Theoretical uncertainties in the supersymmetric landscape

Supersymmetry (SUSY) is a long-lauded extension to the Standard Model of particle physics that potentially answers many of the Standard Model's outstanding questions. As the experimental endeavor progresses, large swaths of the SUSY parameter space are being constrained and ruled out, particularly within the low mass regions and the simplest incarnations of SUSY. However, many areas of SUSY remain potentially valid, with experimental tests still forthcoming. These higher mass and less simplified versions are also the ones that have received the least amount of focus hitherto. As we move to this new frontier, we must ask the question: How well do we understand the uncertainties inherent in the predictions of SUSY? In this dissertation, I present results that help answer this question. To do so, I consider benchmark scenarios from several different SUSY frameworks and evaluate these benchmark points through different analysis pipelines. I and that the theoretical uncertainties in the calculation of the SUSY sparticle spectrum, tame in the electroweak (low scale) SUSY models, are significant in high scale theories. In all cases, even small differences lead to substantial uncertainties in dark matter observables. Furthermore, important uncertainties can be introduced simply by the combination of tools included in one's analysis pipeline. Finally, I will discuss future work on aver physics observables which we expect to exhibit similar sensitivities to the details of SUSY calculations. As the search for physics beyond the Standard Model ramps up and we hope that exciting discoveries are around the corner, how well we understand predictions from theory { how well we understand the uncertainties in the calculations of any model of new physics, including SUSY { is increasingly crucial.

THEORETICAL UNCERTAINTIES IN THE SUPERSYMMETRIC LANDSCAPE by Paul Bergeron A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah August 2019 Copyright c Paul Bergeron 2019 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL This dissertation of Paul Bergeron has been approved by the following supervisory committee members: , Chair Pearl Sandick April 18, 2019 Date Approved , Member Stephan LeBohec April 18, 2019 Date Approved Douglas Bergman , Member April 18, 2019 Date Approved Paolo Gondolo , Member April 18, 2019 Date Approved Zvonimir Rakamaric , Member Date Approved and by the Department of Peter Trapa Physics and Astronomy David B. Kieda of the Graduate School. , Chair of and by ABSTRACT Supersymmetry (SUSY) is a long-lauded extension to the Standard Model of particle physics that potentially answers many of the Standard Model’s outstanding questions. As the experimental endeavor progresses, large swaths of the SUSY parameter space are being constrained and ruled out, particularly within the low mass regions and the simplest incarnations of SUSY. However, many areas of SUSY remain potentially valid, with experimental tests still forthcoming. These higher mass and less simplified versions are also the ones that have received the least amount of focus hitherto. As we move to this new frontier, we must ask the question: How well do we understand the uncertainties inherent in the predictions of SUSY? In this dissertation, I present results that help answer this question. To do so, I consider benchmark scenarios from several different SUSY frameworks and evaluate these benchmark points through different analysis pipelines. I find that the theoretical uncertainties in the calculation of the SUSY sparticle spectrum, tame in the electroweak (low scale) SUSY models, are significant in high scale theories. In all cases, even small differences lead to substantial uncertainties in dark matter observables. Furthermore, important uncertainties can be introduced simply by the combination of tools included in one’s analysis pipeline. Finally, I will discuss future work on flavor physics observables which we expect to exhibit similar sensitivities to the details of SUSY calculations. As the search for physics beyond the Standard Model ramps up and we hope that exciting discoveries are around the corner, how well we understand predictions from theory – how well we understand the uncertainties in the calculations of any model of new physics, including SUSY – is increasingly crucial. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTERS 1. 2. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 A Brief History of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Brief Overview of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A Brief Description of SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 A Motivation for SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 SUSY’s Particle Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Symmetry Breaking and SUSY Models . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Timeliness of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 12 12 14 18 20 23 THEORETICAL UNCERTAINTIES IN THE CALCULATION OF SUPERSYMMETRIC DARK MATTER OBSERVABLES . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Methodology of the Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Benchmark Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Pipeline Structure and Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Details of the Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.1 Spectrum Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.2 Higgs Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.3 Dark Matter Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Physics of Neutralino Dark Matter Benchmarks . . . . . . . . . . . . . . . . . . . 2.3.1 Coannihilation of B̃ with Light Scalars . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Well-tempering of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 A-funnel Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Pure Higgsino (h̃) Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Results: PMSSM Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Bino-Stop Coannihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Bino-Squark Coannihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Pure Higgsino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Well-Tempered Neutrlino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 A-funnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Summary: Broad Trends in the PMSSM Analysis . . . . . . . . . . . . . . . . . 37 40 42 42 44 45 46 47 48 51 52 52 53 54 54 57 59 60 61 62 63 2.5 Results: GUT Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Comparison of EWSB Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Dark Matter Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.1 CMSSM Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.2 NUHM Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1 Uncertainties in Flavor Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Physics of the Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1 Radiative B 0 Decays: b → sγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2 Neutral Bs Meson Decay: Bs0 → µ+ µ− . . . . . . . . . . . . . . . . . . . . . 3.1.1.3 The Muon Anomalous Magnetic Moment, aµ . . . . . . . . . . . . . . . . 3.1.2 Comparison Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uncertainties from the Computational Environment . . . . . . . . . . . . . . . . . . . 3.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 64 66 71 71 74 75 89 97 97 97 99 100 101 103 104 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 v LIST OF FIGURES 1.1 Gauge Coupling Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 CfA Survey of Galaxy Distribution and Dark Matter Structure . . . . . . . . . . . . 5 1.3 CfA2 Survey, SDSS, 2dFGRS Galaxy Distributions and Dark Matter Structure 5 1.4 Particle Content of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 The Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 1-Loop Corrections to mh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1 Pipeline Paths for Dark Matter Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 PMSSM Results: Dark Matter Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 CMSSM Parameter-Space Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.4 NUHM Parameter-Space Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.5 CMSSM and NUHM Results: Dark Matter Observables . . . . . . . . . . . . . . . . . . 72 3.1 Radiative Contributions to b → s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2 Contributions to the Leptonic Decay of Bs0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Radiative Vertex Corrections to the Muon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 LIST OF TABLES 1.1 Chiral and Gaugino Supermultiplets and their SM Partners . . . . . . . . . . . . . . . 15 2.1 GUT-Scale Benchmark Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Snowmass and Mastercode Pipeline Version Numbers . . . . . . . . . . . . . . . . . . . 44 2.3 RGE and Radiative Correction Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 Spin-independent & Dependent Form Factor Comparison . . . . . . . . . . . . . . . . 51 2.5 PMSSM Neutralino and Chargino Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6 PMSSM Squark Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 PMSSM Higgs Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.8 CMSSM and NUHM Neutralino and Chargino Masses . . . . . . . . . . . . . . . . . . . 65 2.9 CMSSM and NUHM Squark Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.10 CMSSM and NUHM Higgs Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.11 CMSSM and NUHM Higgsino Mass Parameters (µ) . . . . . . . . . . . . . . . . . . . . . 68 2.12 PMSSM Dark Matter Relic Density, Raw Values . . . . . . . . . . . . . . . . . . . . . . . 76 2.13 PMSSM Dark Matter Relic Density, Percent Differences . . . . . . . . . . . . . . . . . 77 2.14 PMSSM Dark Matter Annihilation Cross Section, Raw Values . . . . . . . . . . . . 78 2.15 PMSSM Dark Matter Annihilation Cross Section, Percent Differences . . . . . . . 79 2.16 PMSSM Dark Matter Scattering Cross Section, Raw Values from FeynHiggs . 80 2.17 PMSSM Dark Matter Scattering Cross Section, Raw Values from SUSY-HD . . . 81 2.18 PMSSM Dark Matter Scattering Cross Section, Percent Differences from FeynHiggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.19 PMSSM Dark Matter Scattering Cross Section, Percent Differences from SUSY-HD 83 2.20 PMSSM Dark Matter Relic Density, Raw Values . . . . . . . . . . . . . . . . . . . . . . . 84 2.21 PMSSM Dark Matter Annihilation Cross Section, Raw Values . . . . . . . . . . . . 85 2.22 PMSSM Dark Matter Scattering Cross section, Raw Values from FeynHiggs . 86 2.23 PMSSM Dark Matter Scattering Cross section, Raw Values from SUSY-HD . . . 87 2.24 CMSSM EWSB Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1 Flavor Observables Computed by each Computational Tool . . . . . . . . . . . . . . . 102 ACKNOWLEDGMENTS I would like to thank family and friends here – Greg, Flo, Henna, JiHee, and Jessica – and afar – David, Alyssa, Amita, Justin, Caleb, Sophia, and Frances – whose support, commiseration, and laughter helped buoy my journey over the past 6 years. I would also like to especially thank my advisor, Professor Pearl Sandick, for her guidance, support, and expertise all of which were paramount in seeing this journey to come to fruition. CHAPTER 1 INTRODUCTION The Standard Model (SM) of particle physics describes the content of the universe through a rich tapestry of subatomic particles intertwined through the 3 fundamental forces of nature. This theory has withstood tests of enormous accuracy [1], predicted exotic matter subsequently found in nature [2], and was finalized with the discovery of the Higgs boson – the predicted capstone tying together the SM [3, 4]. Despite being a lauded success, aspects of the universe that one would hope the SM would solve remain mysteries. These are not problems with the theory of the SM itself so much as indications that the SM is part of a larger, still incomplete, picture of the universe. Some of these problems are aesthetic, such as how to unify the fundamental forces of nature – no indication is given that this must be the case, but, as the energy scale of interactions increases, the current theory places the strengths of the three fundamental forces tantalizingly close to one another [5] as in Figure 1.1. Currently the fundamental forces are thought about in terms of particle interactions, mediated by gauge bosons, but gravity does not fit into this framework; can gravity be recast to operate on the quantum level as interactions mediated by a force carrier, or is it fundamentally different [6]? Other problems exist as well, and are informed through experimentation. For example, the SM is constructed with a symmetry between matter and antimatter – that the two should be made in nearly equal amounts – yet only matter appears in the universe, indicating a very particular imbalance that cannot be explained by the SM [7]. Another problem – one of particular interest to us that we will discuss below – is that myriad astronomical observations suggest that the familiar matter of the SM is not enough to account for all of the mass in the universe. In physics, supersymmetry was first studied about 50 years ago as a curiosity that allowed for a final, algebraic extension to the SM [12–16]. Since then, it has evolved into 2 60 50 E&M 40 Weak α−1 30 20 Strong 10 0 2 Standard Model SUSY 4 6 8 10 12 14 16 18 log(Q [GeV]) Figure 1.1: Gauge Coupling Unification. Plotted here are the couplings of the fundamental forces, α: the electromagnetic force (E&M; orange), the weak nuclear force (cyan), and the strong nuclear force (magenta) as a function of the energy scale of interactions, Q. Dashed lines show the SM values which do not exactly unify to a single point at a high energies. Unification of the fundamental forces could happen with new, beyond the SM physics, as seen by the solid lines converging at a single point for an instance of the minimal supersymmetric standard model [5, 8–11]. a robust theoretical framework that can solve or help solve many of the problems that the SM does not. In short, and as we will discuss below, it introduces a new operator to extend the SM. By doing so, a second, rich tapestry of particle content unfolds alongside the SM’s. This can happen in such a way that the new particles may allow for correct predictions of the Higgs mass or may be natural candidates to explain the universe’s missing mass. Due to such tantalizing, theoretical successes, it has become a mainstream theory for physics beyond the Standard Model (BSM), and is thus the focus herein. 1.1 A Brief History of Dark Matter Currently, the general consensus amongst physicists and astronomers is that roughly 84% of the universe’s matter content is non-luminous matter that is not a part of the SM [17]. Dark matter was first proposed by Fritz Zwicky in 1933 to explain an observed discrepancy between the virial speed of the galaxies (due to their gravitational interactions) in the Coma Cluster and the velocity implied by their observed Doppler shift. His solution was that there must be some non-luminous – dark – matter increasing the mass of the 3 galaxy and thus allowing for greater virial speeds that agree with those implied by the galaxy’s Doppler shifts [18]. Similar discrepancies were indicated by other studies [19, 20], but Zwicky’s Dunkle Materie was largely ignored until further observations [21, 22] were conducted in the 1950’s and did not begin to be embraced until the 1970s [23] when more other avenues of evidence began to emerge.1 First, the rotation rate of stars about their parent galaxy – including Andromeda [25, 26] and Triangulum [27] – were being measured to increasingly large radii and being found to be far larger than would be expected from Keplerian dynamics and visible matter. Second, the ubiquitous shape of spiral galaxies was shown to be unstable and thus should not be observed [28]. In both cases, the majority of the galaxy’s mass was assumed to reside primarily in the central bulge. This assumption is certainly warranted as this is where most of the luminous matter is observed to be. Relaxing this assumption, it became clear that it was necessary to posit some unseen dark matter to resolve the discordant observations [29, 30]. While the early concordance of evidence certainly made the dark matter hypothesis compelling, it by no means points specifically to dark matter, let alone dark matter from any particular BSM scenario. Fundamentally, what has been observed is a discrepancy between the observed mass of stars and galaxies and their observed dynamics. This mass-acceleration problem could, potentially, be just as easily attacked by focusing on the acceleration side of the discrepancy. Here, gravity would need to be modified to allow new dynamics in the low acceleration limit while achieving Newtonian dynamics in the, comparably, large acceleration limit. Such frameworks of modified Newtonian dynamics are known as “MOND” [31–33]. However, while MOND can certainly account for observations, such as galactic rotation curves, numerous contradictions arise when MOND is applied elsewhere, such as in the dynamics of dwarf galaxies [34, 35]. Additionally, experimental evidence suggests that not only is MOND not valid (at least in its current forms) but that some non-luminous matter must be present. For instance, lensing data of the Bullet Cluster places the majority of the mass of each colliding cluster ahead of the gas (which is the primary component of the 1 While his idea was correct, his calculations were not due, to the accuracy of the data then available [24]. 