - 7 6.33 13.33 -233.87 982.235 - 2.456 - 8 5.67 00.67 -423.20 365.611 - 1.464 1.549 9 5.00 -12.00 -600.50 631.122 5.233 1.028 1.072 10 4.33 -24.67 -765.76 1787.77 2.207 0.764 0.815 11 3.67 -37.33 -918.98 3844.54 1.234 0.578 0.626 12 3.00 -50.00 -1060.17 6810.43 0.754 0.435 0.470 13 2.33 -62.67 -1189.31 10694.4 0.468 0.317 0.337 14 1.67 -75.33 -1306.43 15505.6 0.278 0.215 0.224 15 1.00 -88.00 -1411.50 21252.8 0.143 0.123 0.126 16 0.33 -100.67 -1504.54 27945.2 0.042 0.0397 0.398

Table 4.1: The values of the perturbative IR fixed point at two-, three-, and four-loop order for the SU(3) gauge group. the range of quark flavours, in which the β-function at the three-loop level has a positive value for its IR fixed point, is given by

min,3 max Nf < Nf < Nf . (4.15)

The range of Nf values in Eq. (4.15) contains the two-loop version Eq. (4.7) as a subset. If both conditions are satisﬁed, the result for the IR ﬁxed point is less likely to be RS dependent.

Contribution of Four-Loop Order

The four-loop β-function also vanishes away from the origin, but at three diﬀerent values for the running coupling α [106, 107] which are given by the roots of the cubic equation

2 3 β1 + β2α + β3α + β4α = 0. (4.16)

min,3 and then becomes negative for Nf > Nf .

48 4.1. PERTURBATIVE INFRARED FIXED POINT

The reality of the roots depends on the sign of the discriminant quantity ∆

3 2 2 3 2 2 ∆ = 18β1β2β3β4 − 4β3 β1 + β3 β2 − 4β4β2 − 27β4 β1 . (4.17)

Since ∆ depends on the coeﬃcients of the β-function, its sign is determined by the signs of β1, β2, β3 and β4. The sign of β1 and β4 is positive for all max Nf < Nf , while the sign of β2 and β3 depends on Nf . Thus, we separate the range of Nf into three diﬀerent ranges:

min max • I2: Nf < Nf < Nf , where both β2 and β3 are negative.

min,3 min • I3: Nf < Nf < Nf , where β2 is positive and β3 is negative.

min,3 • I4: 0 < Nf < Nf , where both β2 and β3 are positive.

For Nf ∈ I2, both β2 and β3 are negative, and hence the discriminant quantity ∆ is positive. In this case, the cubic equation has three real solutions for α; one solution is negative (unphysical) and the other two are positive; 1 2 1 2 called α(4) and α(4) with α(4) < α(4). The smallest one can be regarded an IR IR 1 1 ﬁxed point at the four-loop level in I2, denoted α(4) = α(4). For Nf ∈ I3 and

Nf ∈ I4, the discriminant quantity ∆ is negative, then the four-loop order of the β-function has one negative and two complex conjugate solutions for α. min In this case, there is no IR fixed point at the four-loop level for all Nf < Nf min except for Nf ≈ Nf from below, see Table 4.1. Again, by comparing the values of the perturbative IR fixed point at different loop orders in I2, we find that IR IR IR α(3) < α(4) < αBZ. (4.18) In the MS scheme, the perturbative IR fixed point is a monotonic decreasing function of Nf to all loop levels. In other words, the value of it becomes more reliable to trust in perturbation theory when Nf approaches its maximum value. The corrections of the higher-loop order become smaller when more and more higher-loop orders are considered. Further, the higher-loop 2 order of the β-function may have an IR fixed point for a small Nf . However,

1 2 The second fixed point α(4) is a non-trivial UV fixed point. 2Note that this IR fixed point may disappear in a different scheme.

49 4.1. PERTURBATIVE INFRARED FIXED POINT this IR fixed point is unphysical because its value is large to observe in perturbation theory. Thus, the BZ IR fixed point is more acceptable perturbative max IR fixed point as long as Nf → Nf . Numerical comparison between the perturbative IR fixed point at different loop orders is listed in Table 4.1 [106] for the SU(3) gauge group within the MS scheme.

4.1.3 Large Limits: Nf ,Nc → ∞ In Chapter2, it is shown that there is no IR fixed point as the number of quark flavours Nf or colours Nc runs to infinity. An idea of getting an IR

fixed point in the large limit of Nf and Nc is presented by fixing Nf /Nc to a finite value [110, 111]. In this case, the perturbative QCD β-function can be rewritten as

´ ´ 2 ´ 3 ´ 4 ´ 5 β(´α) = −β1α´ − β2α´ − β3α´ − β4α´ + O(´α) (4.19) with ´ 1 Nf β1 = β1|Nf ,Nc→∞ = 11 − 2 , 3(4π) Nc 1 N β´ = β | = 34 − 13 f , 2 1 Nf ,Nc→∞ 2 3(4π) Nc 1 N N 2 β´ = β | = 2857 − 1709 f + 112 f , 3 1 Nf ,Nc→∞ 3 2 54(4π) Nc Nc 1 150653 44 β´ = β | = + ζ 4 1 Nf ,Nc→∞ (4π)4 486 9 3 2 3 485513 20 Nf 8654 84 Nf 130 Nf − + ζ3 + + ζ3 2 + 3 . 1944 9 Nc 243 9 Nc 243 Nc (4.20)

In this limit, the condition of the BZ IR ﬁxed point in Eq. (4.7) takes the following form 34 N 11 < f < . (4.21) 13 Nc 2 max As stated above, Nf must be close to the maximum value Nf in order to attain a reliable value for the perturbative IR ﬁxed point to all loop orders.

50 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

´ 2 ´ 3 ´ 4 ´ IR IR IR Nf /Nc (4π)β1 (4π) β2 (4π) β3 (4π) β4 α´BZ α´(3) α´(4) 3.0 1.67 -1.67 -23.37 -5.62 12.57 2.94 2.87 3.5 1.33 -3.83 -32.45 28.80 4.37 1.91 2.02 4.0 1.00 -6.00 -40.50 89.45 2.09 1.25 1.35 4.5 0.67 -8.17 -47.51 176.71 1.03 0.76 0.80 5.0 0.33 -10.33 -53.48 290.98 0.41 0.35 0.36

Table 4.2: The values of the perturbative IR fixed point at different loop orders in large limits Nf ,Nc → ∞ with Nf /Nc → a finite value.

max This statement still hold in the limit of Nf ,Nc → ∞, where Nf = Nf /Nc = 1 11/2. A numerical comparison is presented in Table 4.2 [111], with Nf /Nc held fixed. It is clear from this comparison that the perturbative IR fixed point at the two-loop level is greater than the IR fixed point of higher-loop levels, but the case is different when one compares the result of the three-loop to the four-loop one

IR IR ´ α´(3) < α´(4) if β4 > 0, IR IR ´ α´(3) > α´(4) if β4 < 0. (4.22)

By comparing the values of the perturbative IR fixed point in the finite and infinite limits of Nf and Nc, we find that the finite limit leads to a more reliable result for the perturbative IR fixed point. Thus, one can conclude that the large limit of Nf and Nc is not the best approach to find an acceptable IR fixed point in the framework of perturbation theory.

4.2 Non-Perturbative Infrared Fixed Point

The above section indicated that the value of the perturbative IR fixed point min becomes too large for perturbation theory as Nf → Nf , so it is not physically relevant. This means that a non-perturbative analysis is required to find an acceptable result, in particular for a smaller number of quark flavours.

1At the four-loop level, there are two positive values for α. We list the smallest one as an IR ﬁxed point in the large limit of Nf and Nc, while the second one is an UV ﬁxed point.

51 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

The following sections review the best known non-perturbative deﬁnitions for the running coupling at low-energy scales.

4.2.1 Schr¨odinger Functional Analysis

The Schrödingerfunctional (SF) method was first proposed to compute the running coupling1 for pure non-Abelian gauge theories [113, 114] and then applied to QCD with fermions in [115]. In general, it is used to study the scaling property of theory in a finite volume for a box of size L × L × L with Dirichlet boundary conditions. In lattice QCD, Dirichlet boundary conditions are a set of gauge links:

0 Uk(x)|x0=0 = exp(aCk) Uk(x)|x0=T = exp(aCk) (4.23) with

   0  φ1 0 0 φ1 0 0 C = i  0 φ 0  ,C 0 = i  0 φ0 0  , k = 1, 2, 3. k L  2  k L  2  0 0 0 φ3 0 0 φ3 (4.24)

In the presence of quark ﬁelds, these boundary conditions are given by

Ψ(x)|x0=0 = Ψ(x)|x0=T = 0, Ψ(x)|x0=0 = Ψ(x)|x0=T = 0. (4.25)

0 2 The angles φj and φj for pure Yang-Mills theories are given [15] by π π φ = ω η − φ = ω η φ = ω η + 1 1 3 2 2 3 3 3 0 0 π 0 2π φ = −ω η − π φ = −ω η + φ = −ω η + 1 1 2 2 3 3 3 3 1 1 ω = 1 ω = − + ν ω = − − ν, (4.26) 1 2 2 3 2 1It was ﬁrst used to compute the running coupling for the pure SU(2) Yang-Mills theory [112] and then applied to the SU(3) Yang-Mills theory [15]. 2In the SU(3) gauge group, these angles must be real and their summation equal to zero in order to be elements of the gauge group. They are also chosen to give a stable solution to ﬁeld equations at small η.

52 4.2. NON-PERTURBATIVE INFRARED FIXED POINT where the parameters η and ν are usually used to define some renormalized quantities. In the SF method, η is used to define the running coupling at ν = 0.1 This coupling is defined at distance L by the derivative of the effective action Γ with respect to η at η = ν = 0,

2 ∂Γ K g¯ (L) = , αSF = (4.27) ∂η η=ν=0 4παSF 4π where 1 K = 12 (L/a)2 [sin θ + sin 2θ], θ = (a/L)2. (4.28) 3 The value of K is chosen to ensure that the SF coupling is equal to the bare coupling at the lowest-order in perturbation theory. The β-function associated with the SF running coupling is deﬁned through

∂α (q) βSF (α ) = L SF SF ∂L 2 3 SF 4 5 = −β1αSF − β2αSF − β3 αSF + O(αSF ), (4.29) where q = 1/L. It is important to note that only the ﬁrst two terms are independent of the SF scheme.2 The relationship between the running coupling of the MS and SF scheme is given by

2 3 αMS = αSF + d1αSF + O(αSF ), (4.30) where d1 = 1.255634 and both runnings are calculated at the same energy scale. For pure Yang-Mills theories, L¨uscher et al. [15] demonstrated that in their scheme, the β-function is negative everywhere, and hence there is no

IR ﬁxed point. For Nf 6= 0, the situation is unclear. Conclusions that an IR

ﬁxed point can occur only for large values of Nf (8 or 9 up to 16) tend to be based on assumptions that scaling condensates are absent at that ﬁxed point, i.e. that there is a manifestly “conformal window” [10, 117, 118, 119, 120]

1The statistical errors on the coupling are small in numerical simulations of the SF, hence ν is chosen to be zero [15]. 2 SF The third coeﬃcient β3 is now known, see [116].

53 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

(case 2(a) in Chapter1). A recent lattice analysis [121] indicates that an IR

ﬁxed point in the presence of the quark condensate can occur for Nf = 3.

4.2.2 Dyson-Schwinger Equations Analysis

The Dyson-Schwinger equations (DSEs) is a well established non-perturbative approach used to study the IR behaviour of QCD.1 In this approach, the non- perturbative running is expressed in terms of the dressing functions for the ghost and gluon propagators in the pure Yang-Mills theory.2 This method is based on solving a truncated system of DSEs for the ghost-gluon vertex in Landau gauge [17, 124, 125, 126, 127, 128].3 The deﬁnition of the DSEs for the running coupling is given [17] by mul- tiplying the dressing function of the gluon propagator with the square of the dressing function of the ghost propagator

α(q2) = α(µ2)Z(q2)G2(q2), (4.31) where α(µ2) is the renormalized coupling constant. This deﬁnition depends on the renormalized parameters α(µ), G(q2) and Z(q2) that depend on the ˜ renormalization constants Zg, Z3 and Z3. However, the Slavnov-Taylor identity [131] ensures that the product α(µ2)Z(q2)G2(q2) is renormalization group invariant.4 Moreover, the renormalized running coupling α(µ2) in Eq. (4.31) is cancelled by the renormalized coupling in the product of G2(q2) and Z(q2). In Euclidean momentum space, the DSEs for the gluon and ghost propagators are given by their inverse equations [17]

4 −1 Z d k D−1(q) = Z Dtree (q) − 4παN Z˜ ik D (p)D (k)G (k, p) µν 3 µν c 1 (2π)4 µ G G ν Z d4k +2παN Z Γtree(q, −p, k)D (k)D (p)Γ (−k, p, −q), c 1 (2π)4 µρω ωβ ρσ βσν (4.32)

1Earliest work of DSE [122, 16] is presented in the absence of the ghost ﬁeld. It shows that the strong running coupling is singular in the IR limit, which agrees with the result of the SF method for Yang-Mills theories [15]. 2The inclusion of quark ﬁelds to the DSEs is presented in [19, 20] and recently in [123]. 3Detailed discussions of DSEs calculations can be found in reviews [129, 130]. 4A completed proof is presented in [127, 132, 129].

54 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

Z d4k D−1(q) = −Z˜ q2 + 4παN Z˜ iq D (q − k)G (q, k)D (k), (4.33) G 3 c 1 (2π)4 µ µν ν G ˜ where p = q + k and Z3, Z1 and Z1 are the renormalization constants for the renormalized quantities Dµν, DG and α, respectively. The renormalized quantities are related to their bare quantities through the multiplicative renormalization constant

0 0 ˜ 0 Dµν = Dµν/Z3,DG = DG/Z3, α = α /Zα, (4.34)

3/2 ˜ ˜ 1/2 Z1 = ZαZ3 , Z1 = ZαZ3Z3 , (4.35) ˜ with Z1 = 1 in Landau gauge [131]. The other notations are deﬁned as follows:

tree • Dµν , DG : the tree-level gluon and ghost propagators

tree • Γµρω : the three-gluon vertex

• Gν and Γβσν : the dressed three-point functions.

In Landau gauge, the gluon and ghost propagators are parameterised by their dressing functions Z(k) and G(k), which are related to propagators through

Z(q2) q q G(q2) D (q) = δ − µ ν ,D (q) = − . (4.36) µν q2 µν q2 G q2

By solving the DSEs for the gluon and ghost propagators, the dressing functions are deﬁned as

Z q2 2 4 2 −1 2 α Nc dk 7k 17k 9 2 2 Z (q ) = Z3 +Z1 2 4 − 2 − Z(k )G(k ) 4π 3 0 q 2q 2q 8 2 Z ΛUV 2 2 α Nc dk 7q 2 2 +Z1 2 2 − 7 Z(k )G(k ) 4π 3 q2 k 8k Z q2 2 2 α Nc dk 3k 2 2 1 2 2 + 2 2 G(q )G(k ) − G (q ) 4π 3 0 q 2q 3 2 Z ΛUV 2 α Nc 1 dk 2 2 + 2 G (k ), (4.37) 4π 3 2 q2 k

55 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

" 2 # Z ΛUV 2 −1 2 ˜ 3αNc 1 2 2 dk 2 2 G (q ) = Z3 − Z(q )G(q ) + 2 Z(k )G(k ) . (4.38) 16π 2 q2 k In the IR limit (q2 → 0), the gluon and ghost dressing function yield

4N 1 1 Z(q2) ' c πα(µ2)γ − c2q4κ → 0 as q2 → 0, (4.39) 3 κ 2

4N 1 1−1 G(q2) ' c πα(µ2)γ − c−1q−2κ → ∞ as q2 → 0, (4.40) 3 κ 2 where c is a constant. Here, γ refers to the coeﬃcient of the perturbative anomalous dimension of the ghost ﬁeld (γ = 9/64π2). In addition, it is found that the gluon dressing function in Eq. (4.37) yields

4N 1 12 3 1 1 1 Z(q2) ' c πα(µ2)γ − × − + c2q4κ → 0 as q2 → 0. 3 κ 2 2 2 − κ 3 4κ (4.41) To avoid an unphysical value for the ghost and gluon dressing functions, the κ-constant must be restricted to

0 < κ < 2. (4.42)

It is clear from Eq. (4.40) that the ghost dressing function is singular in the IR limit; however, the product Z(q2)G2(q2),

−1 2 2 2 16π 1 1 Z(q )G (q ) ≈ 2 − , (4.43) 3Ncα(µ ) κ 2 tends to a fixed and finite value within the condition in Eq. (4.42). According to Eq. (4.43), the product Z(q2)G2(q2) and hence the running coupling Eq. (4.31) is finite in the IR limit q2 → 0 [17, 129], indicating the presence of an IR fixed point. However, recent numerical analysis of the ghost DSE [133, 134, 135, 136] shows that the ghost dressing function G(q2) could be either singular (scaling solution) or regular (decoupling solution) in the IR limit.1 The scaling solution is not compatible with the lattice

1In the decoupling solution, the low-energy behaviour of the running coupling is determined by the next-to-leading part of the ghost propagator [137].

