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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¸c˜aodo t´ıtulode Doutor, ao Pro- grama de P´os-Gradua¸c˜aoem 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 Guimar˜aes Tese (Doutorado) – Universidade do Estado do Rio de Janeiro, Instituto de F´ısicaArmando Dias Tavares, Programa de P´os-Gradua¸c˜aoem 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 acadˆemicose cient´ıficos,a reprodu¸c˜aototal 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¸c˜aodo t´ıtulode Doutor, ao Pro- grama de P´os-Gradua¸c˜aoem 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 Pel´aezfor 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
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