4 luminous matter of the galaxies). This suggests that either non-luminous, weakly interacting, dark matter particles are present and unhindered by the collision, or, in MOND’s case, the gravitational potential is advanced, introducing questions of causality [36]. Meanwhile, observations [37] of NGC 7252 show a galaxy resulting from a recent merger where the cores have already merged and relaxed while tidal-tails of gas are still coalescing. Simulations with MOND do not match the observations as the parent galaxies need to be imbedded in an extended distribution of matter in order for their cores to lose their energy and angular momentum in the time observed [38]. Furthermore, lacking any non-baryonic matter, MOND falls into severe tension with baryon acoustic oscillations in the early universe [39]. With no known viable path for addressing the acceleration side of the observed massacceleration discrepancy, the problem reduces to: what is dark matter? For it to have the observed gravitational effects on astrophysical scales and to be non-luminous, the mystery matter must be massive with no net charge under electromagnetism (or at least have extremely faint electromagnetic emissions). Plenty of astrophysical objects can already satisfy these criteria, and are known as massive compact halo objects (MACHOs). These can range from mundane objects such as gas or brown dwarfs to the more exotic such as black holes [40]. However, cosmic microwave background probes such as the Wilkinson Microwave Anisotropy Probe (WMAP) [41] and Planck [17] satellites have set limits on the amount of baryonic matter in the universe. Along with limits from Big Bang Nucleosynthesis [42], these results have excluded the possibility of MACHOs existing in large enough numbers to solve the dark matter problem. While we will not take this perspective, it is worth noting that MACHOs have garnered renewed interest with the Laser Interferometer Gravitational-Wave Observatory’s gravitational wave results [43], as primordial blackholes [44] are a potential dark matter candidate that could satisfy cosmological observations (e.g., References [45– 48]). This leaves non-baryonic matter as the only refuge left to explain dark matter. With the then recent indications of non-zero neutrino mass, Yakov Zel’dovich proposed the neutrino as a candidate for dark matter [58]. Despite fitting all of the criteria (weakly interacting, massive, electrically neutral), problems were quickly noted with structure formation in the universe, as depicted in Figures 1.2 and 1.3, when dark matter is composed of neutrinos or other relativistic particles, models dubbed as “hot dark matter models” (HDM) [59, 60]. 5 Figure 1.2: CfA Survey of Galaxy Distribution and Dark Matter Structure. Reproduced with permission from Reference [49] are equal area projections of the northern sky, bounded by a declination of 0 and galactic latitude +40. The lower left oculus is shows the results of the CfA Redshift Survey [50]. The rest are results of simulation, with the upper left as well as the middle oculi showing results when employing HDM frameworks, while the rightmost oculi depict results when employing CDM frameworks. Figure 1: The galaxy distribution obtained from spectroscopic redshift surveys and from mock catalogues constructed from cosmological simulations. The small slice at the top shows the CfA2 Figure 1.3: CfA2 Survey, SDSS, 2dFGRS Galaxy Distributions and Dark Matter Struc“Great Wall” , with the Coma cluster at the centre. Drawn to the same scale is a small section of the SDSS, in which an even larger “Sloan Great Wall” has been identified . This is one of the largest ture. Reproduced with observed permission from Reference [51] areoverlarge structure comparisons structures in the Universe, containing over 10,000 galaxies and stretching more than scale 1.37 billion light years. The wedge on the left shows one-half of the 2dFGRS, which determined distances of galaxy catalogues andto more simulations. Insouthern blue, seeof 2the CfA2 Redshift Survey’s [52] “Great than 220,000 galaxies in the sky outwe to a depth billion light years. The SDSS has a similar depth but a larger solid angle and currently includes over 650,000 observed redshifts in the northern sky. on At the the bottom and on the right,Cluster mock galaxy surveys constructed using semi-wedge), the Sloan Digital Wall” galaxy filament centered Coma (top, small analytic techniques to simulate the formation and evolution of galaxies within the evolving dark matter distribution of the “Millennium” simulation are shown, selected with matching survey geometries and Sky Survey’s [53] “Sloan Great Wall” galaxy filament (top, large wedge), and the “cosmic magnitude limits. web” of galaxies [54] from the 2-degree Field Galaxy Redshift Survey [55] (left wedge). 28 In red, we see mock survey results obtained from the Millennium Run’s [56] structure formation simulation of using a ΛCDM framework [57] depicting areas with similar structure as observed in the opposing wedges. 3 100 5 6 Figure 1.2 illustrates results from early simulations [49] that show only non-relativistic dark matter, i.e., cold dark matter (CDM), allows for structure formation as observed [50]. Figure 1.3, meanwhile, we see that more recent results from simulations [61, 62] and observations [63] that analyze larger scales over more of the universe’s history indicate that the concordance in favor of CDM models has only grown stronger. 1.2 A Brief Overview of the Standard Model Before focusing on a BSM scenario, let us review the landscape of the SM. The SM particle content can be divided into the fermions (particles with half-integer spin) and bosons (particles with integer spin); see Figure 1.4. The fermions comprise what we would typically think of as matter and are ordered into three generations, each consisting of two quarks (shown in purple) and two leptons (shown in green). The first generation is the most familiar to us, consisting of up and down quarks (u, d) as well as electron (e− ) and electron neutrino (νe ) leptons. The quarks (along with the force mediating gluons) are what comprise both protons and neutrons while electrons are the familiar orbiters of atoms. The ghostly neutrino is less familiar but arises whenever electrons are involved with phenomena of the weak nuclear force, such as in nuclear fission or fusion. The other two generations follow a similar pattern – an up-like quark, a down-like quark, an electron-like particle, and a corresponding neutrino. The leptons are very similar to one another across the generations, with the muon (µ− ) and tau (τ − ) being quite similar to more massive versions of the electron. Each have an associated type of neutrino as well, though the mass hierarchy is not known, with only upper bounds currently placed on each mass by experiments. These particles are clearly different between the generations as the intra-generation flavor is conserved in interactions. For example, if a neutron (0 leptons) decays into a proton, an electron, and a neutrino (n → p, e− , ν̄e ); the electron (+1 electron lepton-number) is balanced by the neutrino being an electron anti -neutrino (−1 electron lepton-number), so the total lepton number adds to 0 and is conserved. Similarly, a muon may decay into an electron, an electron anti-neutrino, and a muon neutrino (µ− → νµ , e− , ν̄e ) – the muon lepton number is conserved in the reaction (+1 before and after) as well as the electron lepton number (0 before and 0 = (+1) + (−1) after). A similar kind of conservation happens for the quarks under the strong and electromag- 7 Figure 1.4: Particle Content of the Standard Model. Reproduced from Reference [64], we show the particle content of the SM organized by particle type: quarks (purple), leptons (green), gauge bosons (orange), and scalar bosons (yellow). Each box gives the common symbol for the particle and lists the particles mass (top), electric charge (middle, in units of elementary charge), and spin (bottom, in units of ~). 8 netic forces. Here, properties such as ‘strangeness’ were invented to explain the relatively long-lived particles observed in certain air showers [65]. As these observed particles were increasingly understood to be composite particles, the conserved properties were understood to correspond to the existence of different flavors of their constituent particles. The possible constituents are the 6 known quarks and their antimatter partners. Together, they fully explain the plethora of observed particles. Next, there are the gauge bosons (shown in orange), which are the force carriers that mediate particle interactions. These gauge bosons are each associated with a force: the photon (γ) for the electromagnetic force, the W and Z bosons (W ± , Z 0 ) for the weak nuclear force, and the gluons (g) for the strong nuclear force. The photon is the most familiar boson, comprising light and being involved with all electric and magnetic phenomena. Though the weak force is so named by how feeble and fleeting its interactions are, its signature in nuclear physics (such as in fusion processes that allows the Sun to shine) makes the W bosons familiar in their effects. Alternatively, the neutral Z boson is unable to mediate changes of quark flavor as the W bosons do, but it does mediate the interactions of the leptons, allowing for the scattering of neutrinos. As the Z boson, unlike the W bosons, is neutral, this leads to the particle physics maxim: at tree level, there are “no flavor changing neutral currents” (FCNC). Lastly, we have the gluons whose presence is confined to the nucleus, binding quarks together to form hadrons such as nucleons, as well as binding the nucleons together within the nucleus. Finally, the SM also includes a single scalar boson (shown in yellow) – the Higgs boson. This boson couples to any massive particle, but does not mediate any fundamental force. Instead it is a manifestation of the Higgs field and arises as a consequence of the mass generating Higgs mechanism, which we will discuss below. The simplest of these interactions to understand is that of electromagnetism. Here, the Lagrangian is seen to be invariant under global phase transformations – adding an additional phase φ by multiplying the wave function by eiφ – but fixing a local phase transformation (eiφ(x) ) breaks this invariance. As is the case throughout the SM, the name of the game is to upgrade to covariant derivatives (∂µ → ∂µ −ξµ ) in order to fix this perceived shortcoming of the theory. This upgrade works if we let ξµ = −igAµ and let Aµ transform as a function of the phase: Aµ (x) → Aµ (x) + ∂µ φ(x). However, this is nothing more than the 9 beloved gauge invariance found in classical electromagnetism. Furthermore, if we identify g as the fundamental charge qf , we see that our covariant derivative is the quantization of electrodynamic’s canonical momentum. The theory’s dynamics and gauge invariance are baked into allowing the Lagrangian to be invariant under local phase transformations – applying an element of the circle group, eiφ . This tells us something deep about the structure of the theory: that the quantization of electromagnetism is described by a U (1) symmetry. A consequence of this symmetry is an associated field excitation, B, which is known as a “gauge boson” due to its relation to the gauge invariance of the force.2 The situation becomes increasingly more interesting as we consider the other observed properties of the SM. The weak nuclear force involves couplings to particles with the spin-like quantity of weak isospin. In mirror nuclei – where one has the same protons as the other has neutrons and vice versa – it has been found that the binding energy of each is the same. This points to a symmetry where protons and neutrons are effectively identical, with the proton being in a spin-up-like state and the neutron being in a spin-down-like state [66]. Such a symmetry is manifest within the different flavors of both the quarks and leptons. For example, (u, d) is a quark doublet under this symmetry as is the lepton doublet (e, νe )L . As with spin, we have a system whose symmetry transformations are described via the Pauli matrices, σi , and, more generally, an SU (2) symmetry group. Unlike the U (1) symmetry of electromagnetism, the transformation operation now goes as a linear combination of Pauli matrices (i.e., φ → α ~ · ~σ /2). Historically, gauge invariance was not guaranteed for the weak interaction, but due to the success it had with electromagnetism and ultimately in describing nature, it was also wanted here. Just as in electromagnetism, we need to upgrade to covariant derivatives, but now ξµ will be a function of α ~ · ~σ . That is, we are left with three field excitations, unlike in the electromagnetic case, due to the number of Pauli matrices – the isospin raising, lowering, and projection operators. Consequently, this leads to two charged (W ± ) and one neutral (W 0 ) field excitations in order to conserve charge in the relevant particle interactions (e.g., isospin rotation of a proton to a neutron). Together with electricity and magnetism, electroweak theory [67–69] is born out of 2 Note: We have mentioned that the photon (γ) is the gauge boson associated with the electromagnetic interaction. As we will see, this is really a simplification of the full situation and requires the inclusion of the weak force to upgrade and mix the B boson state to a physical particle, the photon. 10 the above. In combining electromagnetism with the weak force, the W 0 and B states mix together to give us the observed parity violating Z 0 of the weak force and the parity conserving γ of electricity and magnetism: A Z0 ! = cos(θw ) − sin(θw ) ! ! sin(θw ) B , cos(θw ) W0 (1.1) for the Weinberg mixing angle, θw . At this level in the theory, all the bosons (W ± , Z 0 , and γ) have zero mass. The final fundamental interaction is the strong nuclear force. Despite it being the strongest of the three forces, its complexity made it the last to be understood. This was due to not only the fact that its effects are confined to the minute region of the nucleus, but also that it stipulates the formation of hadrons – never solitary quarks – in experiments regardless of the energy provided; the latter fact is know understood as asymptotic freedom [70–72]. The strong interaction happens between its field excitations, gluons, and the particles with ‘color’ charge in the SM: the quarks. The quarks themselves can take on any of three color states and combine together to form a color-neutral hadron. These charges are exchanged under the SU (3) symmetry of the strong force via the color-anticolor gluons. Since gluons themselves have color charge, they readily couple to one another, causing significant energy to be stored within the gluon fields. This accounts for the relatively large mass of hadrons despite being comprised by quarks with relatively small masses. The capstone to the entire SM is the Higgs mechanism. In constructing the electroweak interaction, the symmetries involved did not give rise to any mass for the gauge bosons. This is contradicted by experiments [73, 74]. Since the mediators of the weak force are massive while the mediator of the electromagnetic force is massless, some way of breaking the underlying symmetry needs to be included. The Higgs mechanism is such a symmetry breaking process that allows for the otherwise massless gauge bosons to acquire mass [75–77]. The Higgs mechanism is characterized by a complex scalar field φ with a potential V (φ) = µ2 φ2 + λφ4 , (1.2) as depicted in Figure 1.5. The key to the mechanism is that the potential, while having a p local maximum at φmax = 0, has its minimum offset at φmin = −µ2 /2λ. To be able to in- 11 Figure 1.5: The Higgs Potential. Here we plot of Equation 1.2 against the real and imaginary components of the Higgs p field φ. While the local maximum is at φmax = 0, a global minimum at φmin = −µ2 /2λ is where the Higgs boson, h, oscillates as a perturbation of the field. teract weakly, an isospin doublet must be formed, and under a convenient parameterization it can be written as 1 φ= √ 2 ! 0 , v+h (1.3) where the vacuum expectation value v 2 = φ2min /2 and h is an excitation of the field around this minimum. Recasting the Higgs potential (Equation 1.2) and focusing on the quadratic term, we find that 1 1 V (φ) = − (4λv 2 )(v 2 + 2vh + h2 ) + λ(v + h)4 . 2 4 (1.4) We can see that we have a term quadratic in the field which we can identify as a mass term (4λv 2 h2 = m2h h2 ), indicating that a physical boson is to be expected. Once a coupling is included to allow interactions with the electroweak fields, the v-term from φmin leads quite naturally to more mass terms for all the fields except for the photon’s. While seemingly ad hoc, the inclusion of the Higgs field provides a natural way to break the otherwise pristine 12 symmetry of electroweak theory. In doing so, the gauge bosons of the weak interaction to gain mass while the photon remains massless without sacrificing the gauge invariance that created it. Furthermore in 2013, the predicted existence of the Higgs boson was experimentally verified [3, 4], showing that the Higgs mechanism does indeed play a role in nature. 