56 4.2. NON-PERTURBATIVE INFRARED FIXED POINT data, making the decoupling solution is the acceptable solution for the ghost dressing function. In this case, the product Z(q2)G2(q2) decreases rapidly to zero as q2 → 0. It is usually claimed that this is evidence against having an IR fixed point, but in fact, it shows that the definition of α(q2) is not monotonic. Therefore, no conclusion about the existence of an IR fixed point can be drawn. The difference between the earlier analysis by Alkofer et al. (see e.g. [17, 129]) and more recent work arises from the contribution of internal ghost loops in the gluon propagator. This was confirmed in recent analysis [133, 134] of the DSE for the ghost propagator over the whole momentum scale.

Lattice simulation

The analysis of a lattice study [18] for the running coupling of the gluon and ghost propagators depends on a choice of definition for the QCD running coupling. Two definitions are considered: a definition αPT based on the Pinch Technique [138, 139, 140], and a second definition coming from the standard gluon-ghost vertex. These definitions are related to the gluon and ghost propagators through

2 2 2 2 2 αPT(q ) = 4π[q + m (q )]R1(q ), (4.44)

2 2 2 2 2 αgh(q ) = 4π[q + m (q )]R2(q ), (4.45) with ∆(q2) R (q2) = α(µ2) , (4.46) 1 [1 + G(q2)]2

2 2 2 2 2 R2(q ) = α(µ )∆(q )F (q ). (4.47)

2 2 1 Here, m (q ) is the running mass of the gluon, which is equal to m0 as q2 → 0, and ∆(q2) is the gluon propagator

13N g q2 + ρm2 ∆−1(q2) = m2 + q2 1 + c f ln , (4.48) 96π2 µ2

1 It is chosen to be consistent with phenomenological studies (m0 = 500 − 600 MeV).

57 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

2 2 where m , gf and ρ are free-ﬁtting parameters. The auxiliary function G(q ) is related to the ghost dressing function F (q2) by

G(q2) = F −1(q2) − L(q2) − 1, (4.49) where L(q2) is a form factor that is measured from the lattice data. It is clear that the gluon propagator depends on the renormalization scale µ2, but 2 2 lattice data shows that R1(q ) and R2(q ) are renormalization group invariant quantities [18]. The value of the renormalized coupling constant α(µ2) is fixed to a finite value. This value is determined from the lattice data at two different renormalization points, using the given definitions for G(q2), ∆(q2), F (q2) 2 2 and L(q )[18, 141]. It is obtained that α1(µ1) = 0.467 for µ1 = 2.5 GeV and 2 2 α2(µ2) = 0.309 for µ2 = 4 GeV. Using these values for α(µ ) and lattice data for G(q2), ∆(q2), F (q2) and L(q2), it is found that the lattice definitions in 2 Eq. (4.44) and (4.45) tend to the same finite value at a small q for µ1 and

µ2.

4.2.3 Anti-de Sitter Analysis

Studying the light-front holographic mapping1 of classical gravity in anti-de Sitter (AdS) space leads to an interesting non-perturbative definition for the effective coupling [147]. The AdS definition for the strong running coupling is expressed in terms of the space-like scale Q2

2 2 2 αs, AdS(Q ) = αs(0) exp[−Q /4k ], (4.50) where the small s refers to the type of the used scheme and k is a constant that is found to be 0.54 GeV [146]. At low-energy scales, it leads to an IR

fixed point, which is equal to αs(0) . The advantage of the AdS effective coupling is that it satisfies some conditions of the proposed physical running coupling: it is an analytic and monotonic function of Q2. However, the AdS running coupling is RS dependent, and hence its IR fixed point is scheme

1For a detailed discussion of the light-front holography mapping approach, see [142, 143, 144, 145, 146].

58 4.2. NON-PERTURBATIVE INFRARED FIXED POINT dependent. The positive fact about the AdS effective coupling is that its shape as a function of the energy scale agrees with other non-perturbative definitions for the running coupling in the IR limit, such as lattice [148], DSEs [149, 17, 19] and constituent model [150] definitions. The AdS effective coupling is calculated in two different renormalization schemes. The best scheme is that obtained from the Bjorken sum rule [151] at which the AdS effective coupling reads

2 2 2 αg1, AdS(Q ) = αg1 (0) exp[−Q /4k ] with αg1 (0) = π, (4.51)

where αs=g1 refers to the value of the coupling constant associated with the Bjorken sum rule. This scheme gives a good value for the AdS eﬀective running compared to other non-perturbative deﬁnitions for the strong coupling, particularly at low-energy scales [152, 153].1 The second scheme is the scheme of Appelquist et al. [154, 155], where its coupling αv is related to αg1 through the commensurate scale relation (CSR) [156, 157]

2 X n αv(Q ) = dnαg1 . (4.52) n=1

Numerically, the value of the AdS IR fixed point is different in these two AdS AdS schemes (αg1 (0) = π, αv (0) = 2.20), which is normal because the AdS effective coupling is scheme dependent. The β-function associated with the AdS effective coupling is given by [147]

dα (Q2) β (α ) = Q2 s, AdS s, AdS s, AdS dQ2 αs = −αs, AdS ln . (4.53) αs, AdS

In the low-energy region, its shape agrees with other non-perturbative deﬁ- nitions for the β-function, including lattice [148] and DSEs [149, 17, 19]. In the scheme of the Bjorken sum rule, the AdS β-function reads

π βg1, AdS(αg1, AdS) = −αs, AdS ln , (4.54) αg1, AdS

1For a summary of this comparison, see Fig.1 in [147].

59 4.2. NON-PERTURBATIVE INFRARED FIXED POINT

Figure 4.2: The shape of the AdS β-function in the scheme of the Bjorken sum rule is completely diﬀerent compared to the perturbative QCD β-function with Nf = Nc = 3. Clearly, the AdS β-function is not compatible with perturbative QCD.

where its IR fixed point is given by αg1, AdS = π. The AdS β-function has a minimum at Q2 ≈ k2 (see Fig. 2 in [147]) and then becomes a negative increasing function until the IR fixed point is reached. In the high-energy region, the AdS effective coupling tends to a small value in agreement with asymptotic freedom, but it falls off faster than the perturbative running coupling because it is an exponential function. The faster falloff makes the AdS effective coupling invalid in the UV limit. This disagreement can be seen by expanding the logarithmic factor in Eq. (4.54) AdS as a power series in αg1 in which the AdS β-function reads

2 3 4 βAdS(αg1 AdS) = −a1αs, AdS + a2αs AdS − a3αs, AdS + O(αs AdS) ∞ X n n = (−1) an(αs AdS). (4.55) n=1

It is clear from Eq. (4.55) that the AdS β-function looks like a linear function at a small eﬀective coupling αs, AdS ∼ 0. The linear property leads to

60 4.2. NON-PERTURBATIVE INFRARED FIXED POINT a different shape for the AdS β-function compared to the shape of the perturbative β-function in the asymptotic freedom region, as shown in Fig. 4.2. Thus, the AdS definition for the QCD effective charge is only valid in the non-perturbative region and cannot be generalised to the asymptotically free region.

61 Chapter 5

The Infrared Behaviour of Proposed Exact β-Functions

The only known example of an exact β-function in four dimensions is the Novikov, Shifman, Vainshtein and Zakharov (NSVZ) result for supersymmetric Yang-Mills theories [158, 159, 160, 161]. Ryttov and Sannino [21, 162] suggested that a similar expression for the β(α)-function could be used in non- supersymmetric QCD; this is known as the Ryttov-Sannino (RS) β-function. Some of these β-functions have a non-trivial fixed point αFP. However, they have a space-like pole at a finite value of the coupling constant. This pole makes the exact β-function undefined for some values of the running coupling α between the origin and this fixed point, 0 < α < αFP. Thus, part of this thesis is dedicated to modifying the RS β-function to obtain a non-singular form valid for all values of the running coupling in the range 0 < α < αFP. This chapter is presented as follows. Section 5.1 reviews the NSVZ supersymmetric β-function. The RS β-function is introduced in Section 5.2, including the dependence of the space-like pole on the number of quark flavours and colours (Section 5.2.1). The location of the pole with respect to the non- trivial fixed point is discussed in Section 5.2.2. A comparison between this fixed point and the Banks-Zaks IR fixed point is presented in Section 5.2.3. In Section 5.3.1, we modify the RS β-function using the idea of renormalization gauge transformation α → α˜. The modified RS β-function is defined for all values of the running couplingα ˜ ∈ (0, α˜IR), Section 5.3.2.

62 5.1. SUPERSYMMETRIC β-FUNCTION

5.1 Supersymmetric β-Function

The exact supersymmetric β-function was first calculated for super Yang- Mills theories [158] and then applied to theories with matter superfields [159, 160, 161] within the framework of perturbation theory and instanton approach. For pure super Yang-Mills theories, the NSVZ β-function is determined entirely by an integer number, such as Nc. In the presence of matter superfields, the exact β-function involves the anomalous dimension of the matter superfields. For N = 1 super gauge theories including matter, the NSVZ supersymmetric β-function takes the form

α2 b − 2T (r)N γ(α ) βNSVZ(α ) = − s s f s , (5.1) s αs 4π 1 − 2π C2(G) where bs and γ(αs) are the ﬁrst coeﬃcient of the perturbative super β-function and super anomalous dimension,

α b = 3C (G) − 2T (r)N , γ(α ) = s C (r) + O(α2), (5.2) s 2 f s 4π2 2 s respectively. It is clear from Eq. (5.1) that the NSVZ β-function has a pole pole 1 at αs = 2π/C2(G). The pole only depends on the Casimir operator of the fundamental representation C2(G), and hence is only controlled by Nc. In other words, the structure of the pole is independent of the matter superfield, which means that the location of the pole is not affected by considering a small or larger number of quark flavours Nf . In the presence of matter superfields, the NSVZ β-function has a non- trivial fixed point if the value of the anomalous dimension at that point satisfies the requirement

bs γ(αs) = . (5.3) βNSVZ=0 2Nf T (r)

1It is found that the pole appears at strong coupling, which can be ignored when one considers perturbation theory. However, the aim of this thesis is to find a physical definition for the running coupling that should work in both perturbative and non-perturbative regions. Thus, attention is given to this pole in this thesis because it affects the physical behaviour of the running coupling (see next section).

63 5.1. SUPERSYMMETRIC β-FUNCTION

Using the perturbative deﬁnition for the super anomalous dimension in Eq. (5.2), one can write this non-trivial ﬁxed point as

2 4π bs αs = , (5.4) βNSVZ=0 2Nf T (r)C2(r) where the higher-order terms of the anomalous dimension have been ignored. This non-trivial fixed point could be either an IR or UV fixed point, depending on the location of the unphysical pole.1 Thus, the pole plays an important role when discussing the behaviour of the exact β-function.2 Given that there is a large class of renormalization schemes, each of which leads to a different definition for the exact β-function, it is not clear in which scheme the NSVZ β-function was derived. Indeed, the NSVZ β-function is obtained without referring to any physical observed quantity [166, 167], like the β-function calculated perturbatively. Some works were undertaken to give an independent calculation of the NSVZ β-function within the Wilsonian renormalization group [168, 169]. The dependence of the NSVZ β-function on the renormalization scheme can be used to remove the pole. For example, assuming that the super gauge coupling transforms under a change of renormalization scheme as

αs → α˜s = F (αs), (5.5) then the NSVZ β-function of pure Yang-Mills theories takes a new form, such as that [170] βNSVZ(α ) βÑSVZ(˜α ) = YM s , (5.6) YM s NSVZ 0 1 + βYM (αs)F /F 0 ÑSVZ where F = ∂α˜s/∂αs. The new NSVZ β-function βYM may or may not have a pole, depending on the choice of the function F (αs). We use this property of scheme dependence to remove the pole of the exact RS β-function for QCD in the following section, but in the presence of matter fields.

1An analogous dependence will be discussed in the next section for the non- supersymmetric β-function. 2It has been shown that the super QCD has an IR ﬁxed point [163, 164] at diﬀerent energy scales. A comparison between the supersymmetric perturbative and exact β-functions is presented in [165].

64 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

5.2 Ryttov-Sannino (RS) β-Function

The βRS-function was proposed by adding the contribution of fermion and gluon positive modes to the NSVZ β-function. For the SU(Nc) gauge group, the βRS-function is given by

α2 b − 2 N γ(α) RS 3 f β (α) = − α 0 , (5.7) 4π 1 − 2π Nc(1 + 2b /b)

∞ 0 Nf X with b = (11N − 2N )/3 = 4πβ , b = N − , γ(α) = γ αa, c f 1 c 2 a a=1 (5.8) 1 where γa are the coeﬃcients of the anomalous dimension. It is important to note that only the lowest order of the anomalous dimension is renormalization scheme independent, and hence it can contribute to the two-loop βRS-function at a small running coupling.

5.2.1 The Singularity of the βRS-Function

As stated above, the NSVZ β-function has a pole at a ﬁnite value of the coupling constant. This pole is still present in the non-supersymmetric case,2 RS but it becomes a function of Nf in addition to Nc. The pole of β -function takes the following form

pole 2πb α = 0 . (5.9) Nc[b + 2b ]

Note that the pole is independent of the anomalous dimension and then its value is the same to all loop orders. Since the location of the pole is controlled by Nf and Nc, it is important to study its behaviour in the ﬁnite and inﬁnite limits of Nf and Nc.

1The explicit expression of these coeﬃcients is listed in AppendixC up to the fourth- loop order within the MS scheme. 2It is shown that the pole of the βRS-function describes two diﬀerent theories, one that is asymptotically free, which is before the pole, and another that is not asymptotically free [171].

65 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

As an explicit function of Nf and Nc, the location of the pole is given by

11N − 2N αpole = 2π c f . (5.10) Nc[17Nc − 5Nf ]

It is clear from Eq. (5.10) that the value of the pole could be either positive (a problem) or negative (irrelevant), depending on Nf and Nc. This dependence can be summarised as follows:

17 • For Nf < 5 Nc: the sign of the pole is positive; hence, there is an unwanted pole in the physical region.1

17 11 • For 5 Nc < Nf < 2 Nc: the pole has a negative sign at which it is irrelevant. This is acceptable because the βRS-function is non-singular for all values of the running coupling in the physical region.

11 • For Nf > 2 Nc: there is a pole in the physical region, but beyond the asymptotically free region.2

11 RS As Nc increases for ﬁxed Nf with Nf < 2 Nc, the pole of the β -function moves to the origin, as shown in Fig. 5.1. The eﬀect of Nc can be seen clearly in pure Yang-Mills theories in which the pole can be written as

pole 22π α = ≈ 0 as Nc → ∞. (5.11) 17Nc

It is easy to keep the pole away from the perturbative region by considering the large Nc limit with Ncα held fixed to a finite value. In this case, the pole becomes independent of Nc, and hence tends to the same value in the non-perturbative region, whatever the number of quark flavours:

22π N αpole ≈ as N → ∞. (5.12) c 17 c 1The physical region refers to the region where the coupling constant α has a real and positive value in the range 0 < α < αFP. 2This situation is beyond the scope of this thesis; however, it is important to note that gauge theories may have a non-trivial UV ﬁxed point in this case, which may be aﬀected by this pole.

66 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

RS Figure 5.1: The behaviour of the β -function’s pole as a function of Nf for ﬁnite values of Nc.

5.2.2 Non-trivial Fixed Point

It is clear from Eq. (5.7) that the βRS-function has a zero away from the origin (a non-trivial fixed point) if the anomalous dimension satisfies the following value1 3b 11N γ(α) = = c − 1. (5.13) βRS(α)=0 2Nf 2Nf This non-trivial fixed point is not always an IR fixed point, but may be an UV point due to the pole. The type of the non-trivial fixed point depends on the location of the pole. If the pole appears first, the non-trivial fixed point is an UV fixed point. However, it is not a physical point since the βRS-function is undefined for some values of the running coupling between the origin and this fixed point. Otherwise, the non-trivial fixed point is an IR point as long as it appears before the pole, where the βRS-function is well defined for all α < αIR. 1A full discussion of the anomalous dimension at this zero is presented in [172]. This includes all orders up to fourth-loops for the SU(3) and SU(2) gauge groups. The value max of the anomalous dimension at this point has its minimum at Nf = Nf . This statement max means that the value of the non-trivial fixed point becomes small as Nf → Nf in agreement with the BZ IR fixed point.

67 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

In terms of the number of quark flavours and colours , the above analysis can be presented as follows. For a fixed Nc, the location of the pole only 17 11 depends on Nf with Nf < 11Nc/2 (see Eq. (5.10)). For 5 Nc < Nf < 2 Nc, the pole is at a negative value (irrelevant) and the non-trivial fixed point of RS 17 the β -function is an IR fixed point. For Nf < 5 Nc, the type of the non- trivial fixed point depends on the location of the pole with respect to this point.

