Universidade Do Estado Do Rio De Janeiro Centro De Tecnologia E Ciˆencias Instituto De F´Isicaarmando Dias Tavares

Universidade do Estado do Rio de Janeiro Centro de Tecnologia e Ciˆencias Instituto de F´ısicaArmando Dias Tavares

Duive Maria van Egmond

Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions.

Rio de Janeiro 2020 Duive Maria van Egmond

Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions.

Tese apresentada, como requisito parcial para obten¸cãodo t´ıtulode Doutor, ao Pro- grama de Pós-Gradua¸cãoem Curso, da Uni- versidade do Estado do Rio de Janeiro.

Orientador: Prof. Dr. Marcelo Santos Guimar˜aes

Rio de Janeiro 2020 CATALOGAC¸ AO˜ NA FONTE UERJ / REDE SIRIUS / BIBLIOTECA CTC/D

D979 van Egmond, Duive Maria Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions. / Duive Maria van Egmond. – Rio de Janeiro, 2020- 26 f.

Orientador: Prof. Dr. Marcelo Santos Guimarães Tese (Doutorado) – Universidade do Estado do Rio de Janeiro, Instituto de F´ısicaArmando Dias Tavares, Programa de Pós-Gradua¸cãoem Curso, 2020.

1. Yang-Mills theories.. 2. gauge-invariance.. 3. Higgs model.. I. Prof. Dr. Marcelo Santos Guimar˜aes. II. Universidade do Estado do Rio de Janeiro. III. Instituto de F´ısicaArmando Dias Tavares. IV. T´ıtulo

CDU 02:141:005.7

Autorizo, apenas para fins acadêmicose cient´ıficos,a reprodu¸cãototal ou parcial desta tese, desde que citada a fonte.

Assinatura Data Duive Maria van Egmond

Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions.

Tese apresentada, como requisito parcial para obten¸cãodo t´ıtulode Doutor, ao Pro- grama de Pós-Gradua¸cãoem Curso, da Uni- versidade do Estado do Rio de Janeiro.

Aprovada em 11 de 8 de 2020. Banca Examinadora:

Prof. Dr. Marcelo Santos Guimar˜aes(Orientador) Instituto de F´ısicaArmando Dias Tavares– UERJ

Prof. Dr. David Dudal KU Leuven- Kampus Kortrijk

Profa. Dra. Tereza Cristina da Rocha Mendes Universidade de S˜aoPaulo

Prof. Dr. Silvio Paolo Sorella Instituto de F´ısicaArmando Dias Tavares– UERJ

Prof. Dr. Rudnei de Oliveira Ramos Instituto de F´ısicaArmando Dias Tavares– UERJ

Prof. Dr. Antonio Duarte Universidade Federal Fluminense

Prof. Dr. Marcio Andr´eLopes Capri Instituto de F´ısicaArmando Dias Tavares– UERJ

Rio de Janeiro 2020 This thesis is dedicated to my father, dr. Jacob Jan van Egmond. He saw this work before I did. A word of thanks

This thesis was supposed to be written during my last months as a PhD student in Rio de Janeiro. Unfortu- nately, the COVID-19 pandemic kept me in my hometown in the Netherlands, where I have been reminiscing about my time in Brazil while writing this thesis. All of this under the fantastic care of my mother Marieke, and for this and many more reasons I want to thank her ﬁrst and foremost.

Since I went to Rio de Janeiro four years ago, I have had nothing but support from my friends and family in the Netherlands. I want to thank my brothers, sister and all my friends in the Netherlands for their love and friendship. All the visits and phone calls have made me feel connected with you throughout this time. I would also like to thank my friends and colleagues at UERJ, with whom I have shared a lot of nice moments, during and after work. A special thanks to Oz´orio,for all the great discussions, for his encouragements and for supporting me during diﬃcult moments.

In the four years of my PhD at UERJ, I have learned what it means to be a researcher. There are four people I would like the especially thank for that. First of all prof. dr. Marcelo Guimar˜aes,who teached me to trust myself and follow my curiosity. Then, prof. dr. Silvio Sorella, from whom I learned the art of writing an article, and whose phrase “fecha os olhos e vai” (close your eyes and go) has helped me in many moments during my research. Prof. dr. David Dudal, with whom I share a passion for the connection between small details and the greater picture. Prof. dr. Urko Reinosa, who teached me that you can never be sure enough about a calculation, and that there is sheer pleasure in that.

Finally, I would like to thank prof. dr. Leticia Palhares and prof. dr. Marcela Peláezfor showing me what women in physics can achieve. “All language is but a poor translation”- Franz Kafka Abstract We analyze different BRST invariant solutions for the introduction of a mass term in Yang-Mills (YM) h theories. First, we analyze the non-local composite gauge-invariant field Aµ(x), which can be localized by the Stueckelberg-like field ξa(x). This enables us to introduce a mass term in the SU(N) YM model, a feature that has been indicated at a non-perturbative level by both analytical and numerical studies. The configuration of Ah(x) , obtained through the minimization of R d4xA2 along the gauge orbit, gives rise to an all orders renormalizable action, a feature which will be illustrated by means of a class of covariant gauge fixings which, as much as ’t Hooft’s Rξ-gauge of spontaneously broken gauge theories, provide a mass for the Stueckelberg-like field. Then, we consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows to disentangle what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. We highlight the role of the tadpole graphs and the gauge choices to get sensible results. We also introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry. We clearly illustrate its non-unitary nature directly from the spectral function viewpoint. This provides a functional analogue of the Ojima observation in the canonical formalism: there are ghost states with nonzero norm in the BRST-invariant states of the Curci-Ferrari model. Finally, the spectral properties of a set of local gauge-invariant composite operators are investigated in the U(1) and SU(2) Higgs model quantized in the ’t Hooft Rξ gauge. These operators enable us to give a gauge-invariant description of the spectrum of the theory, thereby surpassing certain incommodities when using the standard elementary fields. The corresponding two-point correlation functions are evaluated at one-loop order and their spectral functions are obtained explicitly. It is shown that the spectral functions of the elementary fields suffer from a strong unphysical dependence from the gauge parameter ξ, and can even exhibit positivity violating behaviour. In contrast, the BRST invariant local operators exhibit a well defined positive spectral density. Contents

Introduction 11 Fundamental forces and ﬁeld theory: a historical oversight...... 11 Higgs mechanism without spontaneous symmetry breaking...... 13

1 Mass generation within and beyond the Standard Model 18 1.1 The Higgs mechanism in the Standard Model...... 18 1.1.1 The SM Higgs sector...... 19 1.1.2 The SM Higgs mechanism...... 20 1.1.3 Mass generation for gauge bosons...... 21 1.1.4 Custodial symmetry...... 22 1.1.5 Rξ gauge class...... 24 1.2 Higgs beyond the Standard Model...... 25 1.2.1 SU(2) → U(1)...... 25 1.2.2 SU(5) → SU(3) × SU(2) × U(1)...... 26 1.2.3 SU(3) → SU(2) × U(1)...... 27 1.3 Massive solutions for the QCD sector...... 27 1.3.1 Massive Yang-Mills model...... 29 1.3.2 Positivity violation and complex poles...... 30

2 2 On a renormalizable class of gauge fixings for the gauge invariant operator Amin 33 2 2.1 Brief review of the operator Amin and construction of a non-local gauge invariant and transverse h gauge field Aµ ...... 34 h 2.2 A local action for Aµ ...... 34 2.3 Introducing the gauge fixing term Sgf ...... 35 2.4 A look at the propagators of the elementary fields...... 36 2 2.5 Amin versus the conventional Stueckelberg mass term...... 38 2.6 Algebraic characterization of the counterterm...... 38 2.7 Analysis of the counterterm and renormalization factors...... 43 2.8 Conclusion...... 44

3 Some remarks on the spectral functions of the Abelian Higgs Model 46 3.1 Abelian Higgs model: some essentials...... 47 3.1.1 Gauge ﬁxing...... 47 3.2 Photon and Higgs propagators at one-loop...... 48 3.2.1 Corrections to the photon self-energy...... 49 3.2.2 Corrections to the Higgs self-energy...... 51 3.2.3 Results for d = 4 − ...... 52 3.2.4 One-loop propagators for the elementary ﬁelds...... 53 3.3 Spectral properties of the propagators...... 55 3.3.1 Pole mass, residue and Nielsen identities...... 56 3.3.2 Obtaining the spectral function...... 57 3.3.3 Spectral density functions...... 58 3.4 Some subtleties of the Higgs spectral function...... 60 3.4.1 A slightly less correct approximation for the pole mass...... 60 3.4.2 Something more on the branch points...... 61 3.4.3 Asymptotics of the spectral function...... 61 3.5 A non-unitary U(1) model...... 62 3.5.1 CF-like U(1) model: some essentials...... 62 3.5.2 Propagators and spectral functions...... 63

9 3.5.3 Gauge invariant operator...... 64 3.6 Conclusion...... 65

4 Gauge-invariant spectral description of the U(1) Higgs model from local composite operators 67 4.1 The correlation functions hO(x)O(y)i and hVµ(x)Vν (y)i at one loop order...... 68 4.2 Spectral properties of the gauge-invariant local operators (Vµ(x),O(x))...... 77 4.2.1 Obtaining the spectral function...... 77 4.2.2 Spectral properties of the gauge-invariant composite operators Vµ(x) and O(x)...... 77 4.2.3 Pole masses...... 79 4.3 Unitary limit...... 79 4.4 Conclusion...... 80

5 Spectral properties of local BRST invariant composite operators in the SU(2) Yang-Mills- Higgs model 81 5.1 The action and its symmetries...... 82 5.1.1 Gauge fixing and BRST symmetry...... 83 5.1.2 Custodial symmetry...... 84 5.2 One-loop evaluation of the correlation function of the elementary fields...... 84 5.3 One-loop evaluation of the correlation function of the local BRST invariant composite operators 86 5.3.1 Correlation function of the scalar BRST invariant composite operator O(x)...... 86 5.3.2 A little digression on the unitary gauge...... 88 5.3.3 Vectorial composite operators...... 89 5.4 Spectral properties...... 94 5.4.1 Spectral properties of the elementary fields...... 94 5.4.2 Unphysical threshold effects...... 97 5.4.3 Spectral properties for the composite fields...... 97 5.5 Conclusion...... 99

6 Conclusion and Outlook 101

2 A Technical details of the gauge invariant operator Amin 103 A.1 Properties of the functional fA[u]...... 103 A.2 A generalised Slavnov-Taylor identity...... 106

B Technical details to the Abelian Higgs model 110 B.1 Field propagators of the Abelian Higgs model in the Rξ gauge...... 110 B.2 Vertices of the Abelian Higgs model in the Rξ gauge...... 111 B.3 Equivalence between including tadpole diagrams in the self-energies and shifting hϕi ...... 111 B.4 Feynman integrals...... 113 B.5 Asymptotics of the Higgs propagator...... 113 B.6 Contributions to hO(p)O(−p)i ...... 114 B.7 Contributions to hVµ(x)Vν (y)i ...... 116

C Technical details to the SU(2) Higgs model 120 C.1 Propagators and vertices of the SU(2) Higgs model in the Rξ gauge...... 120 C.2 Elementary propagators of the SU(2) Higgs model in the Rξ gauge...... 121 C.2.1 Higgs propagator...... 121 C.2.2 Gauge ﬁeld propagator...... 123 C.3 Contributions to hO(p)O(−p)i ...... 125 C.4 A few comments on the unitary gauge...... 127 Introduction

Fundamental forces and ﬁeld theory: a historical overview

Four fundamental forces seem to constitute nature: Electromagnetic force, Gravitational force and the Strong and Weak nuclear forces.

The electromagnetic force was at the heart of the development of modern physics. Classically, it is described by Maxwell’s equations from 1862 [1] in terms of electric charges that generate an electromagnetic field, which in turn exerts a force on other electric charges within that field. At the end of the nineteenth century, this picture was challenged by several experimental observations such as blackbody radiation and the photoelectric effect, which did not have an explanation within the Maxwell equations. In 1900, Max Planck introduced the idea that energy is quantized in order to derive, heuristically, a formula for the energy emitted by a black body. Planck’s law [2] marked the beginning of quantum physics. Subsequently, Einstein explained the photoelectric effect by stating that the energy of an electromagnetic field is quantized in discrete units that were called photons. The photon is a force carrier or messenger particle, neutrally charged and massless, that mediates the electromagnetic force between electrically charged matter particles. The phenomenological theories of Planck and Einstein were followed by more rigorous descriptions of quantum-mechanical sytems, with the Schrödinger wave equation (1925) as the main postulate of modern quantum physics [3].

Simultaneously to the emergence of quantum physics, Einstein’s theory of special relativity [4], built on Maxwell’s equations, meant a revolution in the perception of space and time. However, the concepts of relativity seemed incompatible with quantum physics: relativity theory treats time and space on an equal footing, while in quantum physics spatial coordinates are promoted to operators, with time a label. In 1928, Dirac succeeded in writing a relativistic wave equation for the electron [5], which however hypothesized negative energy solutions, pointing to the existence of anti-particles (we now know the antiparticle of the electron to be the positron). This eventually led to the development of the first quantum field theory, dubbed Quantum Electrodynamics (QED) by Dirac. In QED, electrons and other matter particles are pictured as excited states of an underlying field, just as photons are excited states of the electromagnetic field. In the associated field operator ϕ(x, t), position and time are both labels, in line with relativity theory. Moreover, QED naturally incorporates negative energy states.

In order to understand the classical electromagnetic field in the light of quantum field theory, we take the classical magnetic vector potential A, whose curl is equal to the magnetic field. This quantity had been known from Maxwell’s equations but within the Schrödingerpicture it overtook the electromagnetic field as the fundamental quantity. By construction, the vector potential can be changed by terms that have a vanishing curl without changing the magnetic field. This “gauge symmetry” became central to QED, where the potential was promoted to a relativistic four-vector Aµ(x, t), the photon field, and interactions of fields were summarized in a Lagrangian that is invariant under transformations of the local U(1) group, called gauge transformations.

QED gives the most accurate predictions quantum physics currently has to offer. For example, the calculation of the electron’s magnetic moment in QED agrees with experimental data up to ten digits. The success of QED can be attributed to the fact that many of the phenomena in our visible (low-energy) world can be approximated by perturbing the electromagnetic quantum vacuum, known as perturbation theory. Feynman diagrams are the visual representation of the perturbative contributions. However, QED is not well-defined as a quantum field theory to arbitrarily high energies, because the coupling constant (interaction strength) runs to infinity at finite energy. This divergence at high energy scale is known as the Landau pole [6]. Other questions concerning the electromagnetic force, such as the existence of a magnetic monopole, still remain open problems today.

QED has served as the model and template for quantum ﬁeld theories that try to describe the other three fundamental forces. Attempts to ﬁt Einstein’s gravity theory of curved spacetime [7] into the concepts of

11 quantum field theory have not been an unqualified success. For example, the hypothetical force carrier of the gravitational force, the graviton, has never been detected. Moreover, the description of gravity as a field theory has been shown to fail at Planck length. Most modern research in quantum gravity is conducted in the framework of String Theory, which takes a different approach to unite quantum phyics and relativity: instead of making time and space both labels, they are both operators. Today, it is even disputed if gravitational force is truly a fundamental force, or an emergent effect of deeper quantum mechanical processes [8].

In 1954, Yang and Mills extended the abelian U(1) gauge group of QED to non-abelian gauge groups [9]. Their goal was to find an explanation for the strong interaction which holds subatomic particles together. Today, we know the Yang-Mills (YM) SU(3) theories as the quantum field theory describing the strong interactions. The matter particles for this force are quarks, while the strong interaction is mediated by force carriers called gluons. Particles that interact under the strong force carry a color charge, and the YM theory of strong interactions is therefore also called Quantum Chromodynamics (QCD). Different from QED, the gauge fields themselves carry a a color and can thus interact with eachother. Therefore the gauge boson Aµ is displayed as AµT , with a the a a color charge and T the SU(3) generators. There are as many Aµ’s as there are generators, and because the number of generators for an SU(N) group is N 2 − 1, we have eight differently colored gluons.

The fact that gluons carry color has dramatic consequences for the relation between QCD interactions and energy scale. In QED, the fact that charged particles interact less when they get further away is explained by charge screening. A charged particle like the electron is surrounded by the vacuum, a cloud of virtual photons and electron-positron pairs continuously popping in and out of existence. Because of the attraction between opposite charges, the virtual positrons tend to be closer to the electron and screen the electron charge. Thus, the eﬀective charge becomes smaller at large distances (low energies, also called the infrared (IR) region), and grow stronger at small distances (high energies, also called the ultraviolet (UV) region). This is completely intuitive to us: the further away, the less interaction. It is also how Feynman diagrams are designed: a free particle comes in from the distance, interacts, and vanishes into the distance again.

In QCD, quark-antiquark pairs screen the color charged particles in the same way as the electron-positron pairs screen the electrically charged particles. However, the QCD vacuum also contains pairs of the charged gluons, which do not only carry color but also an anti-color magnetic moment. The net effect is not the screening of the color charged particle, but the augmentation of its charge. This means that the particles will interact more at larger distances, or lower energies, and vice versa. This was also observed in experiments: when high energies were applied to quarks, they hardly interacted with eachother. This means a theoretical model should have a coupling constant descreasing to zero in the UV limit. In 1973, Gross, Wilczek and Politzer proved that this is indeed the case for YM theories [10,11]. This behaviour, called asymptotic freedom, is seemingly counter- intuitive and can be best explained by imagining particles inside an elastic rod: in rest, there is no force between them, but the further you try to pull it apart, the more they will try to get back together. It also reminds of a superconductor: if a magnetic field is forced to run through the superconductor, the energy associated with the created flux tube grows with distance. This has led to some QCD models based on superconductivity, known as dual superconductor models [12].

Asymptotic freedom means that for QCD, the relation between energy scale and interaction strength is in- verted with reference to QED. This means that it should be possible to use perturbation theory in the UV limit. Indeed, for high energy phenomena the perturbative QCD models are in excellent agreement with the experiments [13–16]. In order to be quantized, these perturbative models demand a gauge-fixing, or choosing a gauge, in the Faddeev-Popov (FP) procedure. This introduces in YM theories so-called ghost fields, which are Grassmann fields that violate spin-statistics and can therefore not be physical. After gauge-fixing there is no longer a local gauge-invariance, but there is still a residual symmetry, known as Becchi-Rouet-Stora-Tyutin (BRST) symmetry. BRST symmetry guarantees the unitarity of the quantized YM theories and can be used to derive the Slavnov-Taylor identities, which are fundamental to renormalization methods such as Algebraic Renormalization [17]. Important is that the BRST symmetry is nilpotent, which means the BRST variation s applied on another BRST variation is zero, s2 = 0. Kugo and Ojima [18, 19] used this to distinguish between two types of states that are annihilated by a BRST transformation: those that trivially do so because they themselves are a BRST transformations of another state, and those that are not BRST transformations of another states. Physical states are of the second type and are said to be in the BRST cohomology. Non-physical states, such as the ghost fields, are of the first type and are outside the BRST cohomology. BRST symmetry therefore is an important tool for the definition of physical space.

The UV regime of QCD is well described by perturbative YM theories with a FP gauge ﬁxing. However, the standard perturbation theory is unable to access the IR regime because it presents a Landau pole. This may be related to the fact that for large coupling constants the FP procedure is not valid because the gauge-

12 fixing is no longer unique, leading to an infinite number of copies of the gauge field in the model. This was first observed by Gribov in 1978, and is called the Gribov ambiguity. Over the years, various attempts have been made to deal with this problem, see for example [20–25]. Until today, no coherent analytical model for the IR region in non-abelian gauge theories is available. Nonetheless, we have learned several things about the IR physics of QCD from lattice simulations. It was found [26–30], in lattice simulations for the minimal Landau gauge, that the spectral function of the gluon propagator is not non-negative everywhere, which means that in the IR there is no physical interpretation for this propagator like there is for the photon propagator in QED. This behaviour of the gluon spectral function is commonly associated with the concept of confinement [31–34], which means that quarks and gluons clump together under the strong force to form composite particles called hadrons. Because they are confined, we do not see isolated quarks and gluons in nature. The non-positivity of the spectral function then becomes a reflection of the inability of the gluon to exist as a free physical particle. A further important observation in lattice QCD is that the gluon propagator shows massive behaviour in the non-perturbative region. This has prompted research into massive analytical models for the QCD confinement region, which we will discuss in further detail in section 1.3.1, as well as in chapter2, where we will discuss a BRST invariant solution for the massive QCD model.

Through experiments in the 1950s the weak interaction, responsible for radioactive decay of the atoms, was predicted to be carried by three gauge bosons: two W bosons and one Z boson. This ﬁts with the three generators of the SU(2) YM model. Because the W bosons carry electric charge, the weak and electromagnetic force are described together in an SU(2) × U(1) theory, called electroweak theory. However, the W, Z bosons are massive, unlike the photon. In the Lagrangian, a mass term for the gauge boson would take the form

1 2 L ⊃ m AµAµ, (1) 2 which is not gauge-invariant and can not be implemented into the Lagrangian by hand. We now know that the solution for this problem is given by the spontaneous symmetry breaking (SSB) due to the non-zero vacuum value of the scalar SU(2) Higgs ﬁeld Φ(x), and the Higgs mechanism1 that gives mass to the gauge bosons. The SU(2) gauge theory with a scalar ﬁeld is referred to as Yang-Mills-Higgs (YMH) theory. The Glashow- Weinberg-Salam (GWS) model of electroweak symmetry breaking [38–40], based on the Higgs mechanism, gives mass to the W, Z bosons, while keeping the photon massless. Together with QCD, the electroweak theory constitutes the Standard Model (SM) of particle physics.

The electroweak sector can be shown to be both unitary and renormalizable by employing a class of gauge- fixing called ’t Hooft or Rξ gauge, which introduces the gauge parameter ξ. Different choices of ξ highlight different properties of the model. In the formal limit ξ → ∞, we end up in the unitary gauge, which is considered the physical gauge as it decouples the non-physical particles. However, this gauge is known to be non-renormalizable [41]. For any finite choice of ξ, the model is renormalizable because the Rξ gauge cancels an unrenormalizable mixing term between the gauge boson and an unphysical Goldstone boson. For ξ → 0, we end up in the Landau gauge, a very useful gauge choice that picks out the transverse part of the gauge boson. Of course, any physical process or quantity should be independent of the ξ parameter. Therefore, the Rξ gauge provides a powerful check on practical calculations.

Higgs mechanism without spontaneous symmetry breaking

In most textbook accounts, the mass generation of gauge bosons through the Higgs mechanism is displayed in terms of the notion of spontaneous symmetry breaking (SSB) of the local gauge symmetry. However, the meaning of SSB in connection to the Higgs mechanism is ambiguous, since it was established already in 1975 by Elitzur [42] that local gauge symmetries can never be spontaneously broken. In this section, I will try to make sense of these contradictory accounts and discuss the solution provided by the gauge-invariant composite local operators of ’t Hooft [43] and Fr¨ohlich, Morchio and Strocchi [44,45], which are the central subject of this thesis.

The term “spontaneous symmetry breaking” originated in the statistical physics of phase transitions. One of the best known examples is that of SSB in a ferromagnet. Above the Curie temperature TC , the ground state of the ferromagnet is rotationally symmetric because of the random spin orientation of the atoms. Below TC however, the ground state consists of spins which are aligned within a certain domain, thus breaking the rotational symmetry. The orientation of this alignment is random in the sense that each direction is equally likely to occur, but nevertheless one direction is chosen. So, even though the system still has a rotational symmetry, the ground state is not invariant under this symmetry and we say the symmetry is spontaneously

1A more accurate name is the Higgs-Brout-Englert-Guralnik-Hagen-Kibble mechanism [35–37]

13 broken. Any situation in physics in which the ground state of a system has less symmetry than the system itself, exhibits SSB. Two different configurations of the ground state are seperated by an energy barrier, and the phenomenon of SSB therefore goes accompanied by a discontinuous change of a physical quantity related to the free energy, called the order parameter, as a function of another quantity, the control parameter. In the case of a ferromagnet, the magnetic susceptibility (order parameter) goes to infinity at the critical temperature (control parameter).

Figure 1: Representation of the potential for the global U(1) symmetry

Symmetry breaking of the ground state is always connected with mass generation: movement of the ground state in direction of the energy barrier has an energy cost and as is therefore “massive”, while a movement in the direction perpendicular to the energy barrier costs zero energy, i.e. it is a “massless” mode. These concepts become more explicit in field theory, where the massive and massless modes correspond to genuine massive and massless bosons. This was first described by Nambu [46] for the spontaneous breaking of the chiral symmetry of massless fermions and was subsequently elucidated by Goldstone [47]. We can illustrate the occurrence of massless and massive SSB bosons in the simple model of a complex scalar field ϕ(x) with U(1) symmetry. The Lagrangian

2 2 † † † λ † v L = ∂µϕ ∂µϕ − V (ϕ ϕ) with V (ϕϕ ) = ϕ ϕ − , λ > 0 (2) 2 2 is invariant under the global U(1) transformation ϕ → eiαϕ. The minimum of the potential is degenerate, hϕi √v eiθ θ because any configuration of the vacuum expectation value (vev) = 2 , with an arbitrary phase, minimizes the potential. This is the famous “Mexican hat” potential depicted in Figure1. Any choice for θ will lead to the same U(1) invariant Langrangian. However, after choosing a particular θ, the vev hϕi is not U(1) invariant since under a global U(1) transformation θ → θ+α. Thus, the U(1) symmetry is spontaneously broken θ hϕi √v by the vev of the scalar field. Let us now choose one vacuum orientation, e.g. = 0 so that = 2 . We ρ x χ x hρi hχi ϕ √1 v ρ e−iχ/v can define two new real scalar field ( ) and ( ) with zero vev, = = 0, and set = 2 ( + ) . Then, in the Lagrangian (2) the ρ field acquires a mass mρ = λv, while the field χ remains massless. The massless boson is called the Nambu-Goldstone (NG) boson, and the U(1) case is the simplest example of the Goldstone theorem, which states that for every broken generator there is one massless Goldstone boson. For the U(1) model, the SSB breaks the complete symmetry, but for more complex systems, SSB can break the symmetry group down to a subgroup under which the ground state is invariant. For example, the SSB of the chiral symmetry in QCD breaks a global SU(3) × SU(3) chiral flavor symmetry down to the diagonal SU(3) group, generating eight NG bosons.

As is well-known, the global U(1) symmetry can be promoted to a local U(1) gauge symmetry by changing the derivative ∂µϕ in (2) to the covariant derivative Dµϕ = ∂µϕ − ieAµϕ, with Aµ the gauge boson. The Lagrangian

2 2 † † † λ † v L = Fµν Fµν + (Dµϕ) Dµϕ − V (ϕ ϕ) with V (ϕϕ ) = ϕ ϕ − , λ > 0, (3) 2 2

ieα(x) with Fµν = ∂µAν −∂ν Aµ, is invariant under the local U(1) transformations ϕ → e ϕ and Aµ → Aµ +∂µα(x). Naturally, the global U(1) transformations form a subgroup of the local U(1) transformations, namely when α(x) is constant for every point in spacetime, α(x) = α. At first glance, it looks like we can repeat the SSB mecha- U hϕi √v ϕ √1 v h e−iχ/v nism described for the global (1) symmetry by taking = 2 . The field configuration = 2 ( + )

14 then will give a mass to the gauge boson, mA = ev. While the Lagrangian is still invariant under a U(1) gauge transformation, the vev of the scalar ﬁeld is not; we seem to have a SSB of the local gauge symmetry. The χ(x) ﬁeld would be the NG boson for the broken U(1) gauge theory, but for gauge theories this is not a physical degree of freedom since we can use the gauge freedom to set χ(x) = 0; this gauge choice is called unitary gauge.

The process described above is how the Higgs mechanism is displayed in most introductory text on the subject. It can easily be generalized to non-abelian gauge theories, and it is used to describe the SU(2)L×U(1)Y → U(1)EM local symmetry breaking which explains mass generation for the W and Z bosons, as well as for several fermion ﬁelds. In section 1.1 we will go further into the details of the Higgs mechanism in the electroweak sector.

Despite the phenomological success, however, there are some important objections to be made against the concept of SSB of a local gauge symmetry. The most important result in this context is Elitzur’s theorem, which states that local gauge symmetries can never be broken spontaneously. The theorem is rigorously proven on the lattice [42], but can be understood by the 2-D Ising model, a simple lattice configuration with two symmetry-breaking ground states: all ↑ and all ↓. In this system, to go from one ground state to the other there is an extensive energy cost, i.e. there is an energy barrier to overcome, because a change in the ground state configuration encompasses the whole system. Therefore, at low temperature the system gets stuck in a particular spin alignment and the symmetry is broken. In constrast, for local symmetries this argument does not apply. Imagine a local transformation, only acting on a very small subset of the system: this will change a ground state configuration into another ground state configuration at zero energy cost. As a consequence, there is no energy barrier between any two ground states, because they are always related by a sequence of local transformations. The system will be able to explore the entire space of ground states and there is no symmetry breaking. In the case of the local gauge symmetry, this means that φ(x), rather than being stuck in a certain gauge configuration, can move around freely around the “gauge orbit”. This means that when you compute the average value of the operator over the group it must be zero; Elitzur’s theorem is also stated as that all gauge non-invariant operators must have a vanishing vev. For a more formal treatment of Elitzur’s theorem in lattice theory, see for example [48].

Of course, local gauge symmetries can be broken explicitly by adding a symmetry breaking term to the La- grangian. In fact, for the quantization of a field theory we have to break the local gauge symmetry by choosing a specific gauge configuration, i.e. by gauge fixing. It seems that gauge fixing offers a way out of Elitzur’s theorem. Say, for example, that in the U(1) model (3) we fix the gauge in the Landau gauge, ∂µAµ = 0. Since the very purpose of a gauge fixing constraint is to restrict the domain of field configurations, the constraint itself is of course not invariant under a local gauge transformation. Exceptions are the so-called Gribov copies, which appear when the gauge fixing constraint is preserved under local gauge transformations. Gribov copies jeopardize the field description of non-abelian theories in the low-energy (non-perturbative) regime. For small coupling constants, however, the Gribov copies can be ignored and it can be shown that e.g. the Landau gauge does not allow for local gauge transformations in the perturbative region. Even so, it is easy to see that ∂µAµ = 0 is still invariant under the global subgroup of the U(1) gauge transformation, α(x) = α. The residual global symmetry which remains after gauge fixing is called the remnant global gauge symmetry [49]. There is no theorem that forbids the SSB of this symmetry. We can then state that the Higgs mechanism is in fact the hφi √v breaking of the remnant global symmetry by setting = 2 after gauge fixing. This is in fact what is done in some modern takes on the Higgs mechanism [50].

However, there are some important distinction to be made between the SSB of the global gauge symmetry and the original formulation of SSB. In the ferromagnet, the SSB of the spin alignment means that out of all physically distinct configurations, one is actually realised. In contrast, the choice of a particular configuration within the global gauge symmetry does not reflect any physical outcome, and we therefore do not expect an accompanying abrupt change in any physical properties, in particular those related to the free energy. This is in agreement with findings on the lattice [51] that there is no order parameter which varies discontinuously along the path going from the “confinement phase” (strong coupling) to the “Higgs phase” (weak coupling), at least in the fundamental representation of the Higgs field. This absence of a phase boundary is known as “Higgs complementarity”.

A further observation about remnant global gauge symmetry is that it is ambiguous because different gauge choices will lead to different global subgroups after fixing the gauge. For example, the aforementioned unitary gauge leaves no global symmetry, while the Coulomb gauge ∂iAi = 0 leaves a large invariance, α(x) = α(t). The question is then to decide which, if any, of these gauge-symmetry breakings is associated with a transition between physically different phases. In [49], it was found on the lattice that for SSB of the global remnant symmetry, the Landau gauge and the Coulomb gauge show distinct transition lines. Moreover, they both show

15 a SSB at points where there is no actual phase change, making their role in the conﬁnement-Higgs transition highly doubtful.

Another, often forgotten problem with the approach of gauge-fixing to surpass Elitzur’s theorem is that choosing a gauge leaves a residual local symmetry: the BRST symmetry. The vacuum should, even after gauge-fixing, always be BRST invariant because it is a physical quantity. It thus seems there is no way around using gauge- invariant fields to represent physical particles. In non-abelian theories, these fields are necessarily composite. In [43], ’t Hooft defined composite operators for the SU(2) Higgs model. These operators are gauge-invariant combinations of the elementary YM fields, the gauge boson and the fermion fields, with the Higgs field. They are constructed in such a way that for a constant value of the Higgs field, we regain the elementary field. The † ± gauge-invariant composite extension of the neutral Z-boson is defined as Φ DµΦ, while that of the charged W 2 † bosons is given by εijΦiDµΦj and its complex conjugate. The Higgs field itself can be obtained from Φ Φ. ’t Hooft’s definitions incorporate the Higgs complementarity, because there is no fundamental difference between the particle in the Higgs phase and the confining phase. In the confinement phase, the composite operators are bound states of the fundamental field with extremely strong confining forces. In the Higgs phase, the perturbative expansion of these composite operators will lead to gauge-invariant descriptions of the expected weakly-interacting degrees of freedom.

In [44, 45], Fröhlich, Morchio and Strocchi (FMS) further formalized the role of gauge-invariant local composite operators in the Higgs mechanism. Because of their gauge invariance, the composite operators cover the whole gauge orbit, and a non-zero vev for these operators cannot break the local or global gauge invariance. Thus, taking for example the gauge-invariant extension of the Higgs field, O = Φ†Φ, we can assign this field a vev 2 hOi v √1 v ρ e−iχ/v = 2 . Now, we can achieve the usual dynamics of the Higgs mechanism by taking Φ = 2 ( + ) , with hρi = hχi = 0, without having to impose hΦi = v. This is the Higgs mechanism without a symmetry breaking order parameter. The FMS mechanism gives a simple relation between the correlation functions of the ˜ a gauge-invariant fields O(x) and the corresponding gauge dependent ones ϕ = (Aµ, h) calculated in the standard perturbation expansion. Expanding the two-point function of the composite operator we find ˜ ˜ hO(x)O(y)i ∼ hϕ(x)ϕ(y)i1-loop + ... , (4) where ... denote higher-order loop corrections, combinations of the elementary fields which make the sum of correlation functions gauge-invariant.

In QCD, the use of composite operators is common in the description of hypothetical composite bound states in the confinement (IR) region. For example, glueballs, bound states solely composed of gluons, have a field 2 theoretical description as the gauge-invariant operator Fµν [52]. There are however some important differences between configurations like glueballs and the local composite gauge-invariant operators. The local composite gauge-invariant operators have a clear connection to elementary fields: the first transforms into the latter when the Higgs field acquires a constant value. As a consequence, these operators can be analyzed in the Higgs phase by perturbative loop calculations.

The gauge-invariant FMS construction has gained little attention throughout the years. In most texts where it appears, it serves mainly as a justiﬁcation of the Higgs mechanism, while for practical matters the SSB of the gauge symmetry is used. The question is whether the FMS mechanism serves a purpose outside this formal role. To try and answer that question, let us look at the gauge boson propagator

hAµ(x)Aµ(y)i , (5) which plays a fundamental role in QED, describing the quantum dynamical path that a photon takes when it travels from x to y, calculated by means of Feynman-diagrams. In U(1) theory, the transversal part of the propagator is gauge-invariant and can therefore be associated with physically observable quantities. However, in SU(N) theory the (transverse) propagator is not gauge-invariant (or BRST invariant after gauge fixing) and it is therefore impossible that this object would describe a physical quantity. Nevertheless, the propagator (5) is a much used quantity in electroweak theory, and succesfully so: the higher order calculations of the pole masses and cross sections derived from the propagator (5) are in very accurate agreement with the experimental data [13–16]. This apparent paradox can be explained by the FMS construction. Looking at eq. (4), we can see that the pole mass of the elementary field ϕ(x) should coincide with the pole mass of the gauge-invariant composite operator O˜(x). This was confirmed in preliminary lattice result [53, 54]. This means the pole mass

2While they do not play a role in this work, fermions can also be given a gauge-invariant extension, e.g. Φ†Ψ for the neutrino and εij ΦiΨj for the electron.

16 of the gauge-dependent propagator (5) is gauge-invariant and can be interpreted as a physical quantity. The gauge-invariance of the pole mass of (5) can also be proven by the so-called Nielsen identities [55–58] which follow from the Slavnov-Taylor identities. The Nielsen identities ensure that the pole masses of the propagators of the transverse gauge bosons and Higgs ﬁeld do not depend on the gauge parameter ξ entering the Rξ gauge ﬁxing condition.

Nevertheless, as one can easily figure out, the use of the non-gauge-invariant fields has its own limitations which show up in several ways. For example, the spectral density function of the elementary two-point correlation function hϕ(x)ϕ(y)i in terms of the Källén-Lehmann(KL) representation is not protected from gauge- dependence, because the higher-order loop correction in (4) will contribute to the gauge-invariance of the O˜(x) spectral density function. These higher-order loop corrections are the main subject of the present work. We will show how the local gauge invariant composite operators can be introduced in the Higgs model and analyzed by perturbative loop calculations. The main goal is to gain insight in the spectral properties of the gauge bosons and Higgs fields. Calculations of spectral properties of elementary fields are plagued by an unphysical gauge-dependency, but through the use of the gauge-invariant composite operators, these feautures can be made visible.

Outline of this thesis

This thesis is organized as follows. In chapter1, we will discuss existing massive solutions for YM theories. First, we will discuss in detail the GWS model of electroweak symmetry breaking of the Standard Model (SM) in section 1.1. We will also discuss other examples of the Higgs mechanism, that are not part of the SM, in 1.2. Massive solutions for the QCD sector are discussed in section 1.3.1. Chapter2 is based on [59] and was also treated in [60]. In this chapter, we will look at renormalizable gauge class for the non-local gauge invariant h configuration Aµ(x), which can be localized by the dimensionless auxiliary Stueckelberg field ξ. In chapter3, which is based on [61], we will discuss the spectral properties of the U(1) Higgs model. The main purpose of this chapter is to establish the dependence of the spectral properties for the elementary fields on the gauge parameter. This analysis allows to disentangle what is physical and what is not at the level of the elementary particle propagators. In chapter4, the gauge-invariant spectral description of the U(1) Higgs model from local gauge-invariant composite operators will be discussed, as was done in [62]. In5, we will extend this analysis to the SU(2) Higgs model, as done in [63]. Chapter6 summarizes our conclusions and discusses some ideas for future projects, based on the results of this work.

Throughout this thesis, we shall work in Euclidean four-dimensional space-time, unless otherwise indicated.

17 Chapter 1

Mass generation within and beyond the Standard Model

1.1 The Higgs mechanism in the Standard Model

The SM describes the strong interactions, weak interactions and hypercharge in an SU(3)c × SU(2)L × U(1)Y gauge theory, where c stands for color, L for the left-handed fermions it couples to and Y for the hypercharge. Here, we will discuss the Georgi-Weinberg-Salam (GWS) model, where SU(2)L × U(1)Y → U(1)EM. Before engaging in the Higgs mechanism for this model, we will ﬁrst discuss the gauge sector and the Higgs sector for the SM. We will not discuss the fermion sector here, but it is important to realize that quark and lepton masses are also generated through coupling with the Higgs ﬁeld, known as Yukawa coupling. For a nice complete overview, see [64].

The SM gauge sector The four-dimensional action of the SM is described in terms of ﬁeld strength tensors by:

Z 4 1 a a 1 a a 1 Sgauge = d x G G + W W + Bµν Bµν , (1.1) 4 µν µν 4 µν µν 4 with repeated indices taken as summed.

For SU(3)c we have the following ﬁeld strength tensor

a a a abc b c Gµν = ∂µGν − ∂ν Gµ + gs f GµGν , (1.2)

abc with gs the strong interaction coupling strength and f the antisymmetric structure constant. There are eight diﬀerent color charges, corresponding to the eight generators T a of SU(3), given in matrix representation by λa/2, with λa the Gell-Mann matrices.

For the SU(2)L interaction, the ﬁeld strength tensor is given by

a a a abc b c Wµν = ∂µWν − ∂ν Wµ + g WµWν , (1.3) with g the weak interaction coupling strength. There are three diﬀerent charges, so the structure constant is equal to the Levi-Civita tensor f abc = abc. The generators ta are given in matrix representation by τ a/2, with τ a the Pauli matrices.

For both SU(3)c and SU(2)L, and any non-abelian group, the relation between the group generators is given by

[ta, tb] = if abctc. (1.4)

Finally, we have the abelian ﬁeld strength tensor for the U(1)Y interaction

Bµν = ∂µBν − ∂ν Bµ. (1.5)

18 The gauge transformations which leave the Lagrangian (1.1) invariant are

−1 i −1 SU(3)c : Gµ → Uc(x)GµUc (x) + Uc(x)∂µUc (x), gc a a Uc = exp(−igcωc (x)T )

i SU(2) : W → U (x)W U −1(x) + U (x)∂ U −1(x), L µ L µ L g L µ L a a Uc = exp(−igωL(x)t )

i U(1) : B → U (x)B U −1(x) + U (x)∂ U −1(x), Y µ Y µ Y g0Y Y µ Y 0 UY (x) = exp (−ig ωY (x)Y ) (1.6) a a a a 0 with Gµ = GµT , Wµ = Wµ t and g the coupling strength of the hypercharge interaction. In inﬁnitesimal form, this gives

a a a abc b c Gµ → Gµ − ∂µωc (x) − gf Gµωc(x)

a a a abc b c Wµ → Wµ − ∂µωL(x) − g WµωL(x)

Bµ → Bµ − ∂µωY (x) (1.7)

1.1.1 The SM Higgs sector Clearly, the gauge sector does not allow for a massive term of the gauge bosons to be inserted in by hand, since this would violate gauge-invariance. To explain the experimentally-observed nonzero mass of the SU(2)L gauge bosons, the SM requires a new ingredient. We therefore introduce a single SU(2)L-doublet scalar ﬁeld, which will lead to a mass generation of the gauge bosons by means of the Higgs mechanism.