1.3 A Brief Description of SUSY 1.3.1 A Motivation for SUSY The famous quality of the Higgs potential is that its minimum is not where the field p √ strength is zero. Instead, its minimum happens at φmin = −µ2 /2λ = v/ 2, which defined a natural quantity to parametrize the field by in Equation 1.3. The vacuum expectation value is what the Higgs mechanism adds to the weak force in order to generate the masses of the weak force’s gauge bosons. Since we know the masses of the weak gauge bosons, we know the vacuum expectation value of the Higgs boson, and that it is about 250 GeV [78]. This implies that the Higgs boson’s mass should be on the order of 100 GeV [79]. The value of the Higgs mass is one quantity about which the SM is mum. With the perspective that the SM is not the full picture of the universe, we might expect that the Higgs mass would be calculable in some grander theoretical framework. Such calculations perturb the particle’s mass by way of considering loop corrections to it that arise from the Higgs boson’s interactions with other particles. If we regulate these calculations by applying a cut-off at the energy scale ΛUV where we expect the SM to break down and new physics to become important, we find that ∆m2h =− SM |λf |2 2 Λ + h.c. . 16π UV (1.5) That is, the mass should diverge quadratically with ΛUV . This implies that the Higgs boson’s mass should be far greater 100 GeV, but experiments have shown that the Higgs’s physical mass is a mere 125 GeV [3, 4]. In order to solve this so-called hierarchy problem – the problem of the Higgs boson’s mass being smaller than we might expect – new physics that balances out the known corrections must exist. One possible way to accomplish this is to note that while bosons commute mathematically, fermions anticommute. This opposite behavior introduces a relative sign between corrections that arise from interactions between the Higgs and fermions 13 and between the Higgs and bosons. However, fundamental particles have masses that differ over a wide range of values. Thus, while the known particle content does provide some cancellations of one another’s corrections, these cancellations are not significant enough to solve the problem at hand. What would solve the problem is if, for every known boson, there was an equivalent, new fermion with equal mass and similarly a new boson for each of the known fermions. All of these new particles would have to come out of some new theory that adds onto the SM. Fortunately, such a doubling of the number of particles is exactly what is predicted by supersymmetry. The most prominent feature of supersymmetry (SUSY) is the existence of superpartners to the known fundamental particles. In even the most minimal versions of SUSY, this doubles the particle content of the universe. While this seems like a large addition to the SM, mathematically it amounts to adding only one new operator3 – the “supersymmetric operator” Q: Q |fermioni ←→ |bosoni . (1.6) Under the action of Q, the pair of states that get transformed into one another form supermultiplets (“super” due to it being a supersymmetric transformation). In this way, each of the SM states get duplicated: A SM fermion will have a bosonic superpartner (named by appending an “s-” prefix to the original, SM name; e.g., “quark” ↔ “squark”) and a SM boson will have a fermionic superpartner (named by appending an “-ino” suffix to the original, SM name; e.g., “Higgs” ↔ “Higgsino”). This leads quite naturally to a solution to the Higgs mass’s hierarchy problem. Without SUSY, we would calculate (expecting a superseding theory) the radiative corrections to the Higgs mass to go as Equation 1.5, for a fermionic particle, f , where λf is the coupling of the Higgs to said fermion. The divergence is now the energy scale at which we expect SUSY to become relevant to the particle interactions. Now, in a SUSY framework, for every diagram that yields this SM divergence – such as the one in Figure 1.6, left – we would then expect a corresponding diagram – such as the one in Figure 1.6, right – from f ’s supersymmetric, bosonic partner f˜, yielding 3 While a small addition to the mathematical machinery, it has a singular importance: this is the only remaining algebraic extension possible to the SM’s Poincaré algebra [80]. 14 h f˜ f h Figure 1.6: 1-Loop Corrections to mh . Shown here are two, 1-loop corrections to mh from a SM fermion, f (left), and an imagined bosonic superpartner, f˜ (right). ∆m2h = SUSY λf˜ 16π Λ2UV + h.c. . (1.7) Thus the radiative corrections to the Higgs mass cancel each other when added together. However, if the corrections really do vanish, then we have λf˜ = |λf |2 , implying that mf and mf˜ – wrapped up inside each of the couplings – are equal. This, decidedly, is not the case as supersymmetric partners are not readily seen in nature while their more pedestrian, SM cousins are. This implies that the masses are not degenerate between a particle and its partner sparticle, meaning that supersymmetry – if a valid description of nature – is a broken symmetry. As the Higgs mechanism leads to the creation of mass within the SM, one may expect this to be the case in SUSY as well, and thus the mass gap between members of a supersymmetric multiplet to be a sign of SUSY being unphysical. This concern is easily remedied by noting that the sfermions are described by complex scalar fields whose mass terms, m2 φ∗ φ, are gauge invariant and thus may appear naturally in the Lagrangian without requiring them to be generated by the Higgs mechanism. In the case of the gauginos, they are described as elements of a real representation of their gauge group, which avoids the need for their masses to be generated [79]. 1.3.2 SUSY’s Particle Content What are all of these new particles generated by the action of Q? In short, as stated above, we get a duplication of all SM particles in parallel to the SM’s structure. Therefore, we would expect that just as electroweak symmetry breaking gives rise to photons and Z bosons via mixing of B and W 0 states (Equation 1.1), we would also have B̃ and W̃ 0 states that mix to yield photinos (γ̃) and zinos (Z̃). Fundamentally, what we really have are duplicates of all of the SM states, as listed in Table 1.1. This particle content is a bit more than merely a supersymmetric extension of the 15 Table 1.1: Chiral and Gauge Supermultiplets and their SM Partners. The generation index is denoted by i but only the first generation (i=1) shown. Names Spin 0 (ũL , d˜L ) Spin 1/2 Spin 1 ũ∗R d˜∗R (uL , dL ) u†R d†R - Squarks & Quarks Qi ūi d¯i Sleptons & Leptons Li ēi (ν̃, ẽL ) ẽ∗R (ν, eL ) e†R - Higgs & Higgsinos Hu Hd (Hu+ , Hu0 ) (Hd0 , Hd− ) (H̃u+ , H̃u0 ) (H̃d0 , H̃d− ) - Gluino & Gluons - g̃ Winos & W Bosons - W̃ ± , Bino & B Boson - g W̃ 0 B̃ 0 W ±, W0 B0 SM: the SM Higgs sector must be extended to include two chiral multiplets, (Hd0 , Hd− ) and (Hu+ , Hu0 ), which in turn have their own supersymmetric Higgsino partner states. In introducing supersymmetry, the usual SM Higgs sector encounters problems without further extension. For one, electroweak theory is protected against gauge anomalies by virtue of the numbers of known quarks and leptons in the SM, allowing for traceless combinations of the third components of hypercharge and isospin (T r[Y32 T ] = T r[Y 3 ] = 0). Adding one fermionic chiral multiplet offsets this balance and therefore requires a second multiplet with an opposite sign to its Yukawa couplings to be introduced [79]. Furthermore, and unlike in the SM, the Yukawa interactions of these Yukawa fermions must be included in the superpotential of the theory, but Hu+ may only give masses to the up-like quarks, forcing the inclusion of a second multiplet. [79, 81] These supermultiplets mix together to form the physical fields. Just as in the SM, we expect these components to yield Higgs bosons and Nambu-Goldstone bosons (G), the latter being absorbed in electroweak symmetry breaking to produce the physical W and Z bosons. These mix together [81] through ! G+ = H+ cos(β) sin(β) − sin(β) cos(β) ! Hd−∗ Hu+ ! to give a set of charged states; through ! ! ! G0 sin(β) − cos(β) Im(Hu0 ) = A cos(β) sin(β) Im(Hd0 ) (1.8) (1.9) 16 to give a neutral Nambu-Goldstone boson and a neutral, pseudoscalar Higgs boson, A; and through h H ! = ! ! cos(α) sin(α) Re(Hu0 ) − sin(α) cos(α) Re(Hd0 ) (1.10) to give two neutral CP-even Higgs bosons (by convention, h is reserved for the lower mass boson and is identified with the SM Higgs boson). Here, the angles denote the ratio of the Higgs vacuum expectation values, tan(β) = v1 /v2 , and the Higgs mixing angle defined by tan(α) = (m2A − MZ2 ) cos(2β) + q (m2A + MZ2 )2 − 4m2A MZ2 cos2 (2β) (m2A + MZ2 ) sin(2β) , (1.11) where mA is the pseudoscalar Higgs mass and MZ is the mass of the Z boson. Lastly, after some supersymmetry breaking process has occurred (as well as the electroweak symmetry breaking) the mass eigenstates may not necessarily be the gauge eigenstates listed in Table 1.1. The neutral gauge states enter into the Lagrangian via  Lneutral = B̃ W̃ 0 H̃d0  B̃  0 W̃   H̃u0 Mneutral   H̃ 0   d H̃u0 (1.12) under the action of √  √ M1 0 −g 0 vd / 2 g 0 vu / 2 √ √    0 M gv / 2 −gv / 2 2 u d  . = −g 0 v /√2 gv /√2 0 −µ  d d   √ √ −µ 0 g 0 vd / 2 −gvu / 2  Mneutral (1.13) Here, B̃ and W̃ 0 are the bino and wino states (partners of the B and W 0 fields of electroweak theory) with masses M1 and M2 , respectively; H̃d0 and H̃u0 are the partners of the neutral components of the down and up-type Higgs fields typically added into SUSY, with their vacuum expectation values vd and vu , respectively; and g & g 0 are the couplings of the weak fields. The off-diagonal nature of this mass matrix informs us that the gauge eigenstates are not the mass eigenstates. That is, just as in the electroweak case, the neutral gauge eigenstates (B̃, W̃ 0 , H̃d0 , H̃u0 ) mix to form the observable neutralinos expected if SUSY is valid. 17 Similarly, the charginos are composed of a mixture of charged gaugino and higgsino states, (W̃ + , H̃u+ , W̃ − , H̃d− ), and enter into the Lagrangian under the action of:  Mcharge  0 0 M2 gvu    0 0 gvd µ   . =  M gv 0 0 2 d   gvu µ 0 0 (1.14) In the neutralino case, 4 nondegenerate mass eigenstates are yielded, while in the chargino case, only 2 non-degenerate mass eigenstates are yielded. In both cases, the mass eigenstates are usually ordered by their mass and denoted as χ̃0i with i ∈ N4 and χ̃+ i with i ∈ N2 for the neutralinos and charginos, respectively. These mixings create the physical particles expected from supersymmetric theories. Previously, we stated that SUSY creates a duplication of all SM particles, which would include a photino – the superpartner to the photon – as a linear combination of B̃ 0 and W̃ 0 states. This linear combination would be expected to be an admixture for the neutralinos as they are already mixed under the action of Mneutral . However, while we expect electroweak symmetry breaking happens for SUSY just as in the SM and will mix these states, the photino and zino guage bosons would be purer instances of neutralinos, where the Higgsino components are nominal at best. As this depends on the specific values of the SUSY defining parameters, we will talk generically of neutralinos and the fractional admixtures of their constituent states. Of particular importance to the discussion of SUSY as a viable dark matter model is the question of stability – in order to form the structures we observe in the universe, whatever dark sector is created must have long-lived particles that can survive from the early universe to today. In the SM, this is achieved by the fact that any operators that violate its baryon-lepton (B − L) symmetry (which keeps the proton stable) must be suppressed by the mass-scale of the symmetry breaking physics in powers linear in the dimension of said operator [82–85]. This is not necessarily true in in SUSY [78]. However, we can impose such a symmetry, which causes all SM particles to attain R = +1 (an even parity) and all SUSY particles to attain R = −1 (an odd parity) for R = (−1)3(B−L)+2S due to the inherent difference in spins, S, arising from the supersymmetric transformation [86, 87]. Consequently, the lightest supersymmetric particle (LSP) is guaranteed to be stable as 18 it cannot decay to a less massive state without an R-parity violating process, which is advantageous to forbid. Altogether, what is described above is the simplest version of SUSY – the generation of superfields from the action of Q on the SM, the introduction of two new Higgs doublets, and the imposition of R-parity. Together with the SM, this simplest version almost gets us to the Minimal version of the Supersymmetric Standard Model (MSSM). The only remaining piece needed to round off the MSSM is a mechanism to break the symmetry between SUSY and the SM. This is the subject to which we now turn. 1.3.3 Symmetry Breaking and SUSY Models Just as the existence of the electroweak gauge boson masses implied a broken symmetry needing an explanation, the lack of SUSY particles degenerate with their SM partners implies a broken symmetry in need of an explanation. Similarly, we must look outside of SUSY for a sufficient mechanism just as the Higgs mechanism came from outside electroweak theory. Introducing further new physics has the added benefit of alleviating SUSY’s typical violations of experimental observations, including lepton number conservation, CP violation, and FCNCs [78]. Many methods exist to accomplish this. For instance, in gauge mediated symmetry breaking (GMSB) mass generation happens from loop effects. This is achieved by the addition of another new sector that contains messenger multiplets that couple to the gauge bosons and gauginos [88–102]. It is also possible that an extra dimension will physically sequester the new, symmetry-breaking sector while the size of the separating dimension suppresses any potential violations of experimental observations [103–114]. Another highly varied method – and typically related to methods involving extra dimensions – generates masses at the loop level through a super-conformal anomaly or through anomalous violations of scale invariance via supergravity (dubbed anomaly mediated symmetry breaking or AMSB) [105, 115–147]. The most common frameworks are those that make use of supergravity (which can overlap with GMSB and AMSB frameworks) [78]. With supergravity, a supersymmetric partner to the graviton, the gravitino, communicates the symmetry-breaking mechanism and it is dependent on the strength of the gravitational interaction [148–157]. Unless the 19 gravitino is the LSP, it plays little role in the model’s phenomenology. For minimal supergravity frameworks (mSUGRA), the aim is to minimize the kinetic terms in the Kähler potential and are motivated by providing a Grand Unified Theory (GUT) model. As such, parameters are defined at a mass scale at which unification is assumed to happen, typically MGUT or MPlanck . Furthermore, many of the 124 independent parameters of the MSSM are related to each other, such that the model can be defined at this high scale by as few as 2 free parameters [158]. At the GUT scale, these relations reduce the sfermion mass matrices and the Higgsino masses to a common scalar mass: 2 MQ = Mu2 = Md2 = ML2 = Me2 = m20 I m2Hd = m2Hu = m20 , (1.15) (1.16) the gaugino masses to a common gaugino mass: M1 = M2 = M3 = m1/2 , (1.17) and the trilinear couplings to a common trilinear term: Au = Ad = Ae = A0 I ; (1.18) in the above, I denotes the 3 × 3 identity matrix. Together with the sign of the Higgs mass parameter and ratio of the vacuum expectation values from the Higgs sector, we have only five model-defining parameters [78, 79, 81]: m0 , m1/2 , A0 , tan(β), sign(µ) , (1.19) defining the Constrained MSSM (CMSSM) [159]. We should note that there is not widespread agreement as to the terms “mSUGRA” and “CMSSM”. For instance, many authors take the CMSSM to be synonymous with mSUGRA or as a catch-all for models with CMSSM-like relations between the SUSY parameters, which was the original sense of “mSUGRA” [78]. We will not follow these conventions in our discussion of the MSSM. Instead, we follow the convention that mSUGRA is the overarching symmetry breaking framework. This is consistent not only with the discussion of symmetry breaking processes, but allows for us to acknowledge that there are other, less constrained SUSY models that fall under the framework of minimal supergravity. 20 The CMSSM is one of the prime models in which we will be interested for this study. Additionally, we will investigate a second set of models that have slightly more freedom in their defining parameters and are still under the GUT framework of mSUGRA. For these, we do not assume universality of the two Higgs doublet masses, as was indicated for the CMSSM in Equation 1.16. For the first Non-Universal Higgs Mass framework (NUHM 1) [160], we take m2Hd = m2Hu 6= m0 and, for the second (NUHM 2) [161], we further relax universality and let m2Hd 6= m2Hu . Thus, the free parameters for NUHM models are m0 , m1/2 , A0 , tan(β), m2Hd , m2Hu , sign(µ) . (1.20) While a handful of free parameters are far more tractable than the MSSM’s 124 parameters, both the CMSSM and NUHM make significant assumptions about the nature of SUSY. Alternatively, we can take a more agnostic approach to studying SUSY and only assume phenomenological signatures of the MSSM. That is, in the pMSSM [162, 163] the model is defined by weak scale values of the particle masses and couplings. Here we define the standard 19 free parameters of the pMSSM to be: the Higgs mass parameter µ; the ratio of the vacuum expectation values tan(β); the pseudoscalar Higgs mass mA ; the 3 gaugino masses M1 , M2 , and M3 ; 5 sfermion masses assumed to be degenerate between the first and second generations; 5 sfermion masses for the third generation; and 3 trilinear couplings for the third generation Ab , At , and Aτ . This incarnation is sometimes explicitly dubbed the “pMSSM19”. 