Leading-Order

In their proposed exact β-function Eq. (5.7), Ryttov and Sannino treat γ(α) perturbatively. To examine this idea, we have to truncate1 the perturbative anomalous dimension to a particular order. Since higher order terms of the anomalous dimension are renormalization scheme dependent, let us retain just the one-loop term in the anomalous dimension Eq. (5.8), so that the non-trivial ﬁxed point of the βRS-function is given by

4πN [11N − 2N ] α = c c f . (5.14) 2 βRS(α)=0 3Nf (Nc − 1)

Comparing the value of the pole and non-trivial ﬁxed point of the βRS- function to the leading-order of the anomalous dimension in the SU(3) gauge group, we ﬁnd that

RS • For 11 6 Nf 6 16, the β -function does not have a pole, and hence the non-trivial ﬁxed point is an IR ﬁxed point, see region III in Fig. 5.2.

• The case is diﬀerent in the region where the βRS-function has a pole.

For 9 6 Nf 6 10, the non-trivial fixed point appears first, and hence it is an IR fixed point, region II in Fig. 5.2. Otherwise, it is an unphysical

UV ﬁxed point for Nf < 9 as the pole appears ﬁrst, region I in Fig. 5.2.

1Another purpose for truncating the anomalous dimension is to compare this non-trivial ﬁxed point to Banks-Zaks IR ﬁxed point. One may also consider the ’t Hooft scheme here to eliminate the scheme dependent terms in the anomalous dimension, ending with the leading-order of anomalous dimension.

68 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

Figure 5.2: Comparison between the value of the pole (solid line) and non- trivial ﬁxed point (dashed line) of the βRS-function at the leading-order of the anomalous dimension as a function of Nf for Nc = 3.

5.2.3 Comparison of the Banks-Zaks and Ryttov-Sannino Fixed Points

It is easy to show that the βRS-function is equivalent to the perturbative β- function at a small running coupling α (see proof 1 in AppendixD). In this proof, we show that the βRS-function is not consistent with the MS scheme. For the perturbative β-function, the first non-trivial fixed point is obtained at two-loop order and an IR fixed point (BZ IR fixed point). For the βRS- function, the first non-trivial fixed point can be formed at the leading-order of the anomalous dimension. Using the fact that the leading-order of the anomalous dimension is renormalization scheme independent, one can compare the BZ IR fixed point to the non-trivial fixed point of the βRS-function. The comparison between the BZ IR fixed point and the RS non-trivial fixed point can be summarised in two points:

1. The perturbative β-function does not have a non-trivial fixed point min RS for a small Nf < Nf , while the β -function has a non-trivial UV fixed point. However, this UV fixed point is unphysical because the

69 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

RS Figure 5.3: For a small Nf , the β -function has a non-trivial UV ﬁxed point UV αEx. (unphysical), while the perturbative β-function does not have any ﬁxed point at two-loop order.

βRS-function is not deﬁned for the all values of the running coupling between the origin and this ﬁxed point, as shown in Fig. 5.3.

2. Both β-functions have an IR ﬁxed point for all Nf in the range

min max Nf < Nf < Nf . (5.15)

At the lower boundary of Nf , the BZ IR fixed point is larger than the max RS IR point, while the opposite is the case for Nf → Nf , as shown in Fig. 5.4. It is clear from this figure that the RS IR point is too large for perturbation theory to be trustworthy. In this case, it is necessary to define the anomalous dimension non-perturbatively.

70 5.2. RYTTOV-SANNINO (RS) β-FUNCTION

IR Figure 5.4: Comparison of the Banks-Zaks IR ﬁxed point αBZ and the IR IR RS min max ﬁxed point αEx. of the β -function for Nf < Nf < Nf , where both are renormalization scheme independent.

71 5.3. MODIFIED βRS-FUNCTION

5.3 Modiﬁed βRS-Function

5.3.1 Renormalization Gauge Transformation

The property of renormalization scheme dependence leads us to redeﬁne the running coupling α in terms of a new couplingα ˜. An interesting renormalization gauge transformation is1

α α˜ = , (5.16) 1 − kα where k is a positive constant. It is clear from this transformation that the new running couplingα ˜ is singular for a ﬁnite value of α; hence, the general structure of the βRS-function is unchanged under such a transformation. The βRS-function can be written in terms of the new couplingα ˜, leading to a new deﬁnition for the exact β-function, as follows

∂α˜ β˜RS(˜α) = βRS(α) . (5.17) ∂α

Using Eq. (5.16) and its inverse, the β˜RS-function can be written as

α˜2 b − 2 N γ˜(˜α) β˜RS(˜α) = − 3 f (1 + kα˜) (5.18) 4π 1 + (k − d)˜α with N 0 d = c (1 + 2b /b), (5.19) 2π whereγ ˜(˜α) is the modiﬁed anomalous dimension. Notice, the constant k ˜RS should satisfy that k > d in order to avoid any singularity in the β -function. In the framework of perturbation theory, it is easy to prove that the perturbative β-function can be reproduced from the modiﬁed β˜RS-function. This proof can be shown by expanding the denominator of Eq. (5.18) for a smallα ˜, where terms containing the constant k are cancelled with each other (see proof 2 in AppendixD).

1The idea of this transformation comes from decoupling top quarks at leading-order: αt = αb/(1+cαb), where c is constant [173]. Such a transformation was used by S. Weinberg to discuss possible behaviours of β-functions, Section 18.3 in [174].

72 5.3. MODIFIED βRS-FUNCTION

In a special case (k = d), the β˜RS-function reads

α˜2 2 ˜RS β (˜α) = − b − Nf γ˜(˜α) (1 + dα˜), (5.20) k=d 4π 3 where N 17N − 5N k = d = c c f . (5.21) 2π 11Nc − 2Nf The special β˜RS-function is also equivalent to the perturbative β-function, which can be seen by settingγ ˜(˜α) to its perturbative deﬁnition (see proof 3 in AppendixD). For pure Yang-Mills theory, one can easily show that the special β˜RS-function reduces to the perturbative two-loop expression:

α˜2 h i 1 17N ˜RS c βYM(˜α) = − b + dbα˜ with k = d = , (5.22) k=d 4π 2π 11 where b = 4πβ1 and bd = 4πβ2. Clearly, no non-trivial fixed point can be obtained in this case, in agreement with the previous results for pure Yang- Mills theory. In the presence of matter fields, there is an extra term in Eq. (5.20) proportional to the anomalous dimension of the matter mass, and hence there is a non-trivial fixed point. Order by order in perturbation theory, a reduction to two-loop order form is still possible (the ’t Hooft scheme), but non-perturbatively, not so: the transformation would be unacceptably singular.

5.3.2 Infrared Fixed point

The β˜RS-function in Eq. (5.18) still has a zero away from the origin at which the modiﬁed anomalous dimension must satisfy

11N γ˜(˜α) = c − 1. (5.23) 2Nf

The value of the anomalous dimension at the non-trivial fixed point is unchanged for both modified and unmodified βRS-functions. However, the non- trivial fixed point of the modified βRS-function is an IR fixed point for all max Nf < Nf due to the absence of the pole in this case. This means that there

73 5.3. MODIFIED βRS-FUNCTION

IR IR Figure 5.5: Comparison between the non-trivial fixed pointsα ˜Ex., αEx. of the modified (k = 1) and unmodified βRS-functions in the SU(3) gauge group RS with Nf = 16. Both β -functions have an IR fixed point at the same value, IR which is larger than the Banks-Zaks IR fixed point αBZ .

RS is an IR fixed point for a small Nf , unlike the perturbative and β -functions max that require Nf → Nf to have an IR fixed point. For the purpose of comparing, we have to truncate the modified anomalous dimensionγ ˜(˜α) to leading-order.1 At this order, the IR fixed point of the β˜RS- function is given by IR 4πNc[11Nc − 2Nf ] α˜ = 2 . (5.24) 3Nf (Nc − 1) IR For a fixed Nc, the IR fixed pointα ˜ has a physical value for all Nf < max Nf . To attain a clear view, let us consider the SU(3) gauge group and then ˜RS RS compare the β -function to the β -function for a small and large Nf with max max Nf < Nf . For Nf close to the maximum value Nf , both modified and unmodified βRS-functions have an IR fixed point at a large value compared to the Banks-Zaks IR fixed point, as shown in Fig. 5.5. On the other hand, if we go below the lower boundary of Nf , the Banks-Zaks IR fixed point

1The eﬀect of our scheme transformation starts when one considers the higher orders of the anomalous dimension. The coeﬃcient of the leading-order is listed in AppendixC.

74 5.3. MODIFIED βRS-FUNCTION

Figure 5.6: Comparison between the modified (k = 1) and βRS-functions for ˜RS the SU(3) gauge group with Nf = 6. In this case, the β -function is well defined, while the βRS-function is undefined for all 0 < α < αFP.

disappears and the non-trivial ﬁxed point of the βRS-function becomes an UV point (unphysical) due to the pole, while the β˜RS-function still has an IR ﬁxed point, as shown in Fig. 5.6.

Special Case

We have shown that the β˜RS-function has a special case when the constant of our transformation in Eq. (5.16) is equal to a particular value (k = d). In this case, the β˜RS-function reads

α˜2 2 ˜RS β (˜α) = − b − Nf γ˜(˜α) (1 + dα˜). (5.25) k=d 4π 3

It is clear from Eq. (5.8) and (5.19) that d is a function of the number of quark

ﬂavours Nf and colours Nc. Thus, the constant d can be either positive or negative,1 depending on these numbers. In the case of a negative d, the special β˜RS-function has two non-trivial ﬁxed points: one is given by Eq. (5.23), while

1For a positive d, the above analysis for Eq. (5.18) is still held.

75 5.3. MODIFIED βRS-FUNCTION

the second one (call it a special ﬁxed pointα ˜sp.) is given by

1 α˜ = − , (5.26) sp. d where d < 0 for all 17Nc/5 < Nf < 11Nc/2. One of these non-trivial fixed points is an IR fixed point, while another becomes an UV fixed point, depending on which appears first. For example, let us again consider the leading order of the anomalous dimension, then zeros of the special case are given by solving the following equation 2 Nf (Nc − 1) β0 − α˜ (1 + dα˜sp.) = 0. (5.27) 4πNc The solutions of this equation are given by Eq. (5.24) and (5.26), respectively.

At the minimum value of Nf (Nf ≈ 17Nc/5), the special ﬁxed pointα ˜sp. is an UV ﬁxed point and the second pointα ˜ is an IR point, as shown in Fig. 5.7.A. The location of these points changes 1 as the number of quark

flavours increases to the maximum value Nf ≈ 11Nc/2: the special fixed point becomes an IR fixed point and another one is an UV fixed point, as shown in Fig. 5.7.B. At a particular value of Nf , these two point merge to provide one fixed point, as shown in Fig. 5.8. Our result agrees with the result of [175, 176] in which the β-function has two fixed points (an IR and UV fixed point) in the conformal window. However, our result shows that these points do not disappear at the edge of the conformal window compared to [175].

1It is important to note that the behaviour of this emigration may be diﬀerent at next-leading order, and so on.

76 5.3. MODIFIED βRS-FUNCTION

Figure 5.7: At k = d, the β˜RS-function develops two non-trivial fixed points that could be an IR or UV fixed point. These fixed points merge with each other as Nf increases from 17Nc/5 to 11Nc/2. The special fixed point is an UV fixed point as Nf → 17Nc/5 (Figure A) and becomes an IR fixed point as Nf → 11Nc/2 (Figure B).

77 5.3. MODIFIED βRS-FUNCTION

˜RS Figure 5.8: At a particular value of Nf , the ﬁxed points of the special β - functions merge with each other. This can be seen for Nf = 14 in the SU(3) gauge group to leading-order.

78 Chapter 6

Chiral Analysis of QCD at Low-Energy Scales

This chapter contains the main part of the original research in this thesis. We propose using chiral perturbation theory (χPT) to derive an expression for the QCD effective charge at low-energy scales. The starting point is to find the values of the Drell-Yan ratio R(q2) and Adler function D(Q2) in the IR limit. Then we examine Grunberg’s definition [11, 22, 80] for the effective charge associated with R(q2) and the analogues space-like charge for D(Q2). For both charges, we find that monotonicity in scale dependence is not possible. This chapter is organised as follows. Section 6.1 reviews the basic con- cepts1 of χPT, chiral symmetry and effective Lagrangians for mesons.2 This section also reviews the construction of the lowest-order effective Lagrangian for the strong interactions. The effect of electromagnetic interactions on the lowest-order effective Lagrangian is discussed in Section 6.2. In this section, we calculate the contributions of pseudoscalar mesons π±,K± (IR limit) and light quarks (UV limit) to the two-point function T hJµ,Jνi of the electromagnetic current density Jµ(x) to leading-order. In Section 6.3, we show that the Drell-Yan ratio and Adler function tend to the same value in the 1A detailed introduction to chiral perturbation theory can be found in lecture notes and reviews, see e.g. [177, 178, 179, 180, 181]. 2It is easy to extend chiral perturbation theory to include baryon; however, this is beyond the scope of this thesis and can be viewed in [182, 183, 184, 185].

79 6.1. CHIRAL PERTURBATION THEORY respective IR limits q2 → 0 and Q2 → 0. Section 6.4 compares the UV and IR results for the eﬀective charges obtained from the Drell-Yan ratio R(q2) and Adler function R(Q2), with the result that neither eﬀective charge can be monotonic.

6.1 Chiral Perturbation Theory

Chiral perturbation theory (χPT) is the standard method [186, 187, 188] used to investigate non-perturbative strong interactions at low-energy scales. It describes strong interactions due to the pseudoscalar octet mesons (π, K, η) and depends on expansions in the meson momenta instead of the running coupling. The symmetry limit of χPT is the chiral SU(3)L × SU(3)R invariance that is spontaneously broken to SU(3)V via quark condensation, giving rise to the eight massless Goldstone bosons. In order to analyse the structure of QCD, it is first important to construct the effective Lagrangian associated with chiral symmetry breakdown. Thus, the purpose of this section is to review the most general effective Lagrangian for the Goldstone bosons, starting with the properties of chiral symmetry.

6.1.1 Chiral Symmetry

The key idea of constructing the effective field theory for the strong interactions at low-energy scales is to understand the symmetries of the QCD Lagrangian within the framework of χPT. For this purpose, it is important to write the invariant QCD Lagrangian Eq. (2.25) in terms of the left- and right-handed quark fields as follows

1 L = iq¯ Dq6 + iq¯ Dq6 − (¯q M q +q ¯ M q ) − Tr [F µνF ] QCD L L R R L q R R q L 2 µν + [heavy quark, gauge ﬁxing and ghost terms] , (6.1) where the left- and right-handed quark ﬁelds are given by

1 1 q = (1 − γ )q, q = (1 + γ )q. (6.2) L 2 5 R 2 5

80 6.1. CHIRAL PERTURBATION THEORY

Here, q refers to the light quark ﬂavours, q = (u, d, s) and Mq is the light quark mass matrix, Mq = diag(mu, md, ms). In the chiral limit, the quark ﬁeld components decouple, and hence the QCD Lagrangian Eq. (6.1) reads

1 µν LQCD → LQCD = iq¯L Dq6 L + iq¯R Dq6 R − Tr [F Fµν] + ...... (6.3) mu,d,s=0 2

As a result, the QCD Lagrangian becomes invariant under the global SU(3)L×

SU(3)R transformations of the left- and right-handed quark ﬁelds

(qL, qR) → (LqL, RqR) with L, R ∈ SU(3)L,R. (6.4)

aµ aµ The conserved vector V and axial-vector A currents for chiral SU(3)L ×

SU(3)R symmetry are given through the linear combinations of the Noether (J aµ) currents as follows

1 1 V aµ = J aµ + J aµ = qγ¯ µλaq, Aaµ = J aµ − J aµ = qγ¯ µγ λaq, (6.5) L R 2 L R 2 5 where the Noether currents are given by

1 J aµ = q¯ γµλaq . (6.6) L,R 2 L,R L,R

The vector and axial-vector charges associated with the chiral symmetry are deﬁned through Z Z a 3 a0 a 3 a0 QV = d xV (x),QA = d xA (x), (6.7) which satisfy the following commutation relations1

a b a b abc c a b abc c [QV ,QV ] = [QA,QA] = if QV , [QV ,QA] = if QA. (6.8)

Although the chiral SU(3)L × SU(3)R symmetry is a good approximation symmetry for the light quark ﬁelds, it is not obvious in the hadronic spec-

1Note that these commutation relations were the starting point of deﬁning the current Lie-Algebra [189, 190].

81 6.1. CHIRAL PERTURBATION THEORY trum since the parity doublets are not observed1; for example, octet scalar analogues of the light pseudoscalar mesons are not seen. Only the parity even subgroup SU(3)V is obvious from the structure of hadronic multiplets. The solution is to suppose that chiral symmetry is realised in the Nambu- Goldstone (NG) mode [191, 192, 193], where it is spontaneously broken:

SU(3)L × SU(3)R → SU(3)V . (6.9)

The axial transformations do not leave the ground state invariant, and hence there are eight pseudoscalar mesons, according to Goldstone’s theorem [193] (see also [194]). These mesons are the lightest degrees of freedom in the hadronic spectrum, namely π±, π0,K±,K0, K¯ 0 and η. It is important to note that they are not completely massless, but have small masses due to the quark masses matrix Mq. In QCD, a candidate for the order operator of spontaneous chiral symmetry breaking is the non-vanishing quark condensate

1 2 h0|[Qa , qγ¯ λbq]|0i = − h0|q¯{λa, λb}q|0i = − δabh0|qq¯ |0i= 6 0. (6.10) A 5 2 3

The low-energy interactions of the Goldstone bosons can be easily anal- ysed through an effective Lagrangian. Thus, the next task is to construct the effective Lagrangian for these pseudoscalar mesons, in particular the lowest- order effective Lagrangian.

