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Dark Monopoles in Grand Unified Theories

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Dark Monopoles in Grand Unified Theories Maria de Lourdes Zamboni Peixoto Deglmann DARK MONOPOLES IN GRAND UNIFIED THEORIES Dissertação submetida ao Programa de Pós-Graduação em Física da Universidade Federal de Santa Catarina para a obtenção do Grau de Mestre em Física. Orientador: Prof. Dr. Paweł Klimas. Coorientador: Prof. Dr. Marco Aurélio Cattacin Kneipp. Florianópolis 2019 Ficha de identificação da obra elaborada pelo autor, através do Programa de Geração Automática da Biblioteca Universitária da UFSC. Deglmann, Maria de Lourdes Zamboni Peixoto Dark Monopoles in Grand Unified Theories / Maria de Lourdes Zamboni Peixoto Deglmann ; orientador, Pawel Klimas, coorientador, Marco Aurélio Cattacin Kneipp, 2019. 102 p. Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Física, Florianópolis, 2019. Inclui referências. 1. Física. 2. Teorias de Gauge. 3. Monopolos Magnéticos. 4. Quebra Espontânea de Simetria. 5. Carga Magnética. I. Klimas, Pawel. II. Kneipp, Marco Aurélio Cattacin. III. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Física. IV. Título. To the people I consider as family. ACKNOWLEDGMENTS I am grateful to Prof. Marco Kneipp and Prof. Paweł Klimas for the trust and support. In particular, I want to show my gratitude to Prof. Kneipp for his guidance, high-quality supervision and patience. Besides that, I would like to thank him for the encouragement and support, especially during the last year. I owe him almost everything I know about theoretical physics and it has been an honor to collaborate with him. I would also like to thank my mother for being the best and most trustful friend a person can ask for. I could have never done it without you. Thank you very much for listening to hours of explanations about the Dark Monopoles and to many more hours of discussions about academia and its challenges. Likewise, I am grateful to my father, who has the talent to make fun of everything, even in the difficult moments. Moreover, I would like to say thank you to my brother for the reasons I am sure he understands. And, since this section must be of finite size, I add an extra “thank you!” to my grandparents for the support and to my furry friends, Lala and Lorentz, for the companionship. With regard to my university colleagues, I’m grateful to my office mates in room 105, “The Field Theory Room”, for all the nice con- versations about physics and courses. In addition, I have enjoyed the presence of my friends from undergraduate studies, which have chosen other areas of physics and mathematics. Additionally, I want to thank Prof. Nelson Canzian for his long-term patience with me and for some personal advices. I am also grateful to Antônio for helping all the students with each bureaucratic doubt we have had. Furthermore, I would like to thank the National Council for Scien- tific and Technological Development (CNPq) for financial support. RESUMO Neste trabalho nós revisamos a construção de soluções de monopo- los magnéticos em teorias de Yang-Mills-Higgs com um campo es- calar na representação adjunta e com um grupo de gauge G simples e simplesmente conexo. Nós revisitamos a solução do monopolo de ’t Hooft-Polyakov e revemos como obter monopolos SU(2) embutidos em teorias com grupos de gauge arbitrários. Nós analisamos alguns as- pectos de monopolos não-abelianos, tais como os padrões de quebra de simetria e o chamado problema de “Global Color”. Na sequên- cia, nós usamos as nossas ferramentas anteriores para construir uma solução de monopolo com grupo de gauge G = SU(n) quebrado em G0 = [SU(p) × SU(n − p) × U(1)]/Z por um campo de Higgs na ad- junta. Nós obtemos soluções de monopolo cujo campo magnético não está na subálgebra de Cartan. E, como esse campo magnético é nulo na direção do gerador do grupo eletromagnético U(1)em, nós chamamos estes monopolos de Monopolos Escuros. Estes Monopolos Escuros de- vem existir em algumas Teorias de Grande Unificação (GUTs) sem a necessidade de introduzirmos um setor escuro. Nós calculamos a hamiltoniana e equações de movimento (EoMs) para estes monopolos, assim como obtemos as soluções aproximadas nos limites r → 0 and r → ∞. Nós também mostramos que sua massa clássica é dada por M = 4πv Ee(λ/e2)/e, onde Ee(λ/e2) é uma função monotonicamente crescente de λ/e2. Para o caso particular de G = SU(5), nós resolve- mos as equações radiais numericamente e obtemos os limites inferior e superior da massa, dados por Ee(0) = 1.294 e Ee(∞) = 3.262. Por fim, nós damos uma interpretação geométrica para a carga magnética não- abeliana destes monopolos e discutimos alguns problemas em aberto. Palavras-chave: Teorias de Gauge, Monopolos Magnéticos, Quebra Espontânea de Simetria, Carga Magnética. RESUMO EXPANDIDO Introdução Neste trabalho nós revisamos a construção de soluções de