1.4 The Timeliness of this Work From the above, it becomes clear that one potential explanation for dark matter is a weakly interacting, massive particle (WIMP) and no such particle exists within the SM. SUSY is especially attractive here as many instances of SUSY models posit that the LSP is stable and uncharged [164–166]. This, along with the fact that many of these models have sufficiently massive particles to fit into the already constrained niche for a CDM model [167–169], has made SUSY an early and favored contender for solving the dark matter problem, and is still attractive today. It is worth noting that other interesting possibilities do survive after astrophysical constraints are applied. For example, axions as a solution to QCD’s strong CP-problem [170], the motion of Kaluza-Klein particles in other dimensions [171], sterile neutrino models [172], 21 or sequestered dark sectors only reachable through some BSM portal (e.g., see References [173–178]) all add to the rich tapestry of potential solutions [179]. That said, in this report we will confine our focus to the standard picture of a WIMP LSP as a particle dark matter candidate. Alternative hypotheses are becoming increasingly attractive as null results and increasingly stringent squark limits from the LHC [180, 181] as well as and precision cosmology [17, 182–186] have put strenuous constraints on supersymmetric dark matter. This has lead to reports of SUSY’s demise, but the situation is much more complicated [187–189]. For one, the null results are consistent with expectations of SUSY [190–194]. That is, while there is yet no positive signal, the results, frustratingly, do not inherently rule out the possibility for a future positive result for SUSY. For another, the constraints largely apply to the low mass versions of the CMSSM and pMSSM, leaving much of the SUSY parameter space untested [195–223]. Of course, the experimental endeavor to test SUSY is not finished. Many next generation experiments are currently being built or considered [224–233], while others are being upgraded or have already begun taking data [185, 186, 234–240]. These should help further illuminate the status of SUSY in describing reality. However, in order to do so, testable predictions are required. The question is not just “what predictions can we make about SUSY?” but also “how well are we able to make these predictions?” Tackling the 124 parameter MSSM is a daunting task. While the simplified frameworks of the CMSSM, NUHM, and pMSSM help, they are still far from trivial calculations. In order to make the necessary predictions, the community has created a suite of computational tools to compute aspects of the theory including the sparticle spectrum [162, 241–246], corrections to the Higgs sector [218, 218, 247–253], dark matter observables [254–257], flavor physics observables [245, 246, 258–262], etc. Much effort has been put into evaluating the efficacy of these codes, and the discrepancies that have been identified have been resolved. While comparisons have focused on the calculation of the sparticle spectrum (both the full spectrum [263–265] or just the Higgs sector [266, 267]), there has been less work done to directly compare the calculations for dark matter observables [268]. Instead, efforts have focused on evaluating the underlying theory used in the dark matter calculations of each code independently [269–271]. This has also been the case for the dedicated Higgs sector 22 calculators [218, 251, 252, 272, 273] as it is only recently that another major public code has become available [253]. All of which has helped ensure concordance in the predictions of SUSY and increase the efficacy of these sophisticated codes as they reduce the uncertainty in their calculations. However, early work has focused on regions of SUSY that are easily accessible to experiments – the same regions that have now been reached and ruled out by experiments. This leaves us at a crossroad in the search for SUSY: as we move away from the most well understood corners of the supersymmetric landscape, how ready are we for the new corners into which experiments are heading? That is, how well do we understand the uncertainties inherent in the predictions of SUSY from the high mass and more nuanced models? In this thesis, I will shed light on the situation by providing a better understanding of the uncertainties in the theoretical predictions from supersymmetric models. The rest of this thesis is organized as follows. In Chapter 2, we will investigate the current level of theoretical uncertainties in the calculation of dark matter observables. In Chapter 3, we will look at future work, including a study of the theoretical uncertainties in the calculation of flavor physics observables. In Chapter 4, we report the conclusions of this thesis. 23 1.5 References [1] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. C. Odom, Phys. Rev. Lett. 97, 030802 (2006), [Erratum: Phys. Rev. Lett.99,039902(2007)]. [2] R. Aaij et al. (LHCb), Phys. Rev. Lett. 115, 072001 (2015), arXiv:1507.03414 [hep-ex]. [3] G. Aad et al. (ATLAS), Phys. Lett. B716, 1 (2012), arXiv:1207.7214 [hep-ex]. [4] S. Chatrchyan et al. (CMS), Phys. Lett. B716, 30 (2012), arXiv:1207.7235 [hep-ex]. [5] P. Langacker, in Proceedings, 1st International Symposium on Particles, Strings and Cosmology (PASCOS’90): Boston, Massachusetts, March 27-31, 1990 (1990) pp. 237–269. [6] L. H. Ford, in Particles and fields. Proceedings, 9th Jorge Andre Swieca Summer School, Campos do Jordao, Brazil, February 16-28, 1997 (1997) pp. 345–388, arXiv:grqc/9707062 [gr-qc]. [7] V. M. Abazov et al. (D0), Phys. Rev. D82, 032001 (2010), arXiv:1005.2757 [hep-ex]. [8] J. R. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B260, 131 (1991). [9] U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. B260, 447 (1991). [10] P. Langacker and M.-x. Luo, Phys. Rev. D44, 817 (1991). [11] C. Giunti, C. W. Kim, and U. W. Lee, Mod. Phys. Lett. A6, 1745 (1991). [12] Yu. A. Golfand and E. P. Likhtman, JETP Lett. 13, 323 (1971), [Pisma Zh. Eksp. Teor. Fiz.13,452(1971)]. [13] D. V. Volkov and V. P. Akulov, Supersymmetry and quantum field theory. Proceedings, D. Volkov Memorial Seminar, Kharkov, Ukraine, January 5-7, 1997, JETP Lett. 16, 438 (1972), [Pisma Zh. Eksp. Teor. Fiz.16,621(1972)]. [14] D. V. Volkov and V. P. Akulov, Phys. Lett. 46B, 109 (1973). [15] D. V. Volkov and V. A. Soroka, Supersymmetry and quantum field theory. Proceedings, D. Volkov Memorial Seminar, Kharkov, Ukraine, January 5-7, 1997, JETP Lett. 18, 312 (1973), [Pisma Zh. Eksp. Teor. Fiz.18,529(1973)]. [16] J. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974), [,24(1974)]. [17] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589 [astro-ph.CO]. [18] F. Zwicky, Helv. Phys. Acta 6, 110 (1933), [Gen. Rel. Grav.41,207(2009)]. [19] S. Smith, ApJ 83, 23 (1936). [20] E. Holmberg, Annals of the Observatory of Lund 6, 3 (1937). [21] T. Page, AJ 64, 53 (1959). 24 [22] F. D. Kahn and L. Woltjer, ApJ 130, 705 (1959). [23] J. G. de Swart, G. Bertone, and J. van Dongen, Nature Astronomy 1, 10.1038/s41550017-0059 (2017). [24] N. Straumann, Swiss Physical Society Communications , 22 (2013), https://www.sps.ch/artikel/geschichte-der-physik/fritz-zwicky-an-extraordinaryastrophysicist-6/. [25] V. C. Rubin and W. K. Ford, Jr., Astrophys. J. 159, 379 (1970). [26] M. S. Roberts and R. N. Whitehurst, ApJ 201, 327 (1975). [27] K. C. Freeman, ApJ 160, 811 (1970). [28] J. P. Ostriker and P. J. E. Peebles, ApJ 186, 467 (1973). [29] J. P. Ostriker, P. J. E. Peebles, and A. Yahil, ApJL 193, L1 (1974). [30] J. Einasto, A. Kaasik, and E. Saar, Natur 250, 309 (1974). [31] M. Milgrom, Astrophys. J. 270, 365 (1983). [32] M. Milgrom, Astrophys. J. 270, 371 (1983). [33] M. Milgrom, Astrophys. J. 270, 384 (1983). [34] G. Lake, ApJ 345, L17 (1989). [35] O. E. Gerhard and D. N. Spergel, ApJ 397, 38 (1992). [36] D. Clowe, A. Gonzalez, and M. Markevitch, Astrophys. J. 604, 596 (2004), arXiv:astro-ph/0312273 [astro-ph]. [37] F. Schweizer, ApJ 252, 455 (1982). [38] J. Binney, in Proceedings, IAU Symposium 220: Dark Matter in Galaxies: Sydney, Australia, July 21-25, 2003 (2003) [Submitted to: ASP Conf. Ser.(2003)], arXiv:astroph/0310219 [astro-ph]. [39] S. Dodelson, International Journal of Modern Physics D 20, 2749 (2011), arXiv:1112.1320 [astro-ph.CO]. [40] S. Torres, J. Camacho, J. Isern, and E. Garcia-Berro, Astron. Astrophys. 486, 427 (2008), arXiv:0804.2207 [astro-ph]. [41] N. Jarosik et al., Astrophys. J. Suppl. 192, 14 (2011), arXiv:1001.4744 [astro-ph.CO]. [42] A. Dar, Dark matter in cosmology, clocks and test of fundamental laws. Proceedings, 30th Rencontres de Moriond, 15th Moriond Workshop, Villars sur Ollon, Switzerland, January 22-29, 1995, Astrophys. J. 449, 550 (1995), [,21(1995)], arXiv:astroph/9504082 [astro-ph]. [43] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc]. 25 [44] Y. B. Zel’dovich and I. D. Novikov, Astronomicheskii Zhurnal 43, 758 (1966). [45] B. C. Lacki and J. F. Beacom, Astrophys. J. 720, L67 (2010), arXiv:1003.3466 [astroph.CO]. [46] A. Kashlinsky, Astrophys. J. 823, L25 (2016), arXiv:1605.04023 [astro-ph.CO]. [47] J. R. Espinosa, D. Racco, and A. Riotto, Phys. Rev. Lett. 120, 121301 (2018), arXiv:1710.11196 [hep-ph]. [48] S. Clesse and J. Garca-Bellido, Phys. Dark Univ. 22, 137 (2018), arXiv:1711.10458 [astro-ph.CO]. [49] S. D. M. White, The evolution of large-scale structure., in IN: Inner space/outer space: the interface between cosmology and particle physics (A87-26233 10-90). Chicago, University of Chicago Press, 1986, p. 228-245., edited by E. W. Kolb, M. S. Turner, D. Lindley, K. Olive, and D. Seckel (University of Chicago Press, 1986) pp. 228–245. [50] J. Huchra, M. Davis, D. Latham, and J. Tonry, Astrophys. J. Suppl. 52, L89 (1983), [Astrophys. J. Suppl.52,89(1983)]. [51] V. Springel, C. S. Frenk, and S. D. M. White, Nature 440, 1137 (2006), arXiv:astroph/0604561 [astro-ph]. [52] M. J. Geller and J. P. Huchra, Science 246, 897 (1989). [53] J. R. Gott, III, M. Juric, D. Schlegel, F. Hoyle, M. Vogeley, M. Tegmark, N. A. Bahcall, and J. Brinkmann, Astrophys. J. 624, 463 (2005), arXiv:astro-ph/0310571 [astro-ph]. [54] J. R. Bond, L. Kofman, and D. Pogosyan, Nature 380, 603 (1996), arXiv:astroph/9512141 [astro-ph]. [55] M. Colless et al. (2DFGRS), Mon. Not. Roy. Astron. Soc. 328, 1039 (2001), arXiv:astro-ph/0106498 [astro-ph]. [56] V. Springel et al., Nature 435, 629 (2005), arXiv:astro-ph/0504097 [astro-ph]. [57] J. P. Ostriker and P. J. Steinhardt, (1995), arXiv:astro-ph/9505066 [astro-ph]. [58] Ya. B. Zeldovich, J. Einasto, and S. F. Shandarin, Nature 300, 407 (1982). [59] S. D. M. White, C. S. Frenk, and M. Davis, Astrophys. J. 274, L1 (1983), [,80(1984)]. [60] J. R. Primack and G. R. Blumenthal, in Beyond the Standard Model. Proceedings, 18th Rencontres de Moriond, La Plagne, France, March 13-19, 1983. Vol. 2 (1983) pp. 445–464, [,163(1983)]. [61] V. Springel, S. D. M. White, A. Jenkins, C. S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, J. A. Peacock, S. Cole, P. Thomas, H. Couchman, A. Evrard, J. Colberg, and F. Pearce, Natur 435, 629 (2005), arXiv:astro-ph/0504097 [astro-ph]. [62] A. A. Klypin, S. Trujillo-Gomez, and J. Primack, ApJ 740, 102 (2011), arXiv:1002.3660 [astro-ph.CO]. 26 [63] C. P. Ahn et al. (SDSS), Astrophys. J. Suppl. 203, 21 (2012), arXiv:1207.7137 [astroph.IM]. [64] User:Cush, Elementary particles included in the standard model, Wikimedia Commons, Public Domain, https://en.wikipedia.org/wiki/Standard Model#/media/File:Standard Model of Ele mentary Particles.svg, accessed: 2019-04-04. [65] G. D. Rochester and C. C. Butler, Nature 160, 855 (1947). [66] W. Heisenberg, Z. Phys. 77, 1 (1932). [67] S. L. Glashow, Nucl. Phys. 10, 107 (1959). [68] A. Salam and J. C. Ward, Nuovo Cim. 11, 568 (1959). [69] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). [70] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973), [,271(1973)]. [71] D. J. Gross and F. Wilczek, Phys. Rev. D8, 3633 (1973). [72] H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973), [,274(1973)]. [73] G. Arnison et al. (UA1), 30 years of weak neutral currents, Phys. Lett. B122, 103 (1983), [,611(1983)]. [74] G. Arnison et al. (UA1), 30 years of weak neutral currents, Phys. Lett. B126, 398 (1983), [,7.55(1983)]. [75] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964), [,160(1964)]. [76] F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964), [,157(1964)]. [77] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964), [,162(1964)]. [78] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, 030001 (2018). [79] S. P. Martin, , 1 (1997), [Adv. Ser. Direct. High Energy Phys.18,1(1998)], arXiv:hepph/9709356 [hep-ph]. [80] R. Haag, J. T. Lopuszanski, and M. Sohnius, Nucl. Phys. B88, 257 (1975), [,257(1974)]. [81] H. Baer and X. Tata, Weak scale supersymmetry: From superfields to scattering events (Cambridge University Press, 2006). [82] S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979). [83] F. Wilczek and A. Zee, Phys. Rev. Lett. 43, 1571 (1979). [84] S. Weinberg, Phys. Rev. D22, 1694 (1980). [85] H. A. Weldon and A. Zee, Nucl. Phys. B173, 269 (1980). 27 [86] P. Fayet, Phys. Lett. 69B, 489 (1977). [87] G. R. Farrar and P. Fayet, Phys. Lett. 76B, 575 (1978). [88] L. Alvarez-Gaume, M. Claudson, and M. B. Wise, Nucl. Phys. B207, 96 (1982). [89] M. Dine and W. Fischler, Phys. Lett. 110B, 227 (1982). [90] M. Dine, W. Fischler, and M. Srednicki, GRAND UNIFICATION. PROCEEDINGS, 2ND WORKSHOP, ANN ARBOR, USA, APRIL 24-26, 1981, Nucl. Phys. B189, 575 (1981). [91] S. Dimopoulos and S. Raby, Proceedings, Third Workshop on Grand Unification: Chapel Hill, North Carolina, April 15-17, 1982, Nucl. Phys. B219, 479 (1983), [Prog. Phys.6(1982)]. [92] C. R. Nappi and B. A. Ovrut, Phys. Lett. 113B, 175 (1982). [93] M. Dine and A. E. Nelson, Phys. Rev. D48, 1277 (1993), arXiv:hep-ph/9303230 [hepph]. [94] M. Dine, A. E. Nelson, and Y. Shirman, Phys. Rev. D51, 1362 (1995), arXiv:hepph/9408384 [hep-ph]. [95] M. Dine, A. E. Nelson, Y. Nir, and Y. Shirman, Phys. Rev. D53, 2658 (1996), arXiv:hep-ph/9507378 [hep-ph]. [96] S. Dimopoulos, G. F. Giudice, and A. Pomarol, Phys. Lett. B389, 37 (1996), arXiv:hep-ph/9607225 [hep-ph]. [97] S. P. Martin, Phys. Rev. D55, 3177 (1997), arXiv:hep-ph/9608224 [hep-ph]. [98] E. Poppitz and S. P. Trivedi, Phys. Lett. B401, 38 (1997), arXiv:hep-ph/9703246 [hep-ph]. [99] P. Meade, N. Seiberg, and D. Shih, 30 years of mathematical methods in high energy physics. Proceedings, International Conference in honor of Professor Tohru Eguchi’s 60th birthday, Kyoto, Japan, March 17-19, 2008, Prog. Theor. Phys. Suppl. 177, 143 (2009), arXiv:0801.3278 [hep-ph]. [100] L. M. Carpenter, M. Dine, G. Festuccia, and J. D. Mason, Phys. Rev. D79, 035002 (2009), arXiv:0805.2944 [hep-ph]. [101] M. Buican, P. Meade, N. Seiberg, and D. Shih, JHEP 03, 016, arXiv:0812.3668 [hepph]. [102] T. T. Dumitrescu, Z. Komargodski, N. Seiberg, and D. Shih, JHEP 05, 096, arXiv:1003.2661 [hep-ph]. [103] V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. 125B, 136 (1983). [104] E. A. Mirabelli and M. E. Peskin, Phys. Rev. D58, 065002 (1998), arXiv:hepth/9712214 [hep-th]. 28 [105] L. Randall and R. Sundrum, Nucl. Phys. B557, 79 (1999), arXiv:hep-th/9810155 [hep-th]. [106] Z. Chacko, M. A. Luty, and E. Ponton, JHEP 07, 036, arXiv:hep-ph/9909248 [hep-ph]. [107] Z. Chacko, M. A. Luty, A. E. Nelson, and E. Ponton, JHEP 01, 003, arXiv:hepph/9911323 [hep-ph]. [108] D. E. Kaplan, G. D. Kribs, and M. Schmaltz, Phys. Rev. D62, 035010 (2000), arXiv:hep-ph/9911293 [hep-ph]. [109] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), arXiv:hep-ph/9905221 [hep-ph]. [110] Z. Chacko and M. A. Luty, JHEP 05, 067, arXiv:hep-ph/0008103 [hep-ph]. [111] T. Gherghetta and A. Pomarol, Nucl. Phys. B602, 3 (2001), arXiv:hep-ph/0012378 [hep-ph]. [112] M. Schmaltz and W. Skiba, Phys. Rev. D62, 095005 (2000), arXiv:hep-ph/0001172 [hep-ph]. [113] H. C. Cheng, D. E. Kaplan, M. Schmaltz, and W. Skiba, Phys. Lett. B515, 395 (2001), arXiv:hep-ph/0106098 [hep-ph]. [114] C. Csaki, J. Erlich, C. Grojean, and G. D. Kribs, Phys. Rev. D65, 015003 (2002), arXiv:hep-ph/0106044 [hep-ph]. [115] E. Cremmer, S. Ferrara, L. Girardello, and A. Van Proeyen, Nucl. Phys. B212, 413 (1983), [,413(1982)]. [116] I. Antoniadis, C. Munoz, and M. Quiros, Nucl. Phys. B397, 515 (1993), arXiv:hepph/9211309 [hep-ph]. [117] M. Dine and D. MacIntire, Phys. Rev. D46, 2594 (1992), arXiv:hep-ph/9205227 [hepph]. [118] G. R. Dvali and M. A. Shifman, Nucl. Phys. B504, 127 (1997), arXiv:hep-th/9611213 [hep-th]. [119] G. F. Giudice, M. A. Luty, H. Murayama, and R. Rattazzi, JHEP 12, 027, arXiv:hepph/9810442 [hep-ph]. [120] J. A. Bagger, T. Moroi, and E. Poppitz, JHEP 04, 009, arXiv:hep-th/9911029 [hep-th]. [121] Z. Chacko, M. A. Luty, I. Maksymyk, and E. Ponton, JHEP 04, 001, arXiv:hepph/9905390 [hep-ph]. [122] E. Katz, Y. Shadmi, and Y. Shirman, JHEP 08, 015, arXiv:hep-ph/9906296 [hep-ph]. [123] A. Pomarol and R. Rattazzi, JHEP 05, 013, arXiv:hep-ph/9903448 [hep-ph]. [124] M. K. Gaillard and B. D. Nelson, Nucl. Phys. B588, 197 (2000), arXiv:hep-th/0004170 [hep-th]. 