6.1.2 Eﬀective Lagrangians for Strong Interactions

6.1.2.1 Transformation Properties of the Goldstone Bosons

In order to understand the eﬀective Lagrangian for strong interactions, it is important to examine the consequences for Goldstone bosons of the spontaneous chiral symmetry breaking of the chiral group

G ≡ SU(3)L × SU(3)R → SU(3)V ≡ H (6.11)

1The parity doublets would require that particles should have partners with the same spin, opposite parity and similar masses.

82 6.1. CHIRAL PERTURBATION THEORY down to a sub-group H conserved by the vacuum state. It is convenient to re- alise Eq. (6.11) non-linearly, where fields for Goldstone bosons are introduced without spurious partner fields, such as scalars. This realisation is usually implemented by the left- and right-handed components of a matrix field u(φ), which parametrise the coset space G/H in terms of the Goldstone bosons φa with (a = 1, ...., 8). These components transform under a chiral symmetry g ≡ (gL, gR) ∈ G as

† † uL(φ) → gLuL(φ)h (φ, g), uR(φ) → gRuR(φ)h (φ, g), (6.12) where h†(φ, g) ∈ H. Since the h†(φ, g) transformation occurs in both left- and right-handed components of the matrix ﬁeld u(φ), these components can be combined to introduce the matrix ﬁeld U(φ)

† U(φ) = uR(φ)uL(φ). (6.13)

The matrix field U(φ) transforms linearly under the chiral symmetry G. Notice that the matrix field U(φ) is not a polynomial function in φ since the left- and right-handed components of the matrix field u(φ) are non-linear under the chiral symmetry G. In the mesonic sector, the standard choice for the matrix field U(φ) is a 3 × 3 unitary matrix

U(φ) = exp[iφ/F ], (6.14) where the Goldstone bosons are deﬁned through φ by the following matrix

 0  √π + √η π± K+ 8 √ 2 6 X a a  0 η  φ = λ φ = 2  π− − √π + √ K0  . (6.15)  2 6  a=1 K− K¯ 0 − √2 η 6

The constant F is deﬁned through the Goldstone matrix element

µ a b ab µ −ipx h0|qγ¯ γ5λ q/2|φ (p)i = iF δ p e (6.16)

in the chiral symmetric limit. It is related to the pion constant decay Fπ

83 6.1. CHIRAL PERTURBATION THEORY through

Fπ = F [1 + O(mq)]. (6.17)

In χPT, this constant is always a parameter and usually measured in pion decay, such as

+ + π → l νl ⇒ F ≈ Fπ = 92.4 MeV, (6.18) where l refers to lepton ﬁelds (e, µ, τ).

6.1.2.2 The Lowest-Order Eﬀective Lagrangian

The eﬀective Lagrangian for the Goldstone bosons is a function of the unitary matrix U(φ). The Lagrangian terms are usually expressed in terms of an increasing number of derivatives. In general, the eﬀective Lagrangian takes a form such as

Leﬀ = L2n = L2 + L4 + L6 + ...... , (6.19)

1 where L2 is the lowest-order (leading-order) [195] effective Lagrangian. Since the matrix field U(φ) is a unitary matrix (UU † = 1), interacting terms should have at least two derivatives in the effective Lagrangian in order to have a non-vanishing term. Thus, the lowest-order effective Lagrangian for the strong interactions of the Goldstone bosons is unique and given by

F 2 h i L = Tr ∂ U †∂µU + M †U + MU † , (6.20) 2 4 µ where M is the matrix of the light quarks mass,

M = 2B diag(mu, md, ms). (6.21)

The constants B and F are related to each other through the quark condensate 2 h0|qq¯ |0i = −F B[1 + O(mq)]. (6.22)

The mass terms in the lowest-order eﬀective Lagrangian lead to some

1 L4 and L6 denote next-to-leading [187, 188] and next-to-next-to-leading [196, 197] orders, which will not be indicated in this thesis.

84 6.2. ELECTROMAGNETIC INTERACTIONS important results. First, by expanding the mass terms in Eq. (6.20) in powers of the Goldstone bosons, it is found that relationships between the mass of the Goldstone bosons and the mass of the light quarks are give by [183]

2 2 Mπ± = B(mu + md),Mπ0 = B(mu + md) − O(), 2 2 MK± = B(mu + ms),MK0 = B(md + ms), B M 2 = (m + m + 4m ) + O(), (6.23) η 3 u d s

2 where = (mu − md) . These relationships are known as the Gell-Mann- Oakes-Renner relations, which state that the mass of the Goldstone bosons 2 is proportional to the mass of light quarks, MGB ∝ mq [198]. In the isospin limit, we can ignore O() terms in the Gell-Mann-Oakes-Renner relations, 2 2 2 2 and hence ﬁnd that Mπ± = Mπ0 and MK± = MK0 . These relations lead to introducing the Gell-Mann-Okubo mass relation for the pseudoscalar mesons [199, 200] 2 2 2 4MK = Mπ + 3Mη , (6.24) which is the second result of including the mass terms into the lowest-order eﬀective Lagrangian in Eq. (6.20).

6.2 Electromagnetic Interactions

6.2.1 External Fields

The technique of the effective field theory becomes more powerful when one extends the effective QCD Lagrangian to include the effect of coupling quark

ﬁelds to external Hermitian matrix-valued ﬁelds, vµ(x), aµ(x), s(x) and p(x) [187, 188]. In this case, the chiral symmetric QCD Lagrangian takes the following form

Ext. µ LQCD → LQCD = LQCD +qγ ¯ (vµ + γ5aµ)q − q¯(s − iγ5p)q, (6.25)

85 6.2. ELECTROMAGNETIC INTERACTIONS

where the external ﬁelds vµ(x) and aµ(x) are usually expressed in terms of left- and right-handed components as follows

lµ ≡ vµ − aµ rµ ≡ vµ + aµ. (6.26)

The new QCD Lagrangian is invariant under the local chiral SU(3)R ×

SU(3)L symmetry at which the external ﬁelds transform as

† † lµ → LlµL + iL∂µL , † † rµ → RrµR + iR∂µR , s + ip → R(s + ip)L†. (6.27)

These transformations lead to generalising the lowest-order chiral eﬀective Lagrangian to include the external ﬁelds by introducing the χPT covariant derivative (∂µ → Dµ) with

∂µU → DµU = ∂µU − irµU + iUlµ, † † † † † ∂µU → DµU = ∂µU + iU rµ − ilµU , (6.28) which transforms locally as

† † † † DµU → RDµUL , DµU → LDµU R . (6.29)

Thus, the local invariant eﬀective Lagrangian for the Goldstone bosons at the lowest-order reads

F 2 h i L → LExt. = Tr D U †DµU + χ†U + χU † , (6.30) 2 2 4 µ with χ = 2B(s + ip). The external fields enable calculation of the most general Green functions of quark currents, including the effects of electromagnetic and semi-leptonic weak interactions. These effects can be seen by replacing the left- and right- handed components of the external fields as follows

e † lµ → eQf Aµ + √ (WµT+ + h.c.), 2 sin θw

86 6.2. ELECTROMAGNETIC INTERACTIONS

rµ → eQf Aµ, (6.31)

1 where Qf is the quark charge matrix, Qf = 3 diag(2, −1, −1). The matrix T+ gives the relevant Cabibbo-Kobayashi-Maskawa (CKM) matrix elements Vij,

  0 Vud Vus   T+ =  0 0 0  . (6.32) 0 0 0

The ﬁrst term in the right-hand side of (6.31) is responsible for electromagnetic interactions, while the second term is responsible for semi-leptonic weak interactions. Furthermore, the explicit chiral symmetry breaking through quark masses can be seen by the following replacement

s(x) → Mq. (6.33)

6.2.2 Eﬀective Lagrangian for Electromagnetic Inter- actions

In order to generate all effects of electromagnetic interactions,1 we should replace the fields lµ and rµ in the chiral covariant derivative by their values in Eq. (6.31) with p(x) = 0. In this case, the lowest-order effective Lagrangian in Eq. (6.30) can be rewritten in terms of sub-Lagrangians

Ext. L2 = Lkinetic + Lmass + LElect. + LWeak. (6.34)

Here, the first two terms are the kinetic energy and mass terms, giving to- gether the lowest-order effective Lagrangian for the strong interactions of the Goldstone bosons Eq. (6.20). The fourth term presents the contribution of the semi-leptonic weak interactions, which is beyond the scope of this thesis. The most important term here is the third one that gives the lowest-order effective Lagrangian for the electromagnetic interactions

1 One can obtain the effect of the photon field by setting the external field vµ(x) to eAµ. However, it is not enough to generate all effects due to virtual photons, see [183] for an example.

87 6.2. ELECTROMAGNETIC INTERACTIONS

Figure 6.1: The contribution of pseudoscalar mesons (π±,K±) to the photon vacuum polarisation in the IR limit.

F 2 L = Tr ieAµ∂ U †[U, Q] + ieA [U †,Q]∂µU − e2A Aµ[U †,Q][U, Q] . Elect. 4 µ µ µ (6.35) By expanding the matrix ﬁeld U as a power series in φ, we can obtain the electromagnetic interactions Lagrangian in terms of pseudoscalar mesons

− + + − 2 µ + − LElect. = ieAµ[φ ∂µφ − φ ∂µφ ] + e AµA [φ φ ], (6.36) where φ± = π±,K±. The task now is to calculate the amplitude of the corresponding Feynman diagrams to the above electromagnetic interactions. There are two one-loop diagrams for each meson, π± and K±, as shown in Fig. 6.1. Via explicit calculation of these diagrams,1 the amplitude of the electromagnetic interactions takes the following form

µν 2 µν 2 µ ν IR 2 Πφ± (q ) = (g q − q q )iΠφ± (q ), (6.37) IR

IR 2 where the scalar function Πφ± (q ) is given by

Z 1 m2 − x(1 − x)q2 IR 2 αQED X 2 φ± Πφ± (q ) = dx(1 − 2x) log 2 . (6.38) 4π m ± π±,K± 0 φ

2 Eq. (6.38) diverges logarithmically as mφ± → 0, while it tends to zero for q2 → 0. It is simple to consider ﬁrst its derivative with respect to q2

1The detailed calculation is shown in AppendixE. An analogue QED calculation is presented in Section 65 of [201].

88 6.2. ELECTROMAGNETIC INTERACTIONS

Figure 6.2: The contribution of the light quark ﬁelds (u, d, s) to the photon vacuum polarisation in the UV limit.

d ΠIR (q2) Z 1 φ± αQED 2 −x(1 − x) 2 = dx(1 − 2x) 2 2 . (6.39) dq 2π 0 mφ± − x(1 − x)q

2 In the chiral limit (mφ± = 0), Eq. (6.39) reads

IR 2 d Π ± (q ) α 1 φ = QED . (6.40) dq2 6π q2

In the UV region, an analogous interaction is obtained by setting the external field v(x) in Eq. (6.25) to eQf Aµ. In this case, the lowest-order effective electromagnetic Lagrangian for quark fields reads

UV µ LElect. = eAµqγ¯ Qf q. (6.41)

Since we deal with the light quarks, there are three corresponding Feynman diagrams, one for each ﬂavour (u, d, s), as shown in Fig. 6.2. Via explicit calculation of this diagram,1 one can write the amplitude of these quark ﬁelds in the UV limit as

µν 2 µν 2 µ ν UV 2 Πqq¯ (q ) = (g q − q q )iΠqq¯ (q ), (6.42) UV

UV 2 where the scalar function Πqq¯ (q ) is given by

Z 1 2 2 2αQED X mf − x(1 − x)q ΠUV(q2) = N Q2 dx(1 − x)x log . (6.43) qq¯ π c f m2 f=u,d,s 0 f

Also, Eq. (6.43) diverges logarithmically, but in both the UV (q2 → ∞) and

1For the detailed calculation, see AppendixE. Note that we make a sum over all colour ﬂavour in Eq. (6.43).

89 6.3. THE PHYSICAL QUANTITY R(Q2)

2 chiral (mf → 0) limits. Again, it is also simple to consider ﬁrst its derivative with respect to q2

UV 2 Z 1 d Πqq¯ (q ) 2αQED X −x(1 − x) = N Q2 dx(1 − x)x , (6.44) dq2 π c f m2 − x(1 − x)q2 f=u,d,s 0 f

2 and then apply the chiral limit of the light quark (mf = 0)

UV 2 d Πqq¯ (q ) αQED X 1 = N Q2 . (6.45) dq2 3π c f q2 f=u,d,s

It is clear now that Eq. (6.40) is singular in the IR limit as q2 → 0, and hence does not lead to any physical conclusion in its current form. To get a reasonable result, let us introduce a new function G(q2),

d Π(q2) G(q2) = q2 . (6.46) dq2

Using our results in Eq.(6.40) and (6.45), we can deﬁne the G(q2)-function in both limits as follows

αQED αQED X GIR(q2) = ,GUV (q2) = N Q2 . (6.47) 6π 3π c f f=u,d,s

2 IR So, the G(q )-function moves from a constant in the IR limit (G /αQED = ± ± UV 1/6π for π ,K ) to another constant in the UV limit (G /αQED = 2/3π for light quarks). It is closely related to the Drell-Yan ration R(q2) and, via continuation q2 → −Q2 to space-like momenta to the Adler function D(Q2).

6.3 The Physical Quantity R(q2)

For intermediate values of s = q2, the Drell-Yan ratio R(s) is a non-perturbative quantity. The ratio R(s) can be deﬁned by calculating the total cross section for a diagram such that in Fig. 6.3 whose imaginary part is deﬁned non-perturbatively.

90 6.3. THE PHYSICAL QUANTITY R(Q2)

Figure 6.3: Feynman diagram whose imaginary part gives the total cross section for e+e− annihilation to hadrons. In the IR limit, the imaginary part yields the mesons loop of π±,K± and the light quarks loop in the perturbative domain.

In general, the imaginary part of the hadronic vacuum polarisation Π(q2), Z Πµν(q2) = i(gµνq2 − qµqν)Π(q2) = −e2 d4x expiq.xh0|TJ µ(x)J ν(0)|0i, (6.48) is related to the total cross section for e+e− → hadrons through

4πα σ(e+e− → hadrons,s) = QED ImΠ(s), (6.49) s with

+ − 2 σ(e e → hadrons,s) + − + − 4παQED R(s) = + − + − , σ0(e e → µ µ , s) = , σ0(e e → µ µ , s) 3s (6.50) where s = q2 is the momentum scale in the time-like region. In order to calculate the total cross section corresponding to the IR and UV limit, we have to compute the imaginary part of Eq. (6.38) and (6.43) in these limits, respectively. First, we calculate the imaginary part of the hadronic intermediate state in the UV limit and apply it to Eq. (6.49) to ﬁnd the value of R(q2) as q2 → ∞. In this limit, the imaginary part of Eq. (6.43) is given by

s 2 2 αQED X 4mf 2mf ImΠUV(q2) = N Q2 1 − 1 + , (6.51) 3 c f q2 3q2 f=u,d,s

91 6.3. THE PHYSICAL QUANTITY R(Q2) which reads in the chiral limit as follows

UV 2 αQED X 2 ImΠ (q ) = Nc Qf . (6.52) m2 =0 3 f f=u,d,s

From Eq. (6.49), (6.50) and (6.52), the value of the Drell-Yan ratio R(q2) for light quarks in the UV limit is given within the chiral limit by

UV 2 X 2 R (q = ∞) = Nc Qf , (6.53) f=u,d,s which agrees with Eq. (2.51) as αs → 0 at high-energy scales. In the IR limit, the corresponding value for the Drell-Yan ratio R(q2) can be found by calculating the imaginary part of Eq. (6.38) in the chiral limit, which is given by α ImΠIR(q2) = QED . (6.54) m2 =0 6 φ± By applying Eq. (6.54) to (6.49) and (6.50), we ﬁnd the value of the Drell-Yan ratio R(q2) in the IR limit

RIR(q2 = 0) = 1/2 for π±,K±. (6.55)

It is important to note that the above analysis is for the time-like region. In the space-like region, the corresponding analysis can be introduce by ﬁnd- ing the values of the Adler function D(Q2) in these limits. In general, the Adler function is given by

Z ∞ 2 2 R(s) D(Q ) = Q ds 2 2 . (6.56) 0 (s + Q )

Applying the values of the Drell-Yan ratio R(q2) in the UV and IR limits into the Adler function in Eq. (6.56) and then forming the integral, we ﬁnd the same values for the space-like quantity D(Q2) in these limits:

2 X 2 2 D(Q ) → Nc Qf as Q → ∞ for light quarks f=u,d,s

92 6.4. QCD EFFECTIVE CHARGES

D(Q2) → 1/2 as Q2 → 0 for π±,K±. (6.57)

6.4 QCD Eﬀective Charges

In analogy with QED, a definition for the effective charges in QCD, can be obtained through the method of effective charges [11, 22], by relating the 2 2 QCD effective charge αeff (Q ) to a physical quantity σ(Q ) as follows

2 2 σ(Q ) = a + b αeﬀ (Q ) (6.58) at which the QCD eﬀective charge reads

σ(Q2) − a α (Q2) = , (6.59) eff b where a and b are constants. In this thesis, we use the time-like R(q2) and space-like D(Q2), as physical quantities, to define the QCD effective charge in both the time-like and space-like regions. In the time-like region, the QCD effective charge associated with R(q2) is given by R(q2) − a α (q2) = . (6.60) eff b In the UV limit, asymptotic freedom requires that the QCD effective charge goes to zero, hence the constant a must be equal to the value of the Drell-Yan ratio in the UV limit, R(∞) = 2 for the light quarks. In other words, comparing the value of the QCD effective charge at high-energy scales Eq. (6.60) to the approximated one-loop definition for R(q2) in Eq. (2.51), we find that

X 2 X 2 a = Nc Qf , b = Nc Qf /π. (6.61) f=u,d,s f=u,d,s

So, the time-like QCD eﬀective charge can be written as

π h i α (q2) = R(q2) − 2 (6.62) eﬀ 2 for the light quark ﬁelds. As R(q2) moves from the UV region with RUV = 2 to the IR region

93 6.4. QCD EFFECTIVE CHARGES

Figure 6.4: As q2 decreases, the Drell-Yan ratio R(q2) associated with the time-like eﬀective charge loses its monotonic property at the maximum of the ﬁrst resonance.