The Higgs ﬁeld Φ is given by an SU(2)L-doublet of complex scalar ﬁelds that can be written as

+ 1 φ 1 φ1 + iφ2 Φ = √ 0 = √ , (1.8) 2 φ 2 φ3 + iφ4 with φi properly normalized real scalar ﬁelds. The Lagrangian for the Higgs sector is given by

† LΦ = (DµΦ) (DµΦ) − V (Φ), (1.9) with the covariant derivative

0 Dµ = ∂µ − ig BµY − igWµ. (1.10) The Lagrangian (1.9) is invariant under the gauge transformations (1.6) in combination with

Φ → UY (x)Φ

Φ → UL(x)Φ, (1.11) with UL(x) and UY (x) as deﬁned in (1.6). We assign a hypercharge Y = 1/2 to the Higgs ﬁeld, and make it a color singlet.

The most general gauge invariant potential energy function is given by

λ v2 2 V (Φ) = Φ†Φ − , (1.12) 2 2 where we will choose λ and v to be real and positive numbers.

19 1.1.2 The SM Higgs mechanism In the introductory chapter, we have seen that the Higgs mechanism cannot be caused by a non-zero vev of the Higgs field Φ(x), because this field is not gauge-invariant. However, this is not contradictory with the statement that the potential (1.12) is minimized by the configuration

1 0 Φ0 = √ , (1.13) 2 v as long as we do not attribute it any physical meaning, such as vacuum expectation values. In fact, eq. (1.13) is eﬀectively expressing the attribution of a non-zero vev to the composite local gauge-invariant operator, namely † † v2 hΦ Φi = Φ0Φ0 = 2 .

We can establish the conﬁguration (1.13) by choosing

φ3,0 = v, φ1,0 = φ2,0 = φ4,0 = 0. (1.14)

We can also define a new scalar field h defined by

φ3 = h + v (1.15) so that h0 = 0, while renaming

φ1 = ρ2, φ2 = ρ1, φ4 = −ρ3. (1.16)

The Higgs ﬁeld then becomes

1 iρ + ρ 1 0 Φ = √ 1 2 = √ ((v + h) 1 + iρaτ a) · . (1.17) 2 v + h − iρ3 2 1 √ Putting this configuration into the potential function (1.12), we find that h gains a mass mh = λv, while the ρa fields are massless. We can rewrite (1.17) up to first order in the fields as

1 iρaτ a 0 Φ = √ exp , (1.18) 2 v v + h and “gauge away” the fields ρa by making the appropriate SU(2) gauge transformation. This means the ρa fields are no physical fields; they are “would-be” Goldstone bosons. This gauge choice is the unitary gauge, and the Higgs field after gauge fixing becomes

1 0 Φ = √ . (1.19) 2 v + h

a The minimizing conﬁguration (1.13) is not invariant under the gauge transformations (1.11), because τ Φ0 =6 0 and 1Φ0 =6 0. Thus, all the generators of SU(2)L × U(1)Y are broken by the conﬁguration (1.13). However, one linear combination of these generators remains unbroken

3 1 3 0 (t + Y )Φ0 = √ t + 1 = 0. (1.20) 2 2 v This is the electric charge operator

1/2 0 1/2 0 1 0 Q = t3 + Y = + = (1.21) 0 −1/2 0 1/2 0 0 and we see from (1.8) that φ+ is charged, while φ0 is uncharged. Thus, the electromagnetic U(1) symmetry group is unbroken, and the Higgs mechanism for the GWS model gives SU(2)L × U(1)Y → U(1)EM .

20 1.1.3 Mass generation for gauge bosons Let us look at the gauge-kinetic term of the Lagrangian (1.9). The terms quadratic in the gauge ﬁelds are 2 2 02 2 0 2 † g v a a g v gg v 3µ (DµΦ) (DµΦ) ⊃ W W + BµBµ − BµW , (1.22) 8 µ µ 8 4 and we can write this in matrix form in the basis (W 1,W 2,W 3,B):

g2 0 0 0  2 2 2 v  0 g 0 0  M =   , (1.23) 4  0 0 g2 −gg0 0 0 −gg0 g02 so that for the pure SU(2) theory, g0 = 0, there is a symmetry under the rotation W 1 ↔ W 2 ↔ W 3, i.e. it is a a abc b c invariant under Wµ → Wµ + ε ω Wµ. This symmetry is called the custodial symmetry, and we will discuss its origin in section 1.1.4.

We can diagonalize the mass matrix by deﬁning

Z cos θ − sin θ W 3 µ = ω ω µ , (1.24) Aµ sin θω cos θω Bµ

g g0 1 2 with cos θω = √ and sin θω = √ . θω is called weak mixing angle. In the basis (W ,W ,Z,A), we g2+g02 g2+g02 have the mass matrix

g2 0 0 0 2 2 2 v  0 g 0 0 M =   . (1.25) 4  0 0 g2 + g02 0 0 0 0 0

We identify Aµ as the photon and Z as the neutral weak boson, while the charged weak bosons are given by the combinations

± W = W1 ∓ iW2. (1.26) The masses of the W and Z bosons are related by

MW = cos θω, (1.27) MZ 0 and for g → 0, we have cos θω = 1 and the custodial symmetry is restored.

The mass matrix (1.25) allows for the masses of W boson and Z boson to be determined in terms of three experimentally well known quantities. First, we have the Fermi coupling constant G = 1.16 × 10−5GeV −2, √ −1/2 which gives the minimizing constant v = G 2 . Then, the parameters g and g0 can be expressed in terms of electric charge and weak mixing angle as g sin θω = g cos θω = e, where e is related to the ﬁne-structure e2 1 constant as α = 4π = 137.04 . The weak mixing angle, ﬁnally, has been determined from neutrino scattering 2 experiments to be sin θω = 0.235 ± 0.005. We have

απ 1/2 1 απ 1/2 2 MW = √ ,MZ = √ (1.28) G 2 sin θω G 2 sin 2θω which gives

MW ≈ 77 GeV,MZ ≈ 88 GeV (1.29) which is a reasonable, but not perfect, approximation of the experimentally determined values

MW = 80.22 ± 0.26 GeV,MZ = 91.17 ± 0.02 GeV. (1.30) The approximation can be further improved by taking into account renormalization corrections.

21 1.1.4 Custodial symmetry We can see the origin of the custodial symmetry by writing out the potential energy function

v2 2 V (φ) = λ φ2 + φ2 + φ2 + φ2 − . (1.31) 1 2 3 4 2

This potential is cleary invariant under a larger symmetry than SU(2)L × U(1)Y : it preserves a global O(4) symmetry under which the vector (φ1, φ2, φ3, φ4) transforms. The global O(4) corresponds to a global SU(2)L × SU(2)R symmetry, as can be seen by writing Φ in the form of a bidoublet

1 φ0∗ φ+ Φ = √ +∗ 0 (1.32) 2 −φ φ so that

2 λ 1 v2 V (φ) = Tr Φ†Φ − , (1.33) 2 2 2 is invariant under the global SU(2)L × SU(2)R symmetry

−1 Φ → MLΦMR −1 Wµ → MLWµMR (1.34)

a σa with MR,L = exp iaR,L 2 two independent global SU(2) matrices. The minimizing conﬁguration for Φ is the bidoublet

1 v 0 Φ0 = √ , (1.35) 2 0 v which breaks the global SU(2)L × SU(2)R symmetry down to subgroup where ML = MR. This is the diagonal subgroup SU(2)diag, and it is preserved thanks to the fact that we were able to write Φ0 proportional to the unit matrix.

In order to have the dynamical term in the Lagrangian invariant under SU(2)L × SU(2)R, we need a covariant derivative such that

−L DµΦ → MLDµΦMR , (1.36) while preserving the form of the dynamical part of the Lagrangian around the vev eq. (1.22). Let us start 0 by switching oﬀ the hypercharge gauge interaction U(1)Y , i.e. g = 0. In this case, we can easily meet the a a requirements (1.22) and (1.36) by promoting SU(2)L to a local gauge symmetry, αL → ωL(x). Thus, the Lagrangian

2 2 † † v LΦ = Tr (DµΦ) (DµΦ) − λ Tr Φ Φ − , (1.37) 2 with

g a a Dµ = ∂µ − i W σ , (1.38) 2 µ has an SU(2)L,local × SU(2)R,global symmetry. The minimizing configuration (1.35) breaks this symmetry down to the global symmetry of the diagonal subgroup SU(2)diag, defined by ML = MR in eq. (1.34). Under this a a abc b c symmetry, the gauge field transforms in infinitesimal form as Wµ → Wµ + ε ω Wµ. The diagonal subgroup SU(2)diag is the custodial symmetry.

Switching on g0, we see from (1.22) that the custodial symmetry is broken. We can see this in the Lagrangian when we try to add the U(1)Y gauge symmetry, while preserving the SU(2)L,local × SU(2)R,global symmetry.

22 If we change the covariant derivative to the full SU(2)L × U(1)Y covariant derivative in eq. (1.10) while using the bioublet Higgs ﬁeld (1.32) , we would get an erroneous, ‘custodial’ result

2 2 02 2 † g v a a g v Tr (DµΦ) (DµΦ) ⊃ W W + BµBµ. (1.39) 8 µ µ 8 We could also deﬁne a new covariant derivative

0 g a a g 3 Dµ = ∂µ − i W σ + i Bµσ , (1.40) 2 µ L 2 R which gives the right result (1.22) at Φ = Φ0. We then have to gauge the third generator of MR and iden- 3 tify it with the hypercharge generator, αR → ωY (x). The generator of the custodial symmetry would be the 3 electric charge Q = t + Y , which remains unbroken because Φ0 is not charged and therefore Q = 0 at this point. However, gauging only one component of a global symmetry violates the global symmetry. Thus, the custodial symmetry is only an approximate global symmetry of the SM, violated by hypercharge gauge interactions.

The custodial symmetry is important because it rules out some possible scalar ﬁeld conﬁgurations. The ρ parameter

2 MW ρ ≡ 2 2 (1.41) cos θωMZ was experimentally measured to be very close to its ‘custodial’ value ρ = 1. Now imagine a complex triplet X with Y = 1 and a minimizing conﬁguration

 0  X0 =  0  (1.42) vX so that

† 2 2 + − 2 2 3 3 02 2 0 2 3µ (DµX) (DµX) ⊃ g vX Wµ Wµ + g vX Wµ Wµ + g vX BµBµ − 2gg vX BµW . (1.43) The mass matrix is therefore, in the basis (W 1,W 2,W 3,B)

g2 0 0 0  2 2 2  0 g 0 0  M = v   , (1.44) X X  0 0 2g2 −2gg0 0 0 −2gg0 2g02 which we can diagonalize this in the same way as the Φ mass matrix by using the weak mixing angle matrix (1.24), so that in the basis (W 1,W 2,Z,A)

g2 0 0 0 2 2 2  0 g 0 0 M = v   . (1.45) X X  0 0 2(g2 + g02) 0 0 0 0 0 The photon is still massless, but the custodial symmetry for g0 → 0 is no longer present.

In the presence of both the Φ doublet and the X triplet we have

g2 g2 + g02 M 2 = (v2 + 4v2 ),M 2 = (v2 + 8v2 ) (1.46) W 4 X Z 4 X so that

2 2 v + 4vX ρ = 2 2 , (1.47) v + 8vX so that vX has to be very small compared to v to meet the experimental value of ρ. Thus, the existence of a scalar boson X with a signiﬁcant minimizing value X0 is ruled out by the custodial symmetry. However, combinations of X with other triplets can restore the custodial symmetry, and make for some interesting beyond-the-SM phenomenology [64].

23 1.1.5 Rξ gauge class In the previous sections, we have restricted ourselves to a specific gauge choice, the unitary gauge. Even though the gauge choice should never affect any physical outcome, different gauge choices help us to understand different aspects of our model. For example, in the unitary gauge all fields are physical, which proves the unitarity of the S-matrix. On the other hand, using e.g. the Landau gauge, we can prove the renormalizability of the Higgs model.

Instead of using different gauge fixing models, we can combine several gauge choices into a gauge class. Gauge classes can go through different gauge choices by means of an unphysical gauge parameter. For example, the Linear Covariant Gauges (LCG) is a gauge class given by ∂µAµ = α b, with b an auxiliary field and α the gauge parameter. For α = 0, we end up in the Landau gauge, while for finite α the gauge field acquires a longitudinal component. Of course, physical quantities should never depend on α; the gauge parameter is therefore also an important check of the physicality of our outcome.

In this section, we will discuss the Rξ gauge class, introduced by ’t Hooft to prove the renormalizability of the Higgs model. It is therefore an important gauge class for the GWS model, but can be understood in the U(1) Higgs model. In the action

Z ( 2 2) 4 1 † λ † v S = d x Fµν Fµν + (Dµϕ) Dµϕ + ϕ ϕ − , (1.48) 4 2 2

ϕ x √1 v h x e−iχ(x)/v χ x we can parametrize ( ) = 2 ( + ( )) , and then make the unitary gauge choice ( ) = 0. This decouples the non-physical “would-be” NG boson. However, expanding the Lagrangian we have for the squared gauge boson terms

1 2 1 1 2 L ⊃ − Aµ∂ Aµ + Aµ∂µ∂ν Aν + m AµAµ, (1.49) 2 2 2 with m = ev. The two-point function for the gauge boson is then given by

1 2 1 2 hAµ(p)Aν (−p)i = Pµν (p ) + Lµν (p ), (1.50) p2 + m2 m2

2 pµpν 2 pµpν with Pµν (p ) = δµν − p2 and L(p ) = p2 the transversal and longitudinal projectors. The longitudinal part does not vanish for large values of the momentum p, and we cannot prove renormalizability.

The problem with renormalizability can be made more explicit by adapting another parametrization of the ϕ x √1 v h x iρ x scalar ﬁeld in eq. (1.48), namely ( ) = 2 ( + ( ) + ( )). This will lead to a term in the Lagrangian

L ⊃ m Aµ∂µρ, (1.51) and this mixing term between the gauge boson and the would-be NG boson will lead to unrenormalizable results. We therefore employ the gauge-ﬁxing action

Z 4 1 2 Sgf = d x (∂µAµ + ξmρ) , (1.52) 2ξ known as the Rξ gauge class with the gauge parameter ξ. The mixing term in (1.52) will cancel exactly the unwanted mixing term (1.51). The two-point function for the gauge boson is now given by

1 2 ξ 2 hAµ(p)Aν (−p)i = Pµν (p ) + Lµν (p ), (1.53) p2 + m2 p2 + ξm2 and we can show that this gauge is renormalizable for all ﬁnite values of ξ. Notice that for ξ = 0 we end up in the Landau gauge, while in the limit ξ → ∞ we ﬁnd back eq. (1.50), the unitary gauge. This means the Rξ-gauge is both unitary and renormalizable. The would-be Goldstone remains in the Lagrangian and has a two-point function

24 1 hρ(p)ρ(−p)i = , (1.54) p2 + ξm2 but the very fact that its mass depends on the ξ-parameter brands it as unphysical. The same thing happens for the ghost ﬁeld.

Another important feature of the Rξ-gauge is that we can show, in non-abelian and abelian gauge theories, that the pole masses of the two-point function of both the gauge boson Aµ(x) and the Higgs ﬁeld h(x) do not depend on the gauge parameter ξ [57]. This property, contained in the Nielsen Identities [55], is important because it allows to give a physical meaning to the polemass of the otherwise gauge-dependent (and therefore unphysical) correlation functions.

a Even though the Rξ-gauge is unique to the Higgs model because it requires the Goldstone boson ρ , the concepts of this gauge class can be used to deﬁne a renormalizable class of gauge-ﬁxing also outside of the Higgs environment, as we will see in chapter2.

1.2 Higgs beyond the Standard Model

The Higgs mechanism for the electroweak sector has been ﬁrmly established as the mechanism that gives mass to the W, Z bosons, as well as some fermions, while leaving the photons massless. In 2012, the Higgs boson was discovered at the CERN Large Hadron Collidor (LHC) [65, 66], in full agreement with the SM predictions.

It is nonetheless important to realize that there is no a priori reason that the Higgs mechanism only occurs in the version SU(2) × U(1) → U(1). The Higgs mechanism can be described for any gauge theory, such as the simple example U(1) → nothing that we discussed in the introduction. However, the photon is massless, so we know that nature did not provide for a U(1) Higgs boson. Still, the U(1) Higgs model is a useful toy model to study several properties of the Higgs model, as we will see in chapter3 and4 . Another useful model is SU(2) → nothing, the Higgs-Yang-Mills (HYM) model which can be achieved by setting g0 = 0 in the electroweak model. This model will be central to chapter5. As we have seen in the previous section, the pure SU(2) model will exhibit a full custodial symmetry.

Since the discovery of the Higgs mechanism, several symmetry breaking models have been proposed besides the GWS model. We will discuss two of them, both proposed by Georgi and Glashow. Then, we will also disuss how the Higgs mechanism would occur in QCD in the presence of an SU(3) Higgs boson.

It is important to mention that while all the examples of the Higgs mechanism in this section are variations of the Higgs mechanism in the electroweak sector, the same mechanism of a non-zero minimizing value is also widely used in other areas such as condensed matter physics, where it was ﬁrst established by Anderson [67] as an analogy to the Landau-Ginzburg eﬀective model of superconductivity.

1.2.1 SU(2) → U(1) This model, proposed in [68], provides an alternative Higgs model in the electroweak sector. Since SU(2) is homeomorphic to SO(3), we will use the Hermitian traceless generators

0 0 0   0 0 i 0 −i 0 1 2 3 τ = 0 0 −i τ =  0 0 0 τ = i 0 0 , (1.55) 0 i 0 −i 0 0 0 0 0 so that (τ a)bc = −iabc. This is a real representation, so the Lagrangian is given by

1 a a 1 2 λ † 22 L = − F F + (Dµφ) − φφ − v , (1.56) 4 µν µν 2 8 and the minimizing value of φ(x) is

0 φ = 0 (1.57) v

25 so that the mass matrix is given by

m2 0 0 2 M =  0 m2 0 , (1.58) 0 0 0

1 2 3 which means two gauge bosons, Aµ and Aµ, are massive, while Aµ is massless.

The above model does not account for the neutral Z-boson and is therefore not the correct theory for the 3 electroweak mass generation. However, as was shown in [69], identifying the photon with the massless Aµ does provide for a radial magnetic ﬁeld which indicates the existence of a magnetic monopole. The magnetic monopole, often hypothesized but never found, is not present in the GWS model.

1.2.2 SU(5) → SU(3) × SU(2) × U(1) It would be aesthetically nice if the standard model was the low-energy phenomenology of a larger gauge theory. This is the idea of a Grand Unified Theory (GUT). Georgi and Glashow proposed a model to this effect in [70]. The required Higgs field transforms under the adjoint representation of SU(5) and can be represented by 5 × 5 hermitian traceless matrix. The Higgs field transforms under the adjoint representation as

Φa → Φa + f abcωbΦc (1.59) or, adding a generator ta on both sides

Φ → Φ − iωa[ta, Φ], (1.60) so that conﬁgurations of Φ that commute with a (sub)group with generators ta are invariant under the gauge transformation related to this (sub)group.

As for any non-abelian gauge theory, the covariant derivative in the adjoint representation is given by

ab ab abc c Dµ = δ ∂µ − gf Aµ (1.61) with N 2 − 1 = 24 diﬀerent color charges, corresponding to the 24 generators ta of SU(5). We can use eq. (1.4) a b δab and the normalization [t , t ] = 2 to write the mass term of the Lagrangian as

2 g abc b c 2 2 a b a b Lm = (f A Φ ) = −g Tr[[t , Φ0][t , Φ0]]A A , (1.62) 2 µ 0 µ µ which means, from eq. (1.60), that gauge bosons corresponding to unbroken symmetries remain massless.

If we choose

2 0 0 0 0  0 2 0 0 0    Φ0 = v 0 0 2 0 0  , (1.63)   0 0 0 −3 0  0 0 0 0 −3 T a 0 0 0 this commutes with the SU(3) subgroup , the SU(2) subgroup and the U(1) subgroup corre- 0 0 0 ta sponding to the generator proportional to Φ0. We then have twelve massless gauge bosons, corresponding to the unbroken generators of SU(3) × SU(2) × U(1), and twelve massive gauge bosons. This adjoint Higgs particle would only accomplish the seperation of the diﬀerent interactions. Another Higgs ﬁeld, in the fundamental representation, would provide electroweak symmetry breaking.

The attractiveness of the SU(5) model, besides its simplicity, lies in the fact that matter particles ﬁt neatly into representations of SU(5). A single generation of the standard model ﬁts perfectly into two irreducible representations of SU(5), the 5 and 10. However, the model predicts many unobserved phenomena such as proton decay, and quark-to-lepton mass ratios that don’t agree with experiment. It also predicts particles so 15 heavy (MGUT ≈ 10 GeV) that they cannot be detected.

26 1.2.3 SU(3) → SU(2) × U(1) The SU(3) Higgs mechanism works in a similar way as that of SU(5). In the adjoint representation, the minimizing conﬁguration

1 0 0  Φ0 = v 0 1 0  , (1.64) 0 0 −2

ta 0 commutes with the SU(2) subgroup , the Gell-Mann matrices λ , and the U(1) subgroup correspond- 0 0 1,2,3 ing to the generator proportional to Φ0, i.e. λ8. This amounts to four massless gauge bosons, corresponding to the unbroken generators of SU(2) × U(1), and four massive gauge bosons, corresponding to λ4,5,6,7. From (1.62) one can then ﬁnd the mass of the gauge bosons, m = 3gv.

It is well-known that gluons in the high-energy (perturbative) regime are massless. Therefore, we do not expect a Higgs boson for the strong interaction. However, in non-perturbative regimes there are indications of massive behaviour, as we will discuss in the next section.

1.3 Massive solutions for the QCD sector

In the high-energy regime, the FP procedure gives the gauge-ﬁxed YM action for the massless QCD sector: Z 4 1 a a a a a ab b S = d x F F + b ∂µA + c ∂µD c , (1.65) 4 µν µν µ µ

ab with the covariant derivative Dµ in the adjoint representation of the gauge group as in eq. (1.61), the ﬁelds (ca, ca) denoting the FP ghosts and ba the Lagrange multiplier implementing the Landau gauge condition a ∂µGµ = 0. The gauge-ﬁxed Lagrangian is invariant under the nilpotent BRST symmetry

a ab b sAµ = −Dµ c 1 sca = gf abccbcc 2 sca = ba sba = 0, (1.66) with s2 = 0. Notice that the Lagrangian in eq. (1.65) describes any SU(N) theory. As we will see in what follows, in some cases an SU(2) theory is used as a simplification of the SU(3) theory to describe the gluon dynamics of the strong interaction. Notice also that we use the denotation for the gluon field strength tensor a a a a 1 Fµν and for the vector field Aµ, instead of Gµν and Gµ from section 1.1 .

The predictions in the QCD sector derived from the Lagrangian in eq. (3.73) through perturbation theory agree with what has been observed in high-energy experiments. However, when extending this analysis to finite energies, the coupling constant gc diverges and hits a Landau pole. In early studies of QCD, this behaviour was seen as responsible for the IR counterpart of asymptotic freedom, the “IR slavery” that keeps the gluons and quarks confined [71]. In other words, confinement was seen as a direct consequence of the Landau pole. How- ever, the modern view based on lattice simulations is that the Landau pole is not an expression of confinement, but rather a sign that some non-perturbative effect, such as the Gribov ambiguity, invalidates the extension of the perturbative results to the IR regime.

The Gribov ambiguity or Gribov problem, first described by Gribov in [72], demonstrates the non-uniqueness of the FP gauge-fixing beyond the perturbative level. To explain this, let us consider the Landau gauge, although a 0a an analogous argument can be cast in other gauges. If two gluon fields Aµ and Aµ connected by a gauge transformation

0a a ab b Aµ = Aµ − Dµ α , (1.67)

1This is in line with most articles written on the subject of YM theories, since for pure YM theories there is no danger of confusion with the photon ﬁeld Aµ.

27 they are said to be on the same gauge orbit. Now if both ﬁelds are satisfying the same gauge ﬁxing condition, 0 i.e. ∂µAµ = ∂µAµ = 0, so that

ab b −∂µDµ α = 0, (1.68) this means that the FP procedure failed to fully eliminate the multiple counting of physical states in the path integral due to gauge invariance. So, when eq. (1.68) has solutions for certain values of the gauge field, so-called zero modes, this means the model still contains different gauge configurations, known as Gribov copies. Notice 2 that for small values of the coupling constant gc, the LHS of eq. (1.68) reduces to ∂ α, which has only positive eigenvalues. Therefore, the Gribov problem does not exist in the UV regime, and the FP procedure is valid there. Over the years, various attempts have been made to deal with this problem in the continuum functional approach, mainly by trying to evaluate the path integral in such a way that it contained no zero modes. The most notable attempts in this direction are the Gribov-Zwanziger (GZ) approach [20, 21, 25] and the Refined Gribov-Zwanziger (RGZ) approach [22, 24, 73, 74], recently formulated in a BRST invariant fashion [24, 75–78]. For a nice overview of the Gribov problem and the (R)GZ approach, see [79].

The breakdown of the FP procedure and the existence of a Landau pole demonstrate the necessity of non- perturbative components that break with the standard FP gauge-fixed YM picture. This has lead to analytical QCD models that fundamentally differ from the model in eq. (1.65) in the IR, but preserves the standard FP predictions in the UV. Interestingly, the problem of the Landau pole in asymptotically free theories is in some cases circumvented by the introduction of an infrared gluon mass. Still, however, we want to recover the massless character of the FP gauge fixing theory in the perturbative UV region. A model which implements an effective mass only in the IR region was first proposed in [80] based on the idea of a momentum-dependent or dynamical gluon mass [81, 82]. For this, the Schwinger-Dyson (SD) equations are employed in order to get a suitable gap equation that governs the evolution of the dynamical gluon mass m(p), which vanishes for p2 → ∞. This setup preserves both renormalizability and gauge invariance.

The SD equations give relations between Green’s functions that go beyond perturbation theory. In principle this would make them the most important analytical tool to get insight in the gluon propagator in the IR regime, and they are often employed as such [83–87]. However, in practice the SD equations are hard to work with because they entail an inﬁnite set of coupled equations for the vertex functions, which somehow needs to be truncated. This requires involved techniques and calculations, with in some cases an important numerical part. Numerically, the IR regime of the gluon propagator is more rigorously described by lattice simulations. One important observation on the lattice is that the gluon propagator reaches a ﬁnite positive value in the deep IR for space-time Euclidean dimensions d > 2, see e.g. [88–102]. This saturation of the gluon propagator for small momenta p, see Figure 1.1, indicates massive behavior of the gluon in the IR regime. Massive-like behavior for the gluon propagator, known as the decoupling solution, has also emerged within other approaches, as the study of the Schwinger-Dyson equations, the Renornalization Group and other techniques, see for instance [103–111] and references therein.

Figure 1.1: The gluon propagator D(p2) as a function of the lattice momenta p, with p given in GeV. Figure from [94].

28 1.3.1 Massive Yang-Mills model The lattice results, as well as the fact that the Landau pole can be circumvented by an eﬀective gluon mass, has stimulated research into eﬀective massive models for the IR region of QCD. The aforementioned RGZ theory is an example of such a model. In this section we will discuss another example: the massive YM model from [112], which is a particular case of the Curci-Ferrari (CF) model [113]. The action for this theory is

Z 4 1 a a a a a ab b 1 2 a a S = d x G G + b ∂µA + c ∂µD c + m A A , (1.69) 4 µν µν µ µ 2 µ µ which is a Landau gauge FP Euclidean Lagrangian for pure gluodynamics, supplemented with a gluon mass term. This term modifies the theory in the IR but preserves the FP perturbation theory for momenta p m. It was argued that this model could be part of a complete gauge-fixing in the Landau gauge, since a CF gluon mass term may arise after the Gribov copies have been accounted for via an averaging procedure [23], see also [114] for a related discussion in a different gauge. The mass term breaks the BRST symmetry of the model, which means the unitarity of the model can be no longer proven [115], although it is debatable whether some non-perturbative effect could restore unitarity. However, since the BRST breaking is soft, it does not spoil renormalizability. The Lagrangian (1.69) turns out to be still invariant under a modified BRST symmetry

a ab b smAµ = −Dµ c

a 1 abc b c smc = gf c c 2 a a smc = b a 2 a smb = im c , (1.70)

2 a which is however not nilpotent since smc =6 0.

The massive YM model is not justiﬁed a priori from ﬁrst principles. Instead, its legitimacy is measured by how well it accounts for lattice results. In [112, 116, 117], it has been shown that, both at one and two-loop order, the model reproduces very well the lattice predictions for the gluon and ghost propagator, see Figure 1.2. It was also shown that with an adequate renormalization scheme, dubbed the Infrared Safe (IS) scheme, there is no Landau pole.

Figure 1.2: The gluon propagator in the IS scheme compared with the lattice result from [100] for one- and two-loop corrections. Figure from [117].

The analysis of the gluon propagator in [112, 116, 117] was done by perturbative loop calculations. How can the non-perturbative region be accessed with perturbative methods? In [116,117], it is claimed that higher loop corrections for this model seem to be rather small, even for a signiﬁcantly high coupling constant (g = 7.5 in [116]). The reason for this, as explaind in [116], lies in the massive gluons. When momenta are much smaller than the gluon mass, all diagrams that include internal gluon lines are suppressed by inverse powers of the gluon mass. This means that higher order loop terms, which naturally possess more internal gluon lines, will be surpressed. Thus, using an eﬀective mass term makes perturbative loop calculations possible in an otherwise non-perturbative region.

29 1.3.2 Positivity violation and complex poles The CF model is capable of reproducing, to high accuracy, the lattice results of the gluon propagator. This is despite the fact that the gluon propagator as derived from eq. (1.69) is not gauge-invariant and the model has no nilpotent BRST invariance. It should be emphasized here, however, that the goal of [112] and follow-up works was not to introduce a theory for massive gauge bosons, but to discuss a relatively simple and useful eﬀective description of some non-perturbative aspects of QCD. Also, in this respect the CF model is not in a worse position than other models that try to go beyond standard perturbation theory, such as the GZ model, which also breaks BRST symmetry. One could even argue that unitarity of the gauge bosons sector, secured by a nilpotent BRST symmetry, is not so much an issue here as one expects the gauge bosons to be undetectable anyhow, due to conﬁnement.

An interesting question is whether the massive YM model is also capable of reproducing other aspects of QCD observed on the lattice. One of the most intriguing observations of lattice QCD in recent years is positivity violation. Positivity violation means that in the KL spectral representation of the propagator

Z ∞ 2 ρ(t) G(p ) = 2 dt, (1.71) 0 t + p the spectral density function ρ(t) is not positive everywhere. The spectral density function displays, for a certain two-point function such as the gluon propagator, the different states (1-particle states, bound states and multiparticle states) associated with different energy values. If the spectral density function violates positivity, the states it describes cannot be part of the physical state space. Positivity violation is therefore attributed to confinement [31–34]: the non-positivity of the spectral function is seen as a reflection of the inability of the gluon to exist as a free physical particle.

Figure 1.3: Lattice results for the real space propagator related to the gluon propagator, for the quenched case (no quarks) and light sea quark masses. The quenched approximation corresponds to the pure gauge theory for QCD. Figure from [118].

Positivity violation for the gluon propagator has been conﬁrmed in both analytical studies of the SD equations [119–121] and in lattice simulations [27, 118]. In [122], a complete overview of evidence for positivity violation is given. On the lattice, positivity violation is detected through the real space propagator C(t) related to the gluon propagator, see Figure 1.3. The real space propagator is deﬁned by

Z ∞ dp C(t) = eiptG(p), (1.72) −∞ 2π that is, C(t) is the Fourier transform of the gluon propagator G(p). It can be shown, see for example [27], that positivity of the real space propagator implies positivity of the spectral density function.

30 In [112], the real space propagator for the gluon propagator in the CF model was obtained by inserting the one-loop gluon propagator derived from the action (1.69) into eq. (1.72), and it was found that the curve of C(t) as observed on the lattice was reproduced, including the positivity violation, see Figure 1.4. On the one hand, it is remarkable that the CF model seems capable of reproducing the lattice results of the real space propagator. On the other hand, we should distinguish between the significance of positivity violation on the lattice and in the CF model. In lattice simulations, one starts out with a unitary (physical) YM model and subsequently observes positivity violation, i.e. non-physical behavior. In the CF model however, one starts out with a model without a nilpotent BRST symmetry. From the Kugo-Ojima criterion [123], it is known that nilpotency of the BRST symmetry is indispensable to formulate suitable conditions for the construction of the states of the BRST invariant physical (Fock) sub-space, providing unitarity of the S-matrix. Indeed, in [124, 125] the existence of negative norm states in the sm-invariant subspace (“the would-be physical subspace”) was confirmed, see also section 3.5 for a detailed example. Therefore, to find non-physical behavior for a model without BRST symmetry is somewhat of a self fulfilling prophecy.

Figure 1.4: Real space propagator related to the one-loop corrected gluon propagator for the massive YM model. Figure from [112].

In fact, the non-physical behaviour of the CF model can be detected in a step prior to the spectral density function. As was established in [126], and recently in [127] in the context of SD equations, the use of the Landau gauge in the massive YM model (1.69) leads to complex pole masses, which will obstruct the calculation of the KL spectral function. Indeed, if at some order in perturbation theory (one-loop as in [126] for example) a pair of Euclidean complex pole masses appear, at higher order these poles will generate branch points in the complex p2-plane at unwanted locations, i.e. away from the negative real axis, deep into the complex plane, thereby invalidating a KL spectral representation. This can be appreciated by rewriting the Feynman integrals in terms of Schwinger or Feynman parameters, whose analytic properties can be studied through the Landau equations [128]. Let us also refer to [129,130] for concrete examples. In chapter2 we will develop some methods to avoid complex poles in the unitary Higgs model.

Finally, it must be pointed out that also lattice simulations of the gluon propagator are not free of “built- in” non-physical features that could attribute to positivity violation. In all studies that show a violation of positivity, both in the context of SD equations and on the lattice, a gauge-dependent and therefore a priori non-physical Landau-gauge gluon propagator is used. Indeed, even while on the lattice it is in principle possible to work only with gauge-invariant quantities, in [27,118] a gauge-dependent environment was created, including a lattice gauge-fixing. This gauge-dependent environment also provided the opportunity to test the naturally gauge-dependent GZ proposal on the lattice, see for example [94, 131–134]. The use of the gauge-dependent propagator is justified in literature because for an unconfined field, the propagator presumably has a normal KL representation [135], in line with the fact that predictions made from the perturbative gluon propagator of the gauge-fixed YM model agree with experimental observations [13–16]. This motivated the hypothesis that gauge dependent propagators could give some piece of information about confinement. Nonetheless, the question can be asked whether the gauge field in its elementary, gauge-dependent form gives a complete representation of the gauge boson in all ranges of the energy scale. It can be hypothesized that these elementary fields are in fact part of a greater gauge-invariant configuration, which shares with the gauge-dependent field some essential, but not all, properties. In the next chapters, we will discuss two proposals to this effect: the non-local composite

31 conﬁguration Ah in chapter2 and the local gauge-invariant composite operators proposed by ’t Hooft [43] in chapters3 and4. The latter are unique to the Higgs model, but they are not less relevant to the above question since the W, Z bosons are also expected to be conﬁned in the low-energy regime.

32 Chapter 2

On a renormalizable class of gauge ﬁxings for the gauge invariant operator 2 Amin

2 In this chapter we pursue the investigation [136] of the dimension two gauge invariant operator Amin, obtained R 4 u u by minimizing the functional Tr d x AµAµ along the gauge orbit of Aµ [137–140], namely Z 2 4 u u Amin ≡ min Tr d x AµAµ , {u} i Au = u†A u + u†∂ u . (2.1) µ µ g µ

2 As highlighted in [136], the functional Amin enables us to introduce a non-local gauge invariant field configura- h tion Aµ [141] which turns out to be helpful to construct renormalizable BRST invariant YM theories which can be employed as effective massive theories to study non-perturbative infrared aspects of confining YM theories 2 in Euclidean space. As we will see, the massive YM model discussed in 1.3.1 is deeply related to Amin, because h Aµ and Aµ are equal in the Landau gauge, see eq. (2.5) below. See also [142] for a supersymmetric extension h of the composition Aµ(x).

2 In the present chapter we extend the analysis of the operator Amin to a general class of covariant gauges which share great similarity with ’t Hooft’s Rζ -gauge discussed in section 1.1.5. In fact, as shown in [136], the 2 h localization procedure for both Amin and Aµ requires the introduction of a dimensionless auxiliary Stueckelberg field ξ which, as much as the Higgs field of ’t Hooft’s Rζ -gauge, will now enter explicitly the gauge condition through the appearance of a gauge massive paramater µ2. This property will enable us to provide a fully BRST invariant mass for the auxiliary field ξ, a feature which might have helpful consequences in explicit loop calculations involving ξ in order to keep control of potential infrared divergences associated to its dimensionless nature. Moreover, as in the case of Rζ -gauge, also the Faddeev-Popov ghosts will acquire a mass through the gauge-fixing. Of course, setting µ2 = 0, the linear covariant gauges discussed in [136] will be recovered.

The chapter is organized as follows. In section 2.1 we give a short presentation of the main properties of 2 h Amin and of the gauge invariant configuration Aµ, reminding to Appendix (A.1) for more specific details. Sec- 2 h tions 2.2,2.3,2.4,2.5 are devoted to the presentation of the local BRST invariant action for Amin and Aµ as well as of the main properties of the aforementioned gauge-fixing. In section 2.6 we establish the set of Ward identities fulfilled by the resulting action. These identities will be employed to characterize the most general allowed invariant counterterm through the procedure of the algebraic renormalization [17]. In section 2.7 a detailed analysis of the counterterm will be presented together with the renormalization factors needed to establish the all order renormalizability of the model. Section 2.8 contains our conclusion. Finally, in Appendix (A.2), a second, equivalent, proof of the renormalizability of the model will be outlined by making use of a generalised gauge fixing and ensuing Ward identities.

33 2 2.1 Brief review of the operator Amin and construction of a non-local h gauge invariant and transverse gauge ﬁeld Aµ

2 Here we will give a short overview of the operator Amin, eq.(2.1), reminding to the more complete Appendix A.1 for details.

In particular, looking at the the stationary condition for the functional (2.1), one gets a non-local transverse field h h configuration Aµ, ∂µAµ = 0, which can be expressed as an infinite series in the gauge field Aµ, see Appendix A.1, i.e.

∂ ∂ Ah = δ − µ ν φ , ∂ Ah = 0 , µ µν ∂2 ν µ µ 1 ig 1 1 3 φν = Aν − ig ∂A, Aν + ∂A, ∂ν ∂A + O(A ) . (2.2) ∂2 2 ∂2 ∂2

h Remarkably, as shown in Appendix A.1, the conﬁguration Aµ turns out to be left invariant by inﬁnitesimal gauge transformations order by order in the gauge coupling g [141]:

h δAµ = 0 ,

δAµ = −∂µω + ig [Aµ, ω] . (2.3)

Making use of (2.2), the gauge invariant nature of expression (2.1) can be made manifest by rewriting it in terms of the ﬁeld strength Fµν . In fact, as proven in [137], it turns out that Z Z 2 4 h h 1 4 1 1 1 1 A = d xA A = − Tr d x Fµν Fµν + 2i Fλµ DκFκλ, Dν Fνµ min µ µ 2 D2 D2 D2 D2 1 1 1 −2i F D F , D F + O(F 4) , (2.4) D2 λµ D2 κ κν D2 ν λµ from which the gauge invariance becomes apparent. The operator (D2)−1 in expression (2.4) denotes the inverse 2 of the Laplacian D = DµDµ with Dµ being the covariant derivative [137]. Let us also underline that, in the h h 2 Landau gauge ∂µAµ = 0, the operator (AµAµ) reduces to the operator A

h,a h,a a a (Aµ Aµ ) = AµAµ . (2.5) Landau

h,a h,a This feature, combined with the gauge invariant nature of (Aµ Aµ ), implies that the anomalous dimension of h,a h,a a a (Aµ Aµ ) equals [136], to all orders, that of the operator AµAµ of the Landau gauge, i.e.

γ(Ah)2 = γA2 . (2.6) Landau

Moreover, as proven in [143], γA2 is not an independent parameter, being given by Landau

2 β(a) Landau g γA2 = + γA (a) , a = , (2.7) Landau a 16π2

Landau where (β(a), γA (a)) denote, respectively, the β-function and the anomalous dimension of the gauge ﬁeld Aµ in the Landau gauge. This relation was conjectured and explicitly veriﬁed up to three-loop order in [144].

h 2.2 A local action for Aµ

h Following [136], a fully local framework for the gauge invariant operator Aµ can be achieved. To that end, we consider the local, BRST invariant, action

Z Z 2 4 1 a a 4 a h,a m h,a h,a a ab h b Sinv = d x F F + d x τ ∂µA + A A +η ¯ ∂µD (A )η , (2.8) 4 µν µν µ 2 µ µ µ where i Ah ≡ Ah,a T a = h†A h + h†∂ h. (2.9) µ µ µ g µ

34 with a a h = eigξ = eigξ T . (2.10) The matrices {T a} are the generators of the gauge group SU(N) and ξa is an auxiliary localizing Stueckelberg ﬁeld.