29 [125] I. Jack and D. R. T. Jones, Phys. Lett. B482, 167 (2000), arXiv:hep-ph/0003081 [hep-ph]. [126] M. Carena, K. Huitu, and T. Kobayashi, Nucl. Phys. B592, 164 (2001), arXiv:hepph/0003187 [hep-ph]. [127] B. C. Allanach and A. Dedes, JHEP 06, 017, arXiv:hep-ph/0003222 [hep-ph]. [128] I. Antoniadis, S. Dimopoulos, A. Pomarol, and M. Quiros, Nucl. Phys. B544, 503 (1999), arXiv:hep-ph/9810410 [hep-ph]. [129] A. Delgado, A. Pomarol, and M. Quiros, Phys. Rev. D60, 095008 (1999), arXiv:hepph/9812489 [hep-ph]. [130] N. Arkani-Hamed, D. E. Kaplan, H. Murayama, and Y. Nomura, JHEP 02, 041, arXiv:hep-ph/0012103 [hep-ph]. [131] D. E. Kaplan and G. D. Kribs, JHEP 09, 048, arXiv:hep-ph/0009195 [hep-ph]. [132] A. Anisimov, M. Dine, M. Graesser, and S. D. Thomas, Phys. Rev. D65, 105011 (2002), arXiv:hep-th/0111235 [hep-th]. [133] Z. Chacko and M. A. Luty, JHEP 05, 047, arXiv:hep-ph/0112172 [hep-ph]. [134] D. Marti and A. Pomarol, Phys. Rev. D64, 105025 (2001), arXiv:hep-th/0106256 [hep-th]. [135] R. Barbieri, L. J. Hall, and Y. Nomura, Nucl. Phys. B624, 63 (2002), arXiv:hepth/0107004 [hep-th]. [136] D. E. Kaplan and N. Weiner, (2001), arXiv:hep-ph/0108001 [hep-ph]. [137] J. Bagger, F. Feruglio, and F. Zwirner, JHEP 02, 010, arXiv:hep-th/0108010 [hep-th]. [138] T. Gherghetta and A. Riotto, Nucl. Phys. B623, 97 (2002), arXiv:hep-th/0110022 [hep-th]. [139] A. Anisimov, M. Dine, M. Graesser, and S. D. Thomas, JHEP 03, 036, arXiv:hepth/0201256 [hep-th]. [140] N. Okada, Phys. Rev. D65, 115009 (2002), arXiv:hep-ph/0202219 [hep-ph]. [141] R. Harnik, H. Murayama, and A. Pierce, JHEP 08, 034, arXiv:hep-ph/0204122 [hepph]. [142] A. E. Nelson and N. J. Weiner, (2002), arXiv:hep-ph/0210288 [hep-ph]. [143] O. C. Anoka, K. S. Babu, and I. Gogoladze, Nucl. Phys. B686, 135 (2004), arXiv:hepph/0312176 [hep-ph]. [144] M. Dine, P. J. Fox, E. Gorbatov, Y. Shadmi, Y. Shirman, and S. D. Thomas, Phys. Rev. D70, 045023 (2004), arXiv:hep-ph/0405159 [hep-ph]. [145] R. Kitano, G. D. Kribs, and H. Murayama, Phys. Rev. D70, 035001 (2004), arXiv:hepph/0402215 [hep-ph]. 30 [146] R. Sundrum, Phys. Rev. D71, 085003 (2005), arXiv:hep-th/0406012 [hep-th]. [147] F. D’Eramo, J. Thaler, and Z. Thomas, JHEP 06, 151, arXiv:1202.1280 [hep-ph]. [148] A. H. Chamseddine, R. L. Arnowitt, and P. Nath, Phys. Rev. Lett. 49, 970 (1982). [149] J. R. Ellis, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 121B, 123 (1983). [150] R. Barbieri, S. Ferrara, and C. A. Savoy, Phys. Lett. 119B, 343 (1982). [151] N. Ohta, Prog. Theor. Phys. 70, 542 (1983). [152] H. P. Nilles, M. Srednicki, and D. Wyler, Phys. Lett. 120B, 346 (1983). [153] L. E. Ibanez, Nucl. Phys. B218, 514 (1983). [154] E. Cremmer, P. Fayet, and L. Girardello, Phys. Lett. 122B, 41 (1983). [155] L. Alvarez-Gaume, J. Polchinski, and M. B. Wise, Nucl. Phys. B221, 495 (1983). [156] L. J. Hall, J. D. Lykken, and S. Weinberg, Phys. Rev. D27, 2359 (1983). [157] H. P. Nilles, Phys. Rept. 110, 1 (1984). [158] J. Polonyi, Hungary Central Research Institute report KFKI-77-93 (1977). [159] G. L. Kane, C. F. Kolda, L. Roszkowski, and J. D. Wells, Phys. Rev. D49, 6173 (1994), arXiv:hep-ph/9312272 [hep-ph]. [160] H. Baer, A. Mustafayev, S. Profumo, A. Belyaev, and X. Tata, Phys. Rev. D71, 095008 (2005), arXiv:hep-ph/0412059 [hep-ph]. [161] H. Baer, A. Mustafayev, S. Profumo, A. Belyaev, and X. Tata, JHEP 07, 065, arXiv:hep-ph/0504001 [hep-ph]. [162] A. Djouadi, J.-L. Kneur, and G. Moultaka, Comput. Phys. Commun. 176, 426 (2007), arXiv:hep-ph/0211331 [hep-ph]. [163] C. F. Berger, J. S. Gainer, J. L. Hewett, and T. G. Rizzo, JHEP 02, 023, arXiv:0812.0980 [hep-ph]. [164] S. M. Faber and J. S. Gallagher, Ann. Rev. Astron. Astrophys. 17, 135 (1979). [165] G. R. Blumenthal, H. Pagels, and J. R. Primack, Nature 299, 37 (1982). [166] J. R. Bond, A. S. Szalay, and M. S. Turner, Phys. Rev. Lett. 48, 1636 (1982). [167] P. J. E. Peebles, Astrophys. J. 263, L1 (1982), [,85(1982)]. [168] G. R. Blumenthal, S. M. Faber, J. R. Primack, and M. J. Rees, Nature 311, 517 (1984), [,96(1984)]. [169] G. R. Blumenthal, S. M. Faber, R. Flores, and J. R. Primack, Astrophys. J. 301, 27 (1986). [170] L. D. Duffy and K. van Bibber, New J. Phys. 11, 105008 (2009), arXiv:0904.3346 [hep-ph]. 31 [171] E. W. Kolb and R. Slansky, Phys. Lett. 135B, 378 (1984). [172] B. Shakya and J. D. Wells, Phys. Rev. D96, 031702 (2017), arXiv:1611.01517 [hep-ph]. [173] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, Phys. Rev. D79, 015014 (2009), arXiv:0810.0713 [hep-ph]. [174] M. Ardid, I. Felis, A. Herrero, and J. A. Martnez-Mora, JCAP 1704 (04), 010, arXiv:1701.08863 [astro-ph.HE]. [175] A. Falkowski, J. Juknevich, and J. Shelton, (2009), arXiv:0908.1790 [hep-ph]. [176] B. Batell, T. Han, and B. Shams Es Haghi, Phys. Rev. D97, 095020 (2018), arXiv:1704.08708 [hep-ph]. [177] B. Patt and F. Wilczek, (2006), arXiv:hep-ph/0605188 [hep-ph]. [178] G. Arcadi, A. Djouadi, and M. Raidal, (2019), arXiv:1903.03616 [hep-ph]. [179] J. L. Feng, Ann. Rev. Astron. Astrophys. 48, 495 (2010), arXiv:1003.0904 [astroph.CO]. [180] The ATLAS collaboration (ATLAS), 54th Rencontres de Moriond on QCD and High Energy Interactions, ATLAS Notes , 32 (2019). [181] CMS Collaboration (CMS), CMS Physics Analysis Summaries 008, 28 (2019). [182] M. G. Aartsen et al. (IceCube), JCAP 1604 (04), 022, arXiv:1601.00653 [hep-ph]. [183] M. L. Ahnen et al. (MAGIC, Fermi-LAT), JCAP 1602 (02), 039, arXiv:1601.06590 [astro-ph.HE]. [184] D. S. Akerib et al. (LUX), Phys. Rev. Lett. 118, 021303 (2017), arXiv:1608.07648 [astro-ph.CO]. [185] E. Aprile et al. (XENON), Phys. Rev. Lett. 121, 111302 (2018), arXiv:1805.12562 [astro-ph.CO]. [186] X. Cui et al. (PandaX-II), Phys. Rev. Lett. 119, 181302 (2017), arXiv:1708.06917 [astro-ph.CO]. [187] S. Akula, B. Altunkaynak, D. Feldman, P. Nath, and G. Peim, Phys. Rev. D85, 075001 (2012), arXiv:1112.3645 [hep-ph]. [188] P. Nath, Annalen Phys. 528, 167 (2016), arXiv:1501.01679 [hep-ph]. [189] J. Ellis, HKUST Jockey Club Institute for Advanced Study: High Energy Physics (HKUST) Hong Kong, China, January 9-26, 2017, Int. J. Mod. Phys. A32, 1746001 (2017), arXiv:1704.02821 [hep-ph]. [190] Y. Okada, M. Yamaguchi, and T. Yanagida, Prog. Theor. Phys. 85, 1 (1991). [191] H. E. Haber and R. Hempfling, Phys. Rev. Lett. 66, 1815 (1991). [192] J. R. Ellis, G. Ridolfi, and F. Zwirner, Phys. Lett. B262, 477 (1991). 32 [193] J. R. Ellis and D. Ross, Phys. Lett. B506, 331 (2001), arXiv:hep-ph/0012067 [hep-ph]. [194] J. R. Ellis, S. Heinemeyer, K. A. Olive, and G. Weiglein, JHEP 01, 006, arXiv:hepph/0211206 [hep-ph]. [195] H. Baer, V. Barger, and A. Mustafayev, Phys. Rev. D85, 075010 (2012), arXiv:1112.3017 [hep-ph]. [196] J. L. Feng, K. T. Matchev, and D. Sanford, Phys. Rev. D85, 075007 (2012), arXiv:1112.3021 [hep-ph]. [197] S. Heinemeyer, O. Stal, and G. Weiglein, Phys. Lett. B710, 201 (2012), arXiv:1112.3026 [hep-ph]. [198] A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi, and J. Quevillon, Phys. Lett. B708, 162 (2012), arXiv:1112.3028 [hep-ph]. [199] P. Draper, P. Meade, M. Reece, and D. Shih, Phys. Rev. D85, 095007 (2012), arXiv:1112.3068 [hep-ph]. [200] O. Buchmueller et al., Eur. Phys. J. C72, 2020 (2012), arXiv:1112.3564 [hep-ph]. [201] M. Kadastik, K. Kannike, A. Racioppi, and M. Raidal, JHEP 05, 061, arXiv:1112.3647 [hep-ph]. [202] M. Ivanov and A. V. Toporensky, Int. J. Mod. Phys. D21, 1250051 (2012), arXiv:1112.4194 [gr-qc]. [203] L. Aparicio, D. G. Cerdeno, and L. E. Ibanez, JHEP 04, 126, arXiv:1202.0822 [hepph]. [204] J. Ellis and K. A. Olive, Eur. Phys. J. C72, 2005 (2012), arXiv:1202.3262 [hep-ph]. [205] H. Baer, V. Barger, and A. Mustafayev, JHEP 05, 091, arXiv:1202.4038 [hep-ph]. [206] P. Bechtle et al., JHEP 06, 098, arXiv:1204.4199 [hep-ph]. [207] C. Balazs, A. Buckley, D. Carter, B. Farmer, and M. White, Eur. Phys. J. C73, 2563 (2013), arXiv:1205.1568 [hep-ph]. [208] D. Ghosh, M. Guchait, S. Raychaudhuri, and D. Sengupta, Phys. Rev. D86, 055007 (2012), arXiv:1205.2283 [hep-ph]. [209] A. Fowlie, M. Kazana, K. Kowalska, S. Munir, L. Roszkowski, E. M. Sessolo, S. Trojanowski, and Y.-L. S. Tsai, Phys. Rev. D86, 075010 (2012), arXiv:1206.0264 [hep-ph]. [210] O. Buchmueller et al., Eur. Phys. J. C72, 2243 (2012), arXiv:1207.7315 [hep-ph]. [211] K. Kowalska, S. Munir, L. Roszkowski, E. M. Sessolo, S. Trojanowski, and Y.-L. S. Tsai, Phys. Rev. D87, 115010 (2013), arXiv:1211.1693 [hep-ph]. [212] C. Strege, G. Bertone, F. Feroz, M. Fornasa, R. Ruiz de Austri, and R. Trotta, JCAP 1304, 013, arXiv:1212.2636 [hep-ph]. 33 [213] M. E. Cabrera, J. A. Casas, and R. Ruiz de Austri, JHEP 07, 182, arXiv:1212.4821 [hep-ph]. [214] P. Draper, J. L. Feng, P. Kant, S. Profumo, and D. Sanford, Phys. Rev. D88, 015025 (2013), arXiv:1304.1159 [hep-ph]. [215] T. Cohen and J. G. Wacker, JHEP 09, 061, arXiv:1305.2914 [hep-ph]. [216] S. Henrot-Versill, R. Lafaye, T. Plehn, M. Rauch, D. Zerwas, S. Plaszczynski, B. Rouill d’Orfeuil, and M. Spinelli, Phys. Rev. D89, 055017 (2014), arXiv:1309.6958 [hep-ph]. [217] P. Bechtle et al., Proceedings, 2013 European Physical Society Conference on High Energy Physics (EPS-HEP 2013): Stockholm, Sweden, July 18-24, 2013, PoS EPSHEP2013, 313 (2013), arXiv:1310.3045 [hep-ph]. [218] T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein, Phys. Rev. Lett. 112, 141801 (2014), arXiv:1312.4937 [hep-ph]. [219] O. Buchmueller et al., Eur. Phys. J. C74, 2922 (2014), arXiv:1312.5250 [hep-ph]. [220] L. Roszkowski, E. M. Sessolo, and A. J. Williams, JHEP 08, 067, arXiv:1405.4289 [hep-ph]. [221] K. Fukushima, C. Kelso, J. Kumar, P. Sandick, and T. Yamamoto, Phys. Rev. D90, 095007 (2014), arXiv:1406.4903 [hep-ph]. [222] O. Buchmueller, M. Citron, J. Ellis, S. Guha, J. Marrouche, K. A. Olive, K. de Vries, and J. Zheng, Eur. Phys. J. C75, 469 (2015), [Erratum: Eur. Phys. J.C76,no.4,190(2016)], arXiv:1505.04702 [hep-ph]. [223] E. A. Bagnaschi et al., Eur. Phys. J. C75, 500 (2015), arXiv:1508.01173 [hep-ph]. [224] T. Behnke, J. E. Brau, B. Foster, J. Fuster, M. Harrison, J. M. Paterson, M. Peskin, M. Stanitzki, N. Walker, and H. Yamamoto, (2013), arXiv:1306.6327 [physics.acc-ph]. [225] J. Tang et al., (2015), arXiv:1507.03224 [physics.acc-ph]. [226] R. Acciarri et al. (DUNE), (2016), arXiv:1601.05471 [physics.ins-det]. [227] M. Benedikt and F. Zimmermann, 19th International Conference on Accelerators and Beam Utilizations (ICABU2015) Gyeongju, Republic of Korea, November 4-6, 2015, J. Korean Phys. Soc. 69, 893 (2016). [228] M. J. Boland et al. (CLIC, arXiv:1608.07537 [physics.acc-ph]. CLICdp) 10.5170/CERN-2016-004 [229] B. J. Mount et al., (2017), arXiv:1703.09144 [physics.ins-det]. [230] G. Adhikari et al., Nature 564 (2018). [231] D. S. Akerib et al. (LUX-ZEPLIN), (2018), arXiv:1802.06039 [astro-ph.IM]. (2016), 34 [232] M. Sanguineti (ANTARES, KM3NeT), Proceedings, 7th International Conference on New Frontiers in Physics (ICNFP 2018): Kolymbari, Crete, Greece, July 4-12, 2018, Universe 5, 65 (2019). [233] Pico 500, http://www.picoexperiment.com/pico500.php, accessed: 2019-05-09. [234] T. Abe et al. (Belle-II), (2010), arXiv:1011.0352 [physics.ins-det]. [235] K. Abe et al. (Hyper-Kamiokande Working Group) (2014) arXiv:1412.4673 [physics.ins-det]. [236] M. G. Aartsen et al. (IceCube), (2014), arXiv:1412.5106 [astro-ph.HE]. [237] J. Grange et al. (Muon g-2), (2015), arXiv:1501.06858 [physics.ins-det]. [238] G. Apollinari, I. Bjar Alonso, O. Brning, P. Fessia, M. Lamont, L. Rossi, and L. Tavian, CERN Yellow Rep. Monogr. 4, 1 (2017). [239] R. Agnese et al. (SuperCDMS), Phys. Rev. D97, 022002 (2018), arXiv:1707.01632 [astro-ph.CO]. [240] J. Amar et al., (2019), arXiv:1903.03973 [astro-ph.IM]. [241] P. Athron, M. Bach, D. Harries, T. Kwasnitza, J.-h. Park, D. Stckinger, A. Voigt, and J. Ziebell, Comput. Phys. Commun. 230, 145 (2018), arXiv:1710.03760 [hep-ph]. [242] B. C. Allanach, Comput. Phys. Commun. 143, 305 (2002), arXiv:hep-ph/0104145 [hep-ph]. [243] W. Porod, Comput. Phys. Commun. 153, 275 (2003), arXiv:hep-ph/0301101 [hep-ph]. [244] W. Porod and F. Staub, Comput. Phys. Commun. 183, 2458 (2012), arXiv:1104.1573 [hep-ph]. [245] H. Baer, F. E. Paige, S. D. Protopopescu, and X. Tata, in Workshop on Physics at Current Accelerators and the Supercollider Argonne, Illinois, June 2-5, 1993 (1993) pp. 0703–720, arXiv:hep-ph/9305342 [hep-ph]. [246] F. E. Paige, S. D. Protopopescu, H. Baer, and X. Tata, (2003), arXiv:hep-ph/0312045 [hep-ph]. [247] S. Heinemeyer, W. Hollik, and G. Weiglein, Comput. Phys. Commun. 124, 76 (2000), arXiv:hep-ph/9812320 [hep-ph]. [248] S. Heinemeyer, W. Hollik, and G. Weiglein, Eur. Phys. J. C9, 343 (1999), arXiv:hepph/9812472 [hep-ph]. [249] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich, and G. Weiglein, Eur. Phys. J. C28, 133 (2003), arXiv:hep-ph/0212020 [hep-ph]. [250] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein, JHEP 02, 047, arXiv:hep-ph/0611326 [hep-ph]. [251] H. Bahl and W. Hollik, Eur. Phys. J. C76, 499 (2016), arXiv:1608.01880 [hep-ph]. 35 [252] H. Bahl, S. Heinemeyer, W. Hollik, and G. Weiglein, Eur. Phys. J. C78, 57 (2018), arXiv:1706.00346 [hep-ph]. [253] J. Pardo Vega and G. Villadoro, JHEP 07, 159, arXiv:1504.05200 [hep-ph]. [254] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 149, 103 (2002), arXiv:hep-ph/0112278 [hep-ph]. [255] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 174, 577 (2006), arXiv:hep-ph/0405253 [hep-ph]. [256] G. Blanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, Comput. Phys. Commun. 231, 173 (2018), arXiv:1801.03509 [hep-ph]. [257] P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke, and E. A. Baltz, JCAP 0407, 008, arXiv:astro-ph/0406204 [astro-ph]. [258] F. Mahmoudi, Comput. Phys. Commun. 178, 745 (2008), arXiv:0710.2067 [hep-ph]. [259] F. Mahmoudi, Comput. Phys. Commun. 180, 1579 (2009), arXiv:0808.3144 [hep-ph]. [260] P. Athron, M. Bach, H. G. Fargnoli, C. Gnendiger, R. Greifenhagen, J.-h. Park, S. Paehr, D. Stckinger, H. Stckinger-Kim, and A. Voigt, Eur. Phys. J. C76, 62 (2016), arXiv:1510.08071 [hep-ph]. [261] G. Degrassi, P. Gambino, and P. Slavich, Comput. Phys. Commun. 179, 759 (2008), arXiv:0712.3265 [hep-ph]. [262] J. R. Ellis, K. A. Olive, and V. C. Spanos, Phys. Lett. B624, 47 (2005), arXiv:hepph/0504196 [hep-ph]. [263] B. Allanach, S. Kraml, and W. Porod, in Supersymmetry and unification of fundamental interactions. Proceedings, 10th International Conference, SUSY’02, Hamburg, Germany, June 17-23, 2002 (2002) pp. 904–910, arXiv:hep-ph/0207314 [hep-ph]. [264] B. C. Allanach, S. Kraml, and W. Porod, JHEP 03, 016, arXiv:hep-ph/0302102 [hepph]. [265] B. C. Allanach, A. Djouadi, J. L. Kneur, W. Porod, and P. Slavich, JHEP 09, 044, arXiv:hep-ph/0406166 [hep-ph]. [266] E. Bagnaschi, G. F. Giudice, P. Slavich, and A. Strumia, JHEP 09, 092, arXiv:1407.4081 [hep-ph]. [267] P. Draper and H. Rzehak, Phys. Rept. 619, 1 (2016), arXiv:1601.01890 [hep-ph]. [268] M. E. Gomez, G. Lazarides, and C. Pallis, Nucl. Phys. B638, 165 (2002), arXiv:hepph/0203131 [hep-ph]. [269] B. C. Allanach, G. Belanger, F. Boudjema, and A. Pukhov, JHEP 12, 020, arXiv:hepph/0410091 [hep-ph]. [270] R. Catena and B. Schwabe, JCAP 1504 (04), 042, arXiv:1501.03729 [hep-ph]. 36 [271] J. Harz, B. Herrmann, M. Klasen, K. Kovarik, and P. Steppeler, Proceedings, 2015 European Physical Society Conference on High Energy Physics (EPS-HEP 2015): Vienna, Austria, July 22-29, 2015, PoS EPS-HEP2015, 410 (2015), arXiv:1510.06295 [hep-ph]. [272] E. Bagnaschi, J. Pardo Vega, and P. Slavich, Eur. Phys. J. C77, 334 (2017), arXiv:1703.08166 [hep-ph]. [273] M. Endo, T. Moroi, and M. M. Nojiri, JHEP 04, 176, arXiv:1502.03959 [hep-ph]. CHAPTER 2 THEORETICAL UNCERTAINTIES IN THE CALCULATION OF SUPERSYMMETRIC DARK MATTER OBSERVABLES In this chapter, we include the article Theoretical uncertainties in the calculation of supersymmetric dark matter observables with permission of the authors and the Journal of High Energy Physics (JHEP), who originally published it in JHEP 1805, 113 (2018). 