Figure 6.5: The Adler function D(Q2) associated with the space-like eﬀective charge loses its monotonic property as well, but at an unknown value.

94 6.4. QCD EFFECTIVE CHARGES

Figure 6.6: The ψ-function has a false zero A and an IR limit at a negative value for the associated effective charge. with RIR = 0.5, the value of the effective charge in Eq. (6.62) increases UV monotonically from αeff = 0 to a maximum value at the first resonance. At this maximum, the time-like effective charge loses its monotonic behaviour, decreasing to negative (unphysical) values in the IR region (Fig. 6.4). Thus, we can conclude from RUV > RIR that the time-like effective charge associated with R(q2) cannot be monotonic for the whole q2. In the space-like region, the analogous analytic QCD effective charge associated with D(Q2) is give by

π α (Q2) = D(Q2) − 2 , (6.63) eff 2 where dΠ(Q2) D(Q2) = Q2 . (6.64) dQ2 It is clear from Eq. (6.63) that the behaviour of the space-like effective charge is similar to the time-like one. As Q2 decreases, the effective charge increases UV monotonically from αeff = 0 to a maximum value where there is a false zero of the ψ-function, and then it falls to −3π/4 in the IR limit, Fig. 6.5 and 6.6. The difference between the time- and space-like charges is that the location of the maximum value of the space-like effective charge is not known and the UV-IR behaviour of the space-like effective charge is analytic throughout.

95 Chapter 7

Conclusion

7.1 Summary of Results

The aim of this thesis has been to investigate the behaviour of QCD at low- energy scales and particularly to explore the conditions under which QCD has a physical IR fixed point. In the low-energy region, a non-perturbative definition for the QCD running coupling is required due to the limitation of the perturbative running coupling to the UV region. This non-perturbative running coupling should be a monotonic and an analytic function of the space-like scale Q2, which should be related to a physical amplitude. Such a definition can be obtained through the effective charges method [11, 22, 80], but so far, we have not found an example which is monotonic over the whole energy scale. As part of our search for an IR fixed point, we discuss proposals for exact QCD β-functions [21]. These exact β-functions have a non-trivial fixed point αFP, but also have a space-like pole at a finite value for the running coupling αpole. The behaviour of the exact β-function is affected by this pole when it appears between α = 0 and α = αFP, making the prescription for the β- function unphysical. Then we modify the proposed exact β-function for QCD by redefining the running coupling α in terms of a new couplingα ˜, in order to end with a non-singular form. The modified exact β˜-function is well defined for allα ˜ ∈ (0, α˜FP) and its non-trivial fixed point is an IR fixed pointα ˜IR.

This IR ﬁxed point occurs for all values of the quark ﬂavours Nf consistent

96 7.1. SUMMARY OF RESULTS

max with asymptotic freedom, i.e. 0 < Nf < Nf . The exact β-function is expressed in terms of the perturbative anomalous dimension, but we observe that its IR fixed pointα ˜IR is too large to be allowed within the framework of perturbation theory. Thus, a non-perturbative definition for the anomalous dimension is required, such that the perturbative result is reproduced for a small α. These proposals can be criticised on the grounds that they have been artificially produced from special definitions or ansaetze and not derived from the non-perturbative properties on which truly physical answers should depend. The main idea of this thesis is to use chiral perturbation theory to discuss the IR behaviour of QCD effective charges related to two physical quantities; the Drell-Yan ratio R(q2) and the Adler-function D(Q2) for the time-like and space-like regions, respectively. We find that both R(q2) and D(Q2) tend to the same value for π±,K± in the IR limit

R(0) = D(0) = 0.5, (7.1) compared with the UV result

X 2 R(∞) = D(∞) = Nc (light quark charges) = 2 (7.2)

for Nc = 3 colours. Since the gluonic correction at q2 ∼ ∞ is positive, it is evident that neither the time-like effective charge nor its analytic space-like version can be monotonic over all q2 or Q2 values. For the time-like charge derived from R(q2), we observe (even without Eq. (7.1)) that it fails to be monotonic at the highest resonance, i.e. the β-function has a false zero there. For the effective charge associated with the space-like quantity D(Q2), Eq. (7.2) implies that it cannot be monotonic as well. There must be a false zero at some intermediate value of Q2; exactly where cannot be determined from our general analysis. In the light of these negative but important results, we considered whether any of the constraints above could be relaxed. Can α be given a non- perturbative definition which is analytic and tends monotonically from its IR finite value to zero in the UV limit, but does not coincide with the per-

97 7.2. WHAT NEXT? turbative deﬁnition of α in that limit? Certainly that can be done, as has been already shown for a running coupling based on holographic anti-de Sit- ter (AdS) mapping [147]. However, that leaves open the non-perturbative question of how to match the non-perturbative deﬁnition to the perturbative one in an analytic fashion consistent with being in the asymptotically region. We found no satisfactory solution for that problem.

7.2 What Next?

During our search for an IR fixed point in the literature, and even in our work in this thesis, we could not find an acceptable physical QCD running coupling for both the perturbative and non-perturbative regions. This meant that we were not able to decide whether a physical IR fixed point exists or not. Future research into this topic will need to take into account the following observations. Consider two different classes of gauge theories:

• Class A does not have an IR ﬁxed point and its non-perturbative running coupling tends to inﬁnity in the IR limit:

2 αA → ∞ as Q → 0. (7.3)

• Class B has an IR ﬁxed point at αIR:

IR 2 αB → α as Q → 0. (7.4)

What if these non-perturbative running couplings are related to each other through αB αA = IR ? (7.5) 1 − αB/α Then the relationship between the corresponding β-functions is

IR 2 βB(αB) = βA(αA)[1 − αB/α ] (7.6)

It is clear from Eq. (7.6) that the βB-function of the class B has a zero away

98 7.2. WHAT NEXT?

from the origin (an IR fixed point) as αB → αIR, while the βA-function is non- singular everywhere except possibly at αA ∼ ∞. We assume that β-functions associated with these two classes are defined non-perturbatively.1 Apparently, the singularity in Eq. (7.5) is so strong that it relates theories with different physics. For class B theories, one expects scale invariance at αIR realised either explicitly or in Nambu-Goldstone mode. There is no reason to suppose that class A theories have anything to do with scale invariance. The reason for this apparent puzzle is that the transformation in Eq. (7.5) is not analytic at the physical point q2 = −Q2 = 0. Consequently, the set of physics preserving transformations must respect analyticity for the space-like region extended to a neighborhood of q2 = 0. Further progress will depend on whether the method of effective charges can be extended to preserve analyticity and achieve monotonicity. Two av- enues of research are evident:

• Consider various examples of measurable amplitudes.

• Replace linear non-perturbative deﬁnitions like

α(q2) R(q2) = 2 1 + (7.7) π

by expressions of the form

R(q2) ≡ 2F (¯α(q2)/π), (7.8)

where F (¯α/π) is an analytic function shifting the requirements F (0) = F 0 = 1.

1 If we use the two-loop prescription for (say) the βA-function, the βB-function does not IR have an IR ﬁxed point as αB → α .

99 Appendix A

Feynman Diagrams

In this appendix, we list the Feynman rules that are used in thesis. In Fig. A.1, we show the Feynman rules for gqq¯ vertex and quark propagator in QCD,1 which are used in the discussion of dimensional regularization in Chapter 2. Fig. A.2 shows the Feynman rules for γφ−φ+, γγφ−φ+ and γqq¯ vertices (φ± = π±,K±) that are obtained from χPT. We use these vertices to calculate the contribution of electromagnetic interactions to the imaginary part of the total cross section of e+e− → hadrons in both the IR and UV limits, Chapter 6.

Figure A.1: The Feynman rules for gqq¯ vertex and quark propagator in QCD.

1The Feynman rules for gluons themselves interactions and ghost-gluon vertices can be viewed in any textbook of quantum ﬁeld theory, see e.g. [32].

100 FEYNMAN DIAGRAMS

Figure A.2: The Feynman rules for γφ−φ+, γγφ−φ+ and γqq¯ vertices in χPT, including the rule of scalar propagator.

101 Appendix B

Reference Formulaes

In this appendix, we list all matrices introduced in Chapter 2 and their important relations. Also, this appendix shows intermediate steps of dimensional regularization.

B.1 Gell-Mann Matrices

The Gell-Mann matrices are a set of 3 × 3 traceless, hermitian and complex matrices that are given by

 0 1 0   0 −i 0   1 0 0        λ1 =  1 0 0  λ2 =  i 0 0  λ3 =  0 −1 0  0 0 0 0 0 0 0 0 0  0 0 1   0 0 −i   0 0 0        λ4 =  0 0 0  λ5 =  0 0 0  λ6 =  0 0 1  1 0 0 i 0 0 0 1 0  0 0 0   1 0 0  λ =  0 0 −i  λ = √1  0 0 0  (B.1) 7   8 3   0 i 0 0 0 −2

These martices satisfy the following relations:

abc Tr[λaλb] = 2δab, [λa, λb] = 2if λc (B.2)

102 B.2. GAMMA MATRICES

B.2 Gamma Matrices

In Dirac representation, the gamma matrices satisfy the following properties:

{γµ, γν} = 2gµν µ γ γν = 4 µ ν ν γ γ γµ = −2γ µ ν ρ νρ γ γ γ γµ = −2g µ ν ρ σ σ ρ ν γ γ γ γ γµ = −2γ γ γ {γµ, γ5} = 0, (B.3) where the traces of them are given by

Tr(1) = 4 Tr(odd numbers ofγµ) = 0 µ µν Tr(γ γν) = 4g µ ρ σ µν ρσ µρ νσ µσ νρ Tr(γ γνγ γ ) = 4[g g − g g + g g ] Tr(γ5) = 0 µ 5 Tr(γ γνγ ) = 0 µ ρ σ 5 µνρσ Tr(γ γνγ γ γ ) = −4i . (B.4)

B.3 Loop Integrals

In dimensional regularization, one can use the following expressions to form loop integrals over the momentum l

Z ddl 1 (−1)ni Γ(n − d/2) 1 n−d/2 = (2π)d (l2 − ∆)n (4π)d/2 Γ(n) ∆ Z ddl l2 (−1)n−1i d Γ(n − d/2 − 1) 1 n−d/2−1 = (2π)d (l2 − ∆)n (4π)d/2 2 Γ(n) ∆ Z ddl l4 (−1)ni d(d + 2) Γ(n − d/2 − 2) 1 n−d/2−2 = (2π)d (l2 − ∆)n (4π)d/2 4 Γ(n) ∆ Z ddl lµlν (−1)n−1i gµν Γ(n − d/2 − 1) 1 n−d/2−1 = (2π)d (l2 − ∆)n (4π)d/2 2 Γ(n) ∆

103 B.4. DIMENSIONAL REGULARIZATION

Z ddl lµlνlρlσ (−1)ni Γ(n − d/2 − 2) 1 n−d/2−2 = (2π)d (l2 − ∆)n (4π)d/2 Γ(n) ∆ 1h i × gµνgρσ − gµρgνσ + gµσgνρ (B.5) 4

If the loop integral converges, then we can set the dimension to 4 (d = 4) from the beginning. Otherwise, the Γ has a pole at d = 4, then we have to expand the Γ and ∆ as

1 n−d/2 = 1 − (2 − d/2)Log∆ + O(2 − d/2), (B.6) ∆ 1 Γ(n − d/2) = − γ + O(2 − d/2), (B.7) (2 − d/2) where γ is the Euler-Mascheroni constant, γ = 0.5772. Thus, the total expression of Γ and ∆ can be written as;

1 n−d/2 1 Γ(n − d/2) = − Log∆ − γ + O(2 − d/2). (B.8) ∆ (2 − d/2)

Note that the above expression for the loop integral is evaluated within Minkowski space and one can evaluated it within Euclidean space by the following replacement 0 0 2 2 l = ilE, l = −lE. (B.9)

B.4 Dimensional Regularization

In order to discuss the idea of the dimensional regularization in the framework of perturbation theory, let us consider the Feynman diagram for the quark loop in Fig. B.1. This loop refers to the strong interaction between quark and gluon at high-energy scales, where perturbation theory can be applied. The corresponding amplitude is deﬁned through

Z 4 µν 2 2 d k µ a i(k + mf ) ν b i(k + q + mf ) Π (q ) = (−1)(ig) 4 Tr γ t 2 2 γ t 2 2 (2π) (k − mf ) ((k + q) − mf ) Z 4 µν 2 µ ν µ ν µ ν 2 a b d k g (mf − k.(k + q)) + 2k k + k q + q k = −4g Tr[t , t ] 4 2 2 2 2 (2π) (k − mf )((k + q) − mf ) (B.10)

104 B.4. DIMENSIONAL REGULARIZATION

Figure B.1: Leading-order contribution of the quark loop to the gluon propagator.

Feynman parameter

To form the above integral, we have to use the Feynman parameter

1 Z 1 1 = dx 2 . (B.11) AB 0 [xA + (1 − x)B]

By replacing each k by l − xq, the integral can be written as

Z 1 Z 4 µν 2 2 a b d l Π (q ) = −4g Tr[t , t ] dx 4 0 (2π) gµν(m2 + x(1 − x)q2) − 2x(1 − x)qµqν + 2lµlν − gµνl2 + O(l) × f , (l2 − ∆)2 (B.12)

2 2 where ∆ = mf −x(1−x)q . In d-dimension, the linear terms in l vanish by the property of symmetric integration. The remaining terms are UV divergent; thus, regularization technique must be used to remove these divergences. To see how the dimensional regularization works, we choose for example the term whose numerator is independent of l and rewrite it as follows [32]

Z 4 Z Z ∞ d−1 d l 1 dΩd l 4 2 2 = d dl 2 2 , (B.13) (2π) (l − ∆) (2π) 0 (l − ∆) where dΩd is the area of a d-dimension that is given by

Z 2πd/2 dΩ = . (B.14) d Γ(d/2)

105 B.4. DIMENSIONAL REGULARIZATION

Using the Euler-function, the second factor in Eq. (B.13) can be written as

Z ∞ ld−1 1 1 2−d/2 Γ(2 − d/2)Γ(d/2) dl 2 2 = . (B.15) 0 (l − ∆) 2 ∆ Γ(2)

Then, the integral in Eq. (B.13) reads

Z d4l 1 i Γ(2 − d/2) 1 2−d/2 = . (B.16) (2π)4 (l2 − ∆)2 (4π)d/2 Γ(2) ∆

In the same manner, the term whose numerator is proportional to l2 is given by Z d4l l2 −i d Γ(1 − d/2) 1 1−d/2 = . (B.17) (2π)4 (l2 − ∆)2 (4π)d/2 2 Γ(2) ∆ Evaluating the remaining terms in Eq. (B.12) and using the formulaes of loop integrals in Eq. (B.5), we end with the following expression for the quark loop

Πµν(q2) = (gµνq2 − qµqν)iΠ(q2), (B.18) where the scalar part Π(q2) is given by

Z 1 2−d/2 2 1 Γ(2 − d/2) 1 2 Π(q ) = −2πα dx(1 − x)x d/2 , α = g /4π. 0 (4π) Γ(2) ∆ (B.19)

106 Appendix C

The β-Function and Anomalous Dimension

In perturbation theory, the renormalization group equation implies the following expressions for the perturbative β-function and anomalous dimension of the fermion ﬁelds

∂α(µ2) β(α) = µ2 = −β α2 − β α3 − β α4 − β α5 − O(α6), ∂µ2 1 2 3 4 ∂lnM(µ2) γ(α) = −µ2 = γ α + γ α2 + γ α3 + O(α4), (C.1) ∂µ2 1 2 3 where M(µ2) is the renormalized mass and α = g2/4π is the renormalized coupling.

