Cosmology and Particle Physics in Heterotic Orbifolds Andreas N. M¨utter Dissertation Physik–Department T75 Technische Universität München Technische Universität München Fakultät für Physik Cosmology and Particle Physics in Heterotic Orbifolds Andreas Nikolaus M¨utter Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigten Dissertation. Vorsitzender: Prof. Dr. Lothar Oberauer Prüfer der Dissertation: 1. Prof. Dr. Andreas Weiler 2. Prof. Dr. Alejandro Ibarra Diese Dissertation wurde am 27.05.2020 bei der Technischen Universität München eingereicht und durch die Fakultät für Physik am 10.09.2020 angenommen. Cosmology and Particle Physics in Heterotic Orbifolds Kosmologie und Teilchenphysik in heterotischen Orbifaltigkeiten Andreas N. Mütter Abstract Orbifold compactifications of the heterotic superstring can yield realistic models of particle physics. This thesis aims to study the properties of the resulting four-dimensional field theories. In particular, we show that the cosmological constant in heterotic orbifold theories motivates = 1 supersymmetric theories. We also provide a rigorous way to study discrete flavorN symmetries as remnants of higher-dimensional gauge symmetries. Finally, we show that massive string states provide a viable candidate for dark matter. Zusammenfassung Heterotische Superstrings, die auf Orbifaltigkeiten kompaktifiziert sind, können realistische Modelle für die Teilchenphysik liefern. Die vorliegende Arbeit untersucht die daraus resul- tierenden, vierdimensionalen Feldtheorien. Wir zeigen, dass die kosmologische Konstante in heterotischen Orbifaltigkeitstheorien = 1 Supersymmetrie motiviert. Darüber hinaus geben wir eine rigorose Methode an, mitN der diskrete (Flavor-)Symmetrien als Überreste einer höherdimensionalen Eichtheorie verstanden werden können. Schließlich zeigen wir, dass schwere Stringzustände einen realistischen Kandidaten für Dunkelmaterie darstellen können. iv Contents 1 Introduction1 2 Heterotic string theory on orbifolds7 2.1 Overview.....................................7 2.2 Heterotic strings in ten dimensions......................8 2.2.1 The worldsheet action.........................8 2.2.2 Heterotic strings............................ 12 2.3 Orbifolds..................................... 16 2.3.1 Geometric construction......................... 16 2.3.2 Closed strings on orbifolds....................... 18 2.3.3 Geometric eigenstates......................... 18 2.3.4 Orbifold partition function....................... 19 2.4 Heterotic strings on orbifolds.......................... 20 2.4.1 Gauge embedding............................ 20 2.4.2 Twisted states.............................. 21 2.4.3 The heterotic orbifold projection................... 21 2.4.4 The heterotic partition function on orbifolds............. 22 2.5 Heterotic model building............................ 23 3 The cosmological constant in non-supersymmetric compactifications 25 3.1 The cosmological constant in heterotic orbifolds............... 26 3.2 Compactification and supersymmetry..................... 28 3.2.1 Representation theory of the geometric point groups......... 28 3.2.2 Action of twists on target-space spinors................ 29 3.2.3 Killing spinors and supersymmetry breaking............. 30 3.3 Vanishing of the partition function...................... 33 3.3.1 Heterotic partition function with twists and gauge embedding... 33 3.3.2 A Riemann identity for vanishing rightmover partition functions.. 36 3.3.3 Types of orbits and a minimal condition............... 39 3.4 Group theoretical non-existence proof..................... 40 3.4.1 Survey of point groups for toroidal orbifolds............. 41 3.4.2 Group-theoretical conditions...................... 42 3.4.3 Non-existence proof by enumeration.................. 43 3.5 Loopholes beyond toroidal symmetric orbifolds................ 45 3.6 Conjecture for general discrete groups..................... 46 3.7 Concluding remarks............................... 48 v vi Contents 4 Discrete gauge symmetries from orbifolds 51 4.1 Introduction................................... 51 4.2 Gauge theories in extra dimensions...................... 53 4.3 Residual gauge symmetries........................... 58 4.3.1 Unbroken continuous gauge symmetries............... 59 4.3.2 Unbroken discrete gauge symmetries................. 60 4.4 Examples and applications........................... 61 4.4.1 Gauge origin of D-parity and left-right parity............ 61 4.4.2 Non-Abelian residual symmetries................... 65 4.5 Discrete remnant symmetries from Weyl reflections............. 72 4.6 Summary.................................... 76 5 String scale interacting dark matter 79 5.1 Introduction................................... 