38 Published for SISSA by Springer Received: January Revised: March Accepted: April Published: May 11, 30, 25, 17, 2018 2018 2018 2018 Paul Bergeron,a Pearl Sandicka and Kuver Sinhab a Department of Physics & Astronomy, University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, U.S.A. b Dept. of Physics and Astronomy, University of Oklahoma, 440 West Brooks Street, Norman, OK 73019, U.S.A. E-mail: [e-mail], [e-mail], [e-mail] Abstract: We estimate the current theoretical uncertainty in supersymmetric dark matter predictions by comparing several state-of-the-art calculations within the minimal supersymmetric standard model (MSSM). We consider standard neutralino dark matter scenarios — coannihilation, well-tempering, pseudoscalar resonance — and benchmark models both in the pMSSM framework and in frameworks with Grand Unified Theory (GUT)-scale unification of supersymmetric mass parameters. The pipelines we consider are constructed from the publicly available software packages SOFTSUSY, SPheno, FeynHiggs, SusyHD, micrOMEGAs, and DarkSUSY. We find that the theoretical uncertainty in the relic density as calculated by different pipelines, in general, far exceeds the statistical errors reported by the Planck collaboration. In GUT models, in particular, the relative discrepancies in the results reported by different pipelines can be as much as a few orders of magnitude. We find that these discrepancies are especially pronounced for cases where the dark matter physics relies critically on calculations related to electroweak symmetry breaking, which we investigate in detail, and for coannihilation models, where there is heightened sensitivity to the sparticle spectrum. The dark matter annihilation cross section today and the scattering cross section with nuclei also suffer appreciable theoretical uncertainties, which, as experiments reach the relevant sensitivities, could lead to uncertainty in conclusions regarding the viability or exclusion of particular models. Keywords: Supersymmetry Phenomenology ArXiv ePrint: 1712.05491 c The Authors. Open Access, ⃝ Article funded by SCOAP3 . https://doi.org/10.1007/JHEP05(2018)113 JHEP05(2018)113 Theoretical uncertainties in the calculation of supersymmetric dark matter observables 39 Contents 1 Introduction 2 4 4 6 7 8 9 10 3 The 3.1 3.2 3.3 3.4 13 14 14 15 16 physics of neutralino dark matter benchmarks Coannihilation of B̃ with light scalars Well-tempering of dark matter A-funnel annihilation Pure Higgsino (h̃) composition 4 Results: pMSSM analysis 4.1 Bino-stop coannihilation 4.2 Bino-squark coannihilation 4.3 Pure higgsino 4.4 Well-tempered neutralino 4.5 A-funnel 4.6 Summary: broad trends in the pMSSM analysis 16 19 21 22 23 24 25 5 Results: GUT analysis 5.1 Comparison of EWSB sectors 5.2 Dark matter observables 5.2.1 CMSSM benchmarks 5.2.2 NUHM benchmarks 26 28 33 33 36 6 Conclusions 37 A Data tables for pMSSM models 38 B Data tables for GUT models 46 –1– JHEP05(2018)113 2 Methodology of the comparison 2.1 Benchmark points 2.2 Pipeline structure and nomenclature 2.3 Details of the calculations 2.3.1 Spectrum calculators 2.3.2 Higgs calculators 2.3.3 Dark matter calculators 40 1 Introduction –2– JHEP05(2018)113 The search for supersymmetry and its connection to dark matter physics have been prominent areas of research in particle phenomenology, both theoretical and experimental, over the last few decades. Experimental results have provided significant constraints on the Minimal Supersymmetric Standard Model (MSSM) via the discovery of the Higgs boson [1, 2], as well as via null results from collider searches for new particles [3, 4] and dark matter direct and indirect detection experiments. The LHC in particular has pushed the limits on squark masses to roughly the TeV range, ruling out much of the constrained MSSM (CMSSM) (eg. [5]). Nevertheless, within the full MSSM as well as more minimal frameworks, much parameter space remains in which the thermal relic abundance of the lightest neutralino explains the astrophysical cold dark matter and a Higgs boson consistent with that discovered at the LHC is predicted [6–33]. Furthermore, it can be argued that, despite not yet having discovered any new supersymmetric (SUSY) partners, the verdict is still out on (even weak-scale) supersymmetry (eg. [34] and [35]). As the mechanism of supersymmetry breaking is far from clear, models may be defined either in the UV, near the so-called Grand Unification (GUT) scale, or in the IR, for example within the well-studied phenomenological MSSM (pMSSM) framework. In either case, one would like to calculate the superpartner spectrum at the weak scale and the masses and couplings in the Higgs sector. Finally, once the Lagrangian is known at the weak scale, it can be used to calculate the dark matter observables, including the relic density, the current annihilation cross section, and the scattering rates, all of which can be compared with experimental results. A plethora of public software packages have been developed to facilitate the analysis of various SUSY models and compare their predictions with experimental results (for example, [36–43]). Our confidence in the accuracy of the calculations done by any package is based on the continual improvements made by its authors and on the agreement of results from different packages. Comparative studies of spectrum generators and Higgs sector calculators have been undertaken before (eg. [44–49]). Differences in the renormalization group running and predictions for sparticle masses within the same supersymmetric model have been observed [44–46], and sensitivities of the Higgs sector have also been explored [47–49]. However, previous studies have focused primarily on models with relatively light (O(100) GeV) sparticles, and the most recent comparison of the full sparticle spectrum was undertaken more than a decade ago [46]. There are several publicly-available software packages that calculate quantities that can be observed at dark matter direct and indirect detection experiments as well as the relic abundance of dark matter within a particular model. micrOMEGAs [42] and DarkSUSY [43] are two examples. Studies have been carried out regarding the accuracy with which a single observable is calculated by an individual software package [50–52], though there are relatively few studies that compare the calculation of dark matter observables by different software packages (eg. [53]), and none of which we are aware that address LHC-era supersymmetric benchmarks. 41 –3– JHEP05(2018)113 In this report, we embark on a comparison study designed with three advancements over previous studies: first, we compare calculations for the sparticle spectrum, the Higgs sector, and the dark matter observables, discussing, when possible, differences in the implementations of the underlying physics of the calculations in each case. Second, we incorporate the various calculators into comprehensive pipelines to study not only the effects of the choice of an individual calculator, but also all downstream effects of those choices on subsequent calculations. Finally, we analyze the above choices and observables in the context of several SUSY benchmark models chosen as representative of models that are interesting in the light of LHC Run-1 and null results from recent dark matter searches as described below. This study is conducted in two parts: to begin, we investigate a set of pMSSM models from ref. [54]. The dark matter scenarios we consider are coannihilation (bino-stop and bino-squark), A-funnel, well-tempered neutralinos, and pure higgsinos. We will see that the spectrum generators can differ by up to 1 - 2 % in their predicted masses for the stop and the first two generations of squarks, and by up to 20% in the gauge composition of the lightest neutralino, for a given pMSSM model. As for the dark matter observables, differences of up to a factor of ∼ 3 − 5 in the relic density and current annihilation cross section, and up to a factor of ∼ 10 in the predicted scattering cross section are possible for the different pipelines. The theoretical uncertainty in the relic density of neutralino dark matter already far exceeds the statistical errors reported by the Planck collaboration, while the dark matter annihilation cross section today and the scattering cross section with nuclei also suffer appreciable theoretical uncertainties, which, as experiments reach the relevant sensitivities, could lead to uncertainty in conclusions regarding the viability or exclusion of particular models. In the second part of our study, we consider four benchmark models defined at the GUT scale — two CMSSM points and two points from models with non-universal Higgs masses (NUHM) [26, 30]. For GUT-scale models, we will find that discrepancies among the various pipelines are often amplified by the renormalization group running. For our CMSSM and NUHM benchmarks, we will see that the spectrum generators can give low energy values of the higgsino mass parameter µ and the pseudoscalar Higgs mass mA that differ by up to 150% - 200% (though the differences can be much greater at larger m0 than the values considered here). This leads to dramatic differences in the annihilation and scattering cross sections computed by the dark matter calculators. Before proceeding, we would like to reflect on whether a study like this is anachronistic at this juncture. With the LHC failing to find new physics yet, and supersymmetric WIMP searches yielding null results, one might ask whether it makes sense to go back to benchmark SUSY scenarios yet again. We remind the reader that the connection of supersymmetry to dark matter physics, while robust from a high level perspective due to the WIMP miracle, was always fragile at the model-building level, at least under the assumption of a standard cosmological history. The dark matter relic density is often obtained in fine-tuned regions of parameter space, which either exhibit compressed spectra, or have suppressed interactions with nuclei. Many of these scenarios are difficult to probe at colliders or direct detection experiments, and also have small annihilation rates in the current Universe. Moreover, their fine-tuned nature means that detailed predictions for physical quantities in these scenarios are particularly sensitive to the approximations 42 2 Methodology of the comparison In this section, we present our benchmark points, the calculator pipelines we study, and the details of the calculations undertaken by each of the calculators. 2.1 Benchmark points We consider two sets of supersymmetric benchmark points, all of which assume that the lightest supersymmetric particle (LSP) is the lightest neutralino, which is therefore the dark matter candidate. For future reference, we describe our notation here. We choose the bino — wino — higgsino basis to write the neutralino mass matrix as ⎛ ⎞ M1 0 −mZ sW cβ mZ sW sβ ⎜ 0 M2 mZ cW cβ −mZ cW sβ ⎟ ⎜ ⎟ MN = ⎜ (2.1) ⎟, ⎝ −mZ sW cβ mZ cW cβ ⎠ 0 −µ mZ sW sβ −mZ cW sβ −µ 0 where we follow the standard notation: sW = sin θW , cW = cos θW , sβ = sin β, cβ = cos β and tan β = v2 /v1 , with v1,2 being the vacuum expectation values of the two Higgs fields H1,2 . The mass matrix can be diagonalized by a unitary mixing matrix N , N ∗ MN N † = diag(mχ̃01 , mχ̃02 , mχ̃03 , mχ̃04 ), (2.2) where the eigenvalues are the neutralino masses. The lightest neutralino mass eigenstate can be written as χ̃01 = N11 B̃ + N12 W̃ + N13 H̃1 + N14 H̃2 . (2.3) As the composition of the neutralino LSP determines much of the dark matter physics, we will often refer to the bino fraction, |N11 |2 , and the higgsino fraction, |N13 |2 + |N14 |2 . –4– JHEP05(2018)113 used. It is not a surprise that theoretical uncertainties in these scenarios can substantially outweigh experimental uncertainties. Given the current precision of the measurement of the dark matter abundance and the dramatically improving sensitivities to dark matternucleon scattering in the era of ton-scale experiments, it could be argued that it is more important than ever to examine the precision and accuracy of the predicted values for observable quantities in supersymmetric models. We also note that our findings may be relevant for some more general (non-supersymmetric) models of dark matter, so long as they share particular characteristics with the benchmarks considered here, for example, models in which the relic abundance is achieved via resonant annihilations. The paper is organized as follows. In section 2 we present the benchmark MSSM points considered here and discuss the calculators and pipelines we study. In section 3 we discuss relevant aspects of the physics of neutralino dark matter. In sections 4 and 5 we present our results for the pMSSM and GUT-scale benchmarks, respectively. Finally, in section 6 we summarize the conclusions of our study. Numerical results for all benchmarks are compiled in appendices A and B. 43 Point CMSSM 1 CMSSM 2 NUHM NUHM A NUHM B m1/2 2098.41 900.00 3416.12 3200 3200 m0 5648.13 785.00 1376.34 1650 2000 A0 781.89 −2882.83 3139.29 3139.29 3139.29 sign(µ) + + + + + tan(β) 51.28 28.36 39.01 39.01 39.01 m2Hu N/A N/A 1.335 · 107 1.335 · 107 1.335 · 107 mt 173.34 173.20 173.24 173.24 173.24 Table 1. Parameters defining the CMSSM and NUHM benchmarks considered in section 5. The second set consists of 4 points of interest defined at the GUT scale. The defining parameters of these 4 points may be found in table 1. Three of these points are based on the MasterCode Collaboration’s post-LHC Run I best fit points from the CMSSM and NUHM 1 (hereafter, “NUHM 1” one will simply be referred to as “NUHM”) [30]. The MasterCode analysis also includes constraints from dark matter direct detection experiments and the observed dark matter abundance. The MasterCode CMSSM best fit point will be denoted CMSSM 1. The final CMSSM point, denoted CMSSM 2, is inspired by the τ̃ coannihilation benchmark point from ref. [26]. Both the MasterCode best fit NUHM point and the CMSSM 2 point are in coannihilation regions of parameter space, the former by virtue of having nearly pure higgsino dark matter, and are therefore extremely sensitive to variations in the RGE running. Significant variations in the running can occur between different spectrum calculators as well as between different versions of the same spectrum calculator. For example, a point that yields the correct dark matter abundance via τ̃ coannihilation may end up with a τ̃ LSP if a different calculator or version is employed. Since the publication of ref. [30] and ref. [26], there have been several updates to SOFTSUSY, which was used to calculate the sparticle spectrum in both studies. Here, we consider two NUHM points inspired by the best fit point in ref. [30], denoted “NUHM A” and “NUHM B”, chosen with the requirement that a valid relic density would be achieved by NUHM A via our SPheno pipelines and NUHM B via our SOFTSUSY pipelines. The original MasterCode NUHM point is included in table 1 for reference. Furthermore, the original τ̃ coannihilation benchmark from ref. [26], calculated with SOFTSUSY 3.3.7, yields a τ̃ LSP in the more contemporary version SOFTSUSY 3.7.3. As such, we consider a similar point where m0 has been increased by about 20 GeV over the original value to avoid a τ̃ LSP. –5– JHEP05(2018)113 The first set of benchmark points consists of 5 pMSSM points from the Snowmass 2013 white paper ref. [54]. These points are representative of the pMSSM landscape and of the primary mechanisms by which the correct relic density of neutralino dark matter is achieved: sfermion coannihilation, rapid annihilation via a pseudo-scalar Higgs resonance, pure higgsino content, and the so-called well-tempered neutralino. We will discuss each of these in section 3. The spectrum for each point can be found at [55]. 44 SOFTSUSY 3.7.3 SPheno 3.3.8 FeynHiggs 2.12.0 SusyHD 1.0.2 micrOMEGAs 4.3.1f DarkSUSY 5.1.2 Spectrum Generator Higgs Calculator Dark Matter Calculator Spectrum Generator Higgs Calculator Dark Matter Calculator Snowmass SOFTSUSY 3.1.7 N/A micrOMEGAs 2.4 MasterCode SOFTSUSY 3.3.9 FeynHiggs 2.10.0 micrOMEGAs 3.2 Snowmass† SOFTSUSY 3.1.7 N/A micrOMEGAs 2.4.5 MasterCode† SOFTSUSY 3.3.9 FeynHiggs 2.10.0 micrOMEGAs 3.5.5 Snowmass∗ SOFTSUSY 3.7.3 N/A micrOMEGAs 4.3.1f MasterCode∗ SOFTSUSY 3.7.3 FeynHiggs 2.12.0 micrOMEGAs 4.3.1f Table 2. Summary of the calculators used in the Snowmass-type and MasterCode-type pipelines used herein. Unadorned names refer to the original work whose results are quoted in our analysis. Daggers (†) denote our reproductions of original results with our implementations of the packages; these are the same versions as the original with the exception of micrOMEGAs 2.4.5 for Snowmass† . Asterisks (∗) denote our updated versions of the pipelines using the same contemporary versions as our pipelines. 2.2 Pipeline structure and nomenclature The pipelines considered here are comprised of a selection of the many publicly-available calculators on the market for studying supersymmetry and dark matter physics. We will refer to the different calculators considered here in terms of their primary functions: • mass spectrum generators, SOFTSUSY [37] and SPheno [38, 56]; • Higgs sector calculators, FeynHiggs [40, 57–60] and SusyHD [41]; and • dark matter observable calculators, micrOMEGAs [42, 61–63] and DarkSUSY [43, 64–67]. As demonstrated in figure 1, each pipeline is composed of 3 calculators, one of each type — spectrum, Higgs, and dark matter. In this way, we consider 8 different pipelines, such as SOFTSUSY-FeynHiggs-micrOMEGAs or SOFTSUSY-SusyHD-DarkSUSY. The inclusion of the Higgs calculator is to ensure that details for the Higgs sector are achieved before computing dark matter observables. Files in SLHA [68] format are used to pass information between each calculator, with the input and output being retained at each stage. We note –6– JHEP05(2018)113 Figure 1. Depiction of the 8 pipelines used in this study. From top to bottom, we show the supersymmetric sparticle mass spectrum generators, the Higgs sector calculators, and the programs that calculate the dark matter observables. 45 2.3 Details of the calculations Here we discuss the details of the calculations performed by each software package. In particular we focus on the contrasting choices underlying the differences between packages, taking each tier of the pipeline in turn. Unless otherwise specified, we take the default settings for each calculator throughout the following analysis. 