C.1 Perturbative Coeﬃcients

Within the MS renormalization scheme, the coeﬃcients of the perturbative β-function are given by [2,3, 38, 13, 39, 40, 41, 42]

1 11 4 β = C − T N , 1 (4π) 3 A 3 f f 1 34 20 β = C2 − 4C T N − C T N , 2 (4π)2 3 A f f f 3 A f f

107 C.1. PERTURBATIVE COEFFICIENTS

1 2857 205 β = C3 + 2C2T N − C C T N 3 (4π)3 54 A f f f 9 f A f f 1415 44 158 − C2 T N + C T 2N 2 + C T 2N 2 , 27 A f f 9 f f f 27 A f f 1 150653 44 39143 136 β = − ζ C4 + − + ζ C3 T N 4 (4π)4 486 9 3 A 81 3 3 A f f 7073 656 + − ζ C2 C T N 243 9 3 A f f f 4204 352 + − + ζ C C2T N + 46C3T N 27 9 3 A f f f f f f 7930 224 1352 704 + − ζ C2 T 2N 2 + − ζ C2T 2N 2 813 9 3 A f f 27 9 3 f f f 17152 448 424 + − ζ C C T 2N 2 + C T 3N 3 243 9 3 A f f f 243 A f f abcd abcd 1232 3 3 dA dA 80 704 + Cf Tf Nf + − + ζ3 243 NA 9 3 abcd abcd df dA 512 1664 +Nf − ζ3 NA 9 3 abcd abcd 2 df df 704 512 +Nf − + ζ3 NA 9 3 (C.2)

In this scheme, the coeﬃcients of the perturbative anomalous dimension are also given by [202]

3 γ = C , 1 4π f 1 3 97 10 γ = C2 + C C − C T N , 2 (4π)2 2 f 6 A f 3 f f f 1 129 129 11413 γ = C3 − C2C + C C2 3 (4π)3 2 f 4 f A 108 f A 140 − C T 2N 2 + C2T N − 46 + 48ζ 27 f f f f f f 3 −556 +C C T N − 48ζ , f A f f 27 3

108 C.1. PERTURBATIVE COEFFICIENTS

1 −1261 15349 γ = − 336ζ C4 + + 316ζ C3C 4 (4π)4 8 3 f 12 3 f A −34045 + − 152ζ + 440ζ C2 C2 36 3 5 A f 70055 1418 + + ζ − 440ζ C3 C 72 9 3 5 A f −280 + + 552ζ − 480ζ C3T N 3 3 5 f f f −664 128 + + ζ C T 3N 3 81 9 3 f f f −8819 + + 368ζ − 264ζ + 80ζ C2C T N 27 3 4 5 f A f f −65459 2684 + − ζ + 264ζ + 400ζ C2 C T N 162 3 3 4 5 A f f f 304 + − 160ζ + 96ζ C2T 2N 2 27 3 4 f f f 1342 + + 160ζ − 96ζ C C T 2N 2 81 3 4 f A f f abcd abcd abcd abcd df dA df df + − 32 + 240ζ3 + Nf 64 − 480ζ3 . Df Df (C.3)

abcd abcd The symbols dA and df are the higher-order constants of the adjoint and fundamental representation, respectively. They are usually expressed in terms of the symmetric tensors as follows

1 h dabcd = Tr CaCbCcCd + CaCbCdCc + CaCcCbCd + CaCcCdCb A 6 i +CaCdCbCc + CaCdCcCb , 1 h dabcd = Tr T aT bT cT d + T aT bT dT c + T aT cT bT d + T aT cT dT b f 6 i +T aT dT bT c + T aT dT cT b . (C.4)

Here, ζ3,4,5 are the Riemann zeta-functions (ζ3 = 1.2020569, ζ4 = 1.0823232,

ζ5 = 1.0369277). Nf and NA are the number of quark ﬂavours and generators of the adjoint representation, respectively. CA and Cf are the quadratic

109 C.2. THE SU(NC ) COLOURED GAUGE GROUP

Casimir operators of the adjoint and fundamental representation, where Tf is the generators of the fundamental representation with dimension Df . These operators are given by

ab acd bcd a a ab a b a abc CAδ = f f ,Cf δij = [T T ]ij,Tf δ = tr(T T ), [C ]bc = −if , (C.5) a where the matrices [C ]bc are the generators of the adjoint representation.

C.2 The SU(Nc) Coloured Gauge Group

For the standard renormalization of the SU(Nc) gauge group, the above colour operators are represented by the following expressions

2 Nc − 1 1 CA = Nc,Cf = ,Tf = , 2Nc 2 dabcddabcd N 2(N 2 + 36) dabcddabcd N (N 2 + 6) A A = c c , f A = c c , NA 24 NA 48 abcd abcd 4 2 df df Nc − 6Nc + 18 = 2 , NA 96Nc dabcddabcd (N 2 − 1)(N 2 + 6) f A = c c , Df 48 abcd abcd 2 4 2 df df (Nc − 1)(Nc − 6Nc + 18) = 3 . (C.6) Df 96Nc

In QCD (Nc = 3), the coeﬃcients of the perturbative QCD β-function read

1 h 2 i β = 11 − N , 1 (4π) 3 f 1 h 38 i β = 102 − N , 2 (4π)2 3 f 1 2857 5033 325 β = − N + N 2 , 3 (4π)3 2 18 f 54 f 1 149753 1078361 6508 β = + 3564ζ − + ζ N 4 (4π)4 6 3 162 27 3 f 50065 6472 1093 + + ζ N 2 + N 3 . (C.7) 162 81 3 f 729 f

110 C.2. THE SU(NC ) COLOURED GAUGE GROUP

Also, the coeﬃcients of the perturbative anomalous dimension for the SU(3) gauge group are give by

1 γ = , 1 π 1 202 20 γ = − N , 2 (4π)2 3 9 f 1 2N 1108 140 γ = 1249 − f + 80ζ − N 2 , 3 (4π)3 3 9 3 81 f 1 4603055 135680 γ = + ζ − 8800ζ 4 (4π)4 162 27 3 5 91723 34192 18400 + − − ζ + 880ζ + ζ N 27 9 3 4 9 5 f 5242 800 160 332 64 + + ζ − ζ N 2 + − + ζ N 2 . 243 9 3 3 4 f 243 27 3 f (C.8)

111 Appendix D

Proposed Exact β-Function at a Small α

In Chapter5, we indicate that the perturbative β-function can be reproduced from the proposed exact β-function for the non-supersymmetric QCD when it is expanded for a small α. Thus, this appendix is introduced to show the complete proof of this agreement.

D.1 Proof 1

In this proof, we show that the exact RS β-function goes to the perturbative form when it is expanded for a small α. For the purpose of the proof, let us recall the RS β-function

α2 b − 2 N γ(α) βRS(α) = − 3 f , (D.1) 4π 1 − dα where

N 0 d = c (1 + 2b /b), b = (11N − 2N )/3, 2π c f 0 N b = N − f , γ(α) = γ α + γ α2 + O(α3). (D.2) c 2 1 2

112 D.2. PROOF 2

For a small coupling α, the denominator of the RS β-function can be expanded as follows (1 − dα)−1 = 1 + dα + d2α2 + d3α3 + O(α4), (D.3) then the RS β-function takes the following form

α2 2 βRS(α) = − b − N γ(α) 1 + dα + d2α2 + d3α3 + O(α4) 4π 3 f α2 2 2d = − b + bdα + bd2α2 − N γ(α) − N αγ(α) + O(α3) 4π 3 f 3 f α2 2 = − b + bd − N γ α 4π 3 f 1 2 2d + bd2 − N γ − N γ α2 + O(α3) 3 f 2 3 f 1 0 2 0 3 0 4 5 = −β1α − β2α − β3α + O(α ) 2 3 0 4 5 = −β1α − β2α − β3α + O(α ), (D.4) where it is easy to show that

0 0 2 β = β = b/4π, β = β = bd − N γ /4π, 1 1 2 2 3 f 1 0 2 2d β 6= β = bd2 − N γ − N γ /4π. (D.5) 3 3 3 f 2 3 f 1

The first two coefficients of the RS β-function is equivalent to the coefficient of the one- and two-loop order of the perturbative β-function , where the difference starts at the third- and higher-loop orders. Note that it is unknown under which scheme the RS β-function was proposed. However, the above proof tells us that the RS β-function is not proposed within the MS scheme since the the third coefficient is not equal to β3.

D.2 Proof 2

The second proof is to show that our modiﬁed RS β-function is also in agreement with the perturbative β-function at a small α. The modiﬁed RS β-

113 D.2. PROOF 2 function is given by

α˜2 b − 2 N γ˜(˜α) β˜RS(˜α) = − 3 f (1 + kα˜), (D.6) 4π 1 + (k − d)˜α where k ≥ d. Notice, the coeﬃcients of the anomalous dimension are scheme dependent, and hence can be written as

2 3 4 γ˜(˜α) = γ1α˜ +γ ˜2α˜ +γ ˜3α˜ + O(˜α ). (D.7)

For the purpose of the proof, let us rewrite the modiﬁed RS β-function as follows

α˜2 2 β˜RS(˜α) = − b − N γ˜(˜α) (1 + kα˜)[1 − (d − k)˜α]−1, (D.8) 4π 3 f then, in the same manner, we can expand it as

α˜2 2 β˜RS(˜α) = − b − N γ˜(˜α) (1 + kα˜)[1 + dα˜ − kα˜ + (d − k)2α˜2 + O(˜α3)] 4π 3 f α˜2 2 = − b − N γ˜(˜α) 1 + dα˜ + (d2 − kd)˜α2 + O(˜α3) 4π 3 f α˜2 2 = − b + bd − N γ α˜ 4π 3 f 1 2 2d + bd2 − bkd − N γ˜ − N γ α˜2 + O(˜α3) 3 f 2 3 f 1 ˜ 2 ˜ 3 ˜ 4 5 = −β1α˜ − β2α˜ − β3α˜ + O(α ) 2 3 ˜ 4 5 = −β1α˜ − β2α˜ − β3α˜ + O(˜α ), (D.9) where the third coeﬃcient in this case is give by

0 2 2d β 6= β 6= β˜ = bd2 − bkd − N γ˜ − N γ /4π. (D.10) 3 3 3 3 f 2 3 f 1

114 D.3. PROOF 3

D.3 Proof 3

In the special case (k = d), the modiﬁed RSβ-function reads

α˜2 2 ˜RS β (˜α) = − b − Nf γ˜(˜α) (1 + dα˜). (D.11) k=d 4π 3

It is more easy to show that the perturbative β-function can be reproduced in this case as well by replacingγ ˜(˜α) with the modiﬁed perturbative anomalous dimension in Eq. (D.7). The proof can be presented as follows

α˜2 2 ˜RS β (˜α) = − b − Nf γ˜(˜α) (1 + dα˜) k=d 4π 3 α˜2 2 2N = − b + bd − N γ α˜ − f γ d +γ ˜ α˜2 + O(˜α3) 4π 3 f 1 3 1 2 ¯ 2 ¯ 3 ¯ 4 5 = −β1α˜ − β2α˜ + β3α˜ + O(α ) 2 3 ¯ 4 5 = −β1α˜ − β2α˜ + β3α˜ + O(α ), (D.12)

0 ˜ ¯ 1 where β3 6= β3 6= β3 6= β3 = 6π Nf (γ1d +γ ˜2).

115 Appendix E

Loop Integrals

In this appendix, we show the detailed calculation of loop integrals that are introduced in Chapter6. There are three loop integrals: two for each pseudoscalar mesons π±,K± and one for each light quark u, d, s.

E.1 Meson-Loop Integral

In the IR limit, there are two loops for each pseudoscalar mesons, as shown in Fig. E.1. The explicit expression for the ﬁrst loop in this ﬁgure is given by

Z 4 µ ν µν 2 2 d k (2k + q) (2k + q) Π1 (q ) = e 4 2 2 2 2 . (E.1) (2π) [(q + k) − mφ± ][k − mφ± ]

Using the Feynman parameter and shifting the momentum k to k − xq, we ﬁnd Z 1 Z 4 µ ν 2 µ ν µν 2 2 d k 4k k + (1 − 2x) q q Π1 (q ) = e dx 4 2 2 (E.2) 0 (2π) (k − ∆) with 2 2 ∆ = mφ± − x(1 − x)q , (E.3) where φ± refers to π±,K±. Note that linear terms in k are cancelled by symmetric integration. In d-dimension, we can write this amplitude as

Z 1 Z d µν 2 2 µ ν µν 2 2 4−d d k (4/d)g k + (2x − 1) q q Π1 (q ) = e dx µ d 2 2 , (E.4) 0 (2π) (k − ∆)

116 E.1. MESON-LOOP INTEGRAL

Figure E.1: Leading-order diagrams for mesons loop (π±,K±) in the IR limit, including momentum labels. where µ is the renormalization scale and kµkν → (1/d)gµνk2. The amplitude of the second loop in Fig. E.1 is expressed by

Z 4 µν 2 2 µν d k 1 Π2 (q ) = −2e g 4 2 2 . (E.5) (2π) k − mφ±

To combine this amplitude to the previous one, we have to multiply the 2 2 numerator and denominator of Eq. (E.5) by (k + q) − mφ± . By following the same steps in the previous loop integral, we ﬁnd that

Z 1 Z d k2 + (x − 1)2q2 − m2 µν 2 2 µν 4−d d k φ± Π2 (q ) = −2e g dx µ d 2 2 . (E.6) 0 (2π) (k − ∆)

Then the combined result reads

µν 2 µν 2 µν 2 Πφ± (q ) = Π1 (q ) + Π2 (q ) Z 1 Z d 2 2 µν 4−d d k k (2/d − 1) = 2e g dx µ d 2 2 0 (2π) (k − ∆) Z 1 2 h 2 µ ν µν 2 2 µν 2 i + e dx (1 − 2x) q q − 2g (x − 1) q + 2g mφ± 0 Z ddk 1 ×µ4−d . (E.7) (2π)d (k2 − ∆)2

Using the loop integral formulaes in Appendix B,

Z ddk k2(2/d − 1) Γ() 1 = −i∆ , (2π)d (k2 − ∆)2 (4π)2− ∆ Z ddk 1 Γ() 1 = i , (E.8) (2π)d (k2 − ∆)2 (4π)2− ∆

117 E.1. MESON-LOOP INTEGRAL the combined amplitude takes the following form

Z 1 µν 2 2 h µν 2 2 µ νi Πφ± (q ) = ie dx 2(2x − 1)(1 − x)g q + (1 − 2x) q q 0 Γ() 1 ×µ4−d , (E.9) (4π)2− ∆ where = 2 − d/2. Since ∆ is symmetric with respect to x ↔ (1 − x) then linear terms in x transform as x → x/2 + (1 − x)/2 = 1/2 [32], hence we ﬁnd

2(2x − 1)(1 − x) → −(1 − 2x)2. (E.10)

Finally, the combined amplitude in Eq. (E.7) can be written as

µν 2 µν 2 µ ν ¯ 2 Πφ± (q ) = (g q − q q )iΠφ± (q ), (E.11) where the scalar part is given by

Z 1 2 2 2 4−d Γ() 1 ¯ ± Πφ (q ) = −e dx(1 − 2x) µ 2− 0 (4π) ∆ α Z 1 1 = − µ4−d dx(1 − 2x)2 − γ + log(4π) − log ∆ + O() , 4π 0 (E.12) where α = e2/4π and γ is the Euler-Mascheroni constant. The scalar function Π(¯ q2) is still divergent due to the 1/ term in the limit of d → 4. The best way to cancel this divergence, making the scalar function ﬁnite, is to subtract its value at q2 = 0. In this case, we can write the ﬁnite scalar function as

2 ¯ 2 ¯ Πφ± (q ) = Πφ± (q ) − Πφ± (0) Z 1 m2 − x(1 − x)q2 α 2 φ± = dx(1 − 2x) log 2 . (E.13) 4π 0 mφ±

The second way of canceling the divergent term in Eq. (E.12) is to take the derivative of the scalar function with respect to q2 as

118 E.2. QUARK-LOOP INTEGRAL

Figure E.2: Leading-order diagram for the light quark loop in the UV limit, including momentum labels.

¯ 2 Z 1 dΠφ± (q ) α 2 −x(1 − x) 2 = dx(1 − 2x) 2 2 , (E.14) dq 4π 0 mφ± − x(1 − x)q which is equal to the derivative of Eq. (E.13) with respect to q2.