By expanding (2.9), one finds an infinite series whose first terms are

h a a a abc b c g abc b c (A ) = A − ∂µξ − gf A ξ − f ξ ∂µξ + higher orders . (2.11) µ µ µ 2

h That the action Sinv gives a local setup for the nonlocal operator Aµ of eq.(2.2) follows by noticing that the Lagrange multiplier τ implements precisely the transversality condition

h ∂µAµ = 0 , (2.12) which, when solved iteratively for the Stueckelberg ﬁeld ξa, gives back the expression (2.2), see Appendix A.1. In addition, the extra ghosts (¯η, η) account for the Jacobian arising from the functional integration over τ h m2 h,a h,a which gives a delta-function of the type δ(∂A ). Finally, the term 2 Aµ Aµ accounts for the inclusion of h,a h,a 2 the gauge invariant operator Aµ Aµ through the mass parameter m which, as mentioned before, can be used as an eﬀective infrared parameter whose value can be estimated through comparison with the available lattice simulations on the two-point gluon correlation function, see [22,73,74,88,89,92,95,97–99,103–112,116,145]

The local action Sinv, eq.(2.8), enjoys an exact BRST symmetry:

sSinv = 0 , (2.13) where the nilpotent BRST transformations are given by

a ab b sAµ = −Dµ c , g sca = f abccbcc , 2 sc¯a = iba , sba = 0 , sτ a = 0 , sη¯a = sηa = 0 , s2 = 0 . (2.14)

For the Stueckelberg ﬁeld one has [146], with i, j indices associated with a generic representation,

ij a a ik kj h a sh = −igc (T ) h , s(A )µ = 0 , (2.15) from which the BRST transformation of the ﬁeld ξa can be evaluated iteratively, yielding

g g2 sξa = gab(ξ)cb = −ca + f abccbξc − f amrf mpqcpξqξr + O(ξ3) . (2.16) 2 12

2.3 Introducing the gauge ﬁxing term Sgf

As it stands, the action (2.8) needs to be equipped with the gauge ﬁxing term, Sgf , which we choose as Z 4 a a 2 a α a a Sgf = d x s c¯ (∂µA − µ ξ ) − i c¯ b µ 2 Z 4 a a α a a 2 a a a ab b 2 a ab b = d x ib ∂µA + b b − iµ b ξ +c ¯ ∂µD (A)c + µ c¯ g (ξ)c . (2.17) µ 2 µ

Besides the traditional gauge parameter α, we have now introduced a second gauge massive parameter µ2. As it will be clear in the next section, this massive parameter will provide a fully BRST invariant regularizing mass for the Stueckelberg field ξa, a feature which has helpful consequences when performing explicit loop calculations involving ξa. Setting µ2 = 0, the gauge fixing (2.17) reduces to that of the usual linear covariant gauge [136]. 2 a Moreover, when µ = α = 0, the Landau gauge, ∂µAµ = 0, is recovered. Nevertheless, it is worth underlining that both µ2 and α appear only in the gauge fixing term, which is an exact BRST variation. As such, µ2 and α are pure gauge parameters which will not affect the correlation functions of local BRST invariant operators.

35 Though, before going any further, let us provide a few remarks related to the explicit presence of the Stueck- elberg field ξa in eq. (2.17). As it is easily realized, the field ξa is a dimensionless field, a feature encoded in a the fact that the invariant action Sinv itself is an infinite series in powers of ξ . As in any local quantum field theory involving dimensionless fields, one has the freedom of performing arbitrary reparametrization of these fields as, for instance, in the case of the two-dimensional non-linear sigma model [147,148] and of N = 1 super YM in superspace [149, 150]. In the present case, this means that we have the freedom of replacing ξa by an arbitrary dimensionless function of ξa, namely

a a a abc b c abcd b c d abcde b c d e ξ → ω (ξ) = ξ + a1 ξ ξ + a2 ξ ξ ξ + a3 ξ ξ ξ ξ + ...... (2.18)

This freedom, inherent to the dimensionless nature of ξa, is clearly evidentiated at the quantum level by the fact that the Stueckelberg ﬁeld renormalizes in a non-linear way [136], i.e. like eq. (2.18), expressing precisely the freedom one has in the choice of a reparametrization for ξa.

In our context, in eq.(2.17), we could have been equally started with a term like

a a a a a a abc b c abcd b c d s (¯c ξ ) → s (¯c ω (ξ)) = s c¯ (ξ + a1 ξ ξ + a2 ξ ξ ξ + ...) . (2.19)

2 abc abcd abcde Of course, as much as µ and α, all coefficients (a1 , a2 , a3 , ...) are gauge parameters, not affecting the correlation functions of the gauge invariant quantities. Equation (2.19) expresses the freedom which one always has when dealing with a gauge fixing term which depends explicitly from a dimensionless field, as the term (2.17). In particular, this freedom will persist through the renormalization analysis, meaning that the renormalization of the gauge fixing itself has to be determined modulo an exact BRST terms of the kind s (¯caωa(ξ)). Alternatively, one could start directly with the generalized gauge-fixing Z gen 4 a a 2 a α a a S = d x s c¯ (∂µA ) − µ ω (ξ)) − i c¯ b gf µ 2 Z a 4 a a α a a 2 a a a ab b 2 a ∂ω (ξ) cd d = d x ib ∂µA + b b − iµ b ω (ξ) +c ¯ ∂µD (A)c + µ c¯ g (ξ)c , µ 2 µ ∂ξc (2.20) and take into account the renormalization of the quantity ωa(ξ), encoded in the infinte set of gauge parameters abc abcd abcde (a1 , a2 , a3 , ...). In the following, we shall make use of the gauge-fixing (2.17) and identify in the final counterterm the term which corresponds to the reparametrization (2.19). Moreover, in the Appendix A.2, we shall provide a second proof of the renormalizability of the model by deriving the generalized Slavnov-Taylor identities corresponding to the gauge fixing term (2.20).

In summary, as starting point, we shall take the local, BRST invariant action

S = Sinv + Sgf , (2.21) with sS = 0 , (2.22) where the BRST transformations are given by eqs.(2.14),(2.15),(2.16).

Let us proceed now by giving a look at the propagators of the elementary ﬁelds.

2.4 A look at the propagators of the elementary ﬁelds

The propagators of the elementary ﬁelds are easily evaluated from the quadratic part of the action, eq.(2.21), i.e.

36 Z 4 1 a a 2 a a α a a a 2 a 2 a a Squad. = d x (∂µA − ∂ν A ) + ib ∂µA + b b +c ¯ ∂ c − µ c¯ c 4 ν µ µ 2 2 2 m a a 2 a a m a a + A A − m A ∂µξ + (∂µξ )(∂µξ ) 2 µ µ µ 2 a a a 2 a a 2 a 2 a a +τ ∂µAµ − τ ∂ ξ +η ¯ ∂ η − iµ b ξ Z 1 = d4x Aa ba ξa τ a × 2 µ  2 2 2   a  −δµν ∂ + ∂µ∂ν + m −i∂µ −m ∂µ −∂µ Aµ 2 a  i∂ν α −iµ 0   b  ×  2 2 2 2 2   a   m ∂ν −iµ −m ∂ −∂   ξ  2 a ∂ν 0 −∂ 0 τ Z + d4x c¯a∂2ca +η ¯a∂2ηa − µ2c¯aca . (2.23)

Thus, for the propagators we get

2 ! a b ab Pµν αp Lµν Aµ (p) Aν (−p) = δ + p2 + m2 (p2 + µ2)2 p Aa (p) bb (−p) = δab µ µ p2 + µ2 ! a b ab −iαpµ Aµ (p) ξ (−p) = δ (p2 + µ2)2 iµ2p Aa (p) τ b (−p) = δab µ µ p2 (p2 + µ2) ba (p) bb (−p) = 0 δabi ba (p) ξb (−p) = p2 + µ2 ba (p) τ b (−p) = 0 δabα ξa (p) ξb (−p) = (p2 + µ2)2 δab ξa (p) τ b (−p) = p2 + µ2 δabm2 τ a (p) τ b (−p) = − p2 δab c¯a (p) cb (−p) = p2 + µ2 δab η¯a (p) ηb (−p) = (2.24) p2

pµpν pµpν where, Pµν = δµν − p2 and Lµν = p2 are the transverse and longitudinal projectors. We see that all propagators have a nice ultraviolet behavior, fully compatible with the power-counting. Moreover, the role of the massive gauge parameter µ2 becomes now apparent: it gives a BRST invariant regularizing mass for the Stueckelberg field ξa. Observe in fact that, when µ2 = 0, the propagator of the Stueckelberg field is given α by hξ(p)ξ(−p)iµ2=0 = p4 which might give rise to potential infrared divergences in some class of Feynman diagrams. Notice also that, as expected, the mass parameter m2 appears in the transverse part of the gluon propagator, a feature which exhibits its physical meaning. In fact, being coupled to the gauge invariant operator h,a h,a 2 (Aµ Aµ ), the parameter m will enter the correlation functions of physical operators, i.e. gauge invariant operators, allowing thus to parametrize in an effective way their infrared behavior.

37 2 2.5 Amin versus the conventional Stueckelberg mass term As done in [136], before facing the analysis of the renormalizability of the action S, eq. (2.21), let us make a short comparison with the standard Stueckelberg mass term [151], corresponding to the action

Z 2 4 1 a a m h,a h,a SStueck = d x F F + A A + Sgf , (2.25) 4 µν µν 2 µ µ where Sgf is given by eq. (2.17). One sees that the conventional Stueckelberg action corresponds to the addition h,a h,a h,a of the gauge invariant operator (Aµ Aµ ) without taking into account the transversality constraint ∂µAµ = 0, implemented in the action (2.21) through the Lagrange multiplier τ a and the corresponding ghosts (¯ηa, ηa). The h,a removal of the constraint ∂µAµ = 0 gives rise to the conventional Stueckelberg propagator, namely

ab 2 ab a b δ p δ α hξ (p) ξ (−p)iStueck = + . (2.26) m2(p2 + µ2)2 (p2 + µ2)2

From this expression one easily understand the cause of the bad ultraviolet behavior of the Stueckelberg mass term, giving rise to its nonrenormalizability [152]. We see in fact that the mass parameter m2 enters the denominator of expression (2.26). As one easily ﬁgures out, this property jepardizes the renormalizability of the standard Stueckelberg formulation [152]. Due to the presence of the parameter m2 in the denominator of expressions (2.26), non-renormalizable divergences in the inverse of the mass m2 will show up, invalidating thus the perturbative loop expansion based on expression (2.25).

R 4 a h,a h,a The role of the term d x τ ∂µAµ , implementing the constraint ∂µAµ = 0, becomes now clear. It gives rise to a deep modiﬁcation of the Stueckelberg propagator, removing precisely the ﬁrst problematic term, ab 2 ab δ p i.e. δ α m2(p2+µ2)2 , from expression (2.26). We are left therefore only with the second piece, (p2+µ2)2 , which does not pause any problem with the ultraviolet power-counting. It is this nice feature which will ensure the all order renormalizability of the action S, eq.(2.21), as we shall discuss in details in the next sections.

2.6 Algebraic characterization of the counterterm

We are now ready to start the analysis of the renormalizability of the action S, eq. (2.21). Following the setup of the algebraic renormalization [17], we proceed by establishing the set of Ward identities which will be employed for the study of the quantum corrections. To that end, we need to introduce a set of external BRST a a a a a a invariant sources (Ωµ,L ,K ) coupled to the non-linear BRST variations of the ﬁelds (Aµ, c , ξ ) as well as a a ha ab h sources (Jµ , Ξµ) coupled to the BRST invariant composite operators (Aµ ,Dµ (A )),

a a a a a sΩµ = sL = sK = sJµ = sΞµ = 0 . (2.27)

We shall thus start with the BRST invariant complete action Σ deﬁned by Z 4 1 a 2 a a a ab b α a 2 ab a b Σ = d x F + ib ∂µA +c ¯ ∂µD c + (b ) − iM b ξ 4 µν µ µ 2 2 ab a b ab a bc c a ab h b m ha ha −N c¯ ξ + M c¯ g (ξ) c +η ¯ ∂µD A η + A A µ 2 µ µ a ha a ab b g abc a b c a ab b a ha +τ ∂µA − Ω D c + f L c c + K g (ξ) c + J A µ µ µ 2 µ µ a ab h b + ΞµDµ A η , (2.28) where, for later convenience, we have also introduced the BRST doublet of external sources (M ab,N ab)

sM ab = N ab , sN ab = 0 , (2.29) so that sΣ = 0 . (2.30) Notice that the invariant action S of eq. (2.21) is immediately recovered from the complete action Σ upon a a a a a ab ab 2 ab setting the external sources (Ωµ = L = K = Jµ = Ξµ = 0) and (M = δ µ ,N = 0).

It turns out that the complete action Σ obeys the following Ward identities:

38 • the Slavnov-Taylor identity Z 4 δΣ δΣ δΣ δΣ a δΣ δΣ δΣ ab δΣ S (Σ) = d x a a + a a + ib a + a a + N ab = 0 , δΩµ δAµ δL δc δc¯ δK δξ δM (2.31)

• the equation of motion of the Lagrange multiplier ba and of the antighostc ¯a δΣ = i∂ Aa + αba − iM abξb , (2.32) δba µ µ

δΣ δΣ ab δΣ ab b a + ∂µ a − M b = N ξ , (2.33) δc¯ δΩµ δK

• the ghost-number Ward identity Z 4 a δΣ a δΣ a δΣ a δΣ a δΣ ab δΣ d x c a − c¯ a − Ωµ a − 2L a − K a + N ab = 0 (2.34) δc δc¯ δΩµ δL δK δN

• the equation of the Lagrange multiplier τ a δΣ δΣ a − ∂µ a = 0 , (2.35) δτ δJµ

• the ηa Ward identity Z 4 δΣ abc b δΣ abc b δΣ d x a + gf η¯ c + gf Ξ c = 0 , (2.36) δη δτ δJµ

• theη ¯a antighost equation δΣ δΣ a − ∂µ a = 0 , (2.37) δη¯ δΞµ

• the (ηa, η¯a) ghost number

Z δΣ δΣ δΣ d4x ηa − η¯a − Ξa = 0 . (2.38) δηa δη¯a δΞa

39 a a a a a a a a Aµ b c c¯ τ η η¯ ξ dim. 1 2 0 2 2 0 2 0 c gh. number 0 0 1 -1 0 0 0 0 η gh. number 0 0 0 0 0 1 -1 0

Table VI A: The quantum numbers of the ﬁelds

a a a a a ab ab Ωµ L K Jµ Ξµ M N dim. 3 4 4 3 2 2 2 c gh. number -1 -2 -1 0 0 0 1 η gh. number 0 0 0 0 -1 0 0

Table VI B: The quantum numbers of the sources

All quantum numbers and dimensions of all ﬁelds and sources are displayed in Tables (VI A) and (VI B).

In order to characterize the most general invariant counterterm which can be freely added to all order in perturbation theory, we follow the setup of the algebraic renormalization [17] and perturb the classical action Σ, eq.(2.28), by adding an integrated local quantity in the fields and sources, Σct, with dimension bounded by four and vanishing ghost number. We demand thus that the perturbed action, (Σ + εΣct), where ε is an expansion parameter, fulfills, to the first order in ε, the same Ward identities obeyed by the classical action Σ, i.e. equations (2.31), (2.32), (2.34), (2.35), (2.36) and (2.37). This amounts to impose the following constraints on Σ: ct BΣΣ = 0 , (2.39)

δΣct = 0 , (2.40) δba

ct ct ct δΣ δΣ ab δΣ a + ∂µ a − M b = 0 , (2.41) δc¯ δΩµ δK δΣct δΣct a − ∂µ a = 0 , (2.42) δτ δJµ Z ct ct ct 4 δΣ abc b δΣ abc b δΣ d x a + gf η¯ c + gf Ξ c = 0 , (2.43) δη δτ δJµ δΣct δΣct a − ∂µ a = 0 , (2.44) δη¯ δΞµ where BΣ is the so-called nilpotent linearized Slavnov-Taylor operator [17], defined as Z 4 δΣ δ δΣ δ δΣ δ δΣ δ δΣ δ BΣ = d x a a + a a + a a + a a + a a δΩµ δAµ δAµ δΩµ δL δc δc δL δK δξ Z δΣ δ δ δ + d4x + iba + N ab , (2.45) δξa δKa δc¯a δM ab with BΣBΣ = 0 . (2.46) The first condition, eq. (2.39), tells us that the counterterm Σct belongs to the cohomology of the operator BΣ in the space of the integrated local polynomials in the fields, sources and parameters, of dimension four and ghost number zero. Owing to the general results on the BRST cohomology of YM theories [17] and taking advantage of the analysis already done in [136], the most general form for Σct can be written as

ct (−1) Σ = ∆cohom + BΣ∆ , (2.47)

(−1) where ∆cohom identiﬁes the cohomolgy of BΣ, i.e. the non-trivial solution of eq.(2.39), and ∆ stands for the exact part, i.e. for the trivial solution of (2.39). Notice that, according to the quantum numbers of the ﬁelds, ∆(−1) is an integrated polynomial of dimension four, c-ghost number -1 and η-number equal to zero.

40 For ∆cohom, we have Z 4 a0 a 2 ha ha ha ha ∆cohom = d x F + a1 ∂µA ∂ν A + a2 ∂µA ∂µA 4 µν µ ν ν ν abcd ha hb hc hd a a a a a a +a3 Aµ Aµ Aν Aν + ∂µτ + Jµ Fµ (A, ξ) + a5 ∂µη¯ + Ξµ (∂µη ) abc a a b c 2 +f ∂µη¯ + Ξµ η Gµ (A, ξ) + m I (A, ξ) , (2.48)

a c a a where Fµ (A, ξ), Gµ (A, ξ) and I (A, ξ) are local functional of Aµ and ξ , with dimension 1, 1 and 2, respectively. To write expression (2.48) we have taken into account the constraints (2.41)–(2.44). Moreover, from condition (2.39) one immediately gets

a c BΣFµ (A, ξ) = BΣGµ (A, ξ) = BΣI (A, ξ) = 0 . (2.49) Proceeding as in [136], equations (2.49) are solved by

a ha c ha ha ha Fµ (A, ξ) = a4Aµ ,Gµ (A, ξ) = a6Aµ ,I (A, ξ) = a7Aµ Aµ , (2.50) where (a4, a6, a7) are free coeﬃcients. Therefore, Z 4 a0 a 2 ha ha ha ha ∆cohom = d x F + a1 ∂µA ∂ν A + a2 ∂µA ∂µA 4 µν µ ν ν ν abcd ha hb hc hd a a ha a a a +a3 Aµ Aµ Aν Aν + a4 ∂µτ + Jµ Aµ + a5 ∂µη¯ + Ξµ (∂µη ) abc a a b hc 2 ha ha +a6f ∂µη¯ + Ξµ η Aµ + a7m Aµ Aµ .

Let us discuss now the exact part of the cohomology of BΣ which, taking into account the quantum numbers of the ﬁelds and sources, can be parametrized as Z (−1) 4 ab a b ab a b ab a b ab a b ∆ = d x f1 (ξ, α) ξ K + f2 (ξ, α) L c + f3 (ξ, α) ξ ∂µΩµ + f4 (ξ, α)(∂µξ )Ωµ.

ab a b ab a b ab a b +f5 (ξ, α) AµΩµ + f6 (ξ, α) Aµ ∂µc¯ + f7 (ξ, α) ∂µAµ c¯ ab a b ab a 2 b ab a b +f8 (ξ, α)(∂µξ ) ∂µc¯ + f9 (ξ, α) ξ ∂ c¯ + f10 (ξ, α)c ¯ b ab a b abc a b c abc a b c +f11 (ξ, α)c ¯ τ + f12 (ξ, α)η ¯ η c¯ + f13 (ξ, α)c ¯ c¯ c abcd ab c d +f14 (ξ, α) M ξ c¯ , where (f1, ..., f14) are arbitrary coeﬃcients. Imposing the constraint (2.40), i.e. δ B ∆(−1) = 0 , (2.51) δbk Σ and making use of the commutation relation δ δ δ δ kl δ k BΣ = BΣ k + i k + ∂µ k − M l , (2.52) δb δb δc¯ δΩµ δK one ﬁnds δ∆(−1) δ∆(−1) δΣ ∂f ak (ξ, α) = f ak (ξ, α)c ¯a ⇒ B = 10 c¯a + if ak (ξ, α) ba . δbk 10 Σ δbk δKm ∂ξm 10 Moreover, from

(−1) (−1) (−1) δ∆ δ∆ kl δ∆ ak a ak a i k + ∂µ k − M l = −i∂µ f6 (ξ, α) Aµ + if7 (ξ, α) ∂µAµ δc¯ δΩµ δK ak a 2 ak a −i∂µ f8 (ξ, α)(∂µξ ) + i∂ f9 (ξ, α) ξ kb b kb b +if10 (ξ, α) b + if11 (ξ, α) τ abk a b kbc b c +if12 (ξ, α)η ¯ η + 2if13 (ξ, α)c ¯ c abck ab c +if14 (ξ, α) M ξ 2 ak a ak a −i∂ f3 (ξ, α) ξ + i∂µ f4 (ξ, α)(∂µξ ) ak a kl al a +i∂µ f5 (ξ, α) Aµ − iM f1 (ξ, α) ξ ,

41 it follows that δ ∂f bk (ξ, α) B ∆(−1) = 0 = 10 gmc (ξ) − 2if kbc (ξ, α) ccc¯b δbk Σ ∂ξm 13 ak ka a +i f10 (ξ, α) + f10 (ξ, α) b ak ak ak a +i −f6 (ξ, α) + f5 (ξ, α) + f7 (ξ, α) ∂µAµ ak ak a −i ∂µf6 (ξ, α) − ∂µf5 (ξ, α) Aµ ak ak ak ak a +i − ∂µf8 (ξ, α) − ∂µf3 (ξ, α) + ∂µf4 (ξ, α) + ∂µf9 (ξ, α) (∂µξ ) ak ak ak ak 2 a +i −f8 (ξ, α) − f3 (ξ, α) + f4 (ξ, α) + f9 (ξ, α) ∂ ξ 2 ak 2 ak a +i − ∂ f3 (ξ, α) + ∂ f9 (ξ, α) ξ kb b abk a b abck ka cb ab c +if11 (ξ, α) τ + if12 (ξ, α)η ¯ η + i f14 (ξ, α) − δ f1 (ξ, α) M ξ , form which we can derive relations among the coeﬃcients (f1, ..., f14). Let us start with

ak ak ab ab ab ∂µf6 (ξ, α) − ∂µf5 (ξ, α) = 0 ⇒ f6 = f5 + δ a , (2.53) where a is a constant. Further

ak ak ak ak ab −f6 (ξ, α) + f5 (ξ, α) + f7 (ξ, α) = 0 ⇒ f7 (ξ, α) = δ a . (2.54) Analogously

2 ak 2 ak ak ak ak − ∂ f3 (ξ, α) + ∂ f9 (ξ, α) = 0 ⇒ f9 (ξ, α) = f3 (ξ, α) + bδ , (2.55) with b a free constant. Next, from

ak ak ak ak − ∂µf8 (ξ, α) − ∂µf3 (ξ, α) + ∂µf4 (ξ, α) + ∂µf9 (ξ, α) , (2.56) we get ak ak ak f8 (ξ, α) = f4 (ξ, α) + cδ , (2.57) with c constant. Finally

ak ak ak ak −f8 (ξ, α) − f3 (ξ, α) + f4 (ξ, α) + f9 (ξ, α) = 0 ⇒ b = c . (2.58) Therefore, ∆(−1) becomes Z (−1) 4 ab a b cb a c ∆ = d x f1 (ξ, α) ξ K + M ξ c¯

ab a b ab a b 2 b +f2 (ξ, α) L c + f3 (ξ, α) ξ ∂µΩµ + ∂ c¯ ab a b b +f4 (ξ, α)(∂µξ ) Ωµ + ∂µc¯ ab a b b +f5 (ξ, α) Aµ Ωµ + ∂µc¯ 1 ∂f ba (ξ, α) +f ab (ξ, α)c ¯abb + 10 gmc (ξ)c ¯ac¯bcc . 10 2i ∂ξm We can now impose the constraint (2.43) Z ct ct ct 4 δΣ mnp n δΣ mnp n δΣ d x m + gf η¯ p + gf Ξ p = 0 , δη δτ δJµ

Z 4 mnp n mnp n hp ⇒ d x (a6 + a4g) f (∂µη¯ + f Ξ ) Aµ = 0 , from which we obtain a6 = −a4g.

As done in [136], we can further reduce the number of parameters entering Σct by observing that, setting a ab ab a a K = M = N = Jµ = Ξµ = m = 0, the complete action Σ, eq.(2.28), reduces to that or ordinary YM theory in the linear covarinat gauges, as integration over τ a, ηa and ηa gives a unity. As a consequence, making use of the well known renormalization of standard YM theory in the linear covariant gauges [17], we get abcd a1 = a2 = a3 = 0, a5 = a4, as well as

ab ab ab ab f2 (ξ, α) = δ d1 (α) , f5 (ξ, α) = δ d2 (α) , (2.59)

42 with (d1, d2) free parameters. In addition, we also have

ab ab ab f3 (ξ, α) = f4 (ξ, α) = f10 (ξ, α) = 0 .

Hence Z 4 a0 a 2 a a ha a a ab h b ∆cohom = d x F + a4 ∂µτ + J A + ∂µη¯ + Ξ D A η 4 µν µ µ µ 2 ha ha +a7m Aµ Aµ , and

Z (−1) 4 ab a b cb a c a a a a a ∆ = d x f1 (ξ, α) ξ K + M ξ c¯ + d1 (α) L c + d2 (α) Aµ Ωµ + (∂µc¯ ) .

Let us end this section by rewriting the ﬁnal expression of the most general invariant counterterm Σct in its parametric form [17], a task that will simplify the analysis of the renormalziation factors, namely

∂Σ ∂Σ ∂Σ Σct = −a g + d (α) 2α + a m2 0 ∂g 2 ∂α 7 ∂m2 Z 4 a δΣ a δΣ a δΣ a δΣ + d x a4 −τ a + Jµ a − η¯ a + Ξµ a δτ δJµ δη¯ δΞµ ∂f kb (ξ, α) δΣ δΣ − f ab (ξ, α) + 1 ξk Kb + f ab (ξ, α) ξa 1 ∂ξa δKa 1 δξb

a δΣ a δΣ a δΣ a δΣ +d2 (α) Aµ a − d2 (α) b a − d2 (α)Ωµ a − d2 (α)c ¯ a δAµ δb δΩµ δc δΣ δΣ δΣ −d (α) ca + d (α) La + −f cb (ξ, α) + d (α) δcb N ab 1 δca 1 δLa 1 2 δN ac δΣ ∂f ab (ξ, α) + d (α) δab − f ab (ξ, α) M cb + 1 M cbξag (ξ)kd cdc¯c . (2.60) 2 1 δM ca ∂ξk

2.7 Analysis of the counterterm and renormalization factors

Having determined the most general form of the local invariant counterterm, eq.(2.60), let us turn to its physical meaning. As already mentioned before, in order to determine the renormalization of the fields, sources and parameters, we have to pay attention to the fact that, due to the explicit dependence of the gauge fixing from the Stueckelberg field ξa, the renormalization of the gauge fixing itself is determined up to an ambiguity of the type of eq.(2.19), which would correspond to the renormalization of the quantity ωa(ξ), i.e. of the abc abcd abcde gauge parameters (a1 , a2 , a3 , ...). To that end, it will be sufficient to analyse the last two terms of the expression for Σct, eq.(2.60), which, upon setting the sources (M ab,N ab) to their physical values, namely (M ab = δabµ2,N ab = 0), becomes ! ∂Σ Z ∂fãb(ξ, α) (d (α) − f (0, α)) µ2 + µ2 d4x fãb(ξ, α)(ibbξa − c¯bgak(ξ)ck) + 1 c¯bξagkd(ξ)cd (2.61) 2 1 ∂µ2 1 ∂ξk where we have set ab ab ãb f1 (ξ, α) = f1 (0, α) + f1 (ξ, α) , (2.62) ab ab a ab with f1 (0, α) = δ f1(0, α) being the first, ξ -independent, term of the Taylor expansion of f1 (ξ, α) in powers of a ãb ab ab ξ and f1 (ξ, α) denoting the ξ-dependent remaining terms. Of course, f1 (0, α) = δ f1(0, α) is just a constant.

Furthermore, we observe that expression (2.61) can be rewritten as

∂Σ Z (d (α) − f (0, α)) µ2 + µ2 d4x s f˜ab(ξ, α)¯cbξa , (2.63) 2 1 ∂µ2 1 or, equivalently ∂Σ Z (d (α) − f (0, α)) µ2 + µ2 d4x s c¯bω˜b(ξ, α) , (2.64) 2 1 ∂µ2

43 b ˜ab a withω ˜ (ξ, α) = f1 (ξ, α)ξ .

aa We are now able to unravel the meaning of this term. First, the term (d2(α) − f1 (0, α)) corresponds to a multiplicative renormalization of the gauge massive parameter µ2. This follows by observing that, being µ2 a space-time independent parameter, its renormalization must be given by a field independent space-time con- aa R 4 b b stant factor, i.e. precisely by (d2(α) − f1 (0, α)). On the other hand, the term d x s c¯ ω˜ (ξ, α) is of the type of eq.(2.19), thus corresponding to the ambiguity inherent to the gauge fixing discussed before. As already mentioned, this term can be handled by starting with the generalised gauge fixing (2.20), whose algebraic renormalization can be faced by employing the Ward identities displayed in Appendix (A.2). Doing so, the term R d4x s c¯bω˜b(ξ, α) will correspond to a renormalization of the gauge fixing function ωa(ξ), i.e. of the gauge abc abcd abcde parameters (a1 , a2 , a3 , ...).

We can now read oﬀ the renormalization factors, i.e.

ct 2 Σ(Φ) + εΣ (Φ) = Σ(Φ0) + O(ε ) , (2.65) with

2 Φ0 = ZΦΦ + O(ε ) , (2.66) where Φ stands for a short-hand notation for all fields, sources and parameters. Specifically, for the renormalization factors one finds:

1/2 1/2 1/2 1/2 A0 = ZA Aµ , b0 = Zb b , c0 = Zc c , c¯0 = Zc¯ c¯ , (2.67) a ab b 1/2 ξ0 = Zξ (ξ)ξ τ0 = Zτ τ , Ω0 = ZΩΩ ,L0 = ZLL (2.68) a ab b 2 2 K0 = ZK (ξ)K , m0 = Zm2 m , J0 = ZJ J , (2.69) 1/2 1/2 g0 = Zgg , α0 = Zαα , η¯0 = Zη¯ η¯ , η0 = Zη η , (2.70) 2 2 Ξ0 = ZΞΞ , µ0 = Zµ2 µ , (2.71) where

a0 Zg = 1 − ε 2 1/2 −1 −1/2 −1/2 1/2 ZA = ZΩ = Zc¯ = Zb = Zα = 1 + εd2(α) ab ab ab Zξ = δ + εf1 (ξ, α) −1/2 ZL = Zc = 1 + εd1(α) 2 1/2 Zη¯ = Zη = ZΞ = Zτ = ZJ = 1 + εa4

Zm2 = 1 + εa7

Zµ2 = 1 + ε(d2 − f2(0, α)) ∂f kb(ξ, α) Zab = δab − ε f ab(ξ, α) + 1 ξk . (2.72) K 1 ∂ξa

Notice that, as expected, the dimensionless ﬁeld ξa renormalizes in a non-linear way through the quantity ab a f1 (ξ, α) which is a power series in ξ . Equations (2.65) and (2.72) establish the renormalizability of the complete action Σ, eq.(2.28), and thus of the invariant action S of expression (2.21), up to a BRST exact unphysical ambiguity of the type of eq.(2.19). As already mentioned, the explicit inclusion of such an ambiguity will be provided in Appendix (A.2).

2.8 Conclusion

2 In this chapter the gauge invariant operator Amin, eq.(2.1), and corresponding gauge invariant transverse field ah configuration Aµ , eq.(2.2), have been investigated in a general class of gauge fixings, eq.(2.17) and eq.(2.20), which share similarities with ’t Hooft’s Rζ -gauge used in the analysis of YM theory with spontaneous symme- 2 ah try breaking. As shown in [136], a local setup can be constructed for both Amin and Aµ , being summarised by the local and BRST invariant action (2.8). The localization procedure makes use of an auxiliary dimensionless Stueckelberg field ξa. However, despite the presence of the field ξa and unlike the conventional non- renormalizable Stueckelberg mass term, the present construction gives rise to a perfectly well behaved model in the ultraviolet which turns out to be renormalizable to all orders, as discussed in details in sections (2.6) and

44 ah (2.7) as well as in Appendix (A.2). In particular, the pivotal role of the transversality constraint ∂µAµ = 0 has been underlined throughout the paper. It is precisely the direct implementation of this constraint in the local action (2.8) which makes a substantial difference with respect to the conventional Stueckelberg theory. In fact, as pointed out in section (2.5), it removes exactly the component of the Stueckelberg propagator which gives rise to non-renormalizable ultraviolet divergences, see eq.(2.26) versus eqs.(2.24). In particular, form eqs.(2.24), one sees that, similar to what happens in the case of ’t Hooft’s Rζ -gauge, the use of the general class of gauge fixings (2.17) and (2.20) provide a mass µ2 for the dimensionless Stueckelberg field ξa. This a welcome feature which can be effectively employed as a fully BRST invariant infrared regularization for ξa in explicit higher loop calculations.

45 Chapter 3

Some remarks on the spectral functions of the Abelian Higgs Model

In section 1.3.2, we have discussed the two main observations in lattice QCD in recent years: massive behavior of the gluon propagator, and positivity violation of its spectral density function. In this context, it is worth- while to investigate the spectral properties of massive gauge models to try and shed some light on the infrared behavior of their fundamental ﬁelds in an analytical way. The direct comparison between a massive model that violates BRST, such as the massive YM model from section 1.3.1, and a model that preserves the original nilpotent BRST symmetry, such as the Higgs model, can be particularly enlightening. In any case, the explicit determination of the spectral properties of Higgs theories and the study of the role played by gauge symmetry there is an interesting pursue on its own.

Most articles on massive YM models employ the renormalizable Landau gauge, although it was noticed that this gauge might not be the preferred gauge in non-perturbative calculations [153]. For the Higgs model, one can fix the gauge by means of ’t Hooft Rξ-gauge, see section 1.1.5. Understanding the different gauges and their influence on the spectral properties is a delicate subject. This gave us further reason to undertake a systematic study of the spectral properties of Higgs models. In this chapter, we present the results for the simplest case: that of the U(1) Abelian Higgs model. In fact, it turned out that this model is already very illuminating on aspects like positivity of the spectral function, gauge-parameter independence of physical quantities and unitarity. Of course, these properties are not unknown in the Abelian case. This chapter should therefore not be seen as giving any new information on the physical properties of the Abelian model. Rather, exactly because these properties are so well-known, we are in a better position to understand the problems that we face when calculating the analytic structure behind some of them within a gauge-fixed setup. This chapter is therefore a first attempt to understand analytically the spectral properties of a Higgs-gauge model in contrast to those of a non-unitary massive model. As such, it is laying the groundwork for the next chapters.

The U(1) Higgs model is known to be unitary [154,155] and renormalizable [156]. In this work, we consider two propagators: that of the photon, and that of the Higgs scalar field. They are obtained through the calculation of the one-loop corrections to the corresponding 1PI two-point functions. After adopting the Rξ-gauge, we are left with an exact BRST nilpotent symmetry. Of course, the correlation function of BRST invariant quantities should be independent of the gauge parameter. Since the transverse component of the photon propagator is gauge invariant, we should find that the one-loop corrected transverse propagator does not depend on the gauge parameter. As a consequence, the photon pole mass will neither. This property has been proven before by the use of the Nielsen identities, [157], see also [55, 57, 158], but never in a direct calculation. The same goes for the Higgs particle propagator: the gauge independence of its pole mass was proven in [157], but never in a direct loop calculation to our knowledge. We underline here the importance of properly taking into account the tadpole contributions [15,16] or, equivalently, the effect on the propagators of quantum corrections of the Higgs vacuum expectation value. Armed with the one-loop results, we are able to calculate the spectral properties of the respective propagators for different values of the gauge parameter. Finally, we compare our results with those of a non-unitary massive Abelian model, to clearly pinpoint at the level of spectral functions the differences (and issues) of both unitary and non-unitary massive vector boson models.

This chapter is organized as follows. In section 3.1, we review the U(1) Higgs model and its gauge fixing, as well as the tree-level field propagators and vertices. In section 3.2, we calculate the one-loop propagator of both the photon field and the Higgs field, showing the gauge-parameter independence of the transverse photon propagator and of the Higgs pole mass up to one-loop order. In section 3.3, we calculate the spectral function

46 of both propagators. In section 3.4 we discuss some subtleties of the Higgs spectral function and in section 3.5 we compare our results with those of a non-unitary massive Abelian model. We also address the residue computation. Section 3.6 collects our conclusions and outlook.

3.1 Abelian Higgs model: some essentials

We start from the Abelian Higgs classical action with a manifest global U(1) symmetry

Z ( 2 2) 4 1 † λ † v S = d x FµvFµv + (Dµϕ) Dµϕ + ϕ ϕ − , (3.1) 4 2 2 where

Fµv = ∂µAv − ∂vAµ,

Dµϕ = ∂µϕ + ieAµϕ (3.2) and the parameter v gives the minimizing value of the scalar field to first order in ~, ϕ0 = v. The spontaneous symmetry breaking is implemented by expressing the scalar field as an expansion around its minimizing value, namely 1 ϕ = √ ((v + h) + iρ), (3.3) 2 where the real part h is identified as the Higgs field and ρ is the (unphysical) Goldstone boson, with hρi = 0. Here we choose to expand around the classical value of the minimizing value, so that hhi is zero at the classical level, but receives loop corrections1 . The action (3.1) now becomes Z 4 1 1 1 S = d x FµvFµv + ∂µh∂µh + ∂µρ∂µρ − e ρ ∂µh Aµ + e (h + v)Aµ∂µρ 4 2 2 1 2 2 2 1 2 2 2 + e Aµ[(h + v) + ρ ]Aµ + λ(h + 2hv + ρ ) (3.4) 2 8 and we notice that both the gauge field and the Higgs field have acquired the following masses

2 2 2 2 2 m = e v , mh = λv . (3.5)

With this parametrization, the Higgs coupling λ and the parameter v can be ﬁxed in terms of m, mh and e, whose values will be suitably chosen later on in the text.

Even in the broken phase, the action (3.4) is left invariant by the following gauge transformations

† † ∂Aµ = −∂µω, ∂ϕ = ieωϕ, ∂ϕ = −ieωϕ , ∂h = −eωρ, ∂ρ = eω(v + h). (3.6) where ω is the gauge parameter.

3.1.1 Gauge fixing Quantization of the theory (3.4) requires a proper gauge fixing. We shall employ the gauge fixing term Z 4 1 2 Sgf = d x (∂µAµ + ξmρ) , (3.7) 2ξ

R 4 known as the ’t Hooft or Rξ-gauge, which has the pleasant property of cancelling the mixed term d x(ev Aµ∂µρ) in the expression (3.4). Of course, (3.7) breaks the gauge invariance of the action. As is well known, the latter

1There is of course an equivalent procedure of ﬁxing hhi to zero at all orders, by expanding ϕ around the full minimizing value: ϕ = √1 ((hϕi + h) + iρ). In the Appendix B.3 we explicitly show that—as expected—both procedures give the same ﬁnal results 2 up to a given order.

47 is replaced by the BRST invariance. In fact, introducing the FP ghost ﬁeldsc, ¯ c as well as the auxiliary ﬁeld b, for the BRST transformations we have

sAµ = −∂µc, sc = 0, sϕ = iecϕ, sϕ† = −iecϕ†, sh = −ecρ, sρ = ec(v + h), sc¯ = ib, sb = 0. (3.8)

Importantly, the operator s is nilpotent, i.e. s2 = 0, allowing to work with the so-called BRST cohomology, a useful concept to prove unitarity and renormalizability of the Abelian Higgs model [123, 156, 159].

We can now introduce the gauge ﬁxing in a BRST invariant way via Z d n o Sgf = s d x −i cb¯ +c ¯(∂µAµ + mρ) , (3.9) 2 Z d ξ 2 2 2 = d x b + ib∂µAµ + ibξmρ +c∂ ¯ c − ξm cc¯ − ξmechc¯ . (3.10) 2

Notice that the ghosts (¯c, c) get a gauge parameter dependent mass, while interacting directly with the Higgs ﬁeld.

The total gauge ﬁxed BRST invariant action then becomes

Z ( 4 1 1 1 1 2 S = d x FµvFµv + ∂µh∂µh + ∂µρ∂µρ − e ρ ∂µh Aµ + e hAµ∂µρ + m AµAµ 4 2 2 2

1 2 2 2 1 2 2 2 2 1 2 2 ξ 2 + e Aµ[h + 2vh + ρ ]Aµ + λ(h + ρ )(h + ρ + 4hv) + m h + mAµ∂µρ + b + ib∂µAµ 2 8 2 h 2 ) + ibξmρ +c ¯(∂2)c − m2cc¯ − mecch¯ , (3.11) with

sS = 0 . (3.12)

In AppendixB we collect the propagators and vertices corresponding to the action (3.11) of the Abelian Higgs model in the Rξ gauge.

3.2 Photon and Higgs propagators at one-loop

In this section we obtain the one-loop corrections to the photon propagator, as well as to the propagator of the Higgs boson. This requires the calculation2, in section 3.2.1 and 3.2.2, of the Feynman diagrams as shown in Figure 3.1 and Figure 3.2. Notice that the last four diagrams in Figure 3.1 and Figure 3.2 vanish for hhi = 0. Since we have chosen to expand the ϕ field around its classical minimizing value v (cf. (3.3)), hhi has loop contributions which are nonzero and the resulting tadpole diagrams have to be included in the quantum corrections for the propagators3. Of course the final result for the propagators would be the same had we chosen to expand the ϕ field around its full minimizing value and required hhi = 0. In fact, including the tadpole diagrams in our formulation has the same effect as shifting the masses of the fields to include the one-loop corrections to the Higgs minimizing value hϕi, calculated by imposing hhi = 0 (see Appendix B.3 for the technical details). These diagrams can actually be seen as a correction to the tree-level mass term: in the

2We have used the techniques of modifying integrals into “master integrals” with momentum-independent numerators from [160]. 3The diagrams with tadpole balloons are not part of the standard definition of one-particle irreducible diagrams that contribute to the self-energies. However, since the momentum flowing in the vertical h-field (dashed) line is zero, they can be effectively included as a momentum-independent term in the self-energies.