1 In addition to micrOMEGAs, MasterCode’s calculation of the relic density is verified by the private code SSARD [69] (information about this code is available from K.A. Olive: it contains important contributions from J. Evans, T. Falk, A. Ferstl, G. Ganis, F. Luo, A. Mustafayev, J. McDonald, K.A. Olive, P. Sandick, Y. Santoso, V. Spanos and M. Srednicki), which is also used to calculate the SI scattering cross sections [30]. –7– JHEP05(2018)113 that two separate input files are necessary for SOFTSUSY and SPheno, as there are minor differences in the expected format of the SLHA input files for the two calculators. An important caveat in the passage of information between the programs is the handling of the branching ratios. While SLHA formatted files do include blocks for detailing particle decays, they are not universally utilized by all spectrum calculators. For example, SPheno does write the decay blocks for its SLHA output files, while SOFTSUSY does not. For SOFTSUSY pipelines, if FeynHiggs is used, Higgs decay widths will be written, but if SusyHD is used, since it only calculates the CP -even Higgs mass, no Higgs decay widths will be recorded. This means that there are no recorded widths in the SOFTSUSY-SusyHD pipelines, which can lead to discrepancies in the calculation of the dark matter abundance, for example, if dark matter annihilates primarily via the psuedoscalar resonance. For our analysis, the versions of the calculators implemented (unless otherwise noted) are SOFTSUSY 3.7.3, SPheno 3.3.8, FeynHiggs 2.12.0, SusyHD 1.0.2, micrOMEGAs 4.3.1f, and DarkSUSY 5.1.2. Since all of the calculators studied here are continuously updated and improved, specifically since the publication of refs. [26, 30, 54], we also include the versions of the pipelines used in the Snowmass and MasterCode studies for proper comparisons with their results, as summarized in table 2. The Snowmass pipeline uses SOFTSUSY 3.1.7 and micrOMEGAs 2.4 and is denoted as “Snowmass”, and the updated pipeline (still without FeynHiggs) is denoted as “Snowmass∗ ,” i.e. SOFTSUSY 3.7.3 and micrOMEGAs 4.3.1f. Alternatively, the MasterCode pipeline utilized1 SOFTSUSY 3.3.9, FeynHiggs 2.10.0, and micrOMEGAs 3.2. Since the updated MasterCode pipeline (MasterCode∗ ) is identical to that of our SOFTSUSY-FeynHiggs-micrOMEGAs pipeline we do not denote it separately from here forward. Furthermore we denote pipelines with a dagger (Snowmass† /MasterCode† ) to indicate when we have reproduced the calculation of the original pipeline; otherwise the result is quoted as published. That said, not all versions of micrOMEGAs are currently available, in which case we use the closest available version. As will be discussed below, there can be substantial variation in results and calculational techniques between different versions of the same software package. Indeed, the version numbers are critical to the interpretation of the results presented here. In the remainder of this paper, however, for the sake of brevity, we will suppress the version numbers for the packages that compose the pipelines unless otherwise specified, and refer the reader to figure 1 and table 2. 46 SOFTSUSY 2-loop (3-loop) 2-loop, running ̸∋ O(g24 , g22 g12 , g14 ) SPheno 2-loop 1-loop, running 2-loop QCD + 1-loop SUSY 2-loop QCD + 1-loop SUSY 2-loop QCD + 2-loop SUSY O(αs2 ) 2-loop QCD + 2-loop SUSY O(αs2 ) 1-loop + 2-loop O(αi αj , ατ2 ) 1-loop + 2-loop O(αi αj , ατ2 ) 1-loop + 2-loop O(αi αj , αt2 au) 1-loop + 2-loop O(αi αj , αt2 au) 1-loop 1-loop (+ 2-loop SUSY QCD O(αs2 )) 1-loop (+ 2-loop SUSY QCD O(αs2 )) 1-loop (+ 2-loop SUSY QCD O(αs2 )) 1-loop 1-loop 1-loop 1-loop Table 3. Orders of RGE and radiative corrections employed by SOFTSUSY and SPheno. The common terms between both SOFTSUSY and SPheno for the Higgs sector, denoted αi αj , are the members of the set {αt αs , αt2 , αt αb , αb2 , αb αs , αb ατ }. SOFTSUSY’s optional modes are detailed in parentheses: a “high order mode” for 2-loop radiative corrections to the squark and gluino pole masses [70] and a “high accuracy mode” for 3-loop RGEs (requires CLN and GiNaC interfaces) [71]. The default modes are used in our analysis. 2.3.1 Spectrum calculators For the evaluation of the sparticle mass spectrum, we consider SOFTSUSY and SPheno. First, they evaluate the gauge and Yukawa couplings at the electroweak scale before running them to the high scale and applying the soft SUSY breaking boundary conditions. After running back down to the electroweak scale, mZ , the initial tree-level values of the sparticle and Higgs masses are calculated. These masses are used as input for the iterative loop that comprises the calculation. In the iterative step, the current mass spectrum is evolved to a high scale Mx , defined for GUT models to be the scale at which g1 (Mx ) = g2 (Mx ) and √ defined for the pMSSM to be some low scale near MSUSY = mt̃1 mt̃2 , where the soft SUSY breaking parameters are set from the specified boundary conditions. GUT models are then evolved down to MSUSY , where the two cases proceed in the same manner. At MSUSY , electroweak boundary conditions are applied and the sparticle and Higgs pole masses are evaluated at the loop level. These are now input for the next iteration of the loop — beginning with a new, and more accurate, calculation of the gauge and Yukawa couplings. When a stable solution of a given accuracy is reached, the iteration terminates and the spectrum is run down to mZ . The programs, however, do differ in the details of their calculations, as summarized in table 3 (see also table 1 from reference [45]). It is worth reminding that, even when the quoted loop level is the same, the scheme in which the calculation is handled can lead to important differences. This manifests in the choice between M S and DR schemes, where the latter amounts to a higher order correction to the former. While SOFTSUSY and SPheno both employ running DR masses in their calculations, their methods of calculating the DR corrections are different. –8– JHEP05(2018)113 RGEs VEVs Yukawa Couplings hb ht Higgs Sector tadpoles h0 H 0 SUSY masses χ± χ0 t̃ b̃ g̃ 47 This important difference enters in determination of the Yukawa couplings, which are calculated from the quark masses at scale Q: yt (Q) = mt (Q) √ 2, v2 yb,τ (Q) = mb,τ (Q) √ 2. v1 (2.4) and then resummed with the SUSY corrections via mb (mZ )DR MSSM = mb (mZ )DR SM . 1 − ∆bSUSY (mZ ) (2.6) For the top mass, SOFTSUSY employs a similar correction, with the 2-loop QCD corrections as ) ' (* αs 43 3 2 DR MS 2 mt (mZ )SM = mt (mZ )SM 1 − (5 − 3L) − αs 0.538 − L + 2L , (2.7) 3π 24π 2 8π where L = ln(m2t (mZ )/m2Z ) for the top mass. But SPheno uses a modified αs2 term according to the large quark mass expansion in ref. [74], which results in an αs2 coefficient of ' ( 8 2011 16 8ζ(3) 246 22 2 − + + ln(2) − + 2L + 2L . 9 18π 2 9 3π 2 3π π 2.3.2 Higgs calculators The two Higgs mass calculators studied here are FeynHiggs and SusyHD. Prior to FeynHiggs 2.11.3, FeynHiggs consistently yielded SM Higgs masses ∼ 2 − 4 GeV above the value yielded by SusyHD [41]. We found that the differences between the two Higgs sector calculators were enough to present small but noticeable differences in the dark matter observables, particularly for A-funnel points. However, as of FeynHiggs 2.12.0, the differences in the results from FeynHiggs and SusyHD are far better understood, and it is possible to choose flags in FeynHiggs such that the numerical discrepancies are dramatically reduced.2 Most notable are two changes that yielded large shifts [112, 113]. The first S , which induces a change is the inclusion of next-to-next-to-leading order (NNLO) mM t S is used. The downward shift in mh by as much as ∼ 2 GeV relative to when the NLO mM t M second change is the inclusion of electroweak contributions in evaluating mt S , accounting for a downward shift of about 1 GeV. 2 We note that the hybrid approach employed by FeynHiggs has also been analytically compared to results from pure effective field theory, as employed by SusyHD, in ref. [114], which sheds light on the differences between the two approaches. –9– JHEP05(2018)113 In running to the high scale, the quark masses must be shifted from M S to DR, and both programs ultimately follow reference [72] for the 2-loop QCD corrections and reference [73] for the 1-loop SUSY contributions. For the bottom mass the DR value is arrived at in both SOFTSUSY and SPheno by ' ( αs 23αs2 3g22 13g12 MS mb (mZ )DR = m (m ) 1 − − + − , (2.5) Z SM b SM 3π 72π 2 128π 2 1152π 2 48 For the versions employed in our primary pipelines (displayed in figure 1), FeynHiggs and SusyHD are in close agreement when FeynHiggs includes the resummation of large logs at the 2-loop level; note that this is not the default mode of the calculation, but it is used in this study. For this reason, we will focus on pipelines that include FeynHiggs in the remainder of our analysis, though results from the SusyHD pipelines are included in all tables in the appendix. We stress that shifts in the value for mh are significant not just for the calculation of the dark matter observables, but also because they introduce important caveats to previous analyses where the Higgs mass is germane. Dark matter calculators The two dark matter calculators we consider are DarkSUSY and micrOMEGAs. Both are sophisticated programs that analyze dark matter observables and relevant collider observables (eg. b → sγ). We confine our interest to three astrophysical observables: the neutralino relic density, Ωh2 ; the annihilation rate today, ⟨σv⟩; and the spin-independent (SI) scattering cross section with nuclei, σ SI . The relic density is calculated by micrOMEGAs according to the relation Ωh2 |MCO = 2.742 · 108 mχ̃01 1 GeV Y0 , (2.8) where Y0 is the abundance of dark matter today. The same relation is used by DarkSUSY except that the numerical factor is 0.5% larger. The aim of both codes is thus to calculate the abundance of dark matter at the current temperature Y0 ≡ Y (T0 ), where the abundance is defined as the ratio of the number density and entropy density of dark matter Y = n/s. Both programs start with the Boltzmann equation [75] and follow reference [64] to write the differential equation as such that + , dY 2 = A(X) Y 2 (X) − Yeq (X) , dX A(X) = - πg∗ (mχ̃01 /X) mχ̃01 MP l X2 45 ⟨σeff v⟩, (2.9) (2.10) where the temperature has been swapped for the dimensionless quantity X = T /mχ̃01 and MP l is the Planck mass. Yeq is the thermal equilibrium abundance, and is expressed as /m 0 45 . mi i Yeq (T ) = g i K2 , (2.11) 2 T T 4πheff i where heff is the number of effective degrees of freedom in the entropy density and Kn is a Bessel function of the second kind. The parameter g∗ is related to the number of degrees of freedom of the system as ' ( heff T dheff 1/2 g∗ = √ 1+ , (2.12) geff 3heff dT where geff is the number of effective degrees of freedom in the energy density. geff and heff are drawn from hard-coded tables in both programs. In micrOMEGAs, the tables come from – 10 – JHEP05(2018)113 2.3.3 49 ∆Y = Y − Yeq = 1 : ∆Y ≪ Yeq . 2A (2.13) Letting ∆Y (Xf 1 ) = δ Yeq (Xf 1 ) where δ is a small number chosen to be 1.5, micrOMEGAs solves 2 Y ′ (Xf 1 ) = δ(δ + 2)A(Xf 1 )Yeq (Xf 1 ) (2.14) for Xf 1 . This point is used as the starting point for the numerical evaluation of equation (2.9) via the Runge-Kutta method, stopping at a point Xf 2 . This latter point is chosen such that Yeq (Xf 2 ) < 0.01Y (Xf 2 ), and allows for the integration of equation (2.9) to solve for Y0 : 1 X0 1 1 = + A(X)dX . (2.15) Y (X0 ) Y (Xf 2 ) Xf 2 Because T0 = 2.725 K , X0 ∼ 1014 and micrOMEGAs takes the upper bound to be effectively infinity. Alternatively, DarkSUSY chooses to solve equation (2.9) without applying approximations. Stiffness is still a concern, so DarkSUSY solves the problem by first discretizing the function with trapezoids and then numerically solving the differential equation with an adaptive step-size approach to Euler’s method. For the computation of the (co-)annihilation of sparticles contributing to the relic density, both micrOMEGAs and DarkSUSY follow reference [65]. Furthermore, both codes include all 2-body processes4 between neutralinos, charginos, sneutrinos, sleptons, and squarks. micrOMEGAs includes processes with gluons, as well. External programs are incorporated into the distributions of DarkSUSY and micrOMEGAs to calculate the relevant cross sections. micrOMEGAs includes CalcHEP [84] for the evaluation of relevant tree-level annihilation and coannihilation diagrams at run-time for a given model. However, some processes are 3 This description holds for the case of a single dark matter particle. In general, micrOMEGAs can handle models with two component dark matter, which requires a modification of the Boltzmann equation to allow for additional processes and abundances associated with a second dark sector. In the latter case, the Rosenbrock algorithm [79, 80] is used to avoid the stiffness of the ODE. 4 micrOMEGAs does have the option of allowing 2 and 3-body WZ final states, but these are not part of the default calculation [83]. – 11 – JHEP05(2018)113 Olive et al. [76, 77] by default, but there is an option to use the tables from Hindmarsh & Philipsen [78], which is the default in DarkSUSY. The effective degrees of freedom are calculated from their respective extensive quantities, typically by assuming an ideal gas as was done in Olive et al. However, interactions will be significant at in the early universe where temperatures are high, and allowing for interactions requires a modification to the equation of state. As the weak corrections will be suppressed by the W and Z masses, QCD corrections to the effective degrees of freedom will be dominant. Hindmarsh & Philipsen found that incorporating QCD corrections allows for as much as a 3.5% modification to the relic density [78]. From here, the two programs diverge in their treatment of equation (2.9), as care must be taken due to the stiffness of the ODE [42, 43]. To calculate Y (T0 ), micrOMEGAs employs the freeze-out approximation3 [81, 82], writing 50 significantly suppressed so as to be insignificant. micrOMEGAs calculates the Boltzmann suppression factor −Xf K1 ((mi + mj )/Tf ) Bf = ≈e K1 (2mχ̃01 /Tf ) mi +mj −2m 0 χ̃1 m 0 χ̃1 (2.16) 5 See DarkSUSY 5.1.2 src/slha/dsfromslha.F, lines 766-769. – 12 – JHEP05(2018)113 for each channel and, by default, neglects channels where Bf < 10−6 , though this cut-off can be changed by the user [61]. Within DarkSUSY, on the other hand, the programs REDUCE [85] and FORM [86] are used in the evaluation of annihilation and coannihilation cross sections. REDUCE is used for the evaluation of helicity amplitudes for all processes between charginos and neutralinos. This allows for the analytical determination of one type of diagram only once with a numerical sum over all initial and final states performed for the contributing diagrams afterwards. All other processes involving sfermions have their scattering amplitudes evaluated by FORM. One interesting difference between micrOMEGAs and DarkSUSY is in the inclusion of internal bremsstrahlung (IB). micrOMEGAs includes final states with two SM particles plus an additional photon for the evaluation of the annihilation cross section both in the early Universe and today. While DarkSUSY includes IB when considering the gamma ray signature from annihilation in our Galaxy’s halo, it is not incorporated by default in their calculation of the relic density. As we will see below, this will lead to significant differences in the dark matter observables. Next, we consider the differences in approaches used to calculate the SI scattering cross sections. Both programs utilize loop corrections to the scattering amplitudes but follow different frameworks. DarkSUSY follows the effective Lagrangian framework laid out in ref. [67], while micrOMEGAs utilizes the framework of ref. [87]. As discussed, for example, in ref. [87], the effective Lagrangian framework can miss crucial QCD effects, though, with modification, it is capable of reliably reproducing the 1-loop result for most cases. Another important difference between DarkSUSY and micrOMEGAs is in the nucleon form factors used to calculate the expected neutralino-nucleon elastic scattering cross section. The form factors for each calculator are tabulated in table 4. For all quarks, DarkSUSY uses larger form factors than micrOMEGAs. As discussed below, this leads to a difference in the predicted SI neutralino-nucleon scattering cross sections, however, as we will demonstrate, these differences alone are not enough to fully explain the discrepancies in the predictions. It is clear that the details of the loop corrections also play an important role in the calculation, and can provide significant contributions to the SI cross sections. A final difference exists in how each calculator determines the relevant particle widths used in the calculations. When available, micrOMEGAs reads in the decay blocks from the SLHA input file, but otherwise employs their own calculation to find any necessary widths. Alternatively, DarkSUSY does not currently read SLHA decay blocks and always performs their own evaluation of the relevant particle widths, though they note that future versions of their SLHA reader should include this functionality. 5 The importance of including accurate widths in the relic density calculation is particularly relevant for funnel points; previous studies have found O(10%) difference in the calculation of the A-funnel 51 quark micrOMEGAs 4.2.5 DarkSUSY 5.1.2 u 0.0153 0.0230 fTp d 0.0191 0.0340 s 0.0447 0.1400 u 0.0110 0.0190 fTn d 0.0273 0.0410 s 0.0447 0.1400 Table 4. Spin-independent, scattering form-factors used by micrOMEGAs and DarkSUSY for protons (fTp ) and neutrons (fTn ). 3 The physics of neutralino dark matter benchmarks In this section, we provide a brief introduction to aspects of the physics of neutralino dark matter that will be relevant for our analysis of the benchmark points. The challenge in obtaining the correct relic density of neutralino dark matter observed in the Universe is well-known.6 Specifically, one generically obtains an annihilation cross section at freeze-out that is too small, leading to too much neutralino dark matter in the present epoch. The general idea of WIMP dark matter is that, in order to produce the correct relic density, the annihilation cross section of dark matter in the early Universe should have been 6 This challenge is alleviated if one considers a non-thermal history for dark matter [95, 96]. In this work, however, we adhere to a thermal history. – 13 – JHEP05(2018)113 between micrOMEGAs and DarkSUSY [26]. As we will demonstrate, this difference in how the width is included can lead to seemingly inconsistent results among pipelines that end with micrOMEGAs, while results from DarkSUSY pipelines seem more consistent but are less robust. Finally, we would like to mention the important effect of Sommerfeld enhancement. In short, this is an effect from non-relativistic quantum mechanics where scattering is modified by the presence of a potential interaction between the two scattering particles. This enhancement to the scattering cross section goes as 1/v, and has been found to provide substantial enhancements to dark matter annihilation when mχ0 ! 