E.2 Quark-Loop Integral

In the UV limit, there is one loop for each light quark, as shown in Fig. E.2. The related amplitude can be expressed as

Z 4 µν 2 2 2 d k µ (6k + mf ) ν (6k+ 6q + mf ) Πf (q ) = −e Qf 4 T r γ 2 2 γ 2 2 (2π) k − mf (k + q) − mf Z 4 µν 2 µ ν µ ν µ ν 2 2 d k g (mf − k.(k + q)) + 2k k + k q + q k = −4e Qf 4 2 2 2 2 (2π) [k − mf ][(k + q) − mf ] (E.15)

In the same manner of meson loop calculation, using the Feynman parameter and shifting k to k − xq, the amplitude of the light quark takes the following form µν 2 µν 2 µ ν ¯ 2 Πf (q ) = (g q − q q )iΠf (q ), (E.16) where the scalar function is given by

Z 1 ¯ 2 2 2 4−d Γ() 1 Πf (q ) = −8e Qf (1 − x)x µ 2− 0 (4π) ∆ 2α Z 1 h1 i = − µ4−d (1 − x)x − γ + log(4π) − log ∆ + O() , π 0 (E.17)

119 E.2. QUARK-LOOP INTEGRAL

2 2 where ∆ = mf − x(1 − x)q . In the same way, the divergent term (1/) can be cancelled by ﬁnding the diﬀerence between the scalar function as follows

2 ¯ 2 ¯ Πf (q ) = Πf (q ) − Πf (0) Z 1 2 2 2α 2 mf − x(1 − x)q = Qf (1 − x)x log 2 . (E.18) π 0 mf

¯ 2 In equivalent way, we can ﬁnd the ﬁrst derivative of the scalar function Πf (q ) with respect to q2 in order to cancel the divergent term

¯ 2 Z 1 dΠf (q ) 2α 2 −x(1 − x) 2 = Qf dx(1 − x)x 2 2 . (E.19) dq π 0 mf − x(1 − x)q

120 References

[1] H. Fritzsch and M. Gell-Mann, Current Algebra: Quarks and What Else?, Proc. XVI Int. Conf. on High Energy Physics 2, 135 (1972).

[2] D. Gross and F. Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories, Phys. Rev. Lett. 30, 1343 (1973).

[3] H. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30, 1346 (1973).

[4] M. Gell-Mann and F. Low, Quantum Electrodynamics at Small Distances, Phys. Rev. D 95, 1300 (1954).

[5] C. Callan, Broken Scale Invariance in Scalar Field Theory, Phys. Rev. D 2, 1541 (1970).

[6] K. Symanzik, Small Distance Behaviour in Field Theory and Power Counting, Comm. Math. Phys. 18, 227 (1970).

[7] C. Roberts, Looking into the Matter of Light-Quark Hadrons, Few- Body Systems, 52, 345 (2011).

[8] S. Qin, L. Chang, Y. Liu, C. Roberts, and D. Wilson, Interaction Model for the Gap Equation, Phys. Rev. C 84, 042202 (2011).

[9] H. Georgi, Unparticle Physics, Phys. Rev. Lett. 98, 221601 (2007).

[10] L. Debbio, The Conformal Window on the Lattice, Pro. Sci. LAT- TICE 2010, 004 (2010).

121 REFERENCES

[11] G. Grunberg, Renormalization Group Improved Perturbative QCD, Phys. Lett. B 95, 70 (1980).

[12] G. ’t Hooft, Can We Make Sense Out of Quantum Chromodynam- ics?, lectures given at the ”Ettore Majorana” Int. School of Subnuclear Physics, Erice, 1977, Sub. nucl. Ser. 15, 943 (1979).

[13] W. Caswell, Asymptotic Behavior of Non-Abelian Gauge Theories to Two-Loop Order, Phys. Rev. Lett. 33, 244 (1974).

[14] T. Banks and A. Zaks, On the Phase Structure of Vector-like Gauge Theories with Massless Fermions, Nucl. Phys. B 196, 189 (1982).

[15] M. Luscher, R. Sommer, P. Weisz, and U. Wolff, A Precise De- termination of the Running Coupling in the SU(3) Yang-Mills Theory, Nucl. Phys. B 413, 481 (1994).

[16] R. Alkofer, A. Hauck, and L. von Smekal, The Strong Run- ning Coupling from an Approximate Gluon Dyson-Schwinger Equation, [arXiv:hep-ph/9608471v1] (1996).

[17] L. von. Smekal, A. Hauck, and R. Alkofer, Infrared Behavior of Gluon and Ghost Propagators in Landau Gauge QCD, Phys. Rev. Lett. 79, 3591 (1997).

[18] A. Aguilar, D. Binosi, and J. Papavassiliou, QCD Eﬀective Charges from Lattice Data, [arXiv:1004.1105v1] (2010).

[19] J. Bloch, Multiplicative Renormalizability and Quark Propagator, Phys. Rev. D 66, 034032 (2002).

[20] C. Fischer and R. Alkofer, Non-Perturbative Propagators, Run- ning Coupling and the Dynamical Quark Mass of Landau Gauge QCD, Phys. Rev. D 67, 094020 (2003).

[21] T. Ryttov and F. Sannino, Supersymmetry Inspired QCD Beta Function, Phys. Rev. D 78, 065001 (2008).

122 REFERENCES

[22] G. Grunberg, Renormalization-Scheme-Invariant QCD and QED: The Method of Eﬀective Charges, Phys. Rev. D 29, 2315 (1984).

[23] S. Adler, Some Simple Vacuum-Polarization Phenomenology: + − e e → hadrons; the Muonic-Atom X-Ray Discrepancy and gµ-2, Phys. Rev. D 10, 3714 (1974).

[24] L. Faddeev and V. Popov, Feynman Diagrams for the Yang-Mills Field, Phys. Lett. B 25, 29 (1967).

[25] C. Becchi, A. Rouet, and R. Stora, Renormalization of Gauge Theories, Ann. Phys. 98, 287 (1967).

[26] I. Tyutin, Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism, Lebedev Institute preprint N39, [arXiv:0812.0580v2] (1975).

[27] G. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44, 189 (1972).

[28] G. ’t Hooft, Dimensional Regularization and the Renormalization Group, Nucl. Phys. B 61, 455 (1973).

[29] W. Pauli and F. Villars, On the Invariant Regularization in Rela- tivistic Quantum Theory, Rev. Mod. Phys. 21, 434 (1949).

[30] J. Bjorken and S. Drell, Relativistic Quantum Fields, McGraw Hill, USA, 1965.

[31] E. Spear, Generalized Feynman Amplitudes, Princeton University Press, USA, 1969.

[32] M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory, Westview Press, USA, 1995.

[33] T. Cheng and L. Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, USA, 1984.

[34] K. Huang, Quarks, Leptons and Gauge Fields, World Scientiﬁc, Sin- gapore, 1992.

123 REFERENCES

[35] J. Collins, Renormalization, Cambridge University Press, UK, 1984.

[36] W. Bardeen, A. Buras, D. Duke, and T. Mute, Deep-Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories, Phys. Rev. D 18, 3998 (1978).

[37] G. t Hooft, When Was Asymptotic Freedom Discovered? or The Rehabilitation of Quantum Field Theory, Nucl. Phys. Proc. Suppl. 74, 413 (1999).

[38] D. Jones, Two-Loop Diagrams in Yang-Mills Theories, Nucl. Phys. B 75, 531 (1974).

[39] O. Tarasov, A. Vladimirov, and A. Y. Zharkov, The Gell-Mann- Low Function of QCD in the Three-Loop Approximation, Phys. Lett. B 93, 429 (1980).

[40] S. Larin and J. Vermaseren, The Three-Loop QCD β-Function and Anomalous Dimensions, Phys. Lett. B 303, 334 (1993).

[41] T. Ritbergen, J. Vermaseren, and S. Larin, The Four-Loop β-Function in Quantum Chromodynamics, Phys. Lett. B 400, 379 (1997).

[42] M. Czakon, The Four-Loop QCD β-Function and Anomalous Dimen- sions, Nucl. Phys. B 710, 485 (2005).

[43] S. Adler, Adventures in Theoretical Physics: Selected Papers with Commentaries, World Scientiﬁc, Singapore, 2006.

[44] G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72, 461 (1974).

[45] G. ’t Hooft, A Two-Dimensional Model for Mesons, Nucl. Phys. B 75, 461 (1974).

[46] S. Coleman and D. Gross, Price of Asymptotic Freedom, Phys. Rev. Lett. 31, 851 (1973).

124 REFERENCES

[47] A. Zee, Study of the Renormalization Group for Small Coupling Con- stants, Phys. Rev. D 7, 3630 (1973).

[48] R. Crewther, Asymptotic Freedom in Field Theory, Phys. Colloq. 34 C1, 111 (1973).

[49] F. Jegerlehner, The Running Fine Structure Constant α(E) via the Adler Function, Nucl. Phys. B (Proc. Suppl.) 181, 135 (2008).

[50] S. Brodsky, S. Menke, C. Merino, and J. Rathsman, On the

Behavior of the Eﬀective QCD Coupling ατ (s) at Low Scales, Phys. Rev. D 67, 055008 (2003).

4 [51] P. Baikov, K. Chetyrkin, and J. Khn, Order αs QCD Corrections to Z and τ Decays, Phys. Rev. Lett. 101, 012002 (2008).

[52] A. Nesterenko, On the Low energy Behavior of the Adler Function, Nucl. Phys. B (Proc. Suppl.) 186, 207 (2009).

[53] A. Nesterenko, Dispersive Approach to QCD and Inclusive τ Lepton Hadronic Decay, Phys. Rev. D 88, 056009 (2013).

[54] K. Chetyrkin, A. Kataev, and F. Tkachov, Higher-Order Cor- + − rections to σtot(e e → hadrons) in Quantum Chromodynamics, Phys. Lett. B 85, 277 (1979).

[55] M. Dine and J. Sapirstein, Higher-Order Quantum Chromodynamic Corrections in e+e− Annihilation, Phys. Rev. Lett. 43, 668 (1979).

[56] W. Celmaster and R. Gonsalves, Analytic Calculation of Higher- Order Quantum-Chromodynamic Corrections in e+e− Annihilation, Phys. Rev. Lett. 44, 560 (1980).

3 [57] S. Gorishany, A. Kataev, and S. Larin, The O(αs) corrections + − − to σtot(e e → hadrons) and Γ (τ → ντ +hadrons) in QCD, Phys. Lett. B 259, 144 (1991).

[58] L. Surguladze and M. Samuel, Total Hadronic Cross Section in e+e− Annihilation at the Four-Loop Level of Perturbative QCD, Phys. Rev. Lett. 66, 560 (1991).

125 REFERENCES

3 [59] K. Chetyrkin, Corrections of Order αs to Rhad in pQCD with Light Gluinos, Phys. Lett. B 391, 402 (1997).

[60] G. Prosperi, M. Raciti, and C. Simolo, On the Running Coupling Constant in QCD, Prog. in Part. and Nucl. Phys. 58, 387 (2007).

[61] M. Crentz, Quarks, Gluons and Lattice, Cambridge University Press, New York, 1983.

[62] R. Gupta, Introduction to Lattice QCD, [hep-lat/9807028v1] (1998).

[63] G. Munster and M. Walzl, Lattice Gauge Theory, [ariXiv:hep- lat/0012005v1] (2000).

[64] C. Davies, Lattice QCD, [arXiv:hep-ph/0205181v1] (2002).

[65] H. Rothe, Lattice Gauge Theories: An Introduction, World Scientiﬁc, Singspore, 2012.

[66] K. Wilson, Conﬁnement of Quarks, Phys. Rev. D 10, 2445 (1974).

[67] F. Dyson, The S Matrix in Quantum Electrodynamics, Phys. Rev. 75, 1736 (1949).

[68] J. Schwinger, On Green’s Functions of Quantized Fields I, Proc. Nat. Acad. Sci. 37, 452 (1951).

[69] J. Schwinger, On Green’s Functions of Quantized Fields II, Proc. Nat. Acad. Sci. 37, 455 (1951).

[70] G. Ecker, Eﬀective Field Theories, Ency. of Math. Phys., 139 (2006).

[71] T. Appelquist and J. Carazzone, Infrared Singularities and Mas- sive Fields, Phys. Rev. D 11, 2856 (1975).

[72] H. Georgi, Eﬀective Field Theory, Annu. Rev. Nucl. Part. Sci. 43, 209 (1993).

126 REFERENCES

[73] A. Pich, Eﬀective Field Theory, Lectures at the 1997 Les Houches Summer School ”Probing the Standard Model of Particle Interactions”, [arXiv:hep-ph/9806303v1] (1998).

[74] U. van Kolck, L. Abu-Raddad, and D. Cardamone, Introduc- tion to Eﬀective Field Theories in QCD, [arXiv:nucl-th/0205058v1] (2002).

[75] C. Itzykson and J. Zuber, Quantum Field Theory, McGraw-Hill Inc., USA, 1980.

[76] W. Celmaster and R. Gonsalves, Renormalization-Prescription Dependence of the Quantum-Chromodynamic Coupling Constant, Phys. Rev. D 20, 1420 (1979).

[77] W. Celmaster and D. Sivers, Studies in the Renormalization- Prescription Dependence of Perturbative Calculations, Phys. Rev. D 23, 27 (1981).

[78] S. Brodsky, G. Lepage, and P. Mackenzie, On the Elimination of Scale Ambiguities in Perturbative Quantum Chromodynamics, Phys. Rev. D 28, 228 (1983).

[79] S. Gorishny, A. Kataev, S. Larin, and L. Surguladze, Scheme Dependence of the Next-to-Next-to-Leading QCD Corrections to Γtot(H0 hadrons) and the Spurious QCD Infrared Fixed Point, Phys. Rev. D 43, 1633 (1991).

[80] G. Grunberg, On Some Ambiguities in the Method of Eﬀective Charges, Phys. Rev. D 40, 680 (1989).

[81] A. Radyushkin, Optimized Λ-Parametrization for the QCD Running Coupling Constant in Space-Like and Time-Like Regions, JINR Rapid Commun. 78, 96 (1996).

[82] A. Bakulev, A. Radyushkin, and N. Stefanis, Form Factors and QCD in Space-like and Time-like Region, Phys. Rev. D 62, 113001 (2000).

127 REFERENCES

[83] B. Geshkenbein and B. Ioffe, Improvement of Perturbation Theory + − in QCD for e e → hadrons and the Problem of αs freezing, JETP LETTERS 70, 161 (1999).

[84] T. Ryttov and R. Shrock, Scheme Transformations in the Vicinity of an Infrared Fixed Point, Phys. Rev. D 86, 0650321 (2012).

[85] T. Ryttov and R. Shrock, An Analysis of Scheme Transformations in the Vicinity of an Infrared Fixed Point, Phys. Rev. D 86, 085005 (2012).

[86] R. Shrock, Studies on the UV to IR Evolution of Gauge Theories and Quasiconformal Behavior, [arXiv:1305.4572v1] (2013).

[87] G. Grunberg, Interpretation of the Brodsky-Lepage-Mackenzie Cri- terium, Phys. Lett. B 135, 455 (1984).

[88] G. Grunberg, Method of Eﬀective Charges and Brodsky-Lepage- Mackenzie Criterion, Phys. Rev. D 46, 2228 (1992).

[89] P. Stevenson, Optimized Perturbation Theory, Phys. Rev. D 23, 2916 (1981).

[90] P. Stevenson, Resolution of the Renormalization-Scheme Ambiguity in Perturbative QCD, Phys. Lett. B 100, 61 (1981).

[91] P. Stevenson, Sense and Non-Sense in the Renormalization-Scheme- Dependence Problem, Nucl. Phys. B 203, 472 (1982).

[92] S. Brodsky, C. Ji, A. Pang, and D. Robertson, Optimal Renor- malization Scale and Scheme for Exclusive Processes, Phys. Rev. D 57, 245 (1998).

[93] A. Mattingly and P. Stevenson, Optimization of Re+e− and Freez- ing of the QCD Couplant at Low Energies, Phys. Rev. D 49, 437 (1994).

[94] A. Mattingly and P. Stevenson, QCD Perturbation Theory at Low Energies, Phys. Rev. Lett. 69, 1320 (1992).

128 REFERENCES

[95] J. Kubo, S. Sakakibara, and P. Stevenson, Renormalization- Scheme Ambiguity and Perturbation Theory near a Fixed Point, Phys. Rev. D 29, 1682 (1984).

[96] J. Chyla, A. Kataev, and S. Larin, Renormalization Scheme De- pendence in e+e annihilation and τ-Lepton Decay at the Next-to-Next- to-Leading Order of Perturbative QCD, Phys. Lett. B 267, 269 (1991).

[97] I. Suslov, On ’t Hooft’s Representation of the β-Function, [arXiv:hep-ph/0605115v3] (2008).

[98] S. Coleman and E.Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7, 1888 (1973).

[99] S. Brodsky, C. Merino, and J. Pelaez, Measuring the QCD Gell- Mann-Low Ψ-Function, Phys. Rev. D 60, 114007 (1999).

1 [100] P. Stevenson, The 16 2 − Nf Expansion and the Infrared Fixed Point in Perturbative QCD, Phys. Lett. B 331, 187 (1994).

[101] Y. Dokshitzer, G. Marchesini, and B.Webber, Dispersive Ap- proach to Power-Behaved Contributions in QCD Hard Processes, Nucl. Phys. B 469, 93 (1996).

