A new mechanism for symmetry breaking from nilmanifolds
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Springer
Received: March 10, 2020 Accepted: May 3, 2020 Published: May 25, 2020
David Andriot,a Alan Cornell,b Aldo Deandrea,c Fabio Dogliottic and Dimitrios Tsimpisc a
Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria b Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa. c IP2I, Universit´e de Lyon, UCBL, UMR 5822, CNRS/IN2P3 4 rue Enrico Fermi, 69622 Villeurbanne Cedex, France
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We present a method to obtain a scalar potential at tree level from a pure gauge theory on nilmanifolds, a class of negatively-curved compact spaces, and discuss the spontaneous symmetry breaking mechanism induced in the residual Minkowski space after compactification at low energy. We show that the scalar potential is completely determined by the gauge symmetries and the geometry of the compact manifold. In order to allow for simple analytic calculations we consider three extra space dimensions as the minimal example of a nilmanifold, therefore considering a pure Yang-Mills theory in seven dimensions. Keywords: Beyond Standard Model, Field Theories in Higher Dimensions, Large Extra Dimensions ArXiv ePrint: 2002.11128
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)122
JHEP05(2020)122
A new mechanism for symmetry breaking from nilmanifolds
Contents 1
2 Reduction of 7-dimensional Yang-Mills theory to 4 dimensions 2.1 Nilmanifolds 2.2 The truncation
3 3 6
3 The vacuum structure
8
4 From SU(3) to SU(2) × U(1)
9
5 Other examples 5.1 The case of SU(5) 5.2 The case of SU(6)
12 12 13
6 Conclusions
15
1
Introduction
The number of theories beyond the Standard Model (BSM) has remained incredibly diverse, even in light of the range of experimental results of the past decade. Despite the attractiveness of the idea of a high energy theory allowing to obtain all the fundamental interactions, and the efforts towards reaching such a goal, it is not yet possible to obtain in a unique and well defined way the Standard Model (SM) of particle physics from fundamental principles alone. Starting instead from the low energy side allows us to implement in the model building the known experimental facts, but typically lacks the uniqueness and the predictivity of a complete fundamental theory. Somewhere in the middle, and taking inspiration from string theory, compactification of the extra space dimensions can allow us to partially investigate some of the questions which are relegated to free parameters in an effective theory at lower energy. A familiar example are extra compact dimensions which can be large with respect to the Planck scale, and which may lead to some measurable effects [1–5]. Compact dimensions used in this setup are usually flat. Also positivelycurved compact spaces are used, such as the D-sphere and discrete quotients thereof (see for example [6]). A second
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