48 Figure 3.1: Contributions to one-loop photon self-energy in the Abelian Higgs Model, including tadpole contributions in the second line. Wavy lines represent the photon field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field. spontaneously broken phase the gauge boson mass is given by m = ehϕi, depending thus on hϕi that receives quantum corrections order by order. Therefore, the full inverse photon propagator can be written as

−1 2 2 2 2 GAA(p ) = p + e v + (1PI diagrams) + (diagrams with tadpoles) = p2 + e2hϕi2 + (1PI diagrams) , (3.13) where the equalities are to be understood up to a given order in perturbation theory and a similar reasoning can be drawn for the Higgs propagator. In what follows, we shall proceed with the expansion adopted in eq. (3.3) and include the tadpole diagrams explicitly in our self-energy results. The calculations are done for arbitrary dimension d. In section 3.2.3 we will analyze the results for d = 4 − , making use of the techniques of dimensional regularization in the MS scheme.

3.2.1 Corrections to the photon self-energy The first diagram contributing to the photon self-energy is the Higgs boson snail (first diagram in the first line of Figure 3.1) and gives a contribution

2 d−2 2 −4e Γ(2 − d/2) mh ΓA A ,1(p ) = δµv. (3.14) µ v (4π)d/2 2 − d 2 The second diagram is the Goldstone boson snail (second diagram in the ﬁrst line of Figure 3.1)

2 2 d/2−1 2 −4e Γ(2 − d/2) (m ) ΓA A ,2(p ) = δµv. (3.15) µ v (4π)d/2 2 − d 2 Being momentum-independent, the only effect of these first two diagrams is to renormalize the mass parameters 2 2 (mh, m ). The third term contributing to the photon propagator is the Higgs-Goldstone sunset (third diagram in first line of Figure 3.1) " 4e2 Γ(2 − d/2) Z 1 p2 dx K m2 , ξm2 P K m2 , ξm2 ΓAµAv ,3( ) = d/2 d/2−1( h ) µv + d/2−1( h ) (4π) 2 − d 0 ! # (2 − d) 2 2 2 + (1 − 4x(1 − x))p K (m , ξm ) Lµv , (3.16) 4 d/2−2 h where we used the definitions α 2 2 2 2 2 Kα(m1, m2) ≡ p x(1 − x) + xm1 + (1 − x)m2 , (3.17)

49 and p p P = δ − µ v , (3.18) µv µv p2 p p L = µ v , (3.19) µv p2 which are the transversal and longitudinal projectors, respectively. The fourth term contributing to the photon propagator is the Higgs-photon sunset (fourth diagram in ﬁrst line of Figure 3.1) " 4e2 Γ(2 − d/2) Z 1 p2 dx − d m2K m2 , m2 K m2 , m2 ΓAµAv ,4( ) = d/2 (2 ) d/2−2( h ) + d/2−1( h ) (4π) 2 − d 0

2 2 2 2 2 − Kd/2−1(mh, ξm ) Pµv + (2 − d)m Kd/2−2(mh, m ) 2 2 2 2 + Kd/2−1(mh, m ) − Kd/2−1(mh, ξm ) # 2 2 2 2 2 2 +(2 − d)p x (Kd/2−2(mh, m ) − Kd/2−1(mh, ξm )) Lµv . (3.20)

Finally, we have four tadpole (balloon) diagrams. The Higgs boson balloon (ﬁrst diagram of the last line in Figure 3.1)

2 2 4e Γ(2 − d/2) 3 d/2−1 ΓA A ,5(p ) = m δµv, (3.21) µ v (4π)d/2 (2 − d) 2 h the Goldstone boson balloon (second diagram of the last line in Figure 3.1)

2 2 4e Γ(2 − d/2) 1 d/2−1 ΓA A ,6(p ) = (ξm) δµv, (3.22) µ v (4π)d/2 (2 − d) 2 the photon balloon (third diagram of the last line in Figure 3.1)

2 Z d 2 2 m d k 1 ξ ΓAµAv ,7(p ) = 2e 2 d 2 2 (d − 1) + 2 2 δµv, (3.23) mh (2π) k + m k + ξm and ﬁnally, the ghost balloon (fourth diagram of the last line in Figure 3.1)

2 Z d 2 2 m d k ξ ΓAµAv ,8(p ) = −2e 2 d 2 2 δµv. (3.24) mh (2π) k + ξm Combining all these contributions (3.14)-(3.24), we ﬁnd

4e2 Γ(2 − d/2) Z 1 p2 dx − d m2K m2, m2 K m2, m2 ΓAµAv( ) = d/2 (2 ) d/2−2( h) + d/2−1( h) (4π) 2 − d 0 d d−2 m + mh + 2 (d − 1) Pµv mh

4e2 Γ(2 − d/2) Z 1 2 − d dx − x p2K m2 , ξm2 + d/2 (1 4 ) d/2−2( h ) (4π) 2 − d 0 4 2 2 2 2 2 + (2 − d)(m + p x )Kd/2−2(mh, m ) d ! 2 2 d−2 m + Kd/2−1(mh, m ) + mh + 2 (d − 1) Lµv. (3.25) mh

Deﬁning

⊥ 2 k 2 ΓAµAv = ΠAA(p )Pµv + ΠAA(p )Lµv, (3.26) it follows that

⊥ ∂ξΠAA = 0. (3.27)

As expected, eq.(3.27) expresses the gauge parameter independence of the gauge invariant transverse component of the photon propagator [157].

50 Figure 3.2: Contributions to the one-loop Higgs self-energy. Line representations are as in Figure 1.

3.2.2 Corrections to the Higgs self-energy The ﬁrst diagrams contributing to the Higgs self-energy are of the snail type, renormalizing the masses of the internal ﬁelds.

The Higgs boson snail (ﬁrst diagram in the ﬁrst line of Figure 3.2)

2 λ Γ(2 − d/2) d−2 Γhh,1(p ) = −3 m , (3.28) (4π)d/2 (2 − d) h the Goldstone boson snail (second diagram in the ﬁrst line of Figure 3.2)

2 λ Γ(2 − d/2) 2 d/2−1 Γhh,2(p ) = − (m ) (3.29) (4π)d/2 (2 − d) and the photon snail (third diagram in the first line of Figure 3.2) 2 2 e Γ(2 − d/2) d−2 2 d/2−1 Γhh,3(p ) = −2 (d − 1)m + (m ) . (3.30) (4π)d/2 (2 − d) Next, we meet a couple of sunset diagrams. The Higgs boson sunset (fourth diagram in the first line of Figure 3.2): 9 λ Γ(2 − d/2) Z 1 p2 − d m2 dxK m2 , m2 , Γhh,4( ) = d/2 (2 ) h d/2−2( h h) (3.31) 2 (4π) (2 − d) 0 the photon sunset (first diagram in the second line of Figure 3.2): " Γ(2 − d/2) 1 Z 1 p4 p2 e2 dx − d m2 d − p2 K m2, m2 Γhh,5( ) = d/2 (2 ) 2 ( 1) + 2 + 2 d/2−2( ) 2 − d (4π) 0 2m p4 − (2 − d) 2p2 + + ξ2m2 + 2p2ξ − 2ξm2 + m2 K (m2, ξm2) m2 d/2−2 p4 + (2 − d) 2ξp2 + 2ξ2m2 + K (ξm2, ξm2) 2m2 d/2−2 i + 2(ξ − 1)(m2)d/2−1 + 2(1 − ξ)(ξm2)d/2−1 , (3.32) the ghost sunset (second diagram in the second line of Figure 3.2): e2 Γ(2 − d/2) Z 1 p2 − − d m22 dxK m2, m2 , Γhh,6( ) = d/2 (2 ) d/2−2( ) (3.33) (4π) (2 − d) 0

51 the Goldstone boson sunset (third diagram in the second line of Figure 3.2): 1 λ Γ(2 − d/2) Z 1 p2 − d m2 dxK m2, m2 d/2−2 Γhh,7( ) = d/2 (2 ) h d/2−2( ) (3.34) 2 (4π) (2 − d) 0 and a mixed Goldstone-photon sunset (fourth diagram in the second line of Figure 3.2): " Γ(2 − d/2) 1 Z 1 p4 p2 e2 dx − d p2 ξ2m2 Γhh,8( ) = d/2 (2 ) 2 + 2 + (3.35) 2 − d (4π) 0 m

2 2 2 2 2 + 2p ξ − 2ξm + m Kd/2−2(m , ξm ) p4 − (2 − d) ξ2m2 + + 2p2ξ K (ξm2, ξm2) m2 d/2−2 p2 + 2 1 − ξ − (m2)d/2−1 m2 # p2 + 2 2ξ − 1 + (ξm2)d/2−1 . (3.36) m2

Finally, we have the tadpole diagrams. The Higgs balloon (ﬁrst diagram on the third line of Figure 3.2):

2 λ Γ(2 − d/2) d−2 Γhh,9(p ) = 9 m , (3.37) (4π)d/2 (2 − d) h the photon balloon (second diagram on the third line of Figure 3.2):

2 2 e Γ(2 − d/2) d−2 2 d/2−1 Γhh,10(p ) = 6 (d − 1)m + (m ) , (3.38) (4π)d/2 (2 − d) the Goldstone boson balloon (third diagram on the third line of Figure 3.2):

2 λ Γ(2 − d/2) 2 d/2−1 Γhh,11(p ) = 3 (m ) (3.39) (4π)d/2 (2 − d) the ghost balloon (fourth diagram on the third line of Figure 3.2):

2 2 e Γ(2 − d/2) 2 d/2−1 Γhh,12(p ) = −6 (m ) . (3.40) (4π)d/2 (2 − d) Putting together eqs. (3.28) to (3.40) we ﬁnd the total one-loop correction to the Higgs boson self-energy, " Γ(2 − d/2) 1 Z 1 p4 p2 ≡ p2 dx − d e2 m2 d − p2 K m2, m2 Πhh( ) Γhh( ) = d/2 (2 ) 2 ( 1) + 2 + 2 d/2−2( ) 2 − d (4π) 0 2m 9 + λ(2 − d)m2 K (m2 , m2 ) 2 h d/2−2 h h p2 + e2 −2 + 4(d − 1) (m2)d/2−1 m2 2 d/2−1 + 6λ(mh) p4 λ + (2 − d) − e2 + m2 K (ξm2, ξm2) 2m2 2 h d/2−2 p2 i + 2( e2 + λ)(ξm2)d/2−1 . (3.41) m2 3.2.3 Results for d = 4 − For d = 4, the 2-point functions are divergent. We therefore follow the standard procedure of dimensional regularization, as we have no chiral fermions present. Thus, we choose d = 4 − with an inﬁnitesimal parameter, and analyze the solution in the limit → 0.

Let us start with the photon 2-point function, given for arbitrary dimension d by (3.25). The mass dimension of the coupling constant e is [e] = 2 − d/2 = /2, and redeﬁning e → eµ˜/2 = eµ˜2−d/2 we put the dimension onµ ˜, while e is dimensionless. Using 2 2 4e Γ(2 − d/2) d→4− e 2 = −2 + 1 + ln(µ2) , (3.42) (4π)d/2 2 − d (4π)2

52 where we deﬁned 4πµ˜2 µ2 = , (3.43) eγE we ﬁnd for the divergent part of the transverse photon 2-point function:

2 e2 p2 g2 1 Π⊥ (p2) = + 6( − )m2 + 3m2 (3.44) AA,div (4π)2 3 λ 2 h and these infinities are, following the MS-scheme, cancelled by the corresponding counterterms. The renormalized correlation function is then finite in the limit d → 4 and we find the one-loop correction

2 Z 1 ( 2 e 2 2 ΠAA(p ) = 2 2 dx p x(1 − x) + m x (4π) 0 p2x(1 − x) + m2x + m2 (1 − x) m2 + m2 (1 − x)(1 − ln h ) + m2 (1 − ln h ) h µ2 h µ2 4 2 2 2 2 ) m m 2 p x(1 − x) + m x + mh(1 − x) + 2 (1 − 3 ln 2 ) + 2m ln 2 . (3.45) mh µ µ

In the same way, we ﬁnd the divergent part of the Higgs boson 2-point function:

2 1 1 2 2 2 2 2 Πhh,div(p ) = − e (12p − 4ξp ) + λ(8m − 4ξm ) , (3.46) 2 (4π)2 h which is canceled by the corresponding counterterm. Therefore, the one-loop correction to the Higgs boson propagator reads

Z 1 ( " 2 2 2 2 1 2 2 m p x(1 − x) + m Πhh(p ) = 2 dx e p (1 − ln 2 − 2 ln 2 ) (4π) 0 µ µ # p4 p2x(1 − x) + m2 m2 p2x(1 − x) + m2 − ln − 6m2(1 − ln + ln ) 2m2 µ2 µ2 µ2 h1 m2 p2x(1 − x) + m2 i + λ m2 (−6 + 6 ln h − 9 ln h ) 2 h µ2 µ2 " #) ξm2 p4 m2 p2x(1 − x) + ξm2 − ξ(e2p2 + λm2)(1 − ln ) − (e2 − λ h ) ln . (3.47) µ2 2m2 2 µ2

3.2.4 One-loop propagators for the elementary ﬁelds AA 2 For the photon ﬁeld, the transverse part of the propagator Gµv (p ) up to order ~ is given in momentum space by

1 1 2 2 hAµ(p)Av(−p)i = + ΠAA(p ) + O(~ ) (3.48) p2 + m2 (p2 + m2)2 and we can approach the all-order form factor with the resummed approximation

1 GT (p2) = , (3.49) AA p2 + m2 − Π(p2) shown in Figure 3.3.

53 Gp2

0.8

0.6

0.4

0.2

p 1 2 3 4 5 6

Figure 3.3: Resummed form factor for the photon operator, with p given in units of the energy scale µ, for the 1 parameter values e = 1, v = 1 µ, λ = 5 .

For the Higgs ﬁeld, we ﬁnd

1 1 2 2 hh(p)h(−p)i = 2 2 + 2 2 2 Πhh(p ) + O(~ ). (3.50) p + mh (p + mh) Before making the resummed approximation, we notice that the resummation is based on the assumption that the second term in (3.50) is much smaller than the ﬁrst term. Then, we see that (3.50) contains terms of the 4 2 2 p p x(1−x)+mh order of 2 2 2 ln 2 , which cannot be resummed for big values of p. We therefore use the identity (p +mh) µ

4 2 2 2 4 2 2 p = (p + mh) − mh − 2p mh, (3.51) to rewrite

4 2 2 2 2 4 2 2 2 2 p p x(1 − x) + mh p x(1 − x) + mh (m + 2p m ) p x(1 − x) + mh 2 2 2 ln 2 = ln 2 − 2 2 2 ln 2 . (3.52) (p + mh) µ µ (p + m ) µ The last two terms in (3.52) can be safely resummed. We rewrite

2 ˆ 2 Πhh(p ) Πhh(p ) 2 2 2 2 = 2 2 2 + Chh(p ), (3.53) (p + mh) (p + mh) with Z 1 ( " 2 2 2 ˆ 2 1 2 2 m p x(1 − x) + m Πhh(p ) = 2 dx e p (1 − ln 2 − 2 ln 2 ) (4π) 0 µ µ # (m4 + 2p2m2 ) p2x(1 − x) + m2 m2 p2x(1 − x) + m2 + h h ln h − 6m2(1 − ln + ln ) 2m2 µ2 µ2 µ2 " h1 m2 p2x(1 − x) + m2 i ξm2 + λ m2 (−6 + 6 ln h − 9 ln h ) − ξ(e2p2 + λm2)(1 − ln ) 2 h µ2 µ2 µ2 #) (m4 + 2p2m2 ) m2 p2x(1 − x) + ξm2 + (e2 h h + λ h ) ln . (3.54) 2m2 2 µ2 and 2 Z 1 2 2 2 2 2 e n p x(1 − x) + mh p x(1 − x) + ξm o Chh(p ) = − 2 2 dx ln 2 − ln 2 (3.55) 2m (4π) 0 µ µ

54 and the resummed approximation becomes

2 1 2 Ghh(p ) = + Chh(p ), (3.56) p2 + m2 − Π(ˆ p2) which is shown in Figure 3.4.

Gp2

0 p 0 1 2 3 4 5 6

Figure 3.4: Resummed form factor for the Higgs operator, with p given in units of the energy scale µ, for the 1 parameter values e = 1, v = 1 µ, λ = 5 .

Notice that the dependence on the Feynman parameter x for all integrals is restricted to functions of the type 2 2 2 R 1 p x(1−x)+xm1+(1−x)m2 0 dx ln µ2 . These functions have an analytical solution, depicted in Appendix B.4. Since T the transverse component Aµ of the Abelian gauge ﬁeld is gauge invariant, it turns out that the transverse photon propagator is independent from the gauge parameter ξ, while the Higgs propagator does depend on ξ, in agreement with the Nielsen identities analyzed in [157].

3.3 Spectral properties of the propagators

In this section we will investigate the spectral properties corresponding to the connected propagators of the last section. Strictly speaking, the calculation of the spectral properties should only be done to first order4 in ~, since the one-loop corrections to the propagators have been evaluated up to this order. In practice, however, for small values of the coupling constants the higher-order contributions become negligible, and one could treat the one-loop solution as the all-order solution without a significant numerical difference. Even so, when looking for analytical rather than numerical results—for example a gauge parameter dependence—we should restrict ourselves to the first-order results. We shall see the crucial difference between both approaches.

To plot the spectral properties of our model we choose some specific values of the parameters {m, mh, µ, e}. 2 2 We want to restrict ourselves to the case where the Higgs particle is a stable particle, so we need mh < 4m . Furthermore, given the Abelian nature of the model, and thus a weak coupling regime in the infrared, we can choose an energy scale µ that is sufficiently small w.r.t. the elusive Landau pole (that is exponentially large) and a corresponding small value for the coupling constant e. The particular values chosen per graph are denoted in the figure captions. Notice that by choosing µ and e, we are implicitly fixing the Landau pole Λ, with µ Λ, see [161] for more details. We have checked that results are as good as independent from the choice of µ over a very wide range of µ-values.

We start by calculating the pole mass in section 3.3.1. The pole mass is the actual physical mass of a particle

4This would correspond to ﬁrst order in the gauge coupling e2 and in the Higgs coupling λ neglecting the implicit coupling dependence in the masses.

55 that enters the energy-momentum dispersion relation. It is an observable for both the photon and the Higgs boson and should therefore not depend on the gauge parameter ξ. We will also discuss the residue to ﬁrst order and compare these with the output from the Nielsen identities [157]. In section 3.3.2 we show how to obtain the spectral function to ﬁrst order from the propagator. In section 3.4 we will discuss some more details about the Higgs spectral function.

3.3.1 Pole mass, residue and Nielsen identities The pole mass for any massless or massive ﬁeld excitation is obtained by calculating the pole of the resummed connected propagator 1 G(p2) = , (3.57) p2 + m2 − Π(p2) where Π(p2) is the self-energy correction. The pole of the propagator is thus equivalently deﬁned by the equation

p2 + m2 − Π(p2) = 0 (3.58)

2 2 and its solution defines the pole mass p = −mpole. As consistency requires us to work up to a fixed order in perturbation theory, we should solve eq.(3.58) for the pole mass in an iterative fashion. To first order in ~ , we find

2 2 1−loop 2 2 mpole = m − Π (−m ) + O(~ ), (3.59) where Π1−loop is the ﬁrst order, or one-loop, correction to the propagator. Next, we also want to compute the residue Z, again up to order ~. In principle, the residue is given by Z = lim (p2 + m2 )G(p2). (3.60) 2 2 pole p →−mpole

We write (3.57) in a slightly diﬀerent way 1 G(p2) = p2 + m2 − Π(p2) 1 = p2 + m2 − Π1−loop(−m2) − (Π(p2) − Π1−loop(−m2)) 1 = , (3.61) 2 2 2 p + mpole − Π(e p )

2 2 1−loop 2 2 2 2 where we deﬁned Π(e p ) = Π(p ) − Π (−m ). At one-loop, expanding Π(e p ) around p = −mpole = −m2 + O(~) gives the residue

1 2 2 2 2 2 Z = 2 = 1 + ∂p Π(p )|p =−m + O(~ ). (3.62) 1 − ∂p2 Π(p )|p2=−m2

In [157], for the Abelian Higgs model, the Nielsen identities were obtained for both the photon and the Higgs boson. It was found that for the photon propagator, the transverse part is explicitly independent of ξ to all orders of perturbation theory, giving the Nielsen identity:

⊥ −1 2 ∂ξ(GAA) (p ) = 0 (3.63) and consequently

⊥ −1 2 ∂ξ∂ 2 (G ) (p )| 2 2 = 0, p AA p =−mpole ⊥ −1 2 ∂ξ(GAA) (−mpole) = 0, (3.64) conﬁrming the gauge independence of the residue and the pole mass. Of course, this is not unexpected since the transverse part of an Abelian gauge ﬁeld propagator can be written as

T T T Pµν hAµAν iconn ∝ hAµ Aµ i ,Aµ = Pµν Aν (3.65)

T and the transverse component Aµ is gauge invariant under Abelian gauge transformations.

We can now compare the outcome of the Nielsen identities with our one-loop calculation (3.49). Indeed,

56 Z 1.003

1.002

1.001

1.000

20 40 60 80 100 ξ

Figure 3.5: Gauge dependence of the residue of the pole for the Higgs field, for the parameter values m = 2 1 1 GeV, mh = 2 GeV, µ = 10 GeV, e = 10 . to the first order, eq.(3.49) is an explicit demonstration of the identity (3.63). For the Higgs boson, the Nielsen identity is a bit more complicated, and is given by ∂ G−1 p2 −∂ G−1 p2 G−1 p2 , ξ hh ( ) = χ Y1h( ) hh ( ) (3.66) G−1 p2 PI where Y1h( ) stands for a non-vanishing 1 Green function which can be obtained from the extended BRST symmetry which also acts on the gauge parameter [158]. To be more precise, Y1 is a local source coupled to the BRST variation of the Higgs field (see (3.8)), while χ is coupled to the integrated composite operator R 4 i d x − 2 cb¯ + mcρ . Acting with ∂χ inserts the latter composite operator with zero momentum flow into the PI h sh hi G−1 p2 1 Green function ( ) , see [157] for the explicit expression of Y1h( ) in terms of Feynman diagrams. As 2 a consequence, the Higgs propagator Ghh(p ) is not gauge independent, in agreement with our results (3.47). From (3.66) we further find −1 2 −1 2 −1 2 ∂ξ∂ 2 G (p )| 2 2 = −∂χG (−m )∂ 2 G (p )| 2 2 , (3.67) p hh p =−mpole Y1h pole p hh p =−mpole G−1 p2 which means that the residue is not gauge independent, as Y1h( ) does not necessarily vanish at the pole. We can confirm this for the one-loop calculation, see Figure 5.5.

Furthermore, we do have −1 2 ∂ξGhh (−mpole) = 0, (3.68) so that the Higgs pole mass is indeed gauge independent, the expected result for the physical (observable) Higgs mass. This can be conﬁrmed to one-loop order by using eq. (3.59), see also Figure 3.8. Explicitly, in eq. (3.56) −1 2 for Ghh (−mh) all the gauge parameter dependence drops out, which means that the Higgs pole mass is gauge independent to ﬁrst order in ~.

3.3.2 Obtaining the spectral function We can try to determine the spectral function itself to first order. To do so, we compare the Källén-Lehmann spectral representation for the propagator Z ∞ 2 ρ(t) G(p ) = dt 2 , (3.69) 0 t + p where ρ(t) is the spectral density function, with the propagator (3.61) to first order, written as Z G(p2) = 2 2 2 (p + mpole − Π(e p ))Z Z = 2 2 2 2 2 ∂Π(e p2) p + mpole − Π(e p ) + (p + mpole) ∂p2 |p2=−m2 2  2 2 2 ∂Π(e p )  Z Π(e p ) − (p + mpole) ∂p2 |p2=−m2 = 2 2 + Z  2 2 2  , (3.70) p + mpole (p + mpole)

57 where in the last line we used a ﬁrst-order Taylor expansion so that the propagator has an isolated pole at 2 2 p = −mpole. In (3.69) we can isolate this pole in the same way, by deﬁning the spectral density function as 2 ρ(t) = Zδ(t − mpole) + ρe(t), giving Z ∞ 2 Z ρe(t) G(p ) = 2 2 + dt 2 (3.71) p + mpole 0 t + p and we identify the second term in each of the representations (3.70) and (3.71) as the reduced propagator

2 2 Z Ge(p ) ≡ G(p ) − 2 2 , (3.72) p + mpole so that

 2  ∞ 2 2 2 ∂Π(e p ) Z Π(p ) − (p + m ) 2 | 2 2 2 ρe(t) e pole ∂p p =−m Ge(p ) = dt 2 = Z  2 2 2  . (3.73) 0 t + p (p + mpole)

2 Finally, using Cauchy’s integral theorem in complex analysis, we can ﬁnd ρe(t) as a function of Ge(p ), giving 1 ρ(t) = lim Ge(−t − i) − Ge(−t + i) . (3.74) e 2πi →0+

3.3.3 Spectral density functions We are now ready to plot the spectral density function for the photon propagator and the Higgs boson propagator. For this section we shall write all quantities as a function of the renormalization scale µ and choose the e v µ λ 1 m µ m √1 µ parameters = 1, = 1 , = 5 , so that = 1 and h = 5 . For this choice of parameters, all one-loop corrections computed are within 20% of the tree-level results, indicating that our perturbative approximation is under control.

T Since in the Abelian case the transverse component of the gauge field Aµ (x) is gauge invariant, it turns out that the corresponding propagator (3.49) is independent from the gauge parameter ξ, and so are its pole mass, residue and spectral function. Following the steps from section 3.3.1, we find the first-order pole mass of the transverse photon to be

2 2 mpole = 1.05417 µ (3.75) and the ﬁrst-order residue

Z = 0.984983. (3.76)

2 2 These values are small corrections of the tree-level ones, m = µ and Ztree = 1, so that the one-loop approximation appears to be consistent.

2 The spectral function is given in Figure 3.6. We can distinguish a two-particle state threshold at t = (m+mh) = 2.09 µ2, and the spectral density function is positive, adequately describing the physical photon excitation.

58 ˜ 103.ρ(t) 0.020

0.015

0.010

0.005

t 5 10 15 20

Figure 3.6: Spectral function for the transverse part of the reduced photon propagator hA(p)A(−p)iT , with t 2 1 given in units of µ , for the parameter values e = 1, v = 1 µ, λ = 5 .

For the Higgs fields, following the steps from section 3.3.1, we find the pole mass to first order in ~ to be

2 2 2 mh,pole = 0.237987 µ = 1.1899 mh, (3.77) for all values of the parameter ξ. This means that while the Higgs propagator (3.47) is gauge-dependent, the pole mass is gauge-independent. This is in full agreement with the Nielsen identities of the Abelian U(1) Higgs model studied in [157]. For the residue, we distinguish three regions:

1 λ √ • ξ < 20 = 4e2 : for these values mh > 2 ξm, which means the Higgs particle is unstable and can decay into two Goldstone fields ρ(x). Of course, this process is physically impossible because the Goldstone boson itself is not physical. It therefore clearly demonstrates the unphysical nature of the propagator hh(x)h(y)i. For these values of ξ, the pole mass is a real value inside the branch cut created by the two-particle state of Goldstone excitations. This means that we cannot properly define the derivative of the one-loop correction to obtain the corresponding residue through eq. (3.62). • ξ ≤ 3: for these values we find Z > 1. • ξ > 3: for these values we find Z < 1. In Figure 3.7, we find the spectral density functions for three values of ξ : 2, 3, 5. For small t, their behaviour is 2 2 the same, with a two-particle state for the Higgs field at t = (mh + mh) = 0.8 µ , and a two-particle state for 2 2 the photon field, starting at√t = (m√+ m) = 4µ . Then, we see that there is a negative contribution, different for each diagram, at t = ( ξm + ξm)2. This corresponds to the threshold for creation of two (unphysical) Goldstone bosons. This negative contribution eventually overcomes the other ones, leading to a negative regime in the spectral function, independently of the value of ξ. This feature is consistent with the large-momentum behaviour of the Higgs propagator (3.47), for a detailed discussion see Appendix B.5. As one lowers the value of the gauge parameter ξ, this unphysical threshold is shifted towards lower t’s and may occur for momentum λ values lower than the physical two-particle states of two Higgs or two photons. As discussed above, for ξ < 4e2 even the one-particle delta peak becomes located within the unphysical Goldstone production region and the standard interpretation of the spectral properties is completely lost. It is therefore clear that this correlation function does not display the desired spectral properties to describe the Higgs mode in this theory, indicating the necessity of resorting to another operator as we shall do in what follows.

59 10 3.ρ(t) 2.5

2.0

1.5

1.0

0.5 t 5 10 15 20 25 30

Figure 3.7: Spectral function for the reduced Higgs propagator hh(p)h(−p)i, for gauge parameters ξ = 2 (Green, dotted), ξ = 3 (Yellow, dashed), ξ = 5 (Red, Solid), with t given in units of µ2, for the parameter values e = 1, 1 v = 1 µ, λ = 5 .

3.4 Some subtleties of the Higgs spectral function

In this section we discuss some subtleties that arose during the analysis of the spectral function of the Higgs boson.

3.4.1 A slightly less correct approximation for the pole mass

2 m2 Im[m pole] Re[ pole] 0.251667 2.× 10 -8

0.251666 1.× 10 -8

0.251665 0.005 0.010 0.015 0.020 0.025 0.030 ξ

-1.× 10 -8 0.251664

-2.× 10 -8 20 40 60 80 ξ

Figure 3.8: Gauge dependence of the Higgs pole mass obtained iteratively to ﬁrst order (Green) and the 1 1 approximated pole mass (Red), for the parameter values m = 2 GeV, mh = 2 GeV, µ = 10 GeV, e = 10 . Up: real part, down: imaginary part.

In the previous two sections we have obtained strictly ﬁrst-order expressions. In practice, for small values of the coupling parameter e2, we could think about making the approximation 1 1 G(p2) = ≈ , (3.78) p2 + m2 − Π(p2) p2 + m2 − Π1−loop(p2) in which case one can ﬁx the pole mass by locating the root of

p2 + m2 − Π1−loop(p2) = 0. (3.79)

The diﬀerence between the pole masses obtained by the iterative method (3.59) and the approximation (3.79) is very small, of the order 10−6 GeV for our set of parameters. However, it is rather interesting to notice that

60 the pole mass of the Higgs boson becomes gauge dependent in the approximation (3.79). This is no surprise, as the validity of the Nielsen identities is understood either in an exact way, or in a consistent order per order approximation. The previous approximation is neither.

In Figure 3.8 one can see the gauge dependence of the approximated pole mass of the Higgs, in contrast with the ﬁrst order pole mass. Even worse, for very small values of ξ, the approximated pole mass gets complex (conjugate) values. This is due to the fact that the threshold of the branch cut, the branch point, for (3.41) is ξ-dependent, as we will see in the next section.

3.4.2 Something more on the branch points The existence of a diagram with two internal Goldstone lines (see Figure C.1) leads to a term proportional to R 1 2 2 0 dx ln(p x(1 − x) + ξm ) in the Higgs propagator (3.41). This means that for small values of ξ, the threshold for the branch cut of the propagator will be ξ-dependent too. Let us look at the Landau gauge ξ = 0. In this gauge, the above ln-term is proportional to ln(p2), due to the now massless Goldstone bosons. This logarithm has a branch point at p2 = 0, meaning that the pole mass will be lying on the branch cut. Since the first order 1−loop 2 pole mass is real and gauge independent, this means that Πhh (−mh) is a singular real point on the branch cut. In the slightly less correct approximation of the last section, we will find complex conjugate poles as in 2 2 Figure 3.8. This is explained by the fact that for every real value different from p = −mh, we are on the branch cut, see Figure 3.9.

1-loop 2 11 Im[Πhh (mh +a+iy)]*10 2

y*10-8 -1.0 -0.5 0.5 1.0

-1

-2

2 Figure 3.9: Behaviour of the one-loop correction of the Higgs propagator Πhh(p ) around the pole mass, for the values a = −10−6 (Yellow, dashed), a = 0 (Red, dotted), a = 10−6 (Green, solid). The value x is a small 2 imaginary variation of the argument in Πhh(p ). Only for a = 0 we ﬁnd a continuous function at x = 0, meaning that for any other value, we are on the branch cut.

Another consequence of the fact that, for small ξ, the pole mass is a real point inside the branch cut is that 2 2 2 Πhh(p ) is non-diﬀerentiable at p = −mh and we cannot extract a residue for this pole. In order to avoid such a problem, we should move away from the Landau gauge and take a larger value for ξ, so that the threshold 2 2 2 for the branch cut will be smaller than −mh. For this we need that 4ξm > mh, which in the case of our 1 parameters set means to require that ξ > 64 , in accordance with Figure 3.8.

3.4.3 Asymptotics of the spectral function Away from the Landau gauge, we see on Figure 3.7 that for e.g. ξ = 2 the Higgs spectral function is not non-positive everywhere, while for e.g. ξ = 4 it is positive deﬁnite, with a turning point at ξ = 3. How can we explain this diﬀerence? The answer can be related to the UV behaviour of the propagator. For p2 → ∞, the Higgs boson propagator at one-loop behaves as

2 Z Ghh(p ) = , (3.80) 2 p2 p ln µ2 with Z depending on the gauge parameter ξ. Now, one can show (see Appendix B.5) that for Z > 0, ρ(t) becomes negative for a large value of t. For our parameter set, we ﬁnd that for large momenta p2 ln(p2) G−1(p2) → (3 − ξ) , for p2 → ∞, (3.81) hh 1600π2

61 so that for ξ < 3, we indeed find Z > 0. This indicates that the large momentum behaviour of the propagator makes a difference around ξ = 3, and determines the positivity of the spectral function, a known fact [122, 153, 162]. This being said, at the same time we cannot trust the propagator values for p2 → ∞ without taking into account the renormalization group (RG) effects and in particular the running of the coupling, which is problematic for non-asymptotically free gauge theories as the Abelian Higgs model.

3.5 A non-unitary U(1) model

In this section, we will discuss an Abelian model of the Curci-Ferrari (CF) type [163], in order to compare it with the Higgs model (3.4). Both models are massive U(1) models with a BRST symmetry. However, while the BRST operator s of the Higgs model is nilpotent, this is not true for the CF-like model. We know the that the Higgs model is unitary but, by the criterion of [123], the CF model is most probably not.

In section 3.5.1 we discuss some essentials for the CF-like model: the action with the modified BRST symmetry, tree-level propagators and vertices. In section 3.5.2 we discuss the one-loop propagators for the photon and scalar field and extract the spectral function. In section 3.5.3 we introduce a local composite field operator that is left invariant by the modified BRST symmetry ot the CF model. The spectral properties of this composite state’s propagator will tell us something about the (non-)unitarity of the model, since for unitary models, we expect the propagator of a BRST invariant composite operator to be gauge parameter independent, and the spectral function to be positive definite.

3.5.1 CF-like U(1) model: some essentials We start with the action of the CF-like U(1) model

Z 2 d 1 m † 2 † † 2 SCF = d x FµvFµv + AµAµ + (Dµϕ) Dµϕ + m ϕ ϕ + λ(ϕϕ ) 4 2 ϕ 2 b 2 2 − α + b∂µAµ +c∂ ¯ c − αm cc¯ , (3.82) 2

m2 where the mass term 2 AµAµ is put in by hand rather than coming from a spontaneous symmetry breaking, and we have ﬁxed the gauge in the linear covariant gauge with gauge parameter α. The mass term breaks the BRST symmetry (3.8) in a soft way. This Abelian CF action is however invariant under the modiﬁed BRST 5 symmetry, smSCF = 0, with

smAµ = −∂µc,

smc = 0,

smϕ = iecϕ, † † smϕ = −iecϕ ,

smc¯ = b, 2 smb = −m c. (3.83)

2 As noticed before in our Introduction, this modiﬁed BRST symmetry is not nilpotent since smc¯ =6 0. From the quadratic part of (3.82) we ﬁnd the following propagators at tree-level 1 α hAµ(p)Av(−p)i = Pµv + Lµv, p2 + m2 p2 + αm2 pµ hAµ(p)b(−p)i = i , p2 + αm2 m2 hb(p)b(−p)i = − , p2 + αm2 † 1 hϕ (p)ϕ(−p)i = 2 2 , p + mϕ 1 hc¯(p)c(−p)i = − , (3.84) p2 + αm2

5This is the Abelian version of the variation (3.8). For computational purposes, we have also made a rescaling ib → b. Notice that higher order α-dependent terms present in the CF model are absent in the Abelian limit [164, 165].

62 Figure 3.10: Contributions to one-loop CF photon self-energy. Wavy lines represent the photon ﬁeld and dashed lines represent the scalar ﬁeld.

Figure 3.11: Contributions to the one-loop CF scalar self-energy. Line representation as in Figure 3.10. while from the interaction terms we ﬁnd the vertices

† ΓAµϕ ϕ(−p1, −p2, −p3) = e(p3,µ − p2,µ)δ(p1 + p2 + p3), 2 † ΓAµAv ϕ ϕ(−p1, −p2, −p3, −p4) = −2e δµvδ(p1 + p2 + p3 + p4),

Γϕ†ϕϕ†ϕ(−p1, −p2, −p3, −p4) = −4λδ(p1 + p2 + p3 + p4). (3.85)

3.5.2 Propagators and spectral functions The one-loop corrections to the photon and scalar self-energies are given in Figure 3.10 and 3.11. Without going through the calculational details, we will directly give here the propagators in d = 4 and discuss some curiosities. The inverse connected photon propagator,

2 Z 1 2 2 2 ⊥ −1 2 2 2 e 2 2 K(mϕ, mϕ) 2 mϕ (GAA) (p ) = p + m + 2 dxK(mϕ, mϕ)(1 − ln 2 ) − mϕ(1 − ln 2 ), (3.86) (4π) 0 µ µ

∗ 2 is independent of the gauge parameter. The threshold of the branch cut is given by t = −4mϕ, and to avoid 2 2 1 a pole mass lying on the branch cut, we need to choose here m < 4mϕ. Choosing m = 2 GeV, mϕ = 2 GeV, 1 µ = 10 GeV, e = 10 , we ﬁnd a positive spectral function, see Figure 3.12. More interestingly, the scalar propagator

2 Z 1 2 2 −1 2 2 e 2 2 2 2 2 K(mϕ, αm ) Gϕϕ(p) = p + mϕ − 2 dx m − α m − αK(mϕ, m )(1 − 2 ln 2 ) (3.87) (4π) 0 µ p2 K(m2 , 0) p2 K(m2 , m2) + 4K(m2 , 0) (1 − ln ϕ ) − 2K(m2 , m2) (1 − ln ϕ ) ϕ m2 µ2 ϕ m2 µ2 p2 K(m2 , αm2) p4 K(m2 , 0) − 2K(m2 , αm2) (1 − ln ϕ ) + 8 x2 ln ϕ ϕ m2 µ2 m2 µ2 p4 K(m2 , m2) K(m2 , m2) p2 K(m2 , αm2) − 4 x2 ln ϕ − 4p2 ln ϕ − (4αp2x − αp2x2 + 4 x2) ln ϕ m2 µ2 µ2 m2 µ2 m2 αm2 λ m2 − 3m2 ln + αm2 ln + m2 (1 − ln ϕ ), (3.88) µ2 µ2 (4π)2 ϕ µ2

2 2 1−loop 2 is α-dependent, and so is the iterative ﬁrst-order pole mass mϕ,pole = mϕ − Π (−mϕ). This ﬁeld can thus not represent a physical particle. For any value other than the Landau gauge α = 0 we furthermore get complex poles, see Figure 3.13.

From the fact that we find gauge dependent (complex) pole masses for the scalar field, we can already draw the conclusion that the CF model does not describe a physical scalar field. In the next section we will explicitly verify the non-unitary of this model in yet another way. Essentially, our findings so far mean that in the CF setting, the unphysical gauge parameter α plays here a quite important role, just like the coupling: different values of the gauge parameter label dfferent theories. This can also be seen from another example: the one-loop vacuum energy of the model will now not only depend on m, but also on α.

63 ˜ 106.ρ(t) 0.25

0.20

0.15

0.10

0.05 t 50 100 150 200

Figure 3.12: The reduced spectral function of the photon ﬁeld in the Abelian CF model, with t given in GeV2, 1 1 for the parameter values m = 2 GeV, mh = 2 GeV, µ = 10 GeV, e = 10 . The ﬁrst-order pole mass lies at 2 ∗ 2 2 t = 0.25151 GeV . The photon two-particle state starts at t = 4mϕ = 16 GeV .

3.5.3 Gauge invariant operator

b2 2 The Abelian CF model allows us to construct a BRST invariant composite operator 2 + m cc¯ , with 2 b 2 sm + m cc¯ = 0. (3.89) 2 2 Although sm =6 0 and we can therefore no longer introduce the BRST cohomology classes, we can still use the fact that sm is a symmetry generator, thereby defining a would-be physical subspace as the one being annihilated by sm. A Fock space analogue of this operator was introduced in [124], where it was established that it has negative norm. As a consequence, it was shown that the physical subspace relating to the symmetry generator sm was not well-defined, as it contains ghost states. Several more such states were identified later on in [125].

2 2 Re[m φ,pole] Im[m φ,pole]

0.24 -10 -5 5 10 α

0.22 -0.02

0.20 -0.04

0.18 -0.06

-10 -5 5 10 α -0.08

Figure 3.13: Gauge dependence of the ﬁrst order pole mass for the scalar ﬁeld. Left: Real part, Right: Imaginary 1 1 part. The chosen parameter values are m = 2 GeV, mϕ = 2 GeV, µ = 10 GeV, e = 10 .