monopolos magnéticos em teorias de Yang-Mills-Higgs (YMH) com um campo de Higgs φ real na representação adjunta e com um grupo de gauge G simples e simplesmente conexo. Também discutimos alguns aspectos de monopolos não-abelianos, tais como os padrões de quebra de simetria e o problema de Global Color. Na sequência, nós construímos soluções de monopolos com carga magnética na direção abeliana nula. Para isso, consideramos uma teoria de Yang-Mills-Higgs com grupo de gauge G = SU(n), n ≥ 3, quebrado em G0 = [SU(p) × SU(n − p) × U(1)]/Z , por um campo escalar na representação adjunta. Pelo fato de não pos- suírem uma carga magnética na direção abeliana, nossos monopolos não interagem com o campo eletromagnético U(1)em e, por tal motivo, serão chamados de Monopolos Escuros. Por fim, discutimos uma inter- pretação geométrica para a carga magnética não-abeliana dos nossos monopolos e analisamos alguns problemas em aberto. Objetivos Revisar a construção de monopolos magnéticos em teorias de Yang- Mills-Higgs com um campo escalar φ na representação adjunta e com um grupo de gauge G simples e simplesmente conexo. Além disso, pretendemos investigar alguns aspectos das soluções de monopolos em teorias cujo grupo “não-quebrado” G0 é não-abeliano. Contudo, nosso principal objetivo é usar estas ferramentas para construir uma solução de monopolo não-abeliano para o grupo de gauge G = SU(n) com um campo magnético que não esteja na direção da subálgebra de Cartan H, mas sim na direção de operadores escada Eα. Metodologia Durante a realização deste trabalho utilizamos de ferramentas matemáti- cas fornecidas pela teoria de grupos e álgebras de Lie. Também fizemos uso do software MATLAB R para resolver as equações radiais numerica- mente no caso particular dos Monopolos Escuros com G = SU(5). Resultados e Discussões Durante nossa revisão sobre monopolos em teorias de Yang-Mills-Higgs, mostramos que a configuração assintótica das soluções de monopolo pode ser construída ao escolhermos um φ0 = u · H e uma subál- gebra su(2) com geradores Mi, i = 1, 2, 3, tal que M3 ∈ L(G0) e M1,M2 ∈/ L(G0), onde L(G0) denota a álgebra de Lie do grupo não- quebrado G0. A configuração assintótica é obtida a partir de uma (0) configuração de vácuo (φ0,Wi ), por uma transformação de gauge as- sociada ao elemento g ∈ G dado por g(θ, ϕ) = exp (−iϕM3) exp (−iθM2) exp (+iϕM3) . Adicionalmente, construímos um ansatz bastante geral que é usado ao longo de toda a dissertação. Na sequência, discutimos o monopolo de ’t Hooft-Polyakov e analisamos aspectos de soluções de monopolos não-abelianos. Todavia, nossos principais resultados são relacionados aos Monopo- los Escuros construídos para o grupo de gauge G = SU(n). Nós mostramos que estes monopolos podem ser construídos com os ger- adores Mi definidos como combinação linear de operadores escada Eα. Ademais, provamos que estes monopolos não possuem uma carga mag- nética na direção U(1)em. Nós calculamos a hamiltoniana e equações de movimento para estes monopolos, assim como obtivemos as soluções aproximadas nos limites r → 0 e r → ∞. Mostramos que sua massa clássica é dada por 4πv M = E˜(λ/e2) , e onde E˜(λ/e2) é uma função monotonicamente crescente de λ/e2. Para o caso particular dos monopolos escuros com G = SU(5), encontramos os limites inferior e superior de massa dados por E˜(0) = 1.294 e E˜(∞) = 3.262. Por fim, nós damos uma explicação geométrica para a carga magnética não-abeliana destes monopolos, mostramos que ela é quantizada em unidades de 8π/e e está associada à uma simetria ass- intótica das configurações dos campos de gauge e Higgs. Considerações Finais Esta solução de Monopolos Escuros deve existir em algumas Teorias de Grande Unificação (GUTs), onde o campo de Higgs está na adjunta. Além disso, como nosso monopolo não possui uma carga magnética na direção U(1)em, ele pode ter uma pequena contribuição à Matéria Es- cura no Universo. Contudo, ainda que tenhamos construído as soluções de Monopolos Escuros, existem algumas questões em aberto. Elas são referentes à estabilidade dessas soluções e às implicações cosmológicas destes monopolos. Palavras-chave: Teorias de Gauge, Monopolos Magnéticos, Quebra Espontânea de Simetria, Carga Magnética. ABSTRACT In this work we review the construction of magnetic monopole so- lutions in Yang-Mills-Higgs theories with an adjoint scalar field and a simple and simply-connected gauge group G. We revisit the ’t Hooft- Polyakov monopole solution and review how one can construct SU(2)- embedded monopoles in theories with larger gauge groups. We analyze some aspects of non-abelian monopoles, such as the symmetry break- ing pattern and the so-called “Problem of Global Color”. Then, we use our previous tools to construct a monopole solution with gauge group G = SU(n) broken to G0 = [SU(p) × SU(n − p) × U(1)]/Z by a Higgs field in the adjoint representation.
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