[102] J. Gracey, One Loop Gluon form Factor and Freezing of αs in the Gribov-Zwanziger QCD Lagrangian, JHEP 05, 052 (2006).

[103] D. Shirkov and I. Solovtsov, Analytic Model for the QCD Run-

ning Coupling with Universal α¯s(0) Value, Phys. Rev. Lett. 79, 1209 (1997).

[104] A. Aguilar, A. Natale, and P. R. da Silva, Relating a Gluon Mass Scale to an Infrared Fixed Point in Pure Gauge QCD, Phys. Rev. Lett. 90, 152001 (2003).

[105] G. Grunberg, Evidence for Infrared-Finite Coupling in Sudakov Re- summation, Phys. Rev. D 73, 091901 (2006).

129 REFERENCES

[106] T. Ryttov and R. Shrock, Higher-Loop Corrections to the Infrared Evolution of a Gauge Theory with Fermions, Phys. Rev. D 83, 056011 (2011).

[107] R. Shrock, Higher-Loop Structural Properties of the β-Function in Asymptotically Free Vectorial Gauge Theories, Phys. Rev. D 87, 105005 (2013).

[108] J. Gracey, The QCD β-function at O(1/Nf ), Phys. Lett. B 373, 178 (1996).

[109] E. Gardi and M. Karliner, Relations between Observables and the Infrared Fixed Point in QCD, Nucl. Phys. B 529, 383 (1998).

[110] G. Veneziano, Some Aspects of a Unifed Approach to Gauge, Dual and Gribov Theories, Nucl. Phys. B 117, 519 (1976).

[111] R. Shrock, Higher-Loop Calculations of the Ultraviolet to Infrared

Evolution of a Vectorial Gauge Theory in the Limit Nc → ∞, Nf → ∞

with Nf /Nc Fixed, Phys. Rev. D 87, 116007 (2013).

[112] M. Luscher, R. Sommer, P. Weisz, and U. Wolff, Computation of the Running Coupling in the SU(2) Yang-Mills Theory, Nucl. Phys. B 389, 247 (1993).

[113] M. Luscher, P. Weisz, and U. Wolff, A Numerical Method to Compute the Running Coupling in Asymptotically Free Theories, Nucl. Phys. B 359, 221 (1991).

[114] M. Luscher, R. Narayanan, P. Weisz, and U. Wolff, The Schr¨odinger Functional - a Renormalizable Probe for non-Abelian Gauge Theories, Nucl. Phys. B 384, 168 (1992).

[115] S. Sint, On the Schr¨odinger Functional in QCD, Nucl. Phys. B 421, 135 (1994).

[116] A. Bode, P. Weisz, and U. Wolff, (ALPHA Collaboration), Two-Loop Computation of the Schr¨odinger Functional in Lattice QCD, Nucl. Phys. B 576, 517 (2000).

130 REFERENCES

[117] T. Appelquist, G. Fleming, and E. Neil, Lattice Study of the Conformal Window in QCD-like Theories, Phys. Rev. Lett. 100, 171607 (2008).

[118] T. Appelquist, G. Fleming, and E. Neil, Lattice Study of Con- formal Behavior in SU(3) Yang-Mills Theories, Phys. Rev. D 79, 076010 (2009).

[119] S. Aoki, et al. (PACS-CS collaboration), Precise Determination of

the Strong Coupling Constant in Nf = 2 + 1 Lattice QCD with the Schro¨dinger Functional Scheme, JHEP 10, 053 (2009).

[120] F. Tekin, R. Sommer, and U. Wolff, (ALPHA Collaboration), The Running Coupling of QCD with Four Flavors, Nucl. Phys. B 840, 114 (2010).

[121] R. Horsley, H. Perlt, P. Rakow, G. Schierholz, and A. Schiller, The SU(3) Beta Function from Numerical Stochastic Perturbation Theory, Phys. Lett. B 728, 1 (2014).

[122] A. Hauck, L. von Smekal, and R. Alkofer, Solving the Gluon Dyson-Schwinger Equation in the Mandelstam Approximation, Comp. Phys. Comm. 112 [arXiv:hep-ph/9604430v2], 149 (1998).

[123] B. Bloissier, et al., High Statistics Determination of the Strong Cou- pling Constant in Taylor Scheme and its OPE Wilson Coeﬃcient from Lattice QCD with a Dynamical Charm, Phys. Rev. D 89, 014507 (2014).

[124] C. Fischer and R. Alkofer, Infrared Exponents and Running Cou- pling of SU(N) Yang-Mills Theories, Phys. Lett. B 536, 177 (2002).

[125] C. Lerche and L. Smekal, Infrared Exponent for Gluon and Ghost Propagation in Landau gauge QCD, Phys. Rev. D 65, 125006 (2002).

[126] C. Fischer, R. Alkofer, and H. Reinhardt, Elusiveness of In- frared Critical Exponents in Landau Gauge Yang-Mills Theories, Phys. Rev. D 65, 094008 (2002).

131 REFERENCES

[127] R. Alkofer, C. Fischer, and L. von Smekal, The Infrared Be- haviour of the Running Coupling in Landau Gauge QCD, Acta. Phys. Slov. 52, 191 (2002).

[128] R. Alkofer, C. Fischer, and F. Llanes-Estrad, Vertex Functions and Infrared Fixed Point in Landau Gauge SU(N) YangMills Theory, Phys. Lett. B 611, 279 (2005).

[129] L. von Smekal, A. Hauck, and R. Alkofer, A Solution to Coupled Dyson-Schwinger Equations for Gluons and Ghosts in Landau Gauge, Annals Phys. 267, 1 (1998).

[130] C. Fischer, Infrared Properties of QCD From DysonSchwinger Equa- tions, J. Phys. G: Nucl. Part. Phys. 32, R253 (2006).

[131] J. Taylor, Ward Identities and Charge Renormalization of the Yang- Mills Field, Nucl. Phys. B 33, 436 (1971).

[132] A. Hauck, L. von Smekal, and R. Alkofer, Solving a Cou- pled Set of Truncated QCD Dyson-Schwinger Equations, Comp. Phys. Comm. 112, 166 (1998).

[133] P. Boucaud, et al., The Infrared Behaviour of the Pure Yang-Mills Green Functions, arXiv:hep-ph/0507104v4 (2005).

[134] P. Boucaud, et al., Is the QCD Ghost Dressing Function ﬁnite at Zero Momentum?, JHEP 06, 001 (2006).

[135] P. Boucaud, et al., IR Finiteness of the Ghost Dressing Function from Numerical Resolution of the Ghost SD Equation, JHEP 06, 012 (2008).

[136] M. Pennington and D. Wilson, Are the Dressed Gluon and Ghost Propagators in the Landau Gauge Presently Determined in the Conﬁne- ment Regime of QCD?, Phys. Rev. D 84, 094028 (2011).

[137] P. Boucaud, et al., The Infrared Behaviour of the Pure Yang-Mills Green Functions, Few-Body Syst. 53, 387 (2012).

132 REFERENCES

[138] J. Cornwall and J. Papavassiliou, Gauge-Invariant Three-Gluon Vertex in QCD, Phys. Rev. D 40, 3474 (1989).

[139] D. Binosi and J. Papavassiliou, Pinch Technique: Theory and Ap- plications, Phys. Rept. 479, 1 (2009).

[140] N. Watson, The Gauge-Independent QCD Eﬀective Charge, Nucl. Phys. B 494, 388 (1997).

[141] A. Aguilar, D. Binosi, J. Papavassiliou, and J. Rodriguez- Quintero, Non-Perturbative Comparison of QCD Eﬀective Charges, Phy. Rev. D 80, 085018 (2009).

[142] S. Brodsky and G. Teramond, Hadronic Spectra and Light-Front Wave Functions in Holographic QCD, Phys. Rev. Lett. 96, 201601 (2006).

[143] S. Brodsky and G. Teramond, Light-front Dynamics and AdS/QCD Correspondence: The Pion Form Factor in the Space- and Time-like Regions, Phys. Rev. D 77, 056007 (2008).

[144] S. Brodsky and G. Teramond, Light-front Dynamics and AdS/QCD Correspondence: Gravitational Form Factors of Composite Hadrons, Phys. Rev. D 78, 025032 (2008).

[145] G. Teramond and S. Brodsky, Light-Front Holography: A First Approximation to QCD, Phys. Rev. Lett. 102, 081601 (2009).

[146] G. Teramond and S. Brodsky, Light-Front Holography and Gauge/Gravity Duality: The Light Meson and Baryon Spectra, Nucl. Phys. B (Proc. Suppl.) 199, 89 (2010).

[147] S. Brodsky, G. Teramond, and A. Deur, Non-Perturbative QCD Coupling and its β-Function from Light-Front Holography, Phys. Rev. D 81, 096010 (2010).

[148] S. Furui and H. Nakajima, What the Gribov Copy Tells about Conﬁnement and the Theory of Dynamical Chiral Symmetry Breaking, Phys. Rev. D 70, 094504 (2004).

133 REFERENCES

[149] J. Cornwall, Dynamical Mass Generation in Continuum Quantum Chromodynamics, Phys. Rev. D 26, 1453 (1982).

[150] S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32, 189 (1985).

[151] J. Bjorken, Applications of the Chiral U(6)⊗U(6) Algebra of Current Densities, Phys. Rev. 148, 1467 (1966).

[152] A. Deur, V. Burkert, J. Chen, and W. Korsch, Experimental Determination of the Eﬀective Strong Coupling Constant, Phys. Lett. B 650, 244 (2007).

[153] A. Deur, Spin Sum Rules and the Strong Coupling Constant at large distance, AIP Conference Proceedings 1155, 112 (2009).

[154] T. Appelquist, M. Dine, and I. Muzinich, The Static Potential in Quantum Chromodynamics, Phys. Lett. B 69, 231 (1977).

[155] T. Appelquist, M. Dine, and I. Muzinich, Static Limit of Quantum Chromodynamics, Phys. Rev. D 17, 2074 (1978).

[156] S. Brodsky and H. Lu, Commensurate Scale Relations in Quantum Chromodynamics, Phys. Rev. D 51, 3652 (1995).

[157] S. Brodsky, G. Gabadadze, A. Kataev, and H. Lu, The Gener- alized Crewther Relation in QCD and its Experimental Consequences, Phys. Lett. B 372, 133 (1996).

[158] V. Novikov, M. Shifman, A. Vainshtein, and V. Zakharov, Ex- act Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus, Nucl. Phys. B 229, 381 (1983).

[159] V. Novikov, M. Shifman, A. Vainshtein, and V. Zakharov, The Beta Function in Supersymmetric Gauge Theories: Instantons Versus Traditional Approach, Phys. Lett. B 166, 329 (1985).

[160] V. Novikov, M. Shifman, A. Vainshtein, and V. Zakharov, Su- persymmetric Instanton Calculus: Gauge Theories with Matter, Nucl. Phys. B 260, 157 (1985).

134 REFERENCES

[161] M. Shifman and A. Vainshtein, Solution of the Anomaly Puzzle in SUSY Gauge Theories and the Wilson Operator Expansion, Nucl. Phys. B 277, 456 (1986).

[162] C. Pica and F. Sannino, Beta Function and Anomalous Dimensions, Phys. Rev. D 83, 116001 (2011).

[163] N. Seiberg, Electric-Magnetic Duality in Supersymmetric non-Abelian Gauge Theories, Nucl. Phys. B 435, 129 (1995).

[164] K. Intriligator and N. Seiberg, Lectures on Supersymmetric Gauge Theories and Electric-Magnetic Duality, Nucl. Phys. B (Proc. Suppl.) 45, 1 (1996).

[165] T. Ryttov and R. Shrock, Comparison of Some Exact and Per-

turbative Results for a Supersymmetric SU(Nc) Gauge Theory, Phys. Rev. D 85, 076009 (2012).

[166] I. Jack, D. Jones, and C. North, N = 1 Supersymmetry and the Three-Loop Gauge β-Function, Phys. Lett. B 386, 138 (1996).

[167] I. Jack, D. Jones, and C. North, Scheme Dependence and the NSVZ β-Function, Nucl. Phys. B 486, 479 (1997).

[168] N. Arkani-Hamed and H. Murayama, Renormalization Group In- variance of Exact Results in Supersymmetric Gauge Theories, Phys. Rev. D 57, 6638 (1998).

[169] N. Arkani-Hamed and H. Murayama, Holomorphy, Rescaling Anomalies and Exact β-Functions in Supersymmetric Gauge Theories, JHEP 06, 030 (2000).

[170] M. Dine, G. Festuccia, L. Pack, C. Park, L. Ubaldi, and W. Wu, Supersymmetric QCD: Exact Results and Strong Coupling, JHEP 05, 061 (2011).

[171] F. Sannino, Jumping Dynamics, Mod. Phys. Lett. A 28, 1350127 (2013).

135 REFERENCES

[172] C. Pica and F. Sannino, Ultraviolet and Infrared Zeros of Gauge The- ories at the Four-Loop Order and Beyond, Phys. Rev. D 83, 035013 (2011).

[173] R. Crewther, S. Bass, F. Steffens, and A. Thomas, Decoupling heavy particles simultaneously, Nucl. Phys. B (Proc. Suppl.) 141, 159 (2005).

[174] S. Weinberg, The Quantum Theory of Fields II, Cambridge Univer- sity Press, UK, 1996.

[175] Y. Kusafuka and H. Terao, Fixed Point merger in the SN(Nc) Gauge Beta Functions, Phys. Rev. D 84, 125006 (2011).

[176] O. Antipin and K. Tuominen, Resizing the Conformal Window: A β-Function Ansatz, Phys. Rev. D 81, 076011 (2010).

[177] H. Pagels, Departures from Chiral Symmetry: a Review, Phys. Rept. 16, 219 (1975).

[178] A. Pich, Chiral Perturbation Theory, Rep. Prog. Phys. 58, 563 (1995).

[179] G. Ecker, Chiral Perturbation Theory, Prog. Part. Nucl. Phys. 35, 1 (1995).

[180] G. Ecker, Strong Interactions of Light Flavours, Lectures given at the Advanced School on QCD 2000, Benasque, Spain, [arXiv:hep- ph/0011026v1] (2000).

[181] J. Bijnens, Chiral Perturbation Theory Beyond One Loop, Prog. Part. Nucl. Phys. 58, 521 (2007).

[182] S. Scherer, Introduction to Chiral Perturbation Theory, Adv. Nucl. Phys. 27, 277 (2003).

[183] B. Kubis, An Introduction to Chiral Perturbation Theory, Lec- tures Held at the Workshop ”Physics and Astrophysics of Hadrons and Hadronic Matter”, Visva Bharati University, Shantiniketan, India, [arXiv:hep-ph/0703274v1] (2007).

136 REFERENCES

[184] V. Bernard, Chiral Perturbation Theory and Baryon Properties, Prog. Part. Nucl. Phys. 60, 82 (2008).

[185] S. Scherer and M. Schindler, A Primer for Chiral Perturbation Theory, Springer, Heidelberg, 2012.

[186] S. Weinberg, Phenomenological Lagrangians, Physica A 96, 327 (1979).

[187] J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals. Phys. 158, 142 (1984).

[188] J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Ex- pansions in the Mass of the Strange Quark, Nucl. Phys. B 250, 465 (1985).

[189] S. Adler and R. Dashen, Current Algebras, Benjamin, USA, 1968.

[190] V. Alfaro, S. Fubini, G. Furlan, and C. Rossetti, Currents in Hadron Physics, North-Holland, Amsterdam, 1973.

[191] Y. Nambu, Axial Vector Current Conservation in Weak Interactions, Phys. Rev. Lett. 4, 380 (1960).

[192] Y. Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev. 117, 648 (1960).

[193] J. Goldstone, Field Theories with Superconductor Solutions, Nuovo Cim. 19, 154 (1961).

[194] J. Goldstone, A. Salam, and S. Weinberg, Broken Symmetries, Phys. Rev. 127, 965 (1962).

[195] S. Weinberg, Nonlinear Realizations of Chiral Symmetry, Phys. Rev. 166, 1568 (1968).

[196] J. Bijnens, G. Colangelo, and G. Ecker, The Mesonic Chiral Lagrangian of Order p6, JHEP 99, 020 (1999).

137 REFERENCES

[197] J. Bijnens, G. Colangelo, and G. Ecker, Renormalization of Chi- ral Perturbation Theory to Order p6, Annals. Phys. 280, 100 (2000).

[198] M. Gell-Mann, R. Oakes, and B. Renner, Behavior of Current Divergences under SU(3) × SU(3), Phys. Rev. 175, 2195 (1968).

[199] M. Gell-Mann, Model of the Strong Couplings, Phys. Rev. 106, 1296 (1957).

[200] S. Okubo, Note on Unitary Symmetry in Strong Interactions, Prog. Theor. Phys. 27, 949 (1962).

[201] M. Srednicki, Quantum Field Theory, Cambridge University Press, UK, 2007.

[202] J. Vermaseren, S. Larin, and T. Ritbergen, The 4-Loop Quark Mass Anomalous Dimension and the Invariant Quark Mass, Phys. Lett. B 405, 327 (1997).

138