79 5.2 Thermal production of dark matter...................... 81 5.2.1 Boltzmann equation.......................... 81 5.2.2 Freeze-out production......................... 86 5.2.3 Freeze-in production.......................... 87 5.3 Stable dark matter at the string scale..................... 88 5.3.1 Non-contractible cycles and stable particles.............. 88 5.3.2 Stable dark matter in the Z2 Z2–5–1 orbifold........... 90 5.4 Interactions between dark matter and× the standard model......... 91 5.4.1 Kähler potential terms......................... 94 5.4.2 Superpotential terms.......................... 95 5.5 Dark matter production from Kähler potential terms............ 97 5.6 Results...................................... 100 5.7 Conclusions................................... 102 6 Discussion 105 A Jacobi theta-functions: Definitions and useful identities 109 B A more general Riemann identity 111 C Group-theoretical appendix 113 C.1 Vector and spinor representations....................... 113 C.2 An example with non-isomorphic spinor and vector groups......... 114 D D-parity in Pati-Salam from orbifolding 117 E Massive U(1) gauge bosons from string theory 119 Bibliography 123 1 Introduction Today, the standard model of particle physics (SM) is by far one of the most thoroughly tested theories known to mankind. It successfully describes the interactions of the known elementary particles in the language of quantum field theory (QFT), where the fundamental entities of our world are point particles. The predictions of the SM have been tested (and confirmed) experimentally to an astonishing precision, with the capstone being the discovery of a scalar boson at the Large Hadron Collider (LHC) in 2012, whose properties match those of the Higgs boson predicted in the 1960s [1,2]. Despite its tremendous success in explaining observed phenomena in the domain of high-energy particle physics, the standard model is known to have various shortcomings, both of conceptual nature and by being unable to explain some observed phenomena (like, e.g., the origin of dark matter). A commonly agreed viewpoint is that the SM might only be an effective theory valid at low energies, and it is expected that new physics will enter at some energy scale that lies well above the electroweak scale. Throughout the literature, the conceptual shortcomings and their possible solutions have been used as guiding principles on the lookout for signatures of new physics. Among the various conceptual questions being left unanswered by the SM are the following two issues, which we will discuss now in more detail as they will play a major role in the remainder of this thesis. The first issue arises from the following question [3]: due to the fact that the Planck and the electroweak scale are separated by 16 orders of magnitude, the values for the Higgs mass have to be tuned up to an “unnatural” precision in order to obtain the observed value also after including quantum corrections, if new physics effects are expected to enter at a high energy scale. This is commonly referred to as the hierarchy problem. The reason for this fact lies in the corrections to the Higgs mass by its self-interaction, which goes roughly as Λ2, where Λ is the high energy new physics scale. One of the arguably most elegant solutions to this problem is supersymmetry (SUSY). There, it is postulated that all particles (e.g. in the standard model) have a so-called superpartner that has the exact same quantum numbers except that it obeys the opposite spin-statistics. For each bosonic degree of freedom, there must exist a fermionic one in the same representation of the gauge algebra, and vice versa. It has been shown by both direct calculations and by general theorems that this symmetry removes the quadratic dependence of the Higgs boson mass on the cutoff scale, leading to a situation where much less finetuning is needed in order to match observations. Not only does supersymmetry solve the hierarchy problem, it also makes specific predictions of new physics: as the superpartners of the SM particles are charged under the gauge 1 2 Chapter 1. Introduction symmetry, it is expected that they will eventually show up in collider experiments like the LHC. As of today, and unlike the standard model matter, no superparticle has been found by experiment, putting the entire framework of SUSY to question. The second issue is of rather different nature: when examining the renormalization group (RG) evolution of gauge couplings in (supersymmetric extensions of) the standard model, they seem to meet at an energy scale of around 1015 GeV. This fact has been interpreted to indicate that the gauge symmetries of the standard model