Up to leading order, the connected propagator of the composite operator in eq.(3.89) reads 2 2 2 b 2 b 2 1 2 2 4 2 G b +m2cc¯ (p ) = + m cc¯ , + m cc¯ = hb , b i + m hcc,¯ cc¯ i, (3.90) 2 2 2 4 We thus ﬁnd the propagator (3.90) to be Z d 2 3 2 d k 1 1 2 G b +m2cc¯ (p ) = − m d 2 2 2 2 2 4 (2π) k + αm (k − p) + αm 3 1 Z 1 − m2 − d/ dxK αm2, αm2 = d/2 Γ(2 2) d/2−2( ) (3.91) 4 (4π) 0

64 10 3.ρ(t) t 1 2 3 4 5 6 7 -0.2

-0.4

-0.6

-0.8

-1.0

b2 2 Figure 3.14: The spectral function of the composite operator 2 + m cc¯ , for α = 2 (Green, dotted), α = 3 (Red, solid), α = 5 (Yellow, dashed). The chosen parameter values are m = 1 GeV, µ = 10 GeV. The treshold for √ √ 2 the branch cut of the propagator is given by t∗ = ( αm + αm)2, being t∗ = (2 GeV2, 3 GeV2, 5 GeV2) for ξ = (2, 3, 5). and this gives for d = 4, using the MS-scheme

2 Z 1 2 2 2 3 m K(αm , αm ) 2 G b +m2cc¯ (p ) = 2 dx ln 2 . (3.92) 2 4 (4π) 0 µ Clearly, the propagator is depending on the gauge parameter α, a not so welcome feature for a presumably physical object.

We can also find the spectral function immediately from the propagator by again relying on (3.74). In Figure 3.14, one sees that the spectral function is negative for different values of α. Both the α-dependence and the negative-definiteness of the spectral functions demonstrate the non-unitarity of the Abelian CF model. To our knowledge, this is the first time that ghost-dependent invariant operators in the physical subspace of a CF model have been constructed from the functional viewpoint6, complementing the (asymptotic) Fock space analyses of [124, 125].

3.6 Conclusion

In this chapter we have studied the Källén-Lehmannspectral properties of the U(1) Abelian Higgs model in the Rξ gauge, and that of a U(1) Curci-Ferrari (CF) like model. Our main aim was to disentangle in this analytical, gauge-fixed setup what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. Special attention was given to the role played by gauge (in)dependence of different quantities and by the correct implementation of the results up to a given order in perturbation theory. In particular, calculating the spectral function for the Higgs propagator in the U(1) model, it became apparent that an unphysical occurrence of complex poles, as well as a gauge-dependent pole mass, are caused by the use of the resummed (approximate) propagator as being exact. Indeed, for small coupling constants, the one-loop correction gives a good approximation of the all-order loop correction, and this is a much used method to find numerical results [112, 116, 166]. However, for analytical purposes, one should stick to the order at which one has calculated the propagator. As we have illustrated, at least in the U(1) Abelian Higgs model case, one will then find a real and gauge independent pole mass for the Higgs boson, in accordance with what the Nielsen identities dictate [157].

Another issue faced here was the fact that the branch point for the Higgs propagator is ξ-dependent, being located at p2 = 0 for the Landau gauge ξ = 0. For small values of ξ, the pole mass has a real value. However, its value is located on the branch cut, making it impossible to deﬁne a residue at this point, and therefore a spectral function. This means that in order to formulate a spectral function, we should move away from the Landau gauge. These issues with unphysical (gauge-variant) thresholds are nothing new, see for example [167].

6A similar result can be checked to hold for the original non-Abelian CF model, by adding a few higher order terms to the here introduced Abelian operator. This means that a non-Abelian version of the operator (3.89), invariant under the BRST transformation (3.8), can be written down.

65 They reinforce in a natural way the need to work with gauge-invariant ﬁeld operators to correctly describe the observable excitations of a gauge theory.

For the photon, the (transverse) propagator is gauge independent (even BRST invariant), and consequently so are the pole mass, residue and spectral function. For the Higgs boson, the propagator, residue and spectral function are gauge dependent, while the pole mass is gauge independent, in line with the latter being an observable quantity. Notice that the residue of the two-point function does not need to be gauge independent, since this does not follow from the Nielsen identities, as we discussed in our main text. Rather, the Nielsen identities can be used to show that the residues of the pole masses in S-matrix elements are gauge independent, that is the residues of the singularities in observable scattering amplitudes, see [58, 168]. These residues can evidently be different per scattering process (and per different mass pole). The fact that the Higgs propagator is gauge dependent is not surprising, given that the Higgs field is not invariant under the Abelian gauge transformation.

Concluding, in this chapter several tools have been worked out to determine spectral properties in perturbation theory. We worked up to ﬁrst order in ~ , but everything can be consistently extended to higher orders. We paid attention how to avoid problems with complex poles and to the pivotal important role of the Nielsen identities, which are intimately related to the exact nilpotent BRST invariance of the model. These tools will turn out to be quite useful for forthcoming work on the spectral properties of HYM theories. For these theories, the Nielsen identities are well established [57], with supporting lattice data [53], providing thus a solid founda- tion to compare any results with.

In the next chapters we will consider gauge-invariant operators and study their spectral properties using the same techniques of this paper. If the elementary fields are not gauge invariant (like the Higgs field, but also the gluon field in QCD), these aforementioned gauge-invariant operators will turn out to be composite in nature. Such an approach has recently been addressed in [169, 170], based on the seminal observations of Fröhlich- Morchio-Strocchi [44, 45], in which composite operators with the same global quantum numbers (parity, spin, . . . ) as the elementary particles are constructed.7 These composite states will enable us to access directly the physical spectrum of the theory. Moreover, we notice that the spectral properties and the behaviour in the complex momentum plane of a (gauge-invariant) composite operator will nontrivially depend on the spectral properties of its gauge-variant constituents. This gives another motivation why it is meaningful to study spectral properties of gauge-variant propagators. Another nice illustrative example of this interplay is the Bethe-Salpeter study of glueballs in pure gauge theories [172], based on spectral properties of constituent gluons and ghosts [28].

7For a recent discussion of the renormalization properties of higher dimensional gauge invariant operators in HYM models see the recent results by [171].

66 Chapter 4

Gauge-invariant spectral description of the U(1) Higgs model from local composite operators

An essential aspect of gauge theories is that all physical observable quantities have to be gauge invariant [41,43]. Therefore, a formulation of the properties of the elementary excitations in terms of gauge-invariant variables is very welcome. Such an endeavour has been addressed by several authors1 [174–177], who have been able to construct, out of the elementary fields, a set of local gauge invariant composite operators which can effectively implement a gauge invariant framework by using the tools of quantum gauge field theories: renormalizability, locality, Lorentz covariance and BRST exact symmetry.

The aim of this chapter is that of discussing the features of two local gauge invariant operators within the framework of the U(1) Abelian Higgs model discussed in chapter3. Following [174, 175], we shall consider the two local composite operators O(x) and Vµ(x) invariant under (3.6), given by

v2 O(x) = 1/2(h2(x) + 2vh(x) + ρ2(x)) = ϕ†(x)ϕ(x) − , 2 † Vµ(x) = −iϕ (x)(Dµϕ)(x) . (4.1)

The relevance of these operators can be understood by using the expansion (3.3) and retaining the ﬁrst order terms. For the two-point correlator of the scalar operator one ﬁnds (cf. eq. (4.7) for the full expression):

2 3 2 4 hO(x)O(y)i ∼ v hh(x)h(y)i(tree-level) + O(~) + hO h ; hρ ; ρ i , (4.2) while the contributions to the vector operator at lowest order in the ﬁelds read

ev2 Vµ(x) ∼ Aµ(x) + total derivative + higher orders . (4.3) 2

We see therefore that the gauge-invariant operator O(x) is related to the Higgs excitation, while Vµ(x) is associated with the photon.

In this chapter, we shall compute the BRST-invariant two-point correlation functions

hO(x)O(y)i , hVµ(x)Vν (y)i , (4.4) at one-loop order in the ’t Hooft Rξ-gauge and discuss the diﬀerences with respect to the corresponding one-loop elementary propagators hh(x)h(y)i and hAµ(x)Aν (y)i already evaluated in the last chapter. Let us point out that the two local operators (Vµ(x),O(x)) belong to the cohomology of the BRST operator [17], i.e.

sVµ(x) = 0 ,Vµ(x) =6 s∆µ(x) sO(x) = 0 ,O(x) =6 s∆(x) , (4.5) for any local quantities (∆µ(x), ∆(x)).

1See [170, 173] for a general review on this matter.

67 As expected, both correlation functions of eq. (4.4) turn out to be independent from the gauge parameter ξ. Moreover, we shall show that the one-loop pole masses of hVµ(x)Vν (y)iT and hO(x)O(y)i are exactly the same as those of the elementary propagators hAµ(x)Aν (y)iT and hh(x)h(y)i, where hAµ(x)Aν (y)iT stands for the transverse component of hAµ(x)Aν (y)i, i.e.

∂ ∂ hA (x)A (y)i = δ − µ ρ hA (x)A (y)i . (4.6) µ ν T µρ ∂2 ρ ν

This important feature makes apparent the fact that the operators Vµ(x) and O(x) give a gauge invariant picture for the photon and Higgs modes. In addition, the correlation functions hVµ(x)Vν (y)iT and hO(x)O(y)i exhibit a spectral KL representation with positive spectral densities, allowing for a physical interpretation in terms of particles. This property is in sharp contrast with the one-loop spectral density of the elementary non gauge invariant Higgs propagator hh(x)h(y)i, which displays an explicit dependence on the gauge parameter ξ, as established in the last chapter. Moreover, the longitudinal part of the correlator hVµ(x)Vν (y)i – which is independently gauge invariant – is shown to exhibit the pole mass of the Higgs excitation. This last feature reinforces the consistency of the present description of the physical degrees of freedom of the theory, since the only physically expected elementary excitations are indeed the Higgs and the photon ones. Let us also underline that, to our knowledge, this is the ﬁrst explicit one-loop calculation of the gauge-invariant correlators (4.4) and of their analytical properties.

This chapter is organized as follows. In section 4.1 we compute at one-loop order the two-point functions for the composite operators in. In section C.3, we provide the detailed analysis of the spectral properties of the composite operators, and compare them with the spectral properties of the elementary fields. The unitary limit, in which the gauge parameter ξ tends to infinity, is investigated in section 4.3. Section 5.5 collects our conclusion and outlook. The final Appendices contain the derivation of the Feynman rules and of the diagrams contributing to (4.4).

4.1 The correlation functions hO(x)O(y)i and hVµ(x)Vν(y)i at one loop order

We study the two-point correlation functions of the local gauge invariant operators (Vµ(x),O(x)). For the correlator of the scalar composite operator we get:

hO(x)O(y)i = v2 hh(x)h(y)i + v hh(x)ρ(y)2i + v hh(x), h(y)2i + 1 + hh(x)2ρ(y)2i + hh(x)2h(y)2i + hρ(x)2ρ(y)2i . (4.7) 4 Individually, the terms in the expansion (4.7) are not gauge invariant, but their sum is. We can now analyze the connected diagrams for each term, up to one-loop order, through the action (3.4). We calculated the one-loop diagrams in Appendix B.6. Looking at the diagrams in Figure B.1, we can see that the correlation function hO(p)O(−p)i will have the following structure

2 2 2 2 1−loop Afin(p ) + δAdiv(p ) Bfin(p ) + δBdiv(p ) hO(p)O(−p)i = 2 2 2 + 2 2 (p + mh) (p + mh) 2 2 + Cfin(p ) + δCdiv(p ), (4.8) where (Afin,Bfin,Cfin) stand for the ﬁnite parts and (δAdiv, δBdiv, δCdiv) for the purely divergent terms, i.e. 1 the pole terms in obtained by means of the dimensional regularization, namely

2 2 →0 v 2 2 2 2 2 δAdiv(p ) = 2v λ − e (p (−3 + ξ) + v λξ) 8π2 2 4 2 2 2 →0 v (6e − λ + e λξ) δBdiv(p ) = 8π2λ 2 →0 1 δCdiv(p ) = (4.9) 8π2

68 while

2 Z 1 ( " 2 2 2 2 v 2 2 m p x(1 − x) + m Afin(p ) = 2 dx e p (1 − ln 2 − 2 ln 2 ) (4π) 0 µ µ # p4 p2x(1 − x) + m2 m2 p2x(1 − x) + m2 − ln − 6m2(1 − ln + ln ) 2m2 µ2 µ2 µ2 h1 m2 p2x(1 − x) + m2 i + λ m2 (−6 + 6 ln h − 9 ln h ) 2 h µ2 µ2 " #) ξm2 p4 m2 p2x(1 − x) + ξm2 − ξ(e2p2 + λm2)(1 − ln ) − (e2 − λ h ) ln µ2 2m2 2 µ2

Z 1 ( 2 2 2 2 1 2 2 m ξ 2 2 4 mh + p (1 − x)x Bfin(p ) = 2 2 dx − m ξmh ln 2 + m ξmh + mh 3 ln 2 (4π) mh 0 µ µ ) m2ξ + p2(1 − x)x m2 m2 + ln − 3m4 ln h + 3m4 + 2m4 − 6m4 ln µ2 h µ2 h µ2

Z 1 ( 2 2 2 2 ) 2 1 mh + p (1 − x)x m ξ + p (1 − x)x Cfin(p ) = − 2 dx ln 2 + ln 2 . (4.10) 2(4π) 0 µ µ

The divergent terms (δAdiv, δBdiv, δCdiv) can be eliminated by means of the Lagrangian counterterms as well as by suitable counterterms in the external part of the action SJ accounting for the introduction of the composite operator O(x), i.e.

Z " 2 # 4 0 (J(x)) SJ = S + d x (1 + δZ )J(x)O(x) + (1 + δZdiv) , (4.11) div 2 where J(x) is a BRST invariant dimension two source needed to deﬁne the generator Zc(J) of the connected Green function hO(x)O(y)i:

δ2Zc(J) hO(x)O(y)i = |J=0. (4.12) δJ(x)δJ(y) It is worth emphasizing here that we have the freedom of introducing a pure contact BRST invariant term in the external source J(x):

Z α d4x J 2(x), (4.13) 2 which can be arbitrarily added to the action (4.11). Including such a term in (4.11) will have the eﬀect of adding a dimensionless constant to GOO = hO(p)O(−p)i, i.e.

2 2 GOO(p ) → GOO(p ) + α. (4.14)

In particular, α can be chosen to be equal to −GOO(0), implying then that the modiﬁed Green’s function

2 GOO(p ) − GOO(0) (4.15) will obey a one substracted KL representaion .

Inserting the unity

2 2 2 2 2 2 2 2 2 1 = (p + mh)/(p + mh) = ((p + mh)/(p + mh)) , (4.16) for the ﬁnite part of hO(p)O(−p)i, we write

2 2 v ~v 2 2 hO(p)O(−p)ifin = 2 2 + 2 2 2 Π(p ) + O(~ ) (4.17) p + mh (p + mh)

69 where 1 Π (p2) = (A (p2)) + (p2 + m2 )(B (p2)) + (C (p2))(p2 + m2 )2 , OO v2 fin h fin fin h Z 1 ( 2 1 2 4 2 2 2 2 m = 2 2 2 dx − 8mhm − 2m p (mh + 6m ) ln 2 + 32π v mh 0 µ h m2 + p2(1 − x)x m2 + p2(1 − x)x i +m2 − (p2 − 2m2 )2 ln h − (12m4 + 4m2p2 + p4) ln + h h µ2 µ2 ) m2 +2p2(3m4 + m2 m2 + 2m4) − 6m4 p2 ln h , (4.18) h h h µ2

p4 2 and since (4.18) contains terms of the order of p2+m2 ln(p ), we follow the steps (3.51)-(3.53) to ﬁnd the resummed form factor in the one-loop approximation

2 2 v 2 GOO(p ) = + COO(p ) (4.19) 2 2 ˆ 2 p + mh − ΠOO(p ) with

Z 1 ( 2 ˆ 2 1 2 4 2 2 2 2 m ΠOO(p ) = 2 2 2 dx − 8mhm − 2m p (mh + 6m ) ln 2 + 32π v mh 0 µ h m2 + p2(1 − x)x +m2 3(m4 + 2m2 p2) ln h h h h µ2 m2 + p2(1 − x)x i −(12m4 + 4m2p2 − m4 − 2p2m2 ) ln + h h µ2 ) m2 +2p2(3m4 + m2 m2 + 2m4) − 6m4 p2 ln h , (4.20) h h h µ2 and

Z 1 ( 2 2 2 2 ) 2 1 mh + p x(1 − x) m + p x(1 − x) COO(p ) = − 2 dx ln 2 + ln 2 . (4.21) 32π 0 µ µ

2 The form factor is depicted in Figure 4.1. Notice that the Green function GOO(p ) becomes negative for large enough values of the momentum p. As one realizes from expression (4.20), this feature is due to the growing in 2 the UV region of the logarithms contained in the term COO(p ), see eq. (4.21). It is worth mentioning that this behaviour is also present when the parameter v is completely removed from the theory. In fact, setting v = 0, the action S0 in eq. (3.1), reduces to that of massless scalar QED, namely

Z 2 ! 4 Fµν † λ † 2 S0| = d x + (Dµϕ) (Dµϕ) + (ϕ ϕ) , (4.22) v=0 4 2

with

1 ϕ| = √ (h + iρ). (4.23) v=0 2

† † Of course, when v = 0, the operators O = ϕ ϕ and Vµ = −iϕ Dµϕ are still gauge invariant. Though, from eqs. (4.20)-(4.21), computing hO(p)O(−p)iv=0, one immediately gets

1 Z 1 p2x(1 − x) hO(p)O(−p)iv=0 = COO|v=0 = − 2 dx ln 2 . (4.24) 16π 0 µ

This equation precisely shows that the term COO0, and thus the negative behaviour for large enough values of p, is what one usually obtains in a theory for which v = 0, making evident that the presence of COO is not peculiarity of the U(1) Higgs model in the U(1) Higgs model, on the contrary. However, in addition to the term

70 COO and unlike massless scalar QED, the correlation function hO(p)O(−p)i of the U(1) Higgs model exhibits v2 the term 2 2 ˜ , which will play a pivotal role. Indeed, as we shall see later on, this term, originating from p +mh−ΠOO ϕ φ √1 v h iρ the expansion of around the minimum of the Higgs potential, = 2 ( + + ), will enable us to devise a gauge invariant description of the elementary excitations of the model. 2 Let us end the analysis of the correlation function GOO(p ) by displaying the behaviour of its ﬁrst derivative, 2 ∂GOO (p ) 2 ∂p2 , as well as of the one subtracted correlator GOO(p ) − GOO(0), see Figure 4.2. The ﬁrst derivative, as 2 2 2 ∂GOO (p ) expected, is negative while, unlike GOO(p ), decays to zero for p → ∞. The quantity ∂p2 will be helpful when discussing the spectral representation corresponding to hO(p)O(−p)i.

2 GOO(p ) 0.8

0.6

0.4

0.2

0.0 5 10 15 20p -0.2

-0.4

Figure 4.1: Resummed form factor for the scalar composite operator. The momentum p is given in units of the 1 energy scale µ, for the parameter values e = 1, v = 1 µ, λ = 5 .

2 GOO(p )-GOO(0)

0 5 10 15 20p -1 -2 -3 -4 -5 -6

Figure 4.2: The resummed form factor with one subtraction. The momentum p is given in units of the energy 1 scale µ, for the parameter values e = 1, v = 1 µ, λ = 5 .

71 2 ∂G OO p  ∂p 2

0.04

0.02

0.00 20 40 60 80 100p -0.02

-0.04

Figure 4.3: The ﬁrst derivative of the form factor (left). The momentum p is given in units of the energy scale 1 µ, for the parameter values e = 1, v = 1 µ, λ = 5 .

Then, for the vectorial composite operator Vµ(x), we ﬁrst observe that

† Vµ(x) = −iϕ (x)(Dµϕ)(x)

† 1 † 1 † = eϕ (x)Aµ(x)ϕ(x) − iϕ (x)∂µϕ(x) + iϕ(x)∂µϕ (x) − i∂µO(x), (4.25) 2 2 and since we know that the last term is gauge-invariant, the ﬁrst three terms together are also gauge-invariant. We thus deﬁne a new gauge-invariant operator

0 † 1 † 1 † V (x) = eϕ (x)Aµ(x)ϕ(x) − iϕ (x)∂µϕ(x) + iϕ(x)∂µϕ (x), (4.26) µ 2 2 expanding the scalar ﬁeld ϕ(x) we ﬁnd

0 1 2 1 2 V (x) = e(v + h(x)) Aµ(x) + eρ (x)Aµ(x) + (v + h(x))∂µρ(x) − ρ(x)∂µh(x) (4.27) µ 2 2 so that ( ϕ→ √1 (v+h+iρ) 0 0 2 1 2 4 2 3 hV µ(x)V v(y)i = − − e v hAµ(x)Aν (y)i − 4e v hh(x)Aµ(x)Aν (y)i 4 2 2 2 2 2 −2e v hh(x) Aµ(x)Aν (y)i − 4e v hh(x)Aµ(x)h(y)Aν (y)i 2 2 2 2 x −2e v hρ(x) Aµ(x)Aν (y)i − 2ev ∂ µhh(x)ρ(x)Av(y)i 2 x 3 x +4ev h∂ µh(x)ρ(x)Aν (y)i − 2ev ∂ µhρ(x)Aν (y)i 2 x x y −4ev ∂ µhρ(x)h(y)Av(y)i − 2v∂ µ∂ ν hh(x)ρ(x)ρ(y)i y x x y +4v∂ ν h∂ µh(x)ρ(x)ρ(y)i − ∂ µ∂ ν hh(x)ρ(x)h(y)ρ(y)i ) x y 2 x y +4h∂ µh(x)ρ(x)h(y)∂ ν ρ(y)i − v ∂ µ∂ ν hρ(x)ρ(y)i

2 +O(~ ), (4.28) where we have discarded the terms that do not have one-loop contributions. In momentum space, we can split the two-point function into transverse and longitudinal parts in the usual way:

0 0 0 0 T 0 0 L hV µ(p)V ν (−p)i = hV (p)V (−p)i Pµν + hV (p)V (−p)i Lµν , (4.29)

72 where we have introduced the transverse and longitudinal projectors, given respectively by p p P (p) = δ − µ ν , µν µν p2 p p L (p) = µ ν . (4.30) µν p2 At tree-level, we ﬁnd in momentum space

0 0 1 2 4 2 hV (p)V (−p)itree = − −e v hAµ(p)Aν (−p)i − v pµpν hρ(p)ρ(−p)i µ ν 4 2 1 2 4 1 2 4 ξ 2 p = e v Pµν + e v Lµν + v Lµν 4 p2 + m2 p2 + ξm2 p2 + ξm2 2 4 e v 1 2 = Pµν + v Lµν . (4.31) 4 p2 + m2 We can now analyze the connected diagrams for each term, up to one-loop order, through the action (3.4). We calculated the one-loop diagrams in Appendix B.7. Let us start with the transverse part. Looking at the diagrams in Figure B.2, we can see that the one-loop correlation function will have the following structure V 2 V 2 V 2 V 2 T,1−loop A (p ) + δA (p ) B (p ) + δB (p ) hV 0(p)V 0(−p)i = fin div + fin div (p2 + m2)2 (p2 + m2) V 2 V 2 + Cfin(p ) + δCdiv(p ) (4.32) V V V V V V where (Afin,Bfin,Cfin) stand for the ﬁnite parts and (δAdiv, δBdiv, δCdiv) for the purely divergent terms, i.e. 1 the pole terms in obtained by means of the dimensional regularization, namely

4 4 2 →0 e v 1 e 1 δAV = p2 − 6( − )e2v2 + 3λv2 div 2(4π)2 3 λ 2 2 6 2 2 2 →0 v e v e p δBV = (6 − 3e4v2 − + 3e2λv2) div (4π)2 λ 3 →0 1 δCV = (9e2v2 − p2 − 3λv2) (4.33) div 6(4π)2 and 4 4 Z 1 ( V e v 2 2 Afin = 2 dx p x(1 − x) + m x 2(4π) 0 p2x(1 − x) + m2x + m2 (1 − x) m2 + m2 (1 − x)(1 − ln h ) + m2 (1 − ln h ) h µ2 h µ2 4 2 2 2 2 ) m m 2 p x(1 − x) + m x + mh(1 − x) + 2 (1 − 3 ln 2 ) + 2m ln 2 mh µ µ 2 Z 1 ( 2 V m 4 2 2 2 mh Bfin = 2 2 2 dx 3mh mh − m − 7p ln 2 18mhp (4π) 0 µ xm2 + m2(1 − x) + p2(1 − x)x − 3m2 2p2 m2 − 5m2 + m2 − m2 2 + p4 ln h h h h µ2 3 2 2 2 2 2 4 4 4 2 − 3 mh − m mh + 9p m mh + 3mh + 2m + 2p mh ) m2 − 3m2 m2 p2 − m2 + m4 + 18m2p2 ln h h µ2

Z 1 ( 2 V 1 2 2 2 2 m Cfin = 2 2 dx 3m mh − m + p ln 2 36(4π) p 0 µ m2 + 3m2 −m2 + m2 + p2 ln h + 6m2 p2 − m2 − 5p2 3m2 − 9m2 + p2 h h µ2 h h p2x(1 − x) + xm2 + (1 − x)m2 + 3 2m2 p2 − m2 + m4 + m4 − 10m2p2 + p4 ln h h h µ2 ) 4 4 2 2 4 + 3mh + 3m − 54m p + 3p . (4.34)

73 V V V The divergent terms (δAdiv, δBdiv, δCdiv) can be eliminated by means of the Lagrangian counterterms as well as V by suitable counterterms in the external part of the action SJ accounting for the introduction of the composite 0 operator Vµ(x), i.e.

Z " # V 4 V,0 V Jµ(x)Jµ(x) S = S + d x (1 + δZ )Jµ(x)Vµ(x) + (1 + δZ ) , (4.35) J div div 2

c where Jµ(x) is a BRST invariant dimension one source needed to deﬁne the generator Z (J) of the connected 0 0 Green function hVµ(x)Vv (y)i:

2 c 0 0 δ Z (J) hVµ(x)Vv (y)i = |J=0, (4.36) δJµ(x)δJv(y) and like in the scalar case, we have the freedom of introducing a pure contact BRST invariant term in the external source Jµ(x):

Z 4 1 2 2 2 d x (βv Jµ(x)Jµ(x) + γJµ(x)∂ Jµ(x) + σ(∂µJµ(x)) ), (4.37) 2 which can be arbitrarily added to the action (4.35). Including such a term in (4.35) will have the eﬀect of T 2 0 0 T adding a dimensionless constant to GVV (p ) = hV (p)V (−p) i, i.e.

T 2 T 2 2 2 GVV (p ) → GVV (p ) + βv + γp , (4.38) where we notice that the last term in (4.37) does not contribute to the transversal part of the propagator. In particular, β and γ can be choosen so that (4.38) becomes

∂GT (p2) GT (p2) − GT (0) − p2 VV | , (4.39) VV VV ∂p2 p=0 see Figure 4.5. It is a Green’s function that obeys a two substracted KL representation, see section C.3. Following the steps in the same way as for the scalar composite ﬁeld, (4.11)-(5.52), we ﬁnd

2 4 2 4 T 2 0 0 T e v 1 ~e v ΠVV (p ) 2 hV (p)V (−p)i = + + O(~ ), (4.40) 4 p2 + m2 4 (p2 + m2)2 with

Z 1 ( T 2 1 4 4 4 2 4 2 2 ΠVV (p ) = − 2 2 4 2 dx − 18m (mh + m ) + 9mhp (mh + m ) 9(4π) e v mh 0 m2 (1 − x) + m2x + p2(1 − x)x − 3m2 p22p2(m2 − 5m2) + (m2 − m2)2 + p4 ln h + 2m2 p6 h h h µ2 h m2 + 3m4 ln h p2(m2 + 11m2) + 6m4 − p4 h µ2 h m2 + 3m2 − m2 p4 + p2(−m4 + m2 m2 + 36m4) + 18m6 ln h h h µ2 ) 2 6 4 2 2 4 6 − 3p (mh + 10mhm + mhm + 12m ) , (4.41) and following the steps (3.51)-(3.53), we ﬁnd

e2v4 1 GT C p2 VV = 2 2 T 2 + VV ( ) (4.42) 4 p + m − ΠVV (p ) with

74 Z 1 ( ˆ T 2 1 4 4 4 2 4 2 2 ΠVV (p ) = − 2 2 4 2 dx − 18m (mh + m ) + 9mhp (mh + m ) 9(4π) e v mh 0 2 2 2 4 2 2 − 3mh 2(−2m p − m )(mh − 5m ) (4.43) m2 (1 − x) + m2x + p2(1 − x)x + p2(m2 − m2)2 − 2m2p2 − m4 ln h h µ2 m2 + 2m2 p6 + 3m4 ln h p2(m2 + 11m2) + 6m4 − p4 h h µ2 h m2 + 3m2 − m2 p4 + p2(−m4 + m2 m2 + 36m4) + 18m6 ln h h h µ2 ) 2 6 4 2 2 4 6 − 3p (mh + 10mhm + mhm + 12m ) , (4.44) and

Z 1 ( 2 2 2 ) 2 1 2 2 mh(1 − x) + m x + p (1 − x)x CVV (p ) = 2 dx (2mh + p ) ln 2 . (4.45) 12(4π) 0 µ

The resummed form factor (4.42) is depicted in Figure 4.4, as well as the modiﬁed version (4.39) and their second derivative, which is imported for the spectral analysis in section C.3.

T 2 G VV(p ) 0.8

0.6

0.4

0.2

0.0 p 0 5 10 15 20

Figure 4.4: Resummed form factor for the vector composite operator (left), the resummed form factor with one subtraction (middle), and the ﬁrst derivative of the form factor (left). The momentum p is given in units of the 1 energy scale µ, for the parameter values e = 1, v = 1 µ, λ = 5 .

75 T 2 T 2 T G VV(p )-G VV(0)-p G' VV(0) 100

0 p 5 10 15 20

Figure 4.5: Resummed form factor for the vector composite operator. The momentum p is given in units of the 1 energy scale µ, for the parameter values e = 1, v = 1 µ, λ = 5 .

2 T 2 ∂ G VV p  ∂p 22

0.5

0.4

0.3

0.2

0.1

0.0 p 0 5 10 15 20

Figure 4.6: Second derivative of the form factor. The momentum p is given in units of the energy scale µ, for 1 the parameter values e = 1, v = 1 µ, λ = 5 .

For the longitudinal part of the propagator (see appendix B.7 for details) , we ﬁnd the divergent part

2 4 2 →0 v 3e + λ hV 0(p)V 0(−p)iL = − (4.46) div (4π)2λ and the total ﬁnite correction up to ﬁrst order in ~ is given by

2 2 4 4 mh 4 4 m ! m − m ln 2 + m − 3m ln 2 0 0 L 2 h h µ µ hV (p)V (−p)ifin = v − 2 2 32π mh (4.47)

0 0 which means the longitudinal part of the propagator hVµ(x),Vν (y)i is not propagating.

76 4.2 Spectral properties of the gauge-invariant local operators (Vµ(x),O(x))

In this section, we will calculate the spectral properties associated with the correlation function obtained in the last section. In 4.2.1, we will shortly review the techniques employed in the last chapter to obtain the pole mass, residue and spectral density up to ﬁrst order in ~. In 4.2.2, the spectral properties of the composite operators (Vµ(x),O(x)) are discussed.

4.2.1 Obtaining the spectral function For elementary fields we obtained the spectral density function by comparing the Källén-Lehmannspectral representation for the propagator of a generic field Oe(p) Z ∞ 2 ρ(t) hOe(p)Oe(−p)i = G(p ) = dt 2 , (4.48) 0 t + p where ρ(t) is the spectral density function and G(p2) stands for the resummed propagator 1 G(p2) = . (4.49) p2 + m2 − Π(p2)

For higher-dimensional operators, the resummed propagator acquires an overall (dimensionful) factor identical to the one appearing in its tree level result, as we have seen in section 4.1. We also note that in the case of higher dimensional operators, the spectral representation, eq.(4.48), might require appropriate subtraction terms in order to ensure convergence. A standard way to cure this problem is to subtract from G(p2) the first few terms of its Taylor expansion at p = 0 [178], making the integral more and more convergent. These subtraction terms are directly related to the renormalization of the composite operators, and one can see that the modified Green’s functions for the composite scalar field (4.15) and for the composite vector field (4.39) are in fact subtractions of the Taylor series to first and second order, respectively. In our theory we can make use of the subtracted equations at p = 0 because all fields are massive in the Rξ-gauge, so there are no divergences at zero momentum. Also, we stress that the spectral function ρ(t) is not affected by the subtraction procedure. Moreover, we can see that these subtractions do not have an influence on the (second) derivative of the propagator. For the scalar composite operator

2 2 Z ∞ ∂(GOO(p ) − GOO(0)) ∂GOO(p ) ρ(t) 2 = 2 = − dt 2 2 , (4.50) ∂p ∂p 0 (t + p ) which means that for a positive spectral function the ﬁrst derivative of GOO is strictly negative, as is conﬁrmed in Figure 4.3. For the vector composite operator

2 2 2 0 2 2 Z ∞ ∂ (GVV (p ) − GVV (0) − p GVV (0)) ∂ GVV (p ) ρ(t) 2 2 = 2 2 = 2 dt 2 3 , (4.51) (∂p ) (∂p ) 0 (t + p ) which should be strictly positive for a positive spectral function, as is shown in Figure 4.6. Remember, however, that we can also obtain the spectral function directly, by the methods developed in section (3.3.2).

4.2.2 Spectral properties of the gauge-invariant composite operators Vµ(x) and O(x) For the scalar composite operator O(x) with two-point function given by expression (4.20), we ﬁnd the ﬁrst-order pole mass for our set of parameter values to be

2 2 mh,pole = 0.213472 µ , (4.52) which is exactly equal to the pole mass of the elementary Higgs field correlator. Following the steps from 3.3.1, 2 we find the first-order residue to correct the tree-level result Ztree = v by ∼ 7%:

2 2 2 Z = v (1 + ∂ 2 ΠOO(p ) 2 2 ) = 1.06577v , (4.53) p p =−mh while the first-order spectral function is shown in Figure 4.7. Similarly as for the spectral function of the Higgs 2 2 field in Figure 3.7, one finds a two-particle threshold for Higgs pair production at t = (mh + mh) = 0.8 µ , and a two-photon state starting at t = (m + m)2 = 4 µ2. The difference is that for this gauge-invariant correlation function we no longer have the unphysical Goldstone two-particle state. Due to the absence of this negative

77 contribution, the spectral function is always positive. Therefore, this quantity is suitable for describing a physical Higgs excitation spectrum as opposed to the elementary propagator hhhi.

Finally, it is interesting to note that below the unphysical threshold the elementary correlator displays the same qualitative spectral properties as this gauge-invariant approach. This means that spectral description of the physical Higgs mode could in principle be successfully encoded in the elementary propagator in the unitary gauge, in which ξ → ∞ and the Goldstone bosons are inﬁnitely heavy. We shall make an explicit comparison in section 4.3. 10 3.ρ(t) 6

0 t 0 5 10 15 20 25 30

Figure 4.7: Spectral function for the reduced propagator of the scalar composite operator, hO(p)O(−p)i, with 2 1 t given in units of µ , for the parameter values e = 1, v = 1 µ, λ = 5 .

T For the transverse vector composite operator Vµ (x), with our set of parameters we ﬁnd the ﬁrst-order pole mass

2 2 mpole = 1.05417µ , (4.54) which is – as expected from the Nielsen identities – exactly the same as the pole mass of the transverse photon field correlator (3.75). Furthermore, we find the first-order residue

2 4 2 4 e v T 2 e v Z = (1 + ∂ 2 Π (p ) 2 2 ) = 1.09332 , (4.55) 4 p VV p =−m 4 and the ﬁrst order spectral density for the reduced propagator is displayed in Figure 4.8. Like the photon 2 2 spectral density in Figure 3.6, we ﬁnd a photon-Higgs two-particle state at t = (mh + m) = 2.09 µ , and the spectral density is positive for all values of t.

78 10 3.ρ(t)

0 t 0 10 20 30 40 50

Figure 4.8: Spectral function for reduced transverse propagator of the vector composite operator hV (p)V (−p)iT , 2 1 with t given in units of µ , for the parameter values e = 1, v = 1 µ, λ = 5 .

4.2.3 Pole masses We can explain the fact that the pole mass of the elementary propagator equals that of its gauge-invariant extension in a qualitative way by looking at the definition of a first-order pole mass, eq. (3.59). When calculating a one-loop corrections to the two-point function of the composite operators O(p) and Rµ(p), we find that

2 2 2 2 2 2 2 2 2 Πcomposite(p ) = Πelementary(p ) + Π1−leg(p )(p + m ) + Π0−leg(p )(p + m ) , (4.56)

2 2 where Π1−leg(p ) and Π0−leg(p ) are the composite one-loop contributions to the correction of the composite ﬁeld’s two-point functions, with one external leg and zero external legs, respectively. From this, we see immediately that

2 2 Πcomposite(−m ) = Πelementary(−m ) (4.57) and therefore, up to ﬁrst order in ~, we ﬁnd

2 2 mpole,composite = mpole,elementary (4.58) which means that the elementary operators and their composite extensions share the same mass. This is an important feature, providing an alternative way to the Nielsen identities, to understand why the pole masses of the elementary particles are gauge invariant and not just gauge parameter independent.

4.3 Unitary limit

It is well-known [41] that for the Higgs model, the unitary gauge represents the physical gauge, as it decouples the unphysical fields, i.e. the ghost field and the Goldstone field. The unitary gauge can be formally obtained in the Rξ-gauges by taking ξ → ∞. However, this gauge is non-renormalizable, as one can see by looking at this limit for the tree-level propagator of the photon field

ξ→∞ 1 1 hAµ(p)Av(−p)itree = Pµv + Lµv. (4.59) p2 + m2 m2 Nonetheless, we can approximate the unitary gauge by taking large values of ξ. This is especially interesting when looking at the spectral function of the elementary Higgs field, which is ξ-dependent. In Figure 4.9 one finds the spectral function for ξ = 1000 for small and large ranges of t. In Figure 4.10 we show the spectral function of the scalar composite field O(x) for the same ranges of t. As one can see, the pictures are qualitatively very similar. This means that when approximating the unitary gauge, the spectral function of the gauge-dependent, elementary field h(x) approximates that of its composite, gauge-invariant counterpart.

79 10 3.ρ(t) 10 3.ρ(t)

3.0 3.0 2.5 2.8 2.0 2.6 1.5 2.4

1.0 2.2

0.5 2.0 0.0 t t 0 1 2 3 4 100 200 300 400

Figure 4.9: Spectral function for the reduced elementary propagator hh(p)h(−p)i for small values of t (left) and 2 1 large values of t (right), with t given in units of µ , for ξ = 1000 the parameter values e = 1, v = 1 µ, λ = 5 .

10 3.ρ(t) 10 3.ρ(t)

5.0 6

5 4.5 4 4.0 3 3.5 2

3.0 1

t 0 t 1 2 3 4 0 100 200 300 400

Figure 4.10: Spectral function for the reduced composite propagator hO(p),O(−p)i for small values of t (left) 2 1 and large values of t (right), with t given in units of µ , for the parameter values e = 1, v = 1 µ, λ = 5 .

4.4 Conclusion

In the present chapter, following the local gauge-invariant setup of [174–177], we have evaluated at one-loop 0 0 order the two-point correlation functions hVµ(x)Vν (y)i, hO(x)O(y)i of the two local gauge-invariant operators 0 † † v Vµ(x) = −iϕ (x)Dµϕ(x) + i∂µO(x) and O(x) = ϕ (x)ϕ(x) − 2 in the U(1) Abelian Higgs model quantized in the Rξ gauge. Our results can be summarized as follows:

0 0 • both hVµ(x)Vν (y)i and hO(x)O(y)i do not depend on the gauge parameter ξ, as expected;

0 0 T • the pole masses of hVµ(x)Vν (y)i and hO(x)O(y)i are exactly the same as those of the correlation functions T of the elementary ﬁelds hAµ(x)Aν (y)i and hh(x)h(y)i, respectively, where h· · · iT stands for the transverse component of the corresponding propagator;

0 0 • the spectral densities of the Källén-Lehmannrepresentation of the correlation functions hVµ(x)Vν (y)i and hO(x)O(y)i turn out to be always positive, in contrast to the one associated with the (gauge-dependent) elementary Higgs field. These important features give us a fully gauge-invariant picture in order to describe the spectrum of elementary excitations of the model, i.e. the massive photon and the Higgs mode.

80 Chapter 5

Spectral properties of local BRST invariant composite operators in the SU(2) Yang-Mills-Higgs model

In this chapter we will extend the techniques of chapter3 and4 to the more complex case of SU(2) Higgs model with a single Higgs field in the fundamental representation. This model is equal to the electroweak model from chapter 1.1 with the coupling constant of the hypercharge interaction taken to zero, g0 → 0. As we have seen for the Abelian gauge theory, the direct use of non-gauge invariant fields displays several limitations in the spectral representation, and this will become more severe in the case of a non-Abelian gauge theory. For instance, in the case of the U(1) Higgs model, the transverse component of the Abelian gauge field Aµ is gauge pµpν invariant, so that the two-point correlation function Pµν (p)hAµ(p)Aν (−p)i, where Pµν (p) = (δµν − p2 ) is the transverse projector, turns out to be independent from the gauge parameter ξ. However, this is no more true in the non-Abelian case, where both Higgs and gauge boson two-point functions, i.e. hh(p)h(−p)i and a b a Pµν (p)hAµ(p)Aν (−p)i, where h stands for the Higgs field and Aµ for the gauge boson field, exhibit a strong gauge dependence from ξ. As a consequence, the understanding of the two-point correlation functions of both a Higgs field h and gauge vector boson Aµ in terms of the Källén-Lehmann(KL) spectral representation is completely jeopardized by an unphysical dependence from the gauge parameter ξ, obscuring a direct interpretation of the above mentioned correlation functions in terms of the elementary excitations of the physical spectrum, namely the Higgs and the vector gauge boson particles.

We also note here that from a lattice perspective, it is expected that the spectrum of a gauge (Higgs) theory should be describable in terms of local gauge invariant operator correlation functions, with concrete physical information hiding in the various (positive and gauge invariant) spectral functions, not only pole masses, decay widths, but also transport coeﬃcients at ﬁnite temperature etc. Clearly, such information will not correctly be encoded in gauge variant, non-positive spectral functions.

As we discussed in section 1.1.4, besides the exact BRST invariance, the quantized theory exhibits a global SU(2) symmetry known as custodial symmetry. In this chapter we will show how the local composite BRST invariant operators corresponding to the gauge bosons transform as a triplet under the custodial symmetry, a property which will imply useful relations for their two-point correlation functions.