2 TeV (e.g. see references [88–91]). This is of particular interest to high mass wino dark matter, where the presence of a highly degenerate wino-like chargino will decrease the relic density [92, 93]. Higgsino-like dark matter would also experience Sommerfeld enhancement, thanks to a triple mass degeneracy between χ̃01 , χ̃02 , and χ̃± 1 [94]. However, higgsino-like dark matter already exhibits efficient s-channel annihilation, reducing the importance of Sommerfeld enhancement [93]. Of the benchmark models we study herein, we expect the pMSSM pure h̃ to be the only model where Sommerfeld enhancement would play a role, albeit a small one as mχ̃01 is barely above 1 TeV. For this reason, and since neither micrOMEGAs nor DarkSUSY include a calculation of the Sommerfeld enhancement, we do not consider it further. We note that for a heavier wino-like dark matter candidate, it would be interesting and important to explore the theoretical uncertainties in the calculation of dark matter observables with the inclusion of the Sommerfeld effect. 52 3.1 Coannihilation of B̃ with light scalars The calculation of the relic density for the coannihilation region is very sensitive to the relative masses of the dark matter candidate and the light scalar(s) that coannihilate with it, as detailed eg. in [75]. Even small discrepancies in the mass spectrum given by different spectrum generators will result in significantly different predictions for the relic density. Regarding the indirect detection prospects for bino-scalar coannihilation models, the annihilation cross section for bino dark matter in the present Universe occurs mainly through t-channel exchange of light scalars. Bino-nucleon scattering generally proceeds via Higgs- or squark-exchange. The Higgs exchange diagram is suppressed for models in which bino-squark coannihilations dominate in the early Universe, due to the pure bino nature of the dark matter in these cases. Furthermore, if the first and second generation squarks are heavy, as is the case in our bino-stop coannihilation scenario, the squark-exchange diagram is suppressed, resulting in very small scattering cross sections with nuclei, as we will see. In the following analysis, three coannihilation scenarios will be relevant: B̃ − τ̃1 , B̃t̃1 ; and B̃-q̃, where q̃ denotes any first or second generation squark. In each case, the composition of the neutralino LSP is ! 99.9% bino. 3.2 Well-tempering of dark matter Well-tempered dark matter has been extensively studied in the context of recent direct, indirect, and collider searches [97–100]. The annihilation cross section depends on the mixture of bino and higgsino states, or, equivalently, on the higgsino fraction |N13 |2 +|N14 |2 . The full expression for the annihilation cross section, as well as various interesting limits, are – 14 – JHEP05(2018)113 around 1 pb. This is approximately the value one obtains if dark matter is a new particle with approximately weak-scale mass and with electroweak couplings, a happy accident called the “WIMP miracle”. However, as has been long appreciated, the details of actual SUSY models are somewhat less attractive than the idea sketched above. Although supersymmetry predicts that the annihilation cross section of two neutralinos should be in the neighborhood of 1 pb, the exact numerical value can span a range that covers orders of magnitude, depending on the composition of the neutralino and the mass spectrum of the other supersymmetric particles. Dark matter that is predominantly higgsino-like (h̃) annihilates to W and Z bosons with a cross section that involves the full strength of the SU(2) gauge coupling, and is moreover enhanced by the presence of spin-one particles in the final state. Higgsino and wino dark matter thus have cross sections that are too large for the observed relic density. On the other hand, binos (B̃) mainly annihilate to quark and lepton pairs, a process that suffers from helicity suppression. Binos therefore typically have a cross section for annihilation that is too low. The regions of supersymmetric parameter space that are compatible with the dark matter relic density thus tend to be fine-tuned. In the following subsections, we will consider the most well-studied regions, characterized by how the observed dark matter relic density is achieved: coannihilation models, well-tempered dark matter, A-funnel annihilation, and pure higgsino composition. 53 available for example in reference [101]. The predicted relic density and indirect detection signals in the current Universe are both sensitive to the higgsino fraction. The dominant neutralino-nucleon scattering occurs via CP-even Higgs exchange, which is a bino-higgsino-Higgs coupling. Since a well-tempered neutralino has sizable higgsino and bino fractions, the scattering rates with nuclei can be relatively large. 3.3 A-funnel annihilation σv ∼ 2 2 yA 3 χ̃χ̃ yAbb s , 2 2 2π (mA − s) + m2A Γ2A (3.1) where s = 4m2χ̃0 (1 − v 2 /4)−1 1 (3.2) is the Mandelstam variable, ΓA = 4.6 GeV is the width of the pseudoscalar at our benchmark, mA = 2042 GeV is its mass, and the Yukawa couplings to the b-quarks and neutralinos are given by imb tan β yAbb = √ 2vew and yAχ̃χ̃ = ig1 N11 (N14 cos β − N13 sin β). (3.3) To obtain the annihilation cross section, one must take a thermal average of eq. (3.1) or its more general equivalent. The resonance occurs in the zero velocity limit in the current Universe, and the annihilation cross section today is given by σv|v→0 . In the early Universe, the non-zero velocity leads to thermal broadening of the resonance, as is clear from the velocity dependence of the Mandelstam variable, eq. (3.2). Small differences in the calculation of mA and v by different generators can affect the relic density significantly. Note that from eq. (3.3), it is clear that to obtain the correct relic density, the A-funnel benchmark requires non-zero values of N11 and either N13 or N14 . Thus, though the dark matter is typically primarily bino, a higgsino component must be retained, i.e. the neutralino χ̃01 must have some bino-higgsino mixture. In this case, the SI scattering cross section with nuclei is mediated primarily by Higgs exchange. This is enabled by the non-zero higgsino component N13 or N14 required to obtain the correct relic density. However, the values of the higgsino fraction required to obtain the correct relic density are typically very small, corresponding to feeble scattering cross sections with nuclei. The direct detection prospects for the A-funnel scenario are thus rather challenging, as we will see. This should be contrasted to the case of the welltempered neutralino, where the large higgsino fraction drives both the relic density and the scattering cross section. – 15 – JHEP05(2018)113 A neutralino LSP can annihilate resonantly by exchanging a pseudoscalar Higgs boson A in the s-channel, provided mA ∼ 2mχ̃01 . Prospects of probing the A-funnel at colliders and the correlation with the observed Higgs mass have been extensively studied [102–106]. Prospects for direct [107, 108] and indirect [109] detection have also been studied. The annihilation cross section can be expressed as [110] 54 Model B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ % Difference < 0.01% −0.07% −0.01% < 0.01% −0.38% −0.13% −0.07% 0.04% −0.04% −0.95% 1 B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ 1729.83 1047.87 1880.61 1558.05 197.31 1732.11 1048.52 1879.80 1558.72 199.13 −0.13% −0.06% 0.04% −0.04% −0.93% Table 5. Masses, in GeV, of the lightest and second-lightest neutralinos and the lighter chargino for the pMSSM benchmark points. The corresponding higgsino fraction is given in parentheses. The percent differences are given relative to the SOFTSUSY values. 3.4 Pure Higgsino (h̃) composition Higgsinos with mass of ∼ 1 TeV can satisfy the relic density constraint, with the dominant mechanism being annihilation to gauge bosons. Coannihilation among charged (χ̃± 1 ) and neutral (χ̃01 and χ̃02 ) higgsinos is also important in this case. We refer to [111] for expressions of the relic density in various limits. The contribution of Higgs exchange diagrams to the scattering cross section with nuclei is suppressed due to the small gaugino-higgsino mixing. The contribution of squark exchange diagrams is also suppressed at our benchmark point since the squark masses are at several TeV. Thus, we generally expect small direct detection signals for this benchmark point. We refer to [97, 98] for detailed calculations of the scattering cross section of pure higgsinos. 4 Results: pMSSM analysis Here we present our pMSSM analysis. We consider the five pMSSM points from the Snowmass 2013 benchmarks [54] as discussed in section 2.1. The spectra in the neutralino, squark, and Higgs sectors obtained using the different spectrum generators we consider are displayed in tables 5, 6, and 7, respectively. For tables 5 and 6, the first column lists the dark matter benchmark scenario, while the next columns display the spectra of relevant sparticles obtained from SOFTSUSY and SPheno. – 16 – JHEP05(2018)113 B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ 2 + N2 ) mχ̃01 (N13 14 SOFTSUSY SPheno 754.08 (< 0.001) 754.06 (< 0.001) 1047.51 (0.999) 1048.29 (0.999) 853.82 (0.001) 853.90 (0.001) 1013.42 (0.002) 1013.41 (0.002) 148.30 (0.364) 148.86 (0.304) 2 + N2 ) mχ̃02 (N23 24 1729.67 (0.003) 1731.95 (0.003) 1047.69 (0.998) 1048.46 (0.998) 1879.64 (0.987) 1878.92 (0.983) 1557.89 (0.042) 1558.50 (0.043) 201.92 (0.992) 203.84 (0.992) mχ̃+ 55 Model B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ 3733.58 3057.72 925.41 1444.67 2580.93 B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ 781.73 2350.51 1181.25 2195.89 2202.00 mũ,L SPheno 3736.76 3068.16 936.09 1458.15 2588.55 md,L ˜ 3737.42 3069.18 932.77 1460.27 2589.71 mt̃,1 760.63 2351.08 1193.17 2197.74 2198.74 % Difference −0.10% −0.37% −1.51% −1.09% −0.34% SOFTSUSY 3998.02 3686.10 889.52 3089.32 1453.62 −0.10% −0.37% −0.80% −1.08% −0.34% 2805.23 1712.47 1939.34 3461.45 1263.56 2.70% −0.02% −1.01% −0.08% 0.15% 2388.06 2654.02 3190.93 2679.19 2512.46 mũ,R SPheno 4000.68 3693.28 897.26 3096.54 1462.64 md,R ˜ 2807.61 1723.26 1943.85 3468.39 1272.99 mt̃,2 2381.02 2658.47 3190.03 2684.07 2509.73 % Difference −0.07% −0.19% −0.87% −0.23% −0.62% −0.08% −0.63% −0.23% −0.20% −0.75% 0.29% −0.17% 0.03% −0.18% 0.11% Table 6. Masses, in GeV, of the squarks for the pMSSM benchmark points. The percent differences are given relative to the SOFTSUSY values. The final column lists the percent difference in the mass obtained from the two generators. In table 7, we compare results for the Higgs sector from FeynHiggs and SusyHD. As table 7 demonstrates, as of FeynHiggs 2.12.0 and SusyHD 1.0.2 these two Higgs sector calculators are in very good agreement. Hereafter we consider only the FeynHiggs branches of the pipelines in figure 1. We pause to elaborate on the Higgs sector, before moving on to analyze the dark matter results. In tables 7a & 7c we show the masses in the Higgs sector for all the dark matter benchmark scenarios studied. The masses corresponding to table 7a & 7b are computed by FeynHiggs, while those corresponding to tables 7c & 7d are computed by SusyHD. As per convention, Snowmass∗ is the updated Snowmass pipeline which amounts to the SOFTSUSY spectrum prior to FeynHiggs, here. We note that SusyHD only corrects mh and, thus, mH , mA , and mH ± are identical to those calculated by the relevant spectrum calculator. From these results, we see that the Higgs sector masses are not significantly affected by the choice of SUSY spectrum generator, with differences amounting to less that 0.01%. However, the inclusion of either FeynHiggs or SusyHD can provide a significant shift in mh from the Higgs mass calculated by the spectrum generator itself. The Snowmass points were constrained by the Higgs mass (126 ±1 GeV), but utilizing FeynHiggs moves the mass down by as much as 2 GeV (1.5%) which puts the benchmark points in strong tension with the measured value of the Higgs mass (125 ± 0.24 GeV). Figure 2 displays our results for the dark matter observables in the pMSSM. The benchmarks are denoted by different shapes: bino-stop coannihilation (diamonds), pure hig- – 17 – JHEP05(2018)113 B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ SOFTSUSY 3732.89 3056.85 922.18 1442.44 2579.86 56 Model B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ Snowmass* 125.48 125.55 125.02 125.99 126.76 mh SOFTSUSY 123.67 123.73 124.31 124.55 125.06 SPheno 123.67 123.73 124.31 124.55 125.06 Snowmass* 3523.47 1769.46 3725.10 2042.76 1399.90 mH SOFTSUSY 3522.48 1769.42 3723.10 2042.81 1398.77 SPheno 3522.49 1769.42 3723.10 2042.81 1398.77 (a) mh and mH as computed via the FeynHiggs branch of the pipeline. B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ Snowmass* 3522.57 1769.39 3725.00 2042.73 1399.88 mA SOFTSUSY 3522.57 1769.39 3725.00 2042.73 1399.88 SPheno 3522.57 1769.39 3725.00 2042.73 1399.88 Snowmass* 3523.51 1771.58 3725.75 2044.64 1402.36 mH ± SOFTSUSY 3520.26 1773.15 3724.19 2046.18 1398.75 SPheno 3520.27 1773.15 3724.19 2046.18 1398.76 (b) mA and mH ± as computed via the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ Snowmass* 125.48 125.55 125.02 125.99 126.76 mh SOFTSUSY 121.53 122.69 123.28 124.53 123.06 SPheno 121.52 122.69 123.061 124.30 123.06 Snowmass* 3523.47 1769.46 3725.10 2042.76 1399.90 mH SOFTSUSY 3523.47 1769.46 3723.10 2044.81 1399.90 SPheno 3523.64 1769.46 3725.10 2042.77 1399.88 (c) mh and mH as computed via the SusyHD branch of the pipeline. Model B̃ − t̃ Coan. Pure h̃ B̃ − q̃ Coan. A funnel Well-tempered χ̃ Snowmass* 3522.57 1769.39 3725.00 2042.73 1399.88 mA SOFTSUSY 3522.57 1769.39 3725.00 2042.73 1399.88 SPheno 3522.57 1769.39 3725.00 2042.73 1399.88 Snowmass* 3523.51 1771.58 3725.75 2044.64 1402.36 mH ± SOFTSUSY 3523.51 1771.58 3725.75 2044.64 1402.36 SPheno 3523.71 1771.55 3726.16 2044.66 1402.36 (d) mA and mH ± as computed via the SusyHD branch of the pipeline. Table 7. pMSSM Higgs masses: the masses, in GeV, of the CP-even Higgses (tables 7a & 7c) and the CP-odd and charged Higgs (tables 7b & 7d). Tables 7a & 7b employ the FeynHiggs pipelines while tables 7c & 7d use SusyHD. “Snowmass*” refers to the updated Snowmass pipeline which, at this level, amounts to the SOFTSUSY spectrum as no Higgs calculator was employed therein. Furthermore, note that SusyHD only provides a correction to the SM Higgs mass, and thus mH , mA , and mH ± in the SusyHD tables are the masses as computed by the relevant spectrum generator. – 18 – JHEP05(2018)113 Model 57 4.1 Bino-stop coannihilation The masses relevant for coannihilation are the lightest neutralino and the stop. From table 5, we see that there is good agreement between SOFTSUSY and SPheno in the neutralino spectrum for these two benchmark scenarios. The dark matter has a mass of ∼ 754 GeV, with very good agreement between the two spectrum generators, and is almost completely bino. On the other hand, from table 6, we see that the stop mass differs by 2.7% between SOFTSUSY and SPheno, although it is generally in the range where coannihilation is operational. This has a significant effect on the relic density, which, in these cases, is exponentially sensitive to the mass difference between the dark matter and the relevant coannihilation partner. The effect of the variation in the mass of the coannihilation partner can be seen in the upper left panel of figure 2. The solid magenta and cyan diamonds correspond to the relic density values computed by micrOMEGAs, for spectra coming from SOFTSUSY and SPheno, respectively. While SOFTSUSY yields a value of Ωh2 = 0.094, SPheno yields a value of Ωh2 = 0.035, and the difference stems entirely from the difference in stop masses computed by the two generators. Similarly, comparing the hollow magenta and cyan diamonds, which correspond to the relic density values computed by DarkSUSY, we see that while SOFTSUSY gives a value of Ωh2 = 0.120, SPheno yields a value of Ωh2 = 0.045. It is also interesting to compare the values yielded by micrOMEGAs and DarkSUSY for the same spectrum. For example, selecting the spectrum from SOFTSUSY and comparing the solid and hollow magenta diamonds, we see that micrOMEGAs gives Ωh2 = 0.094 while – 19 – JHEP05(2018)113 gsino (stars), bino-squark coannihilation (circles), A-funnel (pentagons), and well-tempered (triangles). Filled/unfilled points correspond to the use of micrOMEGAs/DarkSUSY. The pipelines used in generating the results are distinguished by color: we use magenta/cyan to distinguish, SOFTSUSY/SPheno, respectively, and black/green to delineate between the Snowmass/Snowmass∗ pipelines. The dark matter relic density for the pMSSM benchmarks is shown in the upper left panel of figure 2 (and tabulated in table 12). For comparison, we also include the Planck 3-sigma range [115] for the relic density, which is highlighted by the red band. The upper right panel of figure 2 shows the annihilation cross section today (tabulated in table 14), with the Fermi-LAT 6-year limits from dwarf spheroidal galaxies [116] included for comparison (limits on annihilation to τ + τ − in red and bb̄ in blue). In the lower left panel of figure 2 we show the spin-independent neutralino-nucleon elastic scattering cross section, where we plot the per-nucleon cross section averaged for Xe, computed from the proton and neutron values tabulated