This chapter is organized as follows. In section 5.1, we give a review of the SU(2) HYM model with a single Higgs field in the fundamental representation, of the gauge fixing procedure and its ensuing BRST invariance. In section 5.2 we calculate the two-point correlation functions of the elementary fields up to one-loop order. In a section 5.3, we define the BRST invariant local composite operators (O(x),Rµ(x)) corresponding to the BRST a invariant extension of (h, Aµ) and calculate their one-loop correlation functions. In section 5.4, we discuss the spectral properties of both elementary and composite operators. In order to give a more general idea of the behavior of the spectral functions, we shall be using two sets of parameters which we shall refer as to Region I and Region II. To some extent, Region I can be associated to the perturbative weak coupling regime, while in Region II we keep the gauge coupling a little bit larger, while decreasing the minimizing value v of the Higgs field. Section 5.5 is devoted to our conclusion. The technical details are all collected in the appendices.

81 5.1 The action and its symmetries

The YM action with a single Higgs ﬁeld in the fundamental representation is given by Z 4 1 a a ij †j ik k λ †i i 1 2 2 S0 = d x F F + (D Φ )(D Φ ) + (Φ Φ − v ) 4 µν µν µ µ 2 2

= SYM + SHiggs (5.1) with

a a abc b c Fµν = ∂µAν − ∂ν Aµ + g AµAν (5.2) and

ij j i i a ij a j ij j † i† i j† a ji a D Φ = ∂µΦ − g(τ ) A Φ , (D Φ ) = ∂µΦ + gΦ (τ ) A , (5.3) µ 2 µ µ 2 µ with the Pauli matrices τ a(a = 1, 2, 3) and the Levi-Civita tensor abc referring to the gauge symmetry group SU(2). The scalar complex ﬁeld Φi(x) is in the fundamental representation of SU(2), i.e. i, j = 1, 2. Thus, Φ is an SU(2)-doublet of complex scalar ﬁelds that can be written as

+ 1 φ 1 φ1 + iφ2 Φ = √ 0 = √ . (5.4) 2 φ 2 φ3 + iφ4

The conﬁguration which minimizes the Higgs potential in the expression (5.1) is

v hΦi = √1 (5.5) 2 0 and we write down Φ(x) as an expansion around the conﬁguration (5.5), so that

1 v + h + iρ Φ = √ 3 , (5.6) 2 iρ1 − ρ2 where h is the Higgs ﬁeld and ρa, a = 1, 2, 3, the would-be Goldstone bosons. We can use the matrix notation1:

1 1 Φ = √ ((v + h)1 + iρaτ a) · , (5.7) 2 0 so that the second term in eq. (5.1) becomes " ij j † ik k 1 a a ig a a (D Φ ) D Φ = (1, 0) · ∂µh · 1 − i∂µρ τ + τ A (v + h)1 µ µ 2 2 µ # " b b c c ig −iρ τ × ∂µh · 1 + i∂µρ τ − ((v + h)1 2 # 1 −iρdτ d τ cAc · µ 0

= L˜0 + L˜1 + L˜2, (5.8) with L˜i the ith term in powers of Aµ:

1 2 a a L˜0 = (∂µh) + ∂µρ ∂µρ , 2 1 a a a a a a abc a b c L˜1 = − gvA ∂µρ − gA ρ ∂µh + gA (∂µρ )h + g ∂µρ ρ A , 2 µ µ µ µ 2 g a a 2 b b L˜2 = A A (v + h) + ρ ρ , (5.9) 8 µ µ 1This is of course possible thanks to the fact that Φ counts 3 Goldstone modes and that SU(2) has three generators. This “numerology” is essentially what leads to a large custodial symmetry in the SU(2) case.

82 and we have the full action Z ( 4 1 1 a a 1 2 2 a a 2 a a a a a a a a S0 = d x F F + v g A A + (∂µh) + ∂µρ ∂µρ − gvA ∂µρ + gA ρ ∂µh − gA (∂µρ )h 2 2 µν µν 4 µ µ µ µ µ 2 abc a b c g a a 2 b b 2 2 2 a a − g ∂µρ ρ A + A A 2vh + h + ρ ρ + λv h + λvh(h + ρ ρ ) µ 4 µ µ ) λ + (h2 + ρaρa)2 . (5.10) 4

a One sees that both gauge field Aµ and Higgs field h have acquired a mass given, respectively, by 1 m2 = g2v2, m2 = λv2 . (5.11) 4 h 5.1.1 Gauge fixing and BRST symmetry The action (5.1) is invariant under the local ω-parametrized gauge transformations ig ig δAa = −Dabωb, δΦ = − ωaτ aΦ, δΦ† = ωaΦ†τ a, (5.12) µ µ 2 2 which, when written in terms of the fields (h, ρa), become g g δh = ωaρa, δρa = − (ωa(v + h)1 − abcωbρc). (5.13) 2 2

As done in the U(1) case, we shall be using the Rξ-gauge. We add thus need the gauge ﬁxing term Z 4 n ξ a a a a a o Sgf = s d x −i c¯ b +c ¯ (∂µA − ξmρ ) 2 µ Z 1 4 n a a a a a ab b a a = d x ξb b + 2ib ∂µA + 2¯c ∂µD c − 2iξmb ρ 2 µ µ o − 2ξc¯amca − gξc¯amhca − ξgabcc¯acbρc , (5.14) so that the gauge ﬁxed action Sfull = S0 + Sgf , namely

Z ( 4 1 1 a a 1 2 2 a a Sfull = d x F F + v g A A 2 2 µν µν 4 µ µ 2 a a a a a a a a + (∂µh) + ∂µρ ∂µρ − gvAµ∂µρ + gAµρ ∂µh − gAµ(∂µρ )h 2 abc a b c g a a 2 b b 2 2 − g ∂µρ ρ A + A A 2vh + h + ρ ρ + λv h µ 4 µ µ 2 a a λ 2 a a 2 a a a a a ab b a a + λvh(h + ρ ρ ) + (h + ρ ρ ) + ξb b + 2ib ∂µA + 2¯c ∂µD c − 2iξmb ρ 4 µ µ ) − 2ξc¯amca − gξc¯amhca − ξgabcc¯acbρc (5.15) turns out to be left invariant by the BRST transformations g g sAa = −Dabcb, sh = caρa, sρa = − (ca(v + h)1 − abccbρc) µ µ 2 2 1 sca = gabccbcc, sc¯a = iba, sba = 0 , (5.16) 2

sSfull = 0 . (5.17)

83 5.1.2 Custodial symmetry As already mentioned, apart from the BRST symmetry, there is an extra global symmetry, which we shall refer to as the custodial symmetry:

a abc b c δAµ = β Aµ, δρa = abcβbρc, δca = abcβbcc, δca = abcβbcc, δba = abcβbbc, δh = 0 , (5.18)

a a where β is a constant parameter, ∂µβ = 0,

δSfull = 0 . (5.19)

a a a a a One notices that all ﬁelds carrying the index a = 1, 2, 3, i.e. (Aµ, b , c , c¯ , ρ ), undergo a global transformation in the adjoint representation of SU(2). The origin of this symmetry is an SU(2)gauge ×SU(2)global symmetry of the action in the unbroken phase, see section 1.1.4. The exception is the Higgs ﬁeld h, which is left invariant, i.e. it is a singlet. As we shall see in the following, this additional global symmetry will provide useful relationships for the two-point correlation functions of the BRST invariant composite operators.

5.2 One-loop evaluation of the correlation function of the elementary ﬁelds

a For the elementary ﬁelds h(x) and Aµ, the correlation functions are calculated in Appendices C.2.1 and C.2.2. For the Higgs ﬁeld, for the one-loop propagator we get

1 1 2 2 hh(p)h(−p)i = 2 2 + 2 2 2 Πhh(p ) + O(~ ) (5.20) p + mh (p + mh) where

2 Z 1 ( 2 2 2 3g 2 2 m ξ 2 2 2 m Πhh(p ) = 2 dx 2ξ mh + p ln 2 − 2ξmh + 2 6m − p ln 2 8(4π) 0 µ µ p4 m2 + p2(1 − x)x p4 m4 m2ξ + p2(1 − x)x − (12m2 + + 4p2) ln + − h ln m2 µ2 m2 m2 µ2 ) m4 m2 m2 + p2(1 − x)x − 12m2 − 2ξp2 + 2p2 − h −2 ln h + 3 ln h + 2 . (5.21) m2 µ2 µ2

2 Before trying to resum the self-energy Πhh(p ), we notice that this resummation is tacitly assuming that the second term in (5.20) is much smaller than the ﬁrst term. However, we see that eq. (5.20) contains terms of the 2 2 p4 m +p (1−x)x 2 order of 2 2 2 ln 2 which cannot be resummed for big values of p . We therefore proceed as in (p +mh) µ eq. (3.51)-(3.53) and use the identity

4 2 2 2 4 2 2 p = (p + mh) − mh − 2p mh (5.22) to rewrite

4 2 2 2 2 4 2 2 2 2 p p x(1 − x) + m p x(1 − x) + m (mh + 2p mh) p x(1 − x) + m 2 2 2 ln 2 = ln 2 − 2 2 2 ln 2 . (5.23) (p + mh) µ µ (p + mh) µ

The term which has been underlined in eq. (5.23) can be safely resummed, as it decays fast enough for large values of p2. We thence rewrite

2 ˆ 2 Πhh(p ) Πhh(p ) 2 2 2 2 = 2 2 2 + Chh(p ) , (5.24) (p + mh) (p + mh)

84 with

2 Z 1 ( 2 2 ˆ 2 3g 2 2 m ξ 2 2 2 m Πhh(p ) = 2 dx 2ξ mh + p ln 2 − 2ξmh + 2 6m − p ln 2 8(4π) 0 µ µ (m4 + 2p2m2 ) m2 + p2(1 − x)x (2m4 + 2p2m2 ) m2ξ + p2(1 − x)x − (12m2 − h h + 4p2) ln − h h ln m2 µ2 m2 µ2 ) m4 m2 m2 + p2(1 − x)x − 12m2 − 2ξp2 + 2p2 − h − 2 ln h + 3 ln h + 2 (5.25) m2 µ2 µ2 and 2 Z 1 2 2 2 2 2 3g p x(1 − x) + m p x(1 − x) + ξm Chh(p ) = − 2 2 dx ln 2 − ln 2 . (5.26) 8m (4π) 0 µ µ Thus, for the one-loop Higgs propagator, we get

1 2 2 hh(p)h(−p)i = + Chh(p ) + O( ) . (5.27) 2 2 ˆ 2 ~ p + mh − Πhh(p ) For the gauge ﬁeld, we split the two-point function into transverse and longitudinal parts in the usual way

a b a b T a b L hAµ(p)Aν (−p)i = hAµ(p)Aν (−p)i Pµν (p) + hAµ(p)Aν (−p)i Lµν (p), (5.28) where we have introduced the transverse and longitudinal projectors, given respectively by p p p p P (p) = δ − µ ν , L (p) = µ ν . (5.29) µν µν p2 µν p2 We ﬁnd ab ab a b T δ δ 2 2 hAµ(p)Aν (−p)i = + ΠAAT (p ) + O(~ ), (5.30) p2 + m2 (p2 + m2)2

ab 2 Z 1 ( 2 2 2 δ g 4 2 4 mh 6 2 2 m ξ ΠAAT (p ) = − 2 4 2 2 dx − 27m p mh ln 2 − 27m ξp mh ln 2 36(4π) m p mh 0 µ µ m2 + 3m4m4 m2 − m2 + 2p2 ln h + 27m4p2m2 m2 + m2ξ h h µ2 h h m2ξ − 3m4ξm2 2m4(ξ − 1) + m2(4ξ + 7)p2 + 2(ξ + 9)p4 ln h µ2 m2 + 3m4 ln −m2m4 + m2 m4(2ξ − 1) + m2(4ξ + 45)p2 + 2(ξ + 9)p4 − 54m4p2 µ2 h h 4 2 4 2 4 2 2 4 + m 6m mh + mh 3m (2(ξ − 2)ξ + 1) + 3m (ξ − 1)(4ξ − 1)p + 2(3ξ(ξ + 4) − 17)p

6 4 2 − 3mh + 54m p h p2(1 − x)x + m2 (1 − x) + m2x − 3m2 m4 2p2 m2 − 5m2 + m2 − m2 2 + p4 ln h h h h µ2 2 2 2 2 p (1 − x)x + ξm (1 − x) + m x − 2 m2 + p2 m4(ξ − 1)2 + 2m2(ξ − 5)p2 + p4 ln µ2 p2(1 − x)x + ξm2 + p2 p4 − m4 4m2ξ + p2 ln µ2 ) p2(1 − x)x + m2 i + p2 4m2 + p2 12m4 − 20m2p2 + p4 ln . (5.31) µ2

p4 m2+p2(1−x)x p6 m2+p2(1−x)x We see that (5.30) contains again terms of the order (p2+m2)2 ln µ2 and (p2+m2)2 ln µ2 , which cannot be resummed for big values of p2. We use

p4 p2x(1 − x) + m2 p2x(1 − x) + m2 (m4 + 2p2m2) p2x(1 − x) + m2 ln = ln − ln (5.32) (p2 + m2)2 µ2 µ2 (p2 + m2)2 µ2

85 and p6 p2x(1 − x) + m2 p2x(1 − x) + m2 ln = (p2 − 2m2) ln (p2 + m2)2 µ2 µ2 2m6 + 3p2m4 p2x(1 − x) + m2 + ln . (5.33) (p2 + m2)2 µ2

The underlined terms in (5.32) and (5.33) can be safely resummed. We rewrite

2 ˆ 2 ΠAAT (p ) ΠAAT (p ) 2 2 2 2 = 2 2 2 + CAAT (p ), (5.34) (p + m ) (p + mh) with ( δabg2 Z 1 m2 m2ξ ˆ 2 4 2 4 h 6 2 2 ΠAAT (p ) = − 2 4 2 2 dx − 27m p mh ln 2 − 27m ξp mh ln 2 36(4π) m p mh 0 µ µ m2 + 3m4m4 m2 − m2 + 2p2 ln h + 27m4p2m2 m2 + m2ξ h h µ2 h h m2ξ − 3m4ξm2 2m4(ξ − 1) + m2(4ξ + 7)p2 + 2(ξ + 9)p4 ln h µ2 m2 + 3m4 −m2m4 + m2 m4(2ξ − 1) + m2(4ξ + 45)p2 + 2(ξ + 9)p4 − 54m4p2 ln h h µ2 4 2 4 2 4 2 2 4 6 4 2 + m 6m mh + mh 3m (2(ξ − 2)ξ + 1) + 3m (ξ − 1)(4ξ − 1)p + 2(3ξ(ξ + 4) − 17)p − 3mh + 54m p h p2(1 − x)x + (1 − x)m2 + m2x − 3m2 m4 2p2 m2 − 5m2 + m2 − m2 2 + p4 ln h h h h µ2 2 2 2 2 p (1 − x)x + ξm (1 − x) + m x − 2m4(ξ − 1)2 m2 + p2 ln µ2 p2(1 − x)x + ξm2 + −2m4(4ξ − 1)p2 m2 + p2 ln µ2 ) p2(1 − x)x + m2 i + (66m6p2 − 33m4p4) ln . (5.35) µ2 and

ab 2 Z 1 ( 2 δ g 2 2 CAAT (p ) = 2 4 dx (−4m (ξ − 5) − 2p ) 12(4π) m 0 p2(1 − x)x + ξm2(1 − x) + m2x × ln µ2 p2(1 − x)x + ξm2 + 4 ξm2 + p2 − 2m2 ln µ2 ) p2(1 − x)x + m2 + (−18m2 + p2) ln . (5.36) µ2

Finally ! a b T ab 1 2 2 hA (p)A (−p)i = δ + CAAT (p ) + O( ) . (5.37) µ ν 2 2 2 ~ p + m − Πˆ AAT (p )

5.3 One-loop evaluation of the correlation function of the local BRST invariant composite operators 5.3.1 Correlation function of the scalar BRST invariant composite operator O(x) The BRST invariant local scalar composite operator O(x) is given by

v2 O(x) = Φ†Φ − , s O(x) = 0 , (5.38) 2

86 which, after using the expansion (5.7), becomes

1h 1 i v2 O(x) = 1 0 ((v + h)1 − iρaτ a))((v + h)1 + iρbτ b)) − 2 0 2 1 = h2(x) + 2vh(x) + ρa(x)ρa(x) , (5.39) 2 so that 1 1 hO(x)O(y)i = v2 hh(x)h(y)i + v hh(x)ρb(y)ρb(y)i + v hh(x)h(y)2i + hh(x)2ρb(y)ρb(y)i + hh(x)2h(y)2i 4 4 1 + hρa(x)ρa(x)ρb(y)ρb(y)i . (5.40) 4 Looking at the tree level expression of eq. (5.40), one easily obtains

2 2 1 hO(p)O(−p)itree = v hh(p)h(−p)itree = v 2 2 , (5.41) p + mh showing that the BRST invariant scalar operator O(x) is directly linked to the Higgs propagator. Concerning now the one-loop calculation of expression (5.40), after evaluating each term, see Appendix C.3 for details, we find that the two-point correlation function of the scalar composite operator O(x) develops a geometric series in the same way as the elementary field h(x). This allows us the make a resummed approximation. Using dimensional regularization in the MSbar-scheme, we find

2 2 2 v v 2 2 hO(p)O(−p)i (p ) = 2 2 + 2 2 2 ΠOO(p ) + O(~ ), (5.42) p + mh (p + mh)

Z 1 ( 2 2 1 2 4 2 2 2 2 m ΠOO(p ) = 2 2 2 dx − 24mhm − 6m p mh + 6m ln 2 32v π mh 0 µ 2 2 2 m + p x(1 − x) − m2 p2 − 2m2 ln h h h µ2 m2 + p2x(1 − x) − 3m2 12m4 + 4m2p2 + p4 ln h µ2 ) m2 + 6p2 m4 + m2 m2 + 2m4 − 6m4 p2 ln h . (5.43) h h h µ2

p4 2 Since (5.42) contains terms of the order of (p2+m2)2 ln(p ), we follow the steps (3.51)-(3.53) to ﬁnd the resummed correlation function in the one-loop approximation

2 2 v 2 GOO(p ) = + COO(p ) (5.44) 2 2 ˆ 2 p + mh − ΠOO(p ) with Z 1 ( 2 ˆ 2 1 2 4 2 2 2 2 m ΠOO(p ) = 2 2 2 dx − 24mhm − 6m p mh + 6m ln 2 32v π mh 0 µ m2 + p2x(1 − x) − m2 (3m4 − 6m2 p2) ln h h h h µ2 m2 + p2x(1 − x) − 3m2 12m4 + 4m2p2 − m4 − 2p2m2 ln h h h µ2 ) m2 + 6p2 m4 + m2 m2 + 2m4 − 6m4 p2 ln h (5.45) h h h µ2 and Z 1 ( 2 2 2 2 ) 2 1 mh + p x(1 − x) m + p x(1 − x) COO(p ) = − 2 dx ln 2 + 3 ln 2 . (5.46) 32π 0 µ µ

Expressions (5.44) and (5.46) show that, as expected, and unlike the Higgs propagator, eq. (5.27), the correlator hO(p)O(−p)i is independent from the gauge parameter ξ.

87 5.3.2 A little digression on the unitary gauge Of course, since the composite operator O(x) is BRST invariant, any choice for the gauge parameter ξ should 2 give the same expression for the correlation function GOO(p ). One convenient choice is the so-called unitary gauge, which is formally attained by taking ξ → ∞ at the end of the calculation. Though, we take here a diﬀerent 2 route and perform the same calculation done before for GOO(p ) by employing the tree level propagators and other Feynman rules which follow by taking the limit ξ → ∞ at the beginning. In doing this, for the tree level propagators one ﬁnds

ab ab a b δ ab 1 δ pµpv hA (p)A (−p)i = Pµv(p) + δ Lµv(p) = δµν + , µ v p2 + m2 m2 p2 + m2 m2 1 hh(p)h(−p)i = 2 2 (5.47) p + mh with all other propagators, i.e. the Goldstone and Faddeev-Popov ghost propagators, vanishing. Then, eq. (5.40) simpliﬁes to 1 hO(x)O(y)i = v2 hh(x)h(y)i + v hh(x)h(y)2i + hh(x)2h(y)2i , (5.48) unitary unitary unitary 4 unitary with the contributing diagrams shown in Figure 5.1. Making use of the dimensional regularization in the MSbar-scheme and switching to momentum space, we get

Z 1 ( 2 2 2 3 1 4 2 2 4 4 mh 4 2 2 m v hh(p)h(−p)iunitary = 2 dx 2mh + 12m p + 2p + 2mh ln 2 + 2 6m − m p ln 2 32π 0 µ µ p2(1 − x)x + m2 p2(1 − x)x + m2 − 3m4 ln h − 12m4 + 4m2p2 + p4 ln h µ2 µ2 ) 4 2 2 2 1 − 2mh + 2m p − 6m 2 2 2 , (5.49) (mh + p ) Z 1 ( 2 2 2 3 12 4 4 m 4 mh v hh(p)h(−p) iunitary = 2 2 dx m − 6m ln 2 − mh ln 2 16π mh 0 µ µ 2 2 ) 4 mh + p (1 − x)x 4 4 1 + mh ln 2 + mh + 2m 2 2 , (5.50) µ (mh + p ) Z 1 ( 2 2 ) 1 2 2 1 1 1 mh + p (1 − x)x hh(p) h(p) iunitary = 2 dx − ln 2 . (5.51) 4 16π 0 2 µ

Inserting now the unity

2 2 2 2 2 2 2 2 2 1 = (p + mh)/(p + mh) = ((p + mh)/(p + mh)) , (5.52) we find indeed that hO(x)O(y)iunitary = hO(x)O(y)i, showing that the same expression has been re-obtained by starting directly from the tree level propagators of the unitary gauge, ξ → ∞, despite the fact that this gauge is known to be non-renormalizable. Though, at one-loop order, a simple explanation can be found for the previous result, which is due in part to the BRST invariant nature of the correlation function hO(x)O(y)i and to the fact that, at one-loop order, the handling of the overlapping divergences is not required. From two-loop onward these divergences will show up, requiring a fully renormalizable setup. More specifically, in this case, one would need to keep ξ finite and use all Feynman rules of the Rξ-gauge, while taking ξ → ∞ only at the end. 2 Nevertheless, having checked that the one-loop result for GOO(p ) is the same by using both procedures, in the next section, we will use the same simplifying second trick to evaluate the two-point function of the vectorial composite operator at one-loop order.

88 Figure 5.1: One-loop contributions for the propagator hO(x)O(y)i in the unitary gauge: hh(x)h(y)i (first line),hh(x)h(y)2i (second line) and hh(x)2h(y)2i (third line) . Wavy lines represent the gauge field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field. The • indicates the insertion of a composite operator. .

5.3.3 Vectorial composite operators We identify three gauge invariant vector composite operators, following the deﬁnitions of ’t Hooft in [43], namely

3 † Oµ = iφ Dµφ, 0 1 O+ = φT D φ, µ −1 0 µ − + † Oµ = (Oµ ) . (5.53)

3 + The gauge invariance of Oµ is apparent. For Oµ , we can show the gauge invariance by using the following 2 × 2 matrix representation of a generic SU(2) transformation,

a −b? U = (5.54) b a? with determinant |a|2 + |b|2 = 1. Thus, we ﬁnd that under a SU(2) transformation

0 1 O+ → (Uφ)T D (Uφ) µ −1 0 µ 0 1 = φT U T UD φ −1 0 µ 0 1 = φT D φ = O+, (5.55) −1 0 µ µ

+ − which shows the gauge invariance of Oµ and, subsequently, of Oµ . After using the expansion (5.7), the ﬁrst composite operator reads

3 † Oµ = iφ Dµφ

† 1 † a a = iφ ∂µφ + gφ τ A φ 2 µ " i 3 3 a a 1 2 2 1 i 2 3 1 2 2 1 = (v + h)∂µh + i(v + h)∂µρ − iρ ∂µh + ρ ∂µρ + iρ ∂µρ − iρ ∂µρ − g(v + h) A + ig(v + h)(A ρ − A ρ ) 2 2 µ µ µ # i + gρaA3 ρa − igρ3Ab ρb 2 µ µ " 1 3 3 1 2 2 1 1 2 3 1 2 2 1 = − (v + h)∂µρ + ρ ∂µh − ρ ∂µρ + ρ ∂µρ + g(v + h) A − g(v + h)(A ρ − A ρ ) 2 2 µ µ µ # 1 a 3 a 3 b b i − gρ A ρ + gρ A ρ + ∂µO, (5.56) 2 µ µ 2

89 i and since the last term, i.e. 2 ∂µO, is BRST invariant, the sum of the others terms has to be BRST invariant a too. Therefore, we can introduce the following three “reduced” vector composite operators Rµ with a = 1, 2, 3 : i R1 = O+ − O− , µ 2 µ µ 1 R2 = O+ + O− , µ 2 µ µ 3 3 i R = O − ∂µOµ, (5.57) µ µ 2 so that " a 1 a a abc b c 1 2 a abc b c R = − (v + h)∂µρ + ρ ∂µh − ε ρ ∂µρ + g(v + h) A − g(v + h)ε (ρ A ) µ 2 2 µ µ # 1 − gAa ρmρm + gρaAmρm , (5.58) 2 µ µ with a sRµ(x) = 0 . (5.59) a Remarkably, the BRST invariant operators Rµ transform like a triplet under the custodial symmetry (5.18), namely

a abc b c δRµ = ε β Rµ , (5.60) and since the only rank two invariant tensor is δab, we can write, moving to momentum space,

a b ab 2 2 1 a a hR (p) R (−p)i = δ Rµν (p ) → Rµν (p ) = hR (p) R (−p)i , (5.61) µ ν 3 µ ν as well as

2 2 2 Rµν (p ) = R(p )Pµν (p) + L(p )Lµν (p), (5.62) so that in d dimensions,

1 P (p) R(p2) = µν hRa (p) Ra(−p)i , (5.63) 3 (d − 1) µ ν and

2 1 a a L(p ) = Lµν (p) hR (p) R (−p)i . (5.64) 3 µ ν One recognizes that eqs.(5.61)-(5.64) display exactly the same structure of the gauge vector boson correlation a function hAµ(p)Aν (−p)i. a b In the Rξ-gauge, the non-vanishing contributions, up to ﬁrst order in ~, to the correlation function hRµ(p) Rν (−p)i are

a a 1 2 4 a a a a 1 a a hR (p) R (−p)i = g v hA (p) A (−p)i − h(ρ ∂µh)(p)(∂ν ρ h)(−p)i + pµpν h(ρ h)(p)(ρ h)(−p)i µ ν 16 µ ν 4 1 a b a b 1 a b a b + ∂µ∂ν h(ρ ρ )(p)(ρ ρ )(−p)i − h(ρ ∂µρ )(p)(∂ν ρ ρ )(−p)i 8 2 1 2 3 a a i 3 a a + g v hA (p)(A h)(−p)i − gv pν hA (p) ρ (−p)i 4 µ ν 4 µ 1 2 a a 1 2 2 a b a b + v pµpν hρ (x)ρ (y)i + g v h(ρ A )(p)(ρ A )(−p)i 4 6 µ ν 1 1 − g2v2h(ρaρaAb )(p) Ab(−p)i + g2v2h(h2Aa )(p) Aa(−p)i 24 µ 8 µ ν 1 2 2 a a i 2 a a + g v h(hA )(p)(hA )(−p)i + v gpµh(hρ )(p) A (−p)i 4 µ ν 4 ν 1 2 a a i 2 a a + gv h(∂µhρ )(p) A (−p)i + gv pµhρ (p)(hA )(−p)i 2 ν 2 ν 2 1 2 abc a b c igv abc a b c − gv ε hA (p)(ρ ∂µρ )(−p)i − ε pµhρ (p)(ρ A )(−p)i 4 2 ν

90 1 a a a a + vpµpν h(hρ )(p) ρ (−p)i − ivpν h(∂µhρ )(p) ρ (−p)i (5.65) 2

∂ ∂ 2 where we have used the notation ∂µ = ∂xµ and ∂ν = ∂yν . The first term in expression (5.65) is the gauge a a a field propagator hAµ(p)Aµ(−p)i, which means that Rµ can be thought as a kind of BRST invariant extension a of the elementary gauge field Aµ. At tree-level, we find in fact

a a 1 2 4 a a 1 2 a a hR (x) R (y)i = g v hA (p),A (−p)i + v ∂µ∂ν hρ (p), ρ (−p)i µ ν tree 16 µ ν 4 3 2 4 1 3 2 = g v Pµν (p) + v Lµν (p), (5.66) 16 p2 + m2 4

3 2 where we can see that, apart from the constant factor 4 v appearing in the longitudinal sector, the transverse component reproduces exactly the transverse gauge tree-level propagator. a a Since the correlation function hRµ(p),Rν (−p)i is independent from the gauge parameter ξ, due to the a BRST invariant nature of the operator Rµ(x), we shall proceed as in the previous example by making use of hRa p Ra −p i hRa p ,Ra −p i µ( ) ν ( ) unitary = µ( ) ν ( ) and evaluating the correlator at the one-loop order with the propagators given in (5.47), so that

a a 1 2 4 a a 1 2 3 a a hR (p) R (−p)i = g v hA (p) A (−p)iunitary + g v hA (p)(A h)(−p)iunitary µ ν 16 µ ν 4 µ ν 1 2 2 2 a a 1 2 2 a a + g v h(h A )(p) A (−p)iunitary + g v h(hA )(p)(hA )(−p)iunitary (5.67) 8 µ ν 4 µ ν with the contributing diagrams shown in Figure 5.2. Using dimensional regularization in the MSbar-scheme with (d = 4 − ) and switching to momentum space, we ﬁnd

4 4 Z 1 ( 1 2 4 a a g v 1 4 4 2 6 4 2 2 4 6 8 g v hAµ(p) Aν (−p)iunitary = 2 dx − 4 2 9m mh + mh −9m − 83m p − 14m p + p + 54m 16 32(4π) 0 6m mh m2 m2 + h −m2 + m2 + 7p2 ln h 2p2 h µ2 2 1 4 4 2 6 4 2 2 4 6 6 2 m + 2 2 2 m mh − mh m + 47m p + 16m p − 2p + 54m p ln 2 2m p mh µ 1 p2(1 − x)x + (1 − x)m2 + m2x + −2m2 m2 − p2 + m4 + m4 − 10m2p2 + p4 ln h 2p2 h h µ2 1 m2 + p2(1 − x)x + 4m2 + p2 12m4 − 20m2p2 + p4 ln 2m4 µ2 1 4 6 4 6 4 2 + 4 2 2 3m mh − 3mh 2m + 9m p 6m p mh ) P (p) + m2 3m8 − 9m6p2 − 2m4p4 − 26m2p6 − 2p8 − 54m8p2 µν h (p2 + m2)2

4 4 Z 1 ( g v 1 3 2 2 2 4 4 + 2 4 dx 2 mh 3m + p − 3mh − 18m 32(4π) m 0 mh 2 2 3m 2 2 2 4 2 2 m − 2 2 mh p − m + mh − 18m p ln 2 2p mh µ 3m2 m2 + h m2 − m2 + 5p2 ln h 2p2 h µ2 2 2 2 3 2 2 2 2 p (1 − x)x + (1 − x)mh + m x − (mh − m) + p (mh + m) + p ln 2p2 µ2 ) 3 4 2 2 2 4 2 2 6 4 2 − 2 2 mh 5p − 2m + mh m − m p + mh + 6m p Lµν (p), (5.68) 2p mh ( 1 1 Z 1 1 m2 g2v3 hAa p Aah −p i dx m2 p2 − m2 m4 m4 µ( )( )( ) unitary = 2 2 h 9 + 12 h + 54 4 16π 0 mh

2The derivative here is not expressed in momentum space because for composite operators, the derivative will bring down diﬀerent momenta depending on the conﬁguration of the Feynman diagram.

91 m2m2 m2 − h −m2 + m2 + 10p2 ln h 2p2 h µ2 4 2 m 2 2 2 4 2 2 m − 2 2 mh p − m + mh + 54m p ln 2 2p mh µ m2 p2(1 − x)x + (1 − x)m2 + xm2 − 2p2 m2 − 5m2 + m2 − m2 2 + p4 ln h 2p2 h h µ2 2 ) m 4 2 2 2 4 2 2 4 6 4 2 Pµν (p) + 2 2 6mh m + 6p + mh −3m + 9m p + 2p − 3mh + 54m p 2 2 6p mh (m + p ) Z 1 ( 1 1 3 2 2 2 4 4 + 2 dx 2 −mh 3m + p + mh + 18m 16π 0 mh 2 2 3m 2 2 2 4 2 2 m + 2 2 mh p − m + mh − 18m p ln 2 2p mh µ 3m2 m2 − h m2 − m2 + 3p2 ln h 2p2 h µ2 2 2 2 3 2 2 2 2 2 p (1 − x)x + (1 − x)mh + xm + 2 2 mh (m − mh) + p (mh + m) + p ln 2 2p mh µ 3 3 2 2 2 2 2 4 4 + 2 2 mh − m mh + p −m mh + 3mh + 6m 2p mh ) Lµν (p), (5.69)

2 2 Z 1 ( 2 ) 1 2 2 2 a a 3mhm 2 mh 1 g v h(h Aµ)(p) Aν (−p)iunitary = − 2 dx − ln 2 + 1 2 2 Pµν (p) 8 32π 0 µ p + m 1 + L (p) , (5.70) m2 µν

Z 1 ( 1 2 2 a a 1 1 2 2 2 g v h(hAµ)(p)(hAν )(−p)iunitary = 2 dx (−3mh + 9m − p ) 4 32π 0 m2 m2 m2 m2 + h m2 + p2 − m2 ln h + m2 − m2 + p2 ln 2p2 h µ2 2p2 h µ2 1 p2(1 − x)x + (1 − x)m2 + xm2 + 2p2 m2 − 5m2 + m2 − m2 2 + p4 ln h 2p2 h h µ2 ) 1 2 2 2 2 2 2 4 + 3 m − m − 9p m + m − 2p Pµν (p) 6p2 h h Z 1 ( 3 1 2 2 2 + 2 dx −mh + 3m + p 32π 0 m2 m2 m2 m2 + −m2 + m2 − p2 ln + h m2 − m2 + 3p2 ln h 2p2 h µ2 2p2 h µ2 2 2 2 1 2 2 2 2 p x(1 − x) + (1 − x)mh + xm − (m − mh) + p (mh + m) + p ln 2p2 µ2 ) 1 2 2 2 2 2 2 2 − m − m + 3p m − p m Lµν (p) . (5.71) 2p2 h h

Using the unity (5.52), we ﬁnd that the transverse part of the propagator is given by ! 2 1 2 4 1 1 2 2 2 R(p ) = g v + ΠR(p ) + Πdiv(p ) + O(~ ) (5.72) 16 p2 + m2 (p2 + m2)2 where the divergent part is, see also the comments in the Appendix C.4,

2 2 4 4 2 2 2 2 2 6 4 2 2 g h p 9m 9m p h p h p 23p 7p Πdiv(p ) = − + + + + − + + , (5.73) π2 32m4 16h2 8h2 8m2 16 48m4 96m2 8

92 while for the ﬁnite part we get

Z 1 ( 4 2 2 3 4 4 4 p mh 2 2 2 ΠR(p ) = 2 2 4 2 dx 6m mh + 3m − 9mh + 35m + 4p 36π g v mh 0 3 m2 + p2 m4m2 + 10m2m4 + m6 + 36m6 + m4 −p2 m2 + 11m2 − 6m4 + p4 ln h h h h h h µ2 2 2 2 2 p (1 − x)x + (1 − x)m + xm + m2 2p4 m2 − 5m2 + m2 − m2 p2 + p6 ln h h h h µ2 p2(1 − x)x + m2 + m2 48m6 − 68m4p2 − 16m2p4 + p6 ln h µ2 ) m2 + m2 m2 −48m4 − 17m2p2 + 3p4 + p2m4 − 54 m6 + 2m4p2 ln . (5.74) h h µ2

p4 2 p6 2 Since (5.74) contains terms of the order of (p2+m2)2 ln(p ) and (p2+m2)2 ln(p ), we follow the steps (3.51)-(3.53) to ﬁnd the resummed propagator in the one-loop approximation, namely

2 1 2 4 1 2 GR(p ) = g v + CR(p ) (5.75) 2 2 ˆ 2 16 p + mh − ΠR(p ) with ( Z 1 p4m2 ˆ 2 3 4 4 4 h 2 2 2 ΠR(p ) = 2 2 4 2 dx 6m mh + 3m − 9mh + 35m + 4p 36π g v mh 0 3 m2 + p2 m4m2 + 10m2m4 + m6 + 36m6 + m4 −p2 m2 + 11m2 − 6m4 + p4 ln h h h h h h µ2 ! x m2 − p2(x − 1) − (x − 1)m2 + m2 m4 p2 − 2m2 m4 − 6m2 m2p2 + 12m6 + 24m4p2 ln h h h h h µ2 m2 − p2(x − 1)x + 33m2 2m6 − m4p2 ln h µ2 ) m2 + m2 m2 −48m4 − 17m2p2 + 3p4 + p2m4 − 54 m6 + 2m4p2 ln (5.76) h h µ2 and

Z 1 ( 2 2 2 1 2 2 p (1 − x)x + m CR(p ) = 2 dx −18m + p ln 2 12(4π) 0 µ ) p2x(1 − x) + (1 − x)m2 + xm2 + (2 m2 − 6m2 + p2) ln h . (5.77) h µ2

Looking now at the longitudinal part L(p2), it turns out to be

2 2 4 4 mh 4 4 m ! m − 3m ln 2 + 9m − 27m ln 2 4 2 1 2 1 h h µ µ 1 2 m L(p ) = v − 2 2 − mh − 9 2 . (5.78) 4 (4π) 2mh mh

As it happens in the tree-level case, expression (5.78) is independent from the momentum p2, meaning that it does not correspond to the propagation of some physical mode, a feature which is expected to persist at higher orders.

93 a a Figure 5.2: One-loop contributions for the correlation function hOOi in the unitary gauge: hAµ(x)Aν (y)i (first 2 a a a two lines), hA(x)(Ah)(y)i (third line), hh(x) Aµi (fourth line) and h(Aµh)(x)(Aν h)(y)i . Wavy lines represent the gauge field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field. The • indicates the insertion of a composite operator. .

5.4 Spectral properties

In this section, we will calculate the spectral properties associated with the correlation function obtained in the last section. For the employed techniques in obtaining the pole mass, residue and spectral density up to ﬁrst order we refer to section 4.2.1. In subsection 5.4.1, we analyze the spectral properties of the elementary ﬁelds. a In 5.4.3, the spectral properties of the composite operators O(x) and Rµ(x) are discussed.

5.4.1 Spectral properties of the elementary ﬁelds

Region I Region II v 0.8 µ 1 µ g 1.2 0.5 λ 0.3 0.205

Table IV A: Parameter values used in the spectral density functions.

We first discuss the spectral properties of the elementary fields: the scalar Higgs field h(x) and the transverse a part of the gauge field Aµ(x). We will work with two sets of parameters, set out in TableIVA. All values are 2 1 2 2 2 2 2 2 given in units of the energy scale µ. Also, we have that m = 4 g v and mh = λv , so that m = 0.23 µ and 2 2 2 2 2 2 mh = 0.192 µ in Region I and m = 0.625 µ and mh = 0.205 µ in Region II.

For the Higgs fields, following the steps from section 4.2.1, we find the pole mass to first order in ~ to be: for Region I

2 2 mh,pole = 0.207 µ , (5.79) and for Region II

2 2 mh,pole = 0.206 µ , (5.80) for all values of the parameter ξ. This means that while the Higgs propagator (5.21) is gauge dependent, the pole mass is gauge independent. This is in full agreement with the Nielsen identities of the SU(2) Higgs model studied in [56]. The residue, however, is gauge dependent, as is depicted in Figure 5.3. For small values of ξ, including the Landau gauge ξ = 0, the residue is not well-deﬁned, and we cannot determine the spectral density function, as we will explain further in the next section.

94 Z 1.10

1.05

1.00

0.95

0.90 ξ 0 5 10 15 20

Figure 5.3: Dependence of the residue Z for the Higgs ﬁeld propagator on the gauge parameter ξ, for Region I (Blue), and Region II (Orange).

In Figure 5.4, we find the spectral density functions both regions, for three values of ξ : 1, 2, 5. Looking 2 2 at Region I, we see the first two-particle state appearing at t = (mh + mh) = 0.768 µ , followed by another 2 2 two-particle state at t√= (m +√m) = 0.922 µ . Then, we see that there is a negative contribution, different for each diagram, at t = ( ξm + ξm)2. This corresponds to the (unphysical) two-particle state of two Goldstone bosons. For ξ < 3, this leads to a negative contribution for the spectral function, probably due to the large- momentum behaviour of the Higgs propagator (5.21), for a detailed discussion see Appendix B.5 . For Region 2 2 II, we find essentially the same behaviour: a Higgs two-particle state at t = (mh + mh) = 0.81 µ , and a gauge 2 2 field two-particle√ √ state at t = (m+m) = 0.25 µ . We also see a negative contribution different for each diagram at t = ( ξm + ξm)2, corresponding to the (unphysical) two-particle state of two Goldstone bosons.

˜ ˜ 103 ρ(t) 103 ρ(t)

10 3

2 5

0 t 2 4 6 8 10 0 t 0.5 1.0 1.5 2.0

-5 -1

Figure 5.4: Spectral functions for the propagator hh(p)h(−p)i, for ξ = 1 (Green, Dotted), ξ = 3 (Red, Solid), ξ = 5 (Yellow, Dashed), with t given in units of µ2, for Region I (left) and Region II (right) with the parameter values given in TableIVA.

For the gauge field propagator, following the steps from section 4.2.1, we find the first-order pole mass of the transverse gauge field to be: for Region I

2 2 mpole = 0.274 µ (5.81) and for Region II

2 2 mpole = 0.065 µ (5.82) for all values of the parameter ξ, so that the pole mass is gauge independent. The residue is, however, gauge dependent as is depicted in Figure 5.5. For small values of ξ the residue is not well-deﬁned, as we will explain further in the next section.

95 Z 1.20

1.15

1.10

1.05

1.00

0.95 ξ 0 5 10 15 20

Figure 5.5: Dependence of the residue Z for the gauge ﬁeld propagator from the gauge parameter ξ, for Region I (Blue), and Region II (Orange).

In Figure 5.6, we find the spectral density functions for both regions, for three values of ξ: 1, 2, 5. Looking 2 2 at Region I, we see the first two-particle state appearing at t = (mh +m) = 0.843 µ , followed by a two-particle 2 2 state at t = (√m + m) = 0.922 µ . Then, we see that there is a negative contribution, different for each diagram, at t = (m + ξm)2. This corresponds to the (unphysical) two-particle state of a gauge and Goldstone boson. 2 2 For Region II, we find a gauge field two-particle state√ at t = (m + m) = 0.25 µ . We also see a negative contribution different for each diagram at t = (m + ξm)2, corresponding to the (unphysical) two-particle state of two Goldstone bosons.

˜ ˜ 102 ρ(t) 106 ρ(t) 6 4

3 4 2 2 1

t 0 t 0 2 4 6 8 10 0.5 1.0 1.5 2.0 -1 -2 -2

a b T Figure 5.6: Spectral functions for the propagator hAµ(p)Aν (−p)i , for ξ = 1 (Green, Dotted), ξ = 3 (Red, Solid), ξ = 5 (Yellow, Dashed), with t given in units of µ2, for Region I (left) and Region II (right) with the parameter values given in TableIVA.

96 5.4.2 Unphysical threshold eﬀects

Figure 5.7: Possible decays of the gauge boson (above) and the Higgs boson (below).

From the Feynman vertex rules given in Appendix C.1, for certain values of the masses, unphysical threshold eﬀects can occur. These eﬀects imply that for certain values of the (physical and unphysical) parameters, a “decay” occurs of a gauge and Higgs boson into two other particles, see Figure 5.7. We distinguish three cases: √ (1.) Decay of a gauge vector boson in two Goldstone bosons: this happens when m > 2 ξm.

(2.) Decay of a Higgs boson in two gauge vector bosons: this happens when mh > 2m. √ (3.) Decay of a Higgs boson in two Goldstone bosons: this happens when mh > 2 ξm.

1 In order to guarantee the stability of the gauge boson, we therefore need from (1.) that ξ > 4 . This means that for the Landau gauge ξ = 0, the elementary gauge boson is not stable. For the Higgs particle, to guarantee 2 mh stability we need from (2.) that mh < 2m. Then, from (3.) we find that ξ > 4m2 . This is the window in which we can work with a stable model. We can have a look at what happens when we go outside of this window. 2 For the Higgs particle, we see that for mh > 2m, or λ < g , we will find a complex value for the first order pole mass, calculated through (3.58). For λ ≥ g2, we will always find a real pole mass. Since the pole mass is 2 mh 2 gauge invariant, we find that this is true for all values of ξ. However, we do find that for ξ > 4m2 and λ > g , the real value of the pole mass is a real point inside the branch cut. This means that we cannot achieve the usual differentiation around this point. As a consequence, we cannot consistently construct the residue, so that 1 we are unable to obtain a first-order spectral function. For the gauge field, we find the same problem when ξ < 4 .

The foregoing mathematically correct observations clearly show that there is something physically wrong with using the elementary ﬁelds’ spectral functions. Luckily, all of these shortcomings are surpassed by using the gauge invariant composite operators.

5.4.3 Spectral properties for the composite fields For the scalar composite operator O(x) we find the first-order pole mass for Region I

2 2 mOO,pole = 0.207 µ , (5.83) and for Region II

2 2 mOO,pole = 0.206 µ , (5.84)

97 which is equal to the pole mass of the elementary Higgs field in (5.79), as we expect from eq. (4.58). Following the steps from section 4.2.1, we find the first-order residue

Z = 1.11 v2 (5.85) for Region I and

Z = 1.01 v2 (5.86) for Region II. The first order spectral function for hO(x) O(y)i is shown in Figure 5.8. Comparing this result with that of the spectral function of the Higgs field in Figure 5.6, we see a two-particle state for the Higgs field 2 2 at t = (mh + mh) , and a two-particle state for the gauge vector field, starting at t = (m + m) . The difference is that for the gauge invariant correlation function hO(x) O(y)i we no longer have the unphysical Goldstone two-particle state. Due to the absence of this negative contribution, the spectral function is positive throughout the spectrum. In fact, we see that for bigger values of ξ, we find that the spectral function of the elementary Higgs field resembles more and more the spectral function of the composite operator O(x). This makes sense, since for ξ → ∞, we are approaching the unitary gauge which has a more direct link with the physical spectrum of the elementary excitations. In Appendix C.4, one finds a detailed discussion about the unitary gauge limit ξ → ∞ as well as the calculation of the spectral function. The asymptotic (constant) behaviour is directly related to the (classical) dimension of the used composite operator.

˜ ˜ 103 ρ(t) 103 ρ(t) 35 12 30 10 25 8 20 6 15

4 10

2 5 t t 2 4 6 8 10 0.5 1.0 1.5 2.0

Figure 5.8: Spectral function for the two-point correlation function hO(p)O(−p)i, with t given in units of µ2, for the Region I (left) and Region II (right) with parameter values given in TableIVA.

2 For the transverse part of the two-point correlation function GR(p ), eq. (5.76), for our set of parameters we ﬁnd the ﬁrst-order pole mass: in Region I

2 2 mR,pole = 0.274 µ (5.87) and Region II

2 2 mR,pole = 0.065 µ (5.88) which is the same as the pole mass of the transverse gauge field, eq. (5.81), in agreement with eq. (4.58). Following the steps from 4.2.1, we find the first-order residue 1 Z = g2v4 (1.27) (5.89) 16 for Region I and 1 Z = g2v4 (1.05) (5.90) 16 2 for Region II. The first order spectral function for GR(p ) is shown in Figure 5.9. Comparing this to the spectral 2 function of the gauge vector field in Figure 5.6, we see a two-particle state at t = (mh + m) , and a two-particle state for the gauge field, starting at t = (m + m)2. Again, as in the case of the two-point correlation function of the scalar operator O(x), the difference is that for this gauge invariant correlation function we no longer have

98 the unphysical Goldstone/gauge boson two-particle state. Due to the absence of this negative contribution, the spectral function is positive throughout the spectrum. In fact, we see that for bigger values of ξ, we ﬁnd that the spectral function of the elementary gauge ﬁeld resembles more and more the spectral function of the a composite operator Rµ(x). As already mentioned previously, this relies on the fact that in the limit ξ → ∞ we are approaching the unitary gauge, see Appendix C.4. Also here, the linear increase at large t follows from the operator dimension.

˜ ˜ 103 ρ(t) ρ(t) 14 25 12

10 20

8 15 6 10 4 5 2 t t 2 4 6 8 10 0.5 1.0 1.5 2.0

2 Figure 5.9: Spectral function for the transverse two-point correlation function GR(p ), with t given in units of µ2, for the Region I (left) and Region II (right) with parameter values given in TableIVA.

5.5 Conclusion

This chapter is the natural extension of chapter3 and4, where the Abelian U(1) Higgs model has been scruti- nized by employing two local composite BRST invariant operators whose two-point correlation functions provide a fully gauge independent description of the elementary excitations of the model, namely the Higgs and the massive gauge boson.

This formulation generalizes to the non-Abelian Higgs model as, for example, the SU(2) YM theory with a single Higgs in the fundamental representation [43–45]. This is the model which has been considered in the a present analysis. The local BRST invariant composite operators (O(x),Rµ(x)) which generalize their U(1) counterparts are given in eq. (5.39) and in eqs. (5.57),(5.58). The two-point correlation functions hO(x)O(y)i a b T and hRµ(x)Rν (y)i , where the superscript T stands for the transverse component, have been evaluated at one- loop order in the Rξ-gauge and compared with the corresponding correlation functions of the elementary ﬁelds a b T hh(x)h(y)i and hAµ(x)Aν (y)i . It turns out that both hO(x)O(y)i and hh(x)h(y)i share the same gauge independent pole mass, eqs.(5.79),(5.80),(5.83),(5.84), in agreement with both Nielsen identities [55–58,158,179,180] and the BRST invariant nature of O(x). Nevertheless, unlike the residue and spectral function of the elementary correlator hh(x)h(y)i, which exhibit a strong unphysical dependence from the gauge parameter ξ, Figure (5.4), the spectral density of hO(x)O(y)i turns out to be ξ-independent and positive over the whole p2 axis, Figure a b T a b T (5.8). The same features hold for hAµ(x)Aν (y)i and hRµ(x)Rν (y)i . Again, both correlation function share have the same ξ-independent pole mass, eqs. (5.82),(5.82),(5.87),(5.88). Though, unlike the ξ-dependent spec- a b T a b T tral function associated to hAµ(x)Aν (y)i , Figure (5.6), that corresponding to hRµ(x)Rν (y)i , Figure (5.9), turns out to be independent from the gauge parameter ξ and positive. As such, the local composite operators a (O(x),Rµ(x)) provide a fully BRST consistent description of the observable scalar (Higgs) and vector boson particles.

It is worth mentioning here that, besides the BRST invariance of the gauge ﬁxed action, the model exhibits an additional global custodial symmetry, eqs.(5.18),(5.19), according to which all ﬁelds carrying the index a a a a a a = 1, 2, 3, i.e. (Aµ, b , c , c¯ , ρ ), undergo a global transformation in the adjoint representation of SU(2). The a a same feature holds for the composite operators (O(x),Rµ(x)) which transform exactly as h and Aµ. More a precisely, the operator O(x) is a singlet under the custodial symmetry, while the operators Rµ transform like a b a triplet, eq. (5.60), so that the correlation function hRµ(p)Rν (−p)i displays the same SU(2) structure of the a elementary two-point function hAµ(p)Aν (−p)i, eqs.(5.61)-(5.64). Although not being the aim of the present analysis, we expect that the existence of a global custodial symmetry will imply far-reaching consequences for a the renormalization properties of the composite operators (O(x),Rµ(x)) encoded in the corresponding Ward

99 identities. The renormalization properties of the local gauge-invariant composite operators for the U(1) Higgs model have been reported recently in [181].

100 Chapter 6

Conclusion and Outlook

In this work, we have evaluated different BRST-invariant solutions for the introduction of a mass term in YM theories. More specifically, we have looked at gauge-invariant extensions of the elementary gauge field and Higgs boson, and we evaluated the two-point functions and spectral properties for these gauge-invariant local composite fields. While the proposed solutions differ in methods and levels at which the BRST invariance is introduced, they serve a common goal: to access the physical spectrum for the nuclear forces.

For the composite field Ah(x), we have shown that a renormalizable gauge class can be developed for this field, which introduces a mass µ for the Stueckelberg-like field ξa, which is used in the localization procedure for the composite gauge field. This mass ensures the infrared safety of the propagator for this field. The gauge class, with µ as the gauge parameter, gives the opportunity to investigate physical objects related to the gauge field propagator outside of the Landau gauge. For example, in [182] it was shown, by analyzing dimension two condensates, that the instability of the GZ action observed in the Landau gauge persists in this larger gauge class, suggesting a physical meaning for the refinement of the GZ action in the infrared regime. This indicates an important role for the RGZ action in the IR description of the YM theories.

For the gauge-invariant local composite operators in the Higgs model, we have investigated spectral properties of the field propagators. The most important conclusion here is that for truly gauge-invariant objects, we have not observed any non-physical behavior. In particular, we have not observed any non-physical behaviour of the kind that is commonly associated with confinement, such as positivity violation of the spectral density function. We immediately add that this does not mean that positivity violation, as has been observed for the spectral density function of the elementary gauge field operator, should be solely ascribed to a lack of gauge-invariance. Even though we were able to make a direct comparison between gauge-invariant composite operators and the gauge-dependent elementary operators, and we concluded that the first always shows a positive spectral density function, while the second for some values of the gauge parameter shows a negative spectral density function, our methods are not suited to reach the non-perturbative region in which positivity violation has been observed in other studies. Since our study consisted of perturbative loop calculations in the Higgs model, we are far from the IR region where we expect this confinement-like behavior. It is therefore our hope that this work will en- courage others to investigate the spectral properties of the gauge-invariant composite local operators developed in this work with non-perturbative methods, such as lattice simulations. The composite gauge-invariant local operators have already led to some preliminary lattice results [54, 173].

A direct comparison between the two methods of developing gauge-invariant composite fields discussed in h this thesis can be made in the Landau gauge. In this gauge, the composite gauge field Aµ(x) is equal to the elementary field Aµ(x), and the action will be equal to the action of the massive YM model discussed in section 1.3.1. The appearance of complex poles for the propagator of the gauge field, even at a perturbative level, means that there is no physical interpretation for these propagators. Thus, while BRST invariance is established by h the configuration of Aµ(x), this does not seem to ensure the usual features of the definition of physical space that are associated with BRST invariance. Possibly, this is because the BRST invariance is introduced at a non-local level. In contrast, the local composite BRST invariant operators developed for the Higgs model always have a real pole mass. However, for the Landau gauge the residue is not well-defined, and we are unable to subtract a spectral density function of the gauge field for this gauge choice. Therefore, the Landau gauge might not be the most suitable gauge to work with for massive non-Abelian models.

If we want to access the non-perturbative region to investigate further the local composite gauge-invariant operators in the Higgs model, the continuum oﬀers some ways to access this region besides numerical methods. a The BRST invariant nature of (O(x),Rµ(x)) makes them natural candidates to an attempt at facing the chal-

101 lenge of investigating the infrared non-perturbative behaviour of the model, trying to make contact with the analytical lattice predictions of Fradkin and Shenker [51].

For the SU(2) HYM theory one may for example introduce a horizon term, in its BRST-invariant formulation encoded in the RGZ action (cf. [22,75,79,183] and refs. therein) implementing the restriction to the Gribov region Ω [72] in order to take into account the existence of the Gribov copies plaguing the FP quantization procedure. As a consequence, the gauge-invariant pole masses of the non-Abelian generalization of the correlation a a functions hRµ(x)Rν (y)i and hO(x)O(y)i will now show an explicit dependence on the Gribov mass parameter as well as on the dimension-two condensates present in the RGZ action [22, 75, 79, 183]. Thus, extending the framework already outlined in [184], the aforementioned pole masses and further spectral properties could be employed as gauge-invariant probing quantities in order to extract non-perturbative information about the behaviour of the elementary excitations of HYM theories in the light of the Fradkin-Shenker results.

The employment of an order parameter to distinguish between the confinement/deconfinement region can also be done in a YM model with finite temperature. In recent years, much valuable progress has been made towards the understanding of non-abelian gauge theories at finite temperature using background field gauge (BFG) methods [185,186] in the Landau-DeWitt gauge. BFG methods provide an efficient way to describe the confinement/deconfinement order parameter (the Polyakov loop or any of its proxies [187]) because the related center symmetry is explicit at the quantum level and is easily maintained in approximation schemes [188–190]. Several models have been put forward in order to implement the BFG formalism in the Landau-deWitt gauge while restricting the number of Gribov copies. In [189, 191–194], the formalism was used within the massive YM model to compute the background potential and Polyakov loop up to two-loop order, both in pure YM theories and in heavy-quark QCD. Recently, results on the BFG method in the Landau-DeWitt gauge for the GZ model have been reported [195–198]. It would be interesting to see if this method could be combined with the gauge-invariant composite operators developed in this work.

But even without accessing the non-perturbative region, the gauge-invariant composite operators offer several interesting lines of investigation. For example, we could compare different gauge classes that leave different remnant global gauges, such as the Rξ gauge and the Linear Covariant Gauges (LCG). The Rξ gauges break the global gauge symmetry of the model, in contrast to the LCG. We could compare the pole masses of the elementary fields for these two gauge classes. Then, they can be compared with the pole mass of their gauge-invariant composite extensions, which should of course be the same in every gauge class. This is a very interesting ex- ercise as in a sense it will reveal which class of gauges is physical. Another line of investigation is given by the gauge-invariant quadratic Higgs condensate ϕϕ†, similar to the Coleman-Weinberg condensate [199, 200]. In particular, we are curious to see what is the influence of this condensate on the propagators of the gauge- invariant operators, including their pole structure.

Finally, a most promising avenue to further explore is the SU(2) × U(1) setting of the electroweak theory. This sector is of course of special interest because of the experimental data that are available for the masses of the W and Z bosons. We would be the ﬁrst to ever calculate these masses analytically in a gauge-invariant way.

102 Appendix A

Technical details of the gauge invariant 2 operator Amin

A.1 Properties of the functional fA[u].

In this Appendix we recall some useful properties of the functional fA[u] Z Z i i f [u] ≡ Tr d4x AuAu = Tr d4x u†A u + u†∂ u u†A u + u†∂ u . (A.1) A µ µ µ g µ µ g µ

For a given gauge field configuration Aµ, fA[u] is a functional defined on the gauge orbit of Aµ. Let A be the a space of connections Aµ with finite Hilbert norm ||A||, i.e. Z Z 2 4 1 4 a a ||A|| = Tr d x AµAµ = d xA A < +∞ , (A.2) 2 µ µ and let U be the space of local gauge transformations u such that the Hilbert norm ||u†∂u|| is finite too, namely Z † 2 4 † † ||u ∂u|| = Tr d x u ∂µu u ∂µu < +∞ . (A.3)

The following proposition holds [137–140] • Proposition The functional fA[u] achieves its absolute minimum on the gauge orbit of Aµ. This proposition means that there exists a h ∈ U such that

δfA[h] = 0 , (A.4) 2 δ fA[h] ≥ 0 , (A.5)

fA[h] ≤ fA[u] , ∀ u ∈ U . (A.6)

2 The operator Amin is thus given by Z 2 4 u u Amin = min Tr d x AµAµ = fA[h] . (A.7) {u}

2 1 Let us give a look at the two conditions (A.4) and (A.5). To evaluate δfA[h] and δ fA[h] we set

a a v = heigω = heigω T , (A.8)

1 T a,T b = if abc T c , Tr T aT b = δab , (A.9) 2 where ω is an inﬁnitesimal hermitian matrix and we compute the linear and quadratic terms of the expansion v of the functional fA[v] in power series of ω. Let us ﬁrst obtain an expression for Aµ i Av = v†A v + v†∂ v µ µ g µ

1The case of the gauge group SU(N) is considered here.

103 i i = e−igωh†A heigω + e−igω h†∂ h eigω + e−igω∂ eigω µ g µ g µ i = e−igωAheigω + e−igω∂ eigω . (A.10) µ g µ

Expanding up to the order ω2, we get

2 2 2 2 v 2 ω h 2 ω i 2 ω 2 ω A = 1 − igω − g A 1 + igω − g + 1 − igω − g ∂µ 1 + igω − g µ 2 µ 2 g 2 2 ω2 ω2 = 1 − igω − g2 Ah + igAhω − g2Ah + 2 µ µ µ 2 2 2 2 i 2 ω g g + 1 − igω − g ig∂µω − (∂µω) ω − ω (∂µω) g 2 2 2 g2 g2 = Ah + igAhω − Ahω2 − igωAh + g2ωAhω − ω2Ah µ µ 2 µ µ µ 2 µ 2 2 i g g 2 3 + ig∂µω − (∂µω) ω − ω∂µω + g ω∂µω + O(ω ) , (A.11) g 2 2 from which it follows

2 v h h g h g 3 A = A + ig[A , ω] + [[ω, A ], ω] − ∂µω + i [ω, ∂µω] + O(ω ) , (A.12) µ µ µ 2 µ 2 We now evaluate Z 4 u u fA[v] = Tr d xAµAµ Z 2 4 h h g h g 3 = Tr d x A + ig[A , ω] + [[ω, A ], ω] − ∂µω + i [ω, ∂µω] + O(ω ) × µ µ 2 µ 2 2 h h g h g 3 A + ig[A , ω] + [[ω, A ], ω] − ∂µω + i [ω, ∂µω] + O(ω ) µ µ 2 µ 2 Z 2 2 4 h h h h 2 h h g h h 2 g h 2 h h = Tr d x A A + igA [A , ω] + g A ωA ω − A A ω − A ω A − A ∂µω µ µ µ µ µ µ 2 µ µ 2 µ µ µ g h h h 2 h h h 2 h h + i A [ω, ∂µω] + ig[A , ω]A − g [A , ω][A , ω] − ig[A , ω]∂µω + g ωA ωA 2 µ µ µ µ µ µ µ µ 2 2 g h 2 h g 2 h h h h g h 3 − A ω A − ω A A − ∂µωA − ig∂µω[A , ω] + ∂µω∂µω + i [ω, ∂µω]A + O(ω ) 2 µ µ 2 µ µ µ µ 2 µ Z Z 2 2 4 h 4 2 h h g h h 2 g h 2 h = fA[h] − Tr d x A , ∂µω + Tr d x g A ωA ω − A A ω − A ω A µ µ µ 2 µ µ 2 µ µ 2 2 Z 2 h h 2 h h g h 2 h g 2 h h 4 − g [A , ω][A , ω] + g ωA ωA − A ω A − ω A A + Tr d x (∂µω∂µω µ µ µ µ 2 µ µ 2 µ µ g h h h g h 3 + i [ω, ∂µω]A − ig∂µω[A , ω] − ig[A , ω]∂µω + i A [ω, ∂µω] + O(ω ) 2 µ µ µ 2 µ

Z Z 4 h 4 2 h h 2 h h 2 = fA[h] + 2 d x tr ω∂µAµ + d x tr 2g ωAµωAµ − 2g AµAµω Z 2 h h h h 4 g h g h − g A ω − ωA A ω − ωA + d x tr ∂µω∂µω + i ω∂µωA − i ∂µωωA µ µ µ µ 2 µ 2 µ h h h h g h g h 3 − ig∂µωA ω + ig∂µωωA − igA ω∂µω + igωA ∂µω + i A ω∂µω − i A ∂µωω + O(ω ) µ µ µ µ 2 µ 2 µ Z Z 4 h 4 h h = fA[h] + 2Tr d x ω∂µAµ + Tr d x ∂µω∂µω + igω∂µωAµ − ig∂µωωAµ

h h 3 − 2ig∂µωAµω + 2ig∂µωωAµ + O(ω ) . (A.13)

Thus Z Z 4 h 4 h h fA[v] = fA[h] + 2Tr d x ω∂µAµ + Tr d x ∂µω∂µω + igω∂µωAµ − ig∂µωωAµ

h h 3 − ig (∂µω) Aµω + ig (∂µω) ωAµ + O(ω )

104 Z Z 4 h 4 h 3 = fA[h] + 2Tr d x ω∂µAµ + Tr d x ∂µω ∂µω − ig Aµ, ω + O(ω ) . (A.14)

Finally Z Z 4 h 4 h 3 fA[v] = fA[h] + 2Tr d x ω∂µAµ − Tr d x ω∂µDµ(A )ω + O(ω ) , (A.15) so that

h δfA[h] = 0 ⇒ ∂µAµ = 0 , 2 h δ fA[h] > 0 ⇒ −∂µDµ(A ) > 0 . (A.16)

We see therefore that the set of field configurations fulfilling conditions (A.16), i.e. defining relative minima of the functional fA[u], belong to the so called Gribov region Ω, which is defined as

Ω = {Aµ| ∂µAµ = 0 and − ∂µDµ(A) > 0} . (A.17)

h Let us proceed now by showing that the transversality condition, ∂µAµ = 0, can be solved for h = h(A) as a power series in Aµ. We start from i Ah = h†A h + h†∂ h , (A.18) µ µ g µ with a a h = eigφ = eigφ T . (A.19) Let us expand h in powers of φ g2 h = 1 + igφ − φ2 + O(φ3) . (A.20) 2 From equation (A.18) we have

2 2 h 2 g 2 g 2 g 3 A = Aµ + ig[Aµ, φ] + g φAµφ − Aµφ − φ Aµ − ∂µφ + i [φ, ∂µ] + O(φ ) . (A.21) µ 2 2 2

h Thus, condition ∂µAµ = 0, gives

2 2 2 2 ∂ φ = ∂µA + ig[∂µAµ, φ] + ig[Aµ, ∂µφ] + g ∂µφAµφ + g φ∂µAµφ + g φAµ∂µφ 2 2 2 2 2 2 g 2 g g g g g 2 − ∂µAµφ − Aµ∂µφφ − Aµφ∂µφ − ∂µφφAµ − φ∂µφAµ − φ ∂µAµ 2 2 2 2 2 2 g + i [φ, ∂2φ] + O(φ3) . (A.22) 2

This equation can be solved iteratively for φ as a power series in Aµ, namely 1 g ∂A g ∂A i g ∂A 3 φ = ∂µAµ + i ∂A, + i Aµ, ∂µ + , ∂A + O(A ) , (A.23) ∂2 ∂2 ∂2 ∂2 ∂2 2 ∂2 ∂2 so that h 1 ∂µ ∂A g ∂µ 1 A = Aµ − ∂µ∂A − ig Aν , ∂ν − i ∂A, ∂A µ ∂2 ∂2 ∂2 2 ∂2 ∂2 1 g 1 ∂µ 3 + ig Aµ, ∂A + i ∂A, ∂A + O(A ) . (A.24) ∂2 2 ∂2 ∂2

Expression (A.24) can be written in a more useful way, given in eq.(2.2). In fact h ∂µ∂ν 1 ig 1 1 3 A = δµν − Aν − ig ∂A, Aν + ∂A, ∂ν ∂A + O(A ) µ ∂2 ∂2 2 ∂2 ∂2 1 ig 1 1 ∂µ ∂µ 1 = Aµ − ig ∂A, Aµ + ∂A, ∂µ ∂A − ∂A + ig ∂ν ∂A, Aν ∂2 2 ∂2 ∂2 ∂2 ∂2 ∂2 g ∂µ ∂A ∂ν 3 − i ∂ν , ∂A + O(A ) 2 ∂2 ∂2 ∂2 ∂µ 1 ig 1 1 ∂µ ∂ν = Aµ − ∂A + ig Aµ, ∂A + ∂A, ∂µ ∂A + ig ∂A, Aν ∂2 ∂2 2 ∂2 ∂2 ∂2 ∂2

105 g ∂ ∂A + i µ , ∂A + O(A3) (A.25) 2 ∂2 ∂2 which is precisely expression (A.24). The transverse ﬁeld given in eq.(2.2) enjoys the property of being gauge invariant order by order in the coupling constant g. Let us work out the transformation properties of φν under a gauge transformation δAµ = −∂µω + ig[Aµ, ω] . (A.26) We have, up to the order O(g2), 1 g 1 g ∂A 2 δφν = −∂ν ω + ig ∂A, ∂ν ω − i ω, ∂ν ∂A − i , ∂ν ω + O(g ) ∂2 2 ∂2 2 ∂2 g 1 g 1 2 = −∂ν ω + i ∂A, ∂ν ω + i ∂ν ∂A, ω + O(g ) . (A.27) 2 ∂2 2 ∂2

Therefore g ∂A 2 δφν = −∂ν ω − i , ω + O(g ) , (A.28) 2 ∂2 h from which the gauge invariance of Aµ is established.

2 Finally, let us work out the expression of Amin as a power series in Aµ. Z 2 4 h h Amin = Tr d x AµAµ Z ∂ ∂ = Tr d4x φ δ − µ ν φ µ µν ∂2 ν Z 4 1 ig 1 1 = Tr d x Aµ − ig ∂A, Aµ + ∂A, ∂µ ∂A × ∂2 2 ∂2 ∂2 ∂µ∂ν 1 ig 1 1 δµν − Aν − ig ∂A, Aν + ∂A, ∂ν ∂A ∂2 ∂2 2 ∂2 ∂2 Z a b b c 1 4 a ∂µ∂ν a abc ∂ν ∂A ∂A c abc a ∂A ∂ν ∂A 4 = d x A δµν − A − 2gf A − gf A + O(A ) . 2 µ ∂2 ν ∂2 ∂2 ν ν ∂2 ∂2 (A.29)

2 We conclude this Appendix by noting that, due to gauge invariance, Amin can be rewritten in a manifestly invariant way in terms of Fµν and the covariant derivative Dµ [137].

A.2 A generalised Slavnov-Taylor identity

In this Appendix we derive the Ward identities for the generalised gauge fixing of eq.(2.20). Since the quantity ωa(ξ) is now a composite operator, i.e. a product of fields at the same space-time point, we need to define ωa(ξ) by introducing it into the starting action though a suitable external source. In order to maintain BRST invariance, we make use of a BRST doublet of external sources (Qa,Ra), of dimension four and ghost number (−1, 0), sQa = Ra , sRa = 0 , (A.30) and introduce the term Z Z ∂ωa d4x s (Qaωa(ξ)) = d4x Raωa(ξ) − Qa gcd(ξ)cd . (A.31) ∂ξc We start thus with the complete classical action Σ given now by Z 4 a ah a ab h b Σ = Sinv + d x Jµ Aµ + ΞµDµ (A )η Z b 4 a a α a a ab a b ab a b a ab b ab a ∂ω (ξ) cd d + d x ib ∂µA + b b − iM b ω (ξ) − N c¯ ω (ξ) +c ¯ ∂µD c + M c¯ g (ξ)c µ 2 µ ∂ξc Z gf abc ∂ωa + d4x −Ωa Dabcb + La cbcc + Kagab(ξ)cd + Raωa(ξ) − Qa gcd(ξ)cd , (A.32) µ µ 2 ∂ξc with Sinv given by expression (2.8).

The action Σ, eq.(A.32), obeys the following Ward identities:

106 • the Slavnov-Taylor identity Z 4 δΣ δΣ δΣ δΣ δΣ δΣ a δΣ ab δΣ a δΣ d x a a + a a + a a + ib a + N ab + R a = 0 , (A.33) δAµ δΩµ δc δL δξ δK δc¯ δM δQ

• the equation of motion of the Lagrange multiplier ba δΣ δΣ = i∂ Aa + αba − iM ab , (A.34) δba µ µ δRb

• the anti-ghost equation δΣ δΣ ab δΣ ab δΣ a + ∂µ a + M b − N b = 0 , (A.35) δc¯ δΩµ δQ δR • the equation of τ a δΣ δΣ a − ∂µ a = 0 , (A.36) δτ δJµ

• the equation of the ghost ηa Z 4 δΣ abc b δΣ abc b δΣ d x a + gf η¯ c + gf Ξ c = 0 , (A.37) δη δτ δJµ

• the equation of the antighostη ¯a δΣ δΣ a − ∂µ a = 0 . (A.38) δη¯ δΞµ

These Ward identities can be employed for the analysis of the algebraic renormalization when the generalised function ωa(ξ) is explicitly present in the gauge-ﬁxing. In this case, the general counterterm will be reabsorbed through a renormalization of ωa(ξ), corresponding to a renormalization of the inﬁnite set of unphysical gauge abc abcd abcde parameters (a1 , a2 , a3 , ..) of expression (2.18).

Repeating the lengthy discussion of the previous sections, for the most general local invariant counterterm we ﬁnd now

Z ct 4 2 ∂Σ ∂Σ 2 ∂Σ Σ = d x − a0g 2 + d2 (α) 2α + a7m ∂g ∂α ∂m2 a δΣ a δΣ 1 a δΣ 1 a δΣ 1 a δΣ +a4 τ a + Jµ a + η¯ a + η a + Ξµ a δτ δJµ 2 δη¯ 2 δη 2 δΞµ

a δΣ a δΣ a δΣ a δΣ +d2 (α) Aµ a − d2 (α)Ωµ a − d1 (α) c a + d1 (α) L a δAµ δΩµ δc δL δΣ ∂f kb(ξ, α) δΣ +f ab(ξ, α)ξa − f ab(ξ, α) + 1 ξk Kb 1 δξb 1 ∂ξa δKa

δΣ ab δΣ ab δΣ −d2(α)¯c + (d2(α) − f2(0, α)) M + (d2(α) − f2(0, α)) N δc¯ δM ab δN ab δΣ δΣ δΣ −d (α)ba − f (0, α)Qa − f (0, α)Ra 2 δba 2 δQa 2 δRa

abc abc δΣ abcd abcd δΣ + f2(0, α)a1 +a ˜1 abc + f2(0, α)a2 +a ˜2 abcd δa1 δa2 abcde abcde δΣ + f2(0, α)a3 +a ˜3 abcde + ... , (A.39) δa3 where the dots ... denote the reamaining, inﬁnite set, of terms of the kind X δΣ f (0, α)aabcde... +a ˜abcde... , j = 4, ..., ∞ , (A.40) 2 j j δaabcde... j j

The counterterm Σct in eq.(A.39) can be rewritten as

107 Σct = RΣ , (A.41) with ∂ ∂ ∂ R = −a g2 + d (α) 2α + a m2 0 ∂g2 2 ∂α 7 ∂m2 Z 4 a δ a δ 1 a δ 1 a δ 1 a δ + d x a4 τ a + Jµ a + η¯ a + η a + Ξµ a δτ δJµ 2 δη¯ 2 δη 2 δΞµ

a δ a δ a δ a δ +d2 (α) Aµ a − d2 (α)Ωµ a − d1 (α) c a + d1 (α) L a δAµ δΩµ δc δL δ ∂f kb(ξ, α) δ +f ab(ξ, α)ξa − f ab(ξ, α) + 1 ξk Kb 1 δξb 1 ∂ξa δKa

δ ab δ ab δ −d2(α)¯c + (d2(α) − f2(0, α)) M + (d2(α) − f2(0, α)) N δc¯ δM ab δN ab δ δ δ −d (α)ba − f (0, α)Qa − f (0, α)Ra 2 δba 2 δQa 2 δRa

abc abc δ abcd abcd δ + f2(0, α)a1 +a ˜1 abc + f2(0, α)a2 +a ˜2 abcd δa1 δa2 abcde abcde δ + f2(0, α)a3 +a ˜3 abcde + ... . (A.42) δa3 For the renormalization factors, we have now

ct 2 Σ(Φ) + εΣ (Φ) = Σ(Φ) + εRΣ(Φ) = Σ(Φ0) + O(ε ) , (A.43) with

2 Φ0 = ZΦΦ = (1 + εR)Φ + O(ε ) . (A.44) where

1/2 1/2 1/2 1/2 A0 = ZA Aµ , b0 = Zb , c0 = Zc c , c¯0 = Zc¯ c¯ , a ab b 1/2 ξ0 = Zξ (ξ)ξ , τ0 = Zτ τ , Ω0 = ZΩΩ ,L0 = ZLL, a ab b 2 2 K0 = ZK (ξ)K , m0 = Zm2 m , J0 = ZJ J , 1/2 1/2 g0 = Zg , α0 = Zαα , η¯0 = Zη¯ η¯ , η0 = Zη η ,

Ξ0 = ZΞΞ ,M0 = ZM M,

N0 = ZN N,Q0 = ZQQ, R0 = ZRR, (A.45) and

a0 Zg = 1 − ε 2 1/2 −1 −1/2 −1/2 1/2 ZA = ZΩ = Zc¯ = Zb = Zα = 1 + εd2(α) ab ab ab Zξ = δ + εf1 (ξ, α) −1/2 ZL = Zc = 1 + εd1(α) 2 1/2 Zη¯ = Zη = ZΞ = Zτ = ZJ = 1 + εa4

Zm2 = 1 + εa7

ZM = ZN = 1 + ε(d2 − f2(0, α))

ZQ = ZR = 1 − ε(f2(0, α)) ∂f kb(ξ, α) Zab = δab − ε f ab(ξ, α) + 1 ξk , (A.46) K 1 ∂ξa

abc abcd abcde with the addition of a multiplicative renormalization of the inﬁnite set of gauge parameters (a1 , a2 , a3 , ..) of expression (2.18), namely

abc abc abc (a1 )0 = (1 + εf2(0, α))a1 + εa˜1

108 abcd abcd abcd (a2 )0 = (1 + εf2(0, α))a2 + εa˜2 abcde abcde abcde (a3 )0 = (1 + εf2(0, α))a3 + εa˜3 ... . (A.47)

Equations (A.46) and (A.47) show that the inclusion of the ambiguity ωa(ξ) in the generalised gauge fixing of eq.(2.20) gives rise to a standard renormalization of the fields, parameters and sources. Clearly, from eq.(A.47) one sees that the renormalization of ωa(ξ) itself is now encoded into a multiplicative renormalization of the abc abcd abcde infinite set of the unphysical gauge parameters (a1 , a2 , a3 , ..).

109 Appendix B

Technical details to the Abelian Higgs model

B.1 Field propagators of the Abelian Higgs model in the Rξ gauge

The quadratic part of the action (3.11) in the bosonic sector is given by Z quad 1 4 n 2 2 2 2 2 2 2 S = d x Aµ(−∂µv(∂ − m ) + ∂µ∂v)Av − ρ∂ ρ − h(∂ − m )h +c ¯(∂ − m )c bos 2 h 2 o + 2ib∂µAµ + ξb + 2imξbρ + 2mAµ∂µρ . (B.1)

Putting this in a matrix form yields Z quad 1 4 T S = d x Ψ OµvΨv, (B.2) bos 2 µ where   Av T  b  Ψ = Aµ b ρ h , Ψv =   , (B.3) µ  ρ  h and

 2 2  (−∂µv(∂ − m ) + ∂µ∂v) −i∂µ m∂µ 0  i∂v ξ imξ 0  O =  2  , (B.4)  −m∂v imξ −∂ 0  2 2 0 0 0 −(∂ − mh) the tree-level ﬁeld propagators can be read oﬀ from the inverse of O, leading to the following expressions 1 hAµ(p)Av(−p)i = Pµv + Lµv, p2 + m2 p2 + m2 1 hρ(p)ρ(−p)i = , p2 + m2 1 hh(p)h(−p)i = 2 2 , p + mh pµ hAµ(p)b(−p)i = , p2 + ξm2 −im hb(p)ρ(−k)i = , (B.5) p2 + ξm2

pµpv pµpv where Pµv = δµv − p2 and Lµv = p2 are the transversal and longitudinal projectors, respectively. The ghost propagator is 1 hc¯(p)c(−p)i = . (B.6) p2 + m2

110 B.2 Vertices of the Abelian Higgs model in the Rξ gauge

From the action (3.11), we ﬁnd the following vertices

ΓAµρh(−p1, −p2, −p3) = ie(pµ,3 − pµ,2)δ(p1 + p2 + p3), 2 ΓAµAv h(−p1, −p2, −p3) = −2e vδµvδ(p1 + p2 + p3), 2 ΓAµAv hh(−p1, −p2, −p3, −p4) = −2e δµvδ(p1 + p2 + p3 + p4), 2 ΓAµAv ρρ(−p1, −p2, −p3, −p4) = −2e δµvδ(p1 + p2 + p3 + p4),

Γhhhh(−p1, −p2, −p3, −p4) = −3λ δ(p1 + p2 + p3 + p4),

Γhhρρ(−p1, −p2, −p3, −p4) = −λ δ(p1 + p2 + p3 + p4),

Γρρρρ(−p1, −p2, −p3, −p4) = −3λ δ(p1 + p2 + p3 + p4),

Γhhh(−p1, −p2, −p3) = −3λv δ(p1 + p2 + p3),

Γhρρ(−p1, −p2, −p3) = −λv δ(p1 + p2 + p3),

Γchc¯ (−p1, −p2, −p3) = −me δ(p1 + p2 + p3). (B.7)

B.3 Equivalence between including tadpole diagrams in the self- energies and shifting hϕi

There is yet another way to come to (3.41). For this, we do not need to include the balloon type tadpoles in the self-energies, but rather fix the expectation value of the Higgs field hhi = 0 by shifting the vacuum expectation value of the Higgs field to its proper one-loop value. The h field one-point function has the following contributions at one-loop order :

• the gluon contribution

1 2e2v Γ(2 − d/2) − md−2 d − ξ ξm2 d/2−1 , 2 d/2 ( ( 1) + ( ) ) (B.8) mh (4π) (2 − d)

• the Goldstone boson one 1 1 Γ(2 − d/2) − λv m2 d/2−1, 2 d/2 ( ) (B.9) mh (4π) (2 − d)

• the ghost loop

1 e2v Γ(2 − d/2) m2 d/2−1 2 2 d/2 ( ) (B.10) mh (4π) (2 − d)

• the Higgs boson one 1 λv Γ(2 − d/2) − md−2, 3 2 d/2 h (B.11) mh (4π) (2 − d)

Together those four contributions yield 1 Γ(2 − d/2) 1 − e2vmd−2 d − − λv ξm2 d/2−1 − λvmd−2 , Γhhi = d/2 2 ( 2 ( 1) ( ) 3 h ) (B.12) (4π) (2 − d) mh that becomes, for d = 4 − ,

1 1 1 2 2 2 2− 2 1−/2 2− = − 2 2 ( + 1 + ln(µ ))(−2e vm (3 − ) − λv(ξm ) − 3λvmh ) (B.13) 2 mh (4π) 1 1 1 2 2 2 2 2 2 2 2 2 = − 2 2 ( + 1 + ln(µ ))(−2e vm (1 − ln m )(3 − ) − λvξm (1 − ln m ) − 3λvmh(1 − ln mh)). 2 mh (4π) 2 2 2 (B.14)

We can split this in a divergent part

div 1 1 2 2 2 2 Γhhi = 2 (6e m v + 3mhvλ + ξm v), (B.15) mh

111 which we can cancel with the counterterms, and a ﬁnite part that reads

2 2 2 2 fin 1 e 2 m 1 λ v 2 h 2 ξm Γhhi = 2 2 v m (1 − 3 ln 2 ) + 2 2 3mh(1 − ln 2 ) + ξm (1 − ln 2 ) . (B.16) mh (4π) µ mh (4π) 2 µ µ Now, to see how this reflects on the propagator, we can rewrite our scalar field as 1 ϕ = √ ((hϕi + h) + iρ), (B.17) 2 where the vacuum expectation value of the Higgs field has tree-level and one-loop terms:

hϕi = v + ~v1. (B.18) Thus the “classical” potential part of the action becomes

2 2 λ v λ 2 ϕ†ϕ − = hϕi2 − v2 + 2hhϕi + h2 + ρ2 (B.19) 2 2 8 and expanding this, we ﬁnd for the shifted tree level Higgs mass

2 1 2 2 2 mh = λ(3hϕi − v ) = λv + 3~λvv1, (B.20) 2 while the photon mass is

2 2 2 2 2 2 m = e hϕi = e v + 2~e vv1. (B.21) As now per construction hhi = 0, we can ﬁx the one-loop correction1 to the Higgs minimizing value by requiring it to absorb the tadpole contributions:

fin v1 + Γhhi = 0, (B.22) thus 2 2 2 2 1 e 2 m 1 λ v 2 h 2 ξm v1 = − 2 2 v m (1 − 3 ln 2 ) − 2 2 3mh(1 − ln 2 ) + ξm (1 − ln 2 ) . (B.23) mh (4π) µ mh (4π) 2 µ µ Implementing this in the transverse hAAi-propagator, one gets 1 G⊥ p2 , AA( ) = 2 2 2 ⊥ 2 (B.24) p + e (v + 2~v1v) − ΠAA(p )

⊥ 0 where in the correction ΠAA, which is already of O(~), we only include the O(~ ) part of hϕi, i.e. v. ⊥ 2 We can now verify the ξ-independence of the transverse propagator hAAi. The ξ-dependent part of ΠAA(p ) is −2e2 Γ(2 − d/2) Π⊥ (p2) = (m2)d/2−1, (B.25) AA,ξ (4π)d/2 2 − d while we ﬁnd the ξ-dependent part of v1 to be (using (B.12)) 1 Γ(2 − d/2) 1 v λv ξm2 d/2−1 . 1ξ = d/2 2 ( ( ) ) (B.26) (4π) (2 − d) mh In the denominator of (B.24) we now easily see that

2 ⊥ 2 2e v1ξv0 − ΠAA,ξ(p ) = 0, (B.27) thereby establishing the gauge independence of the transverse photon propagator. For the Higgs propagator, we similarly ﬁnd

2 1 Ghh(p ) = 2 2 2 . (B.28) p + λ(v + 3~v1v) − Πhh(p )

Here we observe that the ξ-dependent part of v1 has the same eﬀect as the balloon tadpole of the Goldstone boson, consequently establishing the gauge parameter independence of the Higgs mass pole.

1This procedure is also equivalent to computing hϕi via an eﬀective potential minimization up to the same order.

112 B.4 Feynman integrals

Z 1 2 2 ( 2 2 2 2 K(m1, m2) 1 2 m2 2 m1 2 m1m2 dx ln 2 = 2 m1 ln( 2 ) + m2 ln( 2 ) + p ln( 4 ) 0 µ 2p m1 m2 µ q 4 2 2 2 2 4 2 2 4 − 2 −m1 + 2m1m2 − 2m1p − m2 − 2m2p − p h −m2 + m2 − p2 i × tan−1 1 2 p 4 2 2 2 2 2 2 −m1 + 2m1(m2 − p ) − (m2 + p ) q 4 2 2 2 2 4 2 2 4 + 2 −m1 + 2m1m2 − 2m1p − m2 − 2m2p − p h −m2 + m2 + p2 i × tan−1 1 2 p 4 2 2 2 2 2 2 −m1 + 2m1(m2 − p ) − (m2 + p ) ) − 4p2 (B.29)

B.5 Asymptotics of the Higgs propagator

At one-loop, the Higgs propagator behaves like

2 Z 2 Ghh(p ) = for p → ∞. (B.30) 2 p2 p ln µ2

For Z > 0, this can only be compatible with Z ∞ 2 ρ(t)dt Ghh(p ) = 2 (B.31) 0 t + p if the superconvergence relation [31, 162] R dtρ(t) = 0 holds, which forbids a positive spectral function. Let us support this non-positivity of ρ(t) by using (B.30) to show that ρ(t) is certainly negative for very large t. This argument can also be found in the Appendix of [30]. Since for a KL representation we have: 1 ρ(t) = lim (G(−t − i) − G(−t + i)) , (B.32) 2πi →0+ we ﬁnd for t → +∞ and → 0+,

 −1 −1  −t−i −t+i Z ln µ2 ln µ2 ρ(t) =  −  2πi  −t − i −t + i  " # Z t −1 t −1 = − ln − iπ + ln + iπ 2πit µ2 µ2 " # Z t −1 = Im ln + iπ πt µ2

2 !−1/2 ! Z t 2 π = ln 2 + π sin − arctan t . (B.33) πt µ ln µ2

From the latter expression, we can indeed infer that ρ(t) becomes negative for t large. We ﬁnd

−2 t→∞ Z t ρ(t) = − ln < 0 (B.34) t µ2 for Z > 0, and vice versa for Z < 0.

113 B.6 Contributions to hO(p)O(−p)i

Figure B.1: One-loop contributions to the propagator hO(p)O(−p)i. Wavy lines represent the photon field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field.

We consider each term in the two-point function hO(p)O(−p)i, given by eq. (4.7). The ﬁrst term is the one-loop correction to the Higgs propagator hh(p)h(−p)i known from the last chapter, shown in frame (1) in Figure B.1, which gives

2 4 2 2 4 2 v 2 2 2 2 2 p 2 2 2 mhλ e p v hh(p)h(−p)i = 2 2 + v e η m , m 2(d − 1)m + 2 + 2p + η m ξ, m ξ − 2 mh + p 2m 2 2m 9 1 2p2 e2p2 + m2 λη m2 , m2 + e2χ m2 4(d − 1) − + χ m2ξ + λ 2 h h h 2 m2 m2 2 1 + 3λχ mh . (B.35) 2 2 2 (mh + p ) The second term, the one-loop correction shown in frame (2) of Figure B.1, gives

2 2 2 2 mhη m ξ, m ξ vhh(p)ρ(−p) i = − 2 2 . (B.36) mh + p The third term, the one-loop correction shown in frame (3) of Figure B.1, gives

2 2 2 2 2 2 2 2 2 2 3mhη mh, mh 3χ mh χ m ξ 2(d − 1)m χ m 2m ξχ m ξ vhh(p)h(−p) i = − 2 2 − 2 2 − 2 2 − 2 2 2 − 2 2 2 mh + p mh + p mh + p mh (mh + p ) mh (mh + p ) 2m2ξχ m2ξ + 2 2 2 . (B.37) mh (mh + p )

114 The fourth term has no 1-loop contributions. The ﬁfth term, the one-loop correction shown in frame (4) of Figure B.1, gives

1 1 hh(p)h(p)h(−p)h(−p)i = η m2 , m2 . (B.38) 4 2 h h The sixth term, the one-loop correction shown in frame (5) of Figure B.1, gives

1 1 hρ(p)ρ(p)ρ(−p)ρ(−p)i = η ξm2, ξm2 . (B.39) 4 2 Using the identity (5.52) we are able to write the whole one-loop correlation function hO(−p),O(p)i, up to the order ~, as

2 Z 1 v 1 1 2 2 4 2 2 4 1 2 2 2 2 2 hO(p)O(−p)i = 2 2 + 2 2 2 dx η m , m (4(d − 1)m + 4m p + p ) + (p − 2mh) η mh, mh p + mh (p + mh) 0 2 2 2 2 2 2 ! p χ[m ](2(d − 1)m + mh) 2 2 − 2 − 3p χ[mh] , (B.40) mh

115 B.7 Contributions to hVµ(x)Vν(y)i

Figure B.2: One-loop contributions for the propagator hVµ(p)Vν (−p)i. Wavy lines represent the photon field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field.

We consider each term in the two-point function hVµ(p)Vv(−p)i, given by eq. (4.28). The ﬁrst term is the one-loop correction to the photon propagator hAµ(p)Av(−p)i known from the last chapter, shown in frame (1) in Figure B.2, which gives

" 2 2 2 2 2 2 2 2 2 1 2 4 1 2 4 Pµv(p) 1 2 4 m η m , mh mh − m + p − 4(d − 2)m p e v hAµ(p)Av(−p)i = e v + e v − 4 4 p2 + m2 4 (d − 1)p2v2 2 2 2 2 2 2 2 2 4 m χ m 2(d − 1) m p + mh p − m + mh + 2 2 2 (d − 1)p v mh 2 2 2 2 2# m χ m (2d − 1)p − m + m P (p) + h h µv (d − 1)p2v2 (p2 + m2)2 2 "m2ξ2 −2m2 m2 − p2 + m4 + m2 + p2 η m2, m2 1 2 4 ξ 1 2 4 h h h + e v Lµv(p) + e v 4 p2 + ξm2 4 p2v2

116 m2ξ2 2m2 − 2m2ξ + p2 η m2 , m2ξ − h h v2 2 2 2 2 2 2 2 2 4 m ξ χ m 2(d − 1)m p + mh m − p − mh + 2 2 2 p v mh 2 2 2 2 2 2 2 2 2 # m ξ −m + m − 3p χ m 2m ξ χ m ξ L (p) − h h + µv . (B.41) p2v2 v2 (p2 + ξm2)2

The second term, the one-loop correction shown in frame (2) of Figure B.2, gives

2 eη m2 , m2ξ 2evm4 p2 − m2ξ + evm2 m2ξ + p2 + evm6 2 3 2 3h h h h h e v hhAµ(p)Av(−p)i = e v − 2 2 2 2(d − 1)m p mh 2 2 2 2 2 4 4 4 2 2 6 eη m , mh evmh −2(3 − 2d)m p − m − p + 2evmh m − p − evmh − 2 2 2 2(d − 1)m p mh 2 2 2 2 2 2 2 4 eχ m ξ dep vmh + em ξvmh − 2ep vmh − evmh − 2 2 2 2(d − 1)m p mh) 2 2 2 2 2 2 2 2 2 eχ mh (3dep vmh − em ξvmh + em vmh − 3ep vmh − 2 2 2 2(d − 1)m p mh 2 2 2 2 2 2 2 2 4 eχ m 2(d − 1) em p v − em vmh + ep vmh + evmh i Pµv(p) − 2 2 2 2 2 2(d − 1)m p mh (p + m ) h 1 + e2v3 ξ m2 + m2(−ξ) + p2 2η m2 , m2ξ 2p2v h h 2 2 2 4 2 22 2 2 − −2mh m − p + mh + m + p η m , mh 2 2 2 2 2 2 4 χ m −2(d − 1)m p + mh p − m + mh 2 2 2 2 + 2 − χ m ξ mh + m (−ξ) + 2p mh i L − χ m2 m2(ξ − 1) + 3p2 µν . (B.42) h p2 + ξm2

The third term, the one-loop correction shown in frame (3) of Figure B.2, gives

2 2 1 2 2 2 1 2 2h χ mh i 1 2 2h ξχ mh i e v hh Aµ(p)Av(−p)i = e v Pµv + e v Lµv. (B.43) 2 2 m2 + p2 2 m2ξ + p2

The fourth term, the one-loop correction shown in frame (4) of Figure B.2, gives

2 2 2 2 (mh+m (−ξ)+p ) 2 2 m2p2 + 4ξ η mh, m ξ 2 2 2 2h e v hhAµ(p)hAv(−p)i = e v 4(d − 1) 2 2 2 2 2 2 (mh−m +p ) η m , m 4(d − 2) − 2 2 h m p χ m2ξ m2 + m2(−ξ) + p2 + − h 4(d − 1) 4(d − 1)m2p2 2 2 2 2 2 χ m mh − m + p (ξ − 1)χ mh i + − Pµv 4(d − 1)m2p2 4(d − 1)p2 h 1 + e2v2 − m2 + m2(−ξ) + p2 2η m2 , m2ξ 4m2p2 h h 2 2 2 2 2 2 2 2 2 2 + mh − m + p + 4m p η m , mh + m (ξ − 1)χ mh 2 2 2 2 2 2 2 2 i + χ m ξ mh + m (−ξ) + p + χ m −mh + m − p Lµv. (B.44)

The ﬁfth term, the one-loop correction shown in frame (5) of Figure B.2, gives

2 2 1 2 2 2 1 2 2 χ m ξ 1 2 2 ξχ m ξ e v hρ Aµ(p),Av(−p)i = e v Pµv + e v Lµv. (B.45) 2 2 m2 + p2 2 m2ξ + p2

The sixth term, the one-loop correction shown in frame (6) of Figure B.2, gives

117 2 2 2 2 2 2 i 2 i 2hieξ m ξ − mh η mh, m ξ ieξχ mh ieξχ m ξ i − ev pµhhρ(p)Av(−p)i = − ev − + Lµv. (B.46) 2 2 m2ξ + p2 m2ξ + p2 m2ξ + p2 The seventh term, the one-loop correction shown in frame (7) of Figure B.2, gives

2 2 2 2 2 2 2 2 2 2 2 2 2 x 2h e −mh + m ξ + p + 4p mh η mh, m ξ eχ m ξ mh + m (−ξ) + p ev h∂ µhρ(p)Av(−p)i = ev − + 2(d − 1)p2 (m2 + p2) 2(d − 1)p2 (m2 + p2) 2 2 2 2 eχ mh −mh + m ξ + p i + Pµν 2(d − 1)p2 (m2 + p2) 2 2 2 2 2 2 22 4 2 2 heξ −3p mh − m ξ + p + mh − m ξ + p + 2p η m ξ, mh + ev2 2p2 (m2ξ + p2) 2 2 2 2 2 2 2 eξχ mh mh − m ξ eξχ m ξ −mh + m ξ + 2p i + + Lµv. (B.47) 2p2 (m2ξ + p2) 2p2 (m2ξ + p2) The eighth term, the one-loop correction shown in frame (11) of Figure B.2, gives

" 3 2 2 2 2 2 2 2 2 1 3 1 3 ie ξv mh − m + p + 4m p η m , mh − iev pµhρ(p)Av(−p)i = − iev 2 2 m2 (m2ξ + p2)2 3 2 2 2 3 2 2 2 2 2! 2 2 ie ξvmh mh − m ξ ie ξv mh + p mh + m (−ξ) + p + η mh, m ξ − m2 (m2ξ + p2)2 m2 (m2ξ + p2)2 3 2 2 2! i(d − 1)e3ξp2v ie ξv m − m + p + χ m2 − h 2 2 2 2 2 2 2 2 mh (m ξ + p ) m (m ξ + p ) 3 2 3 3 2 ! 2 ie ξvmh ie ξv 3ie ξp v + χ mh − + m2 (m2ξ + p2)2 (m2ξ + p2)2 2m2 (m2ξ + p2)2

ie3ξ2p2v ie3ξv m2 + p2 + χ m2ξ + h 2 2 2 2 2 2 2 2 mh (m ξ + p ) m (m ξ + p ) 3 2 3 2 2 2 2 !# ie ξvmh ie ξp v ie mξ p − + − Lµv. (B.48) 2 2 2 2 2 2 2 2 2 2 2 2 m (m ξ + p ) 2m (m ξ + p ) mh (m ξ + p ) The ninth term, the one-loop correction shown in frame (12) of Figure B.2, gives

" 2 2 2 2 2 ie mh + p χ m ξ −iev pµhρ(p)hAv(−p)i = −iev − 2m2 (m2ξ + p2) 1 − ie − m2 + p2 m2 + m2(−ξ) + p2 η m2ξ, m2 2m2 (m2ξ + p2) h h h 2 2 2 2 2 2 2 2 2 2 + mh − m + p + 4m p η m , mh − m χ mh # 2 2 2 2 2 2 − mhχ m − p χ m + m χ m Lµv. (B.49)

The tenth term, the one-loop correction shown in frame (14) of Figure B.2, gives

" 2 2 2 2 2 2 1 1 2 e vmhη m ξ, mh (d − 1)e vχ m − vpµpν hhρ(p)ρ(−p)i = − v p − 2 2 2 − 2 2 2 2 2 m (m ξ + p ) mh (m ξ + p ) 2 2 2 2 2 2 2 # 3e vχ mh e ξvχ m ξ e vχ m ξ emξχ m ξ − 2 2 2 − 2 2 2 − 2 2 2 + 2 2 2 L(B.50)µv. 2m (m ξ + p ) mh (m ξ + p ) 2m (m ξ + p ) mh (m ξ + p )

The eleventh term, the one-loop correction shown in frame (15) of Figure B.2, gives

2 2 2 2 2 2 2 2 2 2 2 x hie mhvχ m ξ ie mhv mh − m ξ − p η m ξ, mh + χ mh i ivpν h∂ µhρ(p)ρ(−p)i = iv − L(B.51)µv. 2m4ξ + 2m2p2 2m2 (m2ξ + p2)

118 The twelfth term, the one-loop correction shown in frame (16) of Figure B.2, gives

1 2 2 2 pµpν hhh(p)ρρ(−p)i = p η m ξ, m Lµv. (B.52) 4 h The thirteenth term, the one-loop correction shown in frame (17) of Figure B.2, gives

2 2 2 2 (−mh+m ξ+p ) 2 2 2 p2 + 4mh η mh, m ξ 2 2 2 2 x y h χ m ξ −mh + m ξ − p −h∂ µhρ(p)h∂ ν ρ(−p)i = − + 4(d − 1) 4(d − 1)p2 2 2 2 2 χ mh −mh + m ξ + p i − Pµv 4(d − 1)p2 h m2 + m2(−ξ) − p2 m2 + m2(−ξ) + p2 η m2ξ, m2 − − h h h 4p2 2 2 2 2 2 2 2 2 χ m ξ mh + m (−ξ) − p χ mh mh + m (−ξ) + p i + − Lµv. (B.53) 4p2 4p2

The fourteenth term, the one-loop correction shown in frame (18) of Figure B.2, gives

4 2 4 2 2 2 2 2 2 2 2 2 2 2 1 2 h 2e v mhη m ξ, mh e mh − m + p + 4m p η m , mh v pµpν hρ(p)ρ(−p)i = p + 4 m4 (m2ξ + p2)2 m2 (m2ξ + p2)2 e2 m2 + p2 2η m2 , m2ξ (d − 1)e4v2χ m2 (d − 1)e2χ m2 e4v2m2 χ m2ξ − h h + − + h m2 (m2ξ + p2)2 m2 (m2ξ + p2)2 (m2ξ + p2)2 2m4 (m2ξ + p2)2 3e4v2m2 χ m2 e4ξv2χ m2ξ e3ξvχ m2ξ e2χ m2ξ m2 + m2ξ + p2 + h h + − + h 2m4 (m2ξ + p2)2 m2 (m2ξ + p2)2 m (m2ξ + p2)2 m2 (m2ξ + p2)2 e2χ m2 e2χ m2 m2 − m2 + p2 3e2m2 χ m2ξ e2m2 χ m2 − h − h − h − h h (m2ξ + p2)2 m2 (m2ξ + p2)2 2m2 (m2ξ + p2)2 2m2 (m2ξ + p2)2 2 2 e ξχ m ξ i − Lµv. (B.54) (m2ξ + p2)2

119 Appendix C

Technical details to the SU(2) Higgs model

C.1 Propagators and vertices of the SU(2) Higgs model in the Rξ gauge

The tree level elementary propagators of the ﬁelds are easily computed, being given by

ab a b δ ab ξ hA (p)A (−p)i = Pµv(p) + δ Lµv(p), µ v p2 + m2 p2 + ξm2 δab hρa(p)ρb(−p)i = , p2 + ξm2 1 hh(p)h(−p)i = 2 2 , p + mh p hAa (p)bb(−p)i = δab µ , µ p2 + ξm2 im hba(p)ρb(−k)i = δab (C.1) p2 + ξm2 and δab hc¯a(p)cb(−p)i = (C.2) p2 + ξm2 for the ghost propagator. For all vertices, adopting the convention that the momentum is ﬂowing towards the vertex, we get

g2v ab • AAh a b −p , −p , −p − δ δ δ p p p The -vertex: ΓAµAν h( 1 2 3) = 2 µν ( 1 + 2 + 3).

g abc • ρρA a b c −p , −p , −p i p − p δ p p p The -vertex: Γρ ρ Aµ ( 1 2 3) = 2 ( µ,1 µ,2) ( 1 + 2 + 3).

g ab • Aρh a b −p , −p , −p i δ p − p δ p p p The -vertex: ΓAµρ h( 1 2 3) = 2 ( µ,3 µ,2) ( 1 + 2 + 3).

• The hhh vertex: Γhhh(−p1, −p2, −p3) = −3λv δ(p1 + p2 + p3).

ab • The hρρ vertex: Γhρaρb (−p1, −p2, −p3) = −λvδ δ(p1 + p2 + p3).

abc • AAA a b c −p , −p , −p −igf p − p δ p − p δ p − p δ δ p The -vertex: ΓAµAv Aσ ( 1 2 3) = [( 1 3)ν σµ + ( 3 2)µ vσ + ( 2 1)σ vµ] ( 1+ p2 + p3).

abc • cAc a b c −p , −p , −p igf p δ p p p The -vertex: Γc Aµc ( 1 2 3) = 1,µ ( 1 + 2 + 3).

2 eab ecd eac ebd • AAAA a b c d −p , −p , −p , −p g f f δ δ − δ δ f f δ δ − The -vertex: ΓAµAν AρAσ ( 1 2 3 4) = [ ( µσ νρ µρ νσ) + ( µσ νρ ead ebc δµν δρσ) + f f (δµρδνσ − δµν δρσ)]δ(p1 + p2 + p3 + p4).

1 2 ab • AAhh a b −p , −p , −p , −p − g δ δ The -vertex: ΓAµAv hh( 1 2 3 4) = 2 µv.

1 2 ab cd • AAρρ a b c d −p , −p , −p , −p − g δ δ δ The -vertex: ΓAµAv ρ ρ ( 1 2 3 4) = 2 µv .

120 C.2 Elementary propagators of the SU(2) Higgs model in the Rξ gauge

Here we will calculate the one-loop corrections to the Higgs and gauge field propagator. This requires the calculation 1 of the Feynman diagrams as shown in Figures C.1 and C.2. We will use the following definitions: 1 1 d Z d/2−2 η m , m ≡ − dx p2x − x xm − x m , ( 1 2) d/2 Γ(2 ) (1 ) + 1 + (1 ) 2 (4π) 2 0 1 d d/2−1 χ(m1) ≡ Γ(1 − )m . (C.3) (4π)d/2 2 1 Notice that the last four diagrams for both particles are zero for hhi = 0. In fact, including these diagrams has the same effect as making a shift in the minimizing value of the scalar field Φ to demand hhi = 0, see the Appendix of [61] for the technical details. In the context of the FMS operators, we found it more convenient to expand around the (classical) v that is gauge invariant, and thus to include the tadpoles. Expanding the FMS operator around the quantum corrected vev would lead to cancellations in that quantum vev coming from the propagator loop corrections to render it gauge invariant again, indeed the minimum of the quantum corrected effective Higgs potential is not gauge invariant itself.

C.2.1 Higgs propagator

Figure C.1: One-loop contributions to the propagator hh(p)h(−p)i. Curly lines represent the gauge field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field.

The first diagrams contributing to the Higgs self-energy are of the snail type, renormalizing the masses of the internal fields. The Higgs boson snail (first diagram in the first line of Figure C.1):

2 2 3λχ mh Γhh,1(p ) = − , (C.4) 2 2 2 2 (mh + p ) the Goldstone boson snail (second diagram in the ﬁrst line of Figure C.1):

2 2 3λχ m ξ Γhh,2(p ) = − , (C.5) 2 2 2 2 (mh + p ) and the gauge ﬁeld snail (third diagram in the ﬁrst line of Figure C.1):

2 2 2 2 2 3(d − 1)g χ m 3g ξχ m ξ Γhh,3(p ) = − − . (C.6) 2 2 2 2 2 2 4 (mh + p ) 4 (mh + p ) Next, we meet a couple of sunset diagrams. The Higgs boson sunset (fourth diagram in the ﬁrst line of Figure C.1): Z 1 2 2 2 2 2 n9λ v η mh, mh o Γhh,4(p ) = dx , (C.7) 2 2 2 0 2 (mh + p ) 1We have used from [160] the technique of modifying integrals into “master integrals” without numerators.

121 the gauge ﬁeld sunset (ﬁrst diagram in the second line of Figure C.1):

Z 1 2 2 2 4 2 2 4 2 2 22 2 2 2 n3g η m , m 4(D − 1)m + 4m p + p 3g 2m ξ + p η m ξ, m ξ Γhh,5(p ) = dx + 2 2 2 2 2 2 2 2 0 8m (mh + p ) 8m (mh + p ) 3g2 m4(ξ − 1)2 + 2m2ξp2 + 2m2p2 + p4 η m2, m2ξ 3g2(ξ − 1)χ m2 − + 2 2 2 2 2 2 2 4m (mh + p ) 4 (mh + p ) 3g2(ξ − 1)χ m2ξo − , (C.8) 2 2 2 4 (mh + p ) the ghost sunset (second diagram in the second line of Figure C.1):

Z 1 2 2 2 2 2 2 n3g m ξ η m ξ, m ξ o Γhh,6(p ) = − dx , (C.9) 2 2 2 0 4 (mh + p ) the Goldstone boson sunset (third diagram in the second line of Figure C.1):

Z 1 2 2 2 2 2 n3λ v η m ξ, m ξ o Γhh,7(p ) = dx , (C.10) 2 2 2 0 2 (mh + p ) and a mixed Goldstone-gauge sunset (fourth diagram in the second line of Figure C.1):

2 2 22 2 2 2 2 Z 1 3g m (ξ − 1) + p + 4m p η m , m ξ 2 2 22 2 2 2 n 3g m ξ + p η m ξ, m ξ Γhh,8(p ) = dx − 2 2 2 2 2 2 2 2 0 4m (mh + p ) 4m (mh + p ) 3g2χ m2ξ m2(2ξ − 1) + p2 3g2χ m2 m2(ξ − 1) + p2o + − . (C.11) 2 2 2 2 2 2 2 2 4m (mh + p ) 4m (mh + p ) Finally, we have the tadpole diagrams. The Higgs balloon (ﬁrst diagram on the third line of Figure C.1):

2 2 2 2 9λ v χ mh Γhh,9(p ) = , (C.12) 2 2 2 2 2mh (mh + p ) the gauge balloon (second diagram on the third line of Figure C.1):

2 2 2 9gλmv (d − 1)χ m + ξχ m ξ Γhh,10(p ) = , (C.13) 2 2 2 2 2mh (mh + p ) the Goldstone boson balloon (third diagram on the third line of Figure C.1):

2 2 2 2 9λ v χ m ξ Γhh,11(p ) = , (C.14) 2 2 2 2 2mh (mh + p ) the ghost balloon (fourth diagram on the third line of Figure C.1):

2 2 9gλmξvχ m ξ Γhh,12(p ) = − 2 . (C.15) 2mh

Putting together eqs. (C.4) to (C.15) we ﬁnd the Higgs propagator up to ﬁrst order in ~ Z 1 4 2 2 4 1 2 n3 4(d − 1)m + 4m p + p 2 2 hh(x) h(y)i = 2 2 + g dx 2 η m , m p + mh 0 8m 4 4 2 2 9m4 3 m − p 6(d − 1)m − 3p + h η m2 , m2 + h η m2ξ, m2ξ + χ m2 8m2 h h 8m2 4m2 2 2 2 3 mh + p 2 3mh 2 o 1 + 2 χ m ξ + 2 χ mh 2 2 2 . (C.16) 4m 4m (p + mh) The resummed one-loop Higgs propagator can be now approximated by

Z 1 4 2 2 4 −1 2 2 2 2 n3 4(d − 1)m + 4m p + p 2 2 Ghh (p ) = p + mh − g dx 2 η m , m 0 8m

122 4 4 2 2 9m4 3 m − p 6(d − 1)m − 3p + h η m2 , m2 + h η m2ξ, m2ξ + χ m2 8m2 h h 8m2 4m2 2 2 3 m + p 3m2 o + h χ m2ξ + h χ m2 . (C.17) 4m2 4m2 h For d = 4, the above expression, eq. (C.17), is divergent. Employing the procedure of dimensional regularization, 2 i.e. setting d = 4 − , the divergent part for Ghh(p ) is given by:

4 2 3mh 2 2 2 g 2 − 3ξm − 3ξp + 9p 2 m h Ghh,div(p ) = , (C.18) 32π2 which, following the MSbar-scheme, are re-absorbed by the introduction of suitable local counterterms. We remain thus with the ﬁnite part of the Higgs propagator, given in eq. (5.21).

C.2.2 Gauge ﬁeld propagator

Figure C.2: Contributions to the one-loop gauge ﬁeld self-energy.

The first diagram contributing to transverse part of the gauge field self-energy is the gauge field snail (first diagram in the first line of Figure C.2) and gives a contribution:

2 2 2 2 2 2 2 2 2 2 2g p − d d − 3d + 3 p χ m 2g ξ (d − 2)dp + p χ m ξ ΠAAT ,1(p ) = − . (C.19) (d − 1)dp2 (m2 + p2)2 (d − 1)dp2 (m2 + p2)2 The second diagram is the Goldstone boson snail (second diagram in the ﬁrst line of Figure C.2):

2 2 2 3g χ m ξ ΠAAT ,2(p ) = − . (C.20) 4 (m2 + p2)2 The third diagram is the Higgs boson snail (third diagram in the ﬁrst line of Figure C.2):

2 2 2 g χ mh ΠAAT ,3(p ) = − . (C.21) 4 (m2 + p2)2 The fourth diagram is the gauge ﬁeld sunrise (ﬁrst diagram in the second line of Figure C.2): Z 1 2 2 2 2 4 2 4 2 2 nη m , m ξ 2m p (−2d + ξ + 3) + m (ξ − 1) + p ΠAAT ,4(p ) = g dx 4 2 0 2(d − 1)m p

123 4m2 + p2 η m2, m2 4(d − 1)m4 + 4(3 − 2d)m2p2 + p4 − 4(d − 1)m4 (m2 + p2)2 4m2ξp4 + p6 η m2ξ, m2ξ − 4(d − 1)m4 (m2 + p2)2 χ m2ξ 4d2 m2(ξ + 1)p2 + p4 + d m4(ξ − 1) − m2(6ξ + 7)p2 + (ξ − 7)p4 + 4m2ξp2 + 2(d − 1)dm2p2 (m2 + p2)2 χ m2 4d2p4 + d m4(ξ − 1) + m2(2ξ − 5)p2 + (ξ − 7)p4 + 4m2p2o − . (C.22) 2(d − 1)dm2p2 (m2 + p2)2

The ﬁfth diagram is the ghost sunrise (second diagram in the second line of Figure C.2):

Z 1 2 2 ! 2 2 2 2 2m ξ p ΠAAT ,5(p ) = g dxη m ξ, m ξ + 2 2 2 2 2 2 0 (d − 1) (m + p ) 2(d − 1) (m + p ) χ m2ξ − . (C.23) (d − 1) (m2 + p2)2

The sixth diagram is the Goldstone sunrise (ﬁrst diagram in the third line of Figure C.2):

Z 1 2 2 ! 2 2 2 2 m ξ p ΠAAT ,6(p ) = g dxη m ξ, m ξ − − 2 2 2 2 2 2 0 (d − 1) (m + p ) 4(d − 1) (m + p ) χ m2ξ + . (C.24) 2(d − 1) (m2 + p2)2

The seventh diagram is the mixed Goldstone-Higgs sunrise (second diagram in the third line of Figure C.2):

2 2 22 2 2 2 2 Z 1 −m + m ξ + p + 4m p η m , m ξ 2 2 2 2 2 2 h h h χ mh −mh + m ξ + p ΠAAT ,7(p ) = g dx − + 2 2 2 2 2 2 2 2 0 4(d − 1)p (m + p ) 4(d − 1)p (m + p ) χ m2ξ m2 − m2ξ + p2 + h . (C.25) 4(d − 1)p2 (m2 + p2)2

The eighth diagram is the mixed Goldstone-gauge ﬁeld sunrise (third diagram in the third line of Figure C.2):

2 2 22 2 2 2 2 Z 1 m − m ξ + p + 4m ξp η m , m ξ 2 2 h h ΠAAT ,8(p ) = g dx 2 2 2 2 0 4(D − 1)p (m + p ) 2 2 2 2 2 2 2 2 2 2 η m , mh mh − m + p − 4(D − 2)m p m (ξ − 1)χ m − − h 4(D − 1)p2 (m2 + p2)2 4(D − 1)p2 (m2 + p2)2 χ m2ξ m2 − m2ξ + p2 χ m2 m2 − m2 + p2 − h + h . (C.26) 4(D − 1)p2 (m2 + p2)2 4(D − 1)p2 (m2 + p2)2

Finally, we have four tadpole (balloon) diagrams. The Higgs boson balloon (ﬁrst diagram of the last line in Figure C.2):

2 2 3gmχ mh ΠAAT ,5(p ) = , (C.27) 2v (m2 + p2)2 the Goldstone boson balloon (second diagram of the last line in Figure C.2):

2 2 3gλmvχ m ξ ΠAAT ,6(p ) = , (C.28) 2 2 2 2 2mh (m + p ) The gauge ﬁeld balloon (third diagram of the last line in Figure C.2):

2 2 2 2 2 2 2 3(D − 1)g m χ m 3g m ξχ m ξ ΠAAT ,7(p ) = + (C.29) 2 2 2 2 2 2 2 2 2mh (m + p ) 2mh (m + p )

124 and ﬁnally, the ghost balloon (fourth diagram of the last line in Figure C.2):

2 2 2 2 3g m ξχ m ξ ΓAAT ,8(p ) = − . (C.30) 2 2 2 2 2mh (m + p ) Combining all these contributions (C.19)-(C.30), we ﬁnd the total one-loop correction to the gauge ﬁeld self- energy

ab Z 1 2 2 4 2 2 2 4 4 a b T δ ab 2 n 2(3 − 2d)m p + mh − 2mh m − p + m + p 2 2 hAµ(p)Aν (p)i = 2 2 + δ g dx − 2 η m , mh p + m 0 4(d − 1)p 2 m2 + p2 2m2p2(−2d + ξ + 3) + m4(ξ − 1)2 + p4 + η m2, m2ξ 2(d − 1)m4p2 m4 − p4 4m2ξ + p2 4m2 + p2 4(d − 1)m4 + 4(3 − 2d)m2p2 + p4 + η m2ξ, m2ξ − η m2, m2 4(d − 1)m4 4(d − 1)m4 2 2 2 2 4 4 2 4 2 4 2 mh −m p 8d − 24d + 4ξ + 13 − 2p (4d + ξ − 7) + m (1 − 2ξ) + 6(d − 1) m p + mhm 2 + 2 2 2 χ m 4(d − 1)mhm p (d − 2)p2 + m2 − m2 m2p2(5d + 4ξ − 13) + 2p4(4d + ξ − 7) + 2m4(ξ − 1) − h χ m2 + χ m2ξ 4(d − 1)p2 h 4(d − 1)m2p2 3 3 o 1 + χ m2 + χ m2ξ (C.31) 4 h 4 (p2 + m2)2 and the resummed propagator for the transverse gauge ﬁeld can be approximated, at one-loop order, by

Z 1 2 2 4 2 2 2 4 4 −1 ab 2 2 2 n 2(3 − 2d)m p + mh − 2mh m − p + m + p 2 2 GAAT = δ p + m − g dx − 2 η m , mh 0 4(d − 1)p 2 m2 + p2 2m2p2(−2d + ξ + 3) + m4(ξ − 1)2 + p4 + η m2, m2ξ 2(d − 1)m4p2 m4 − p4 4m2ξ + p2 4m2 + p2 4(d − 1)m4 + 4(3 − 2d)m2p2 + p4 + η m2ξ, m2ξ − η m2, m2 4(d − 1)m4 4(d − 1)m4 2 2 2 2 4 4 2 4 2 4 2 mh −m p 8d − 24d + 4ξ + 13 − 2p (4d + ξ − 7) + m (1 − 2ξ) + 6(d − 1) m p + mhm 2 + 2 2 2 χ m 4(d − 1)mhm p (d − 2)p2 + m2 − m2 m2p2(5d + 4ξ − 13) + 2p4(4d + ξ − 7) + 2m4(ξ − 1) − h χ m2 + χ m2ξ 4(d − 1)p2 h 4(d − 1)m2p2 ! 3 3 o + χ m2 + χ m2ξ . (C.32) 4 h 4

2 For d = 4−, following the procedure of dimensional regularization, we ﬁnd that the divergent part for GAAT (p ) is given by:

2 4 2 2 2 2 2 2 g 9m 3mh m ξ 3m ξp 25p GAAT ,div(p ) = 2 − 2 − − − − + , (C.33) π 16mh 32 8 32 8 48 and these terms can be, following the MSbar-scheme, absorbed by means of appropriate counterterms. We remain with the ﬁnite part of the propagator, given in eq. (5.31).

C.3 Contributions to hO(p)O(−p)i

The diagrams which contribute to the correlation function hO(p)O(−p)i are depicted in Figure(C.3)

125 Figure C.3: One-loop contributions to the correlation function hOOi. Wavy lines represent the gauge field, dashed lines the Higgs field, solid lines the Goldstone boson and double lines the ghost field. The • indicates the insertion of a composite operator. .

The ﬁrst term is v2 times the one-loop correction to the Higgs propagator, given in eq. (C.17). The second term is

2 2 2 a a mhη m ξ, m ξ vhh(p)(ρ ρ )(−p)i = −3 2 2 . (C.34) mh + p The third term is

2 2 2 2 2 2 2 2 2 2 3mhη mh, mh 3χ mh χ m ξ 2(D − 1)m χ m 2m ξχ m ξ vhh(p)h (−p)i = − 2 2 − 2 2 − 2 2 − 2 2 2 − 2 2 2 mh + p mh + p mh + p mh (mh + p ) mh (mh + p ) 2m2ξχ m2ξ + 2 2 2 . (C.35) mh (mh + p ) The fourth term is 1 hm2 (p)m2 (−p)i = η m2 , m2 . (C.36) h h 2 h h The ﬁfth term is 3 h(ρaρa)(p)(ρbρb)(−p)i = η ξm2, ξm2 (C.37) 2 and together these terms give the correlation function of the scalar composite operator O up to ﬁrst order in ~ 2 Z 1 v n3 2 2 4 2 2 4 1 2 2 2 2 2 hO(p)O(−p)i = 2 2 + dx η m , m (4(d − 1)m + 4m p + p ) + (p − 2mh) η mh, mh p + mh 0 2 2 2 2 2 2 3p χ(m )(2(d − 1)m + mh) 2 2 o 1 − 2 − 3p χ(mh) 2 2 2 . (C.38) mh (p + mh)

126 Thus, for the one-loop resummed correlation function, we get

2 2 Z 1 −1 2 p + mh 1 n3 2 2 4 2 2 4 1 2 2 2 2 2 GOO(p ) = 2 − 4 dx η m , m (4(d − 1)m + 4m p + p ) + (p − 2mh) η mh, mh v v 0 2 2 2 2 2 2 3p χ(m )(2(d − 1)m + mh) 2 2 o 1 − 2 − 3p χ(mh) 2 2 2 . (C.39) mh (p + mh) Following the procedure of dimensional regularization for d = 4 − , we ﬁnd that the divergent part of the correlator is given by

1 9g4p2v2 9g4v4 9 1 G−1 = + + g2p2v2 + p4 + λp2v2 + λ2v4 , (C.40) OO,div 4v4π2 16λ 16 8 2 which can be accounted for by appropriate counterterm, following the MSbar-scheme renormalization procedure. We remain with the ﬁnite part of the correlator, given in eq. (5.76).

C.4 A few comments on the unitary gauge

It is well-known that in the unitary gauge the unphysical fields, like the Goldstone and ghost fields, decouple, a feature which allows for a more direct link with the spectrum of the elementary excitations of the model. However, this gauge is known to be non-renormalizable. In fact, working directly with the elementary tree level propagators taken already in the unitary limit, i.e. ξ → ∞, and following the steps of dimensional regularization, we find that the divergent part of the Higgs propagator reads

3g2 m4 + 6m2p2 + p4 G−1 (p2) = h . (C.41) hh,div 64π2m2

p4 In expression (C.41) we clearly see the presence of the term ∼ m2 , signalling the aforementioned issue of the non-renormalizability. Nevertheless, it is interesting to observe that, if we remove the divergent part (C.41) anyway, we obtain the spectral function as shown in Figure C.4. This spectral function is almost identical to that obtained for the composite operator O(x), see Figure 5.8.

˜ ˜ 103 ρ(t) 106 ρ(t) 14 12 3.0 10 2.5 8 2.0 6 1.5 4 1.0 2 0.5 t t 2 4 6 8 10 0.5 1.0 1.5 2.0

Figure C.4: Spectral function for the propagator hh(p)h(−p)i in the unitary gauge, with t given in unity of µ2, for the Region I (left) and Region II (right), with parameter values given in TableIVA.

For the gauge ﬁeld propagator, proceeding in the same way, we ﬁnd the divergent part

2 4 2 6 2 4 2 2 2 2 2 −1 2 1 9g m g p 7g p 3g m 83g p 3λm GAA,div(p ) = − 2 2 − 2 4 + 2 2 + 2 + 2 − 2 (C.42) 16π mh 96π m 48π m 32π 96π 8π

p6 p4 which shows again the non-renormalizability of unitary gauge, through the terms ∼ m4 and ∼ m2 . However, if we remove again those terms anyway, we obtain the spectral function as shown in Figure C.5. Nevertheless, as already remarked in the previous sections, this nice behaviour of the spectral densities for the Higgs and gauge ﬁeld obtained by a direct use of the tree level propagators already taken in the unitary limit, ξ → ∞, can be, to some extent, justiﬁed by the fact that we are working at the one-loop order in perturbation theory.

127 Since overlapping divergences start from one-loop onward, we can easily ﬁgure out that the naive use of the elementary tree level propagators taken already in the unitary limit will run into severe non-renormalizibility issues, making the removal of the (overlapping) divergent parts (C.41), (C.42) quite problematic.

˜ ˜ ρ(t) 106 ρ(t) 0.6 30

0.5 25

0.4 20

0.3 15

0.2 10

0.1 5

t t 2 4 6 8 10 0.5 1.0 1.5 2.0

a b Figure C.5: Spectral function for the propagator hAµ(p)Aν (−p)i in the unitary gauge, with t given in unity of µ2, for the Region I (left) and Region II (right), with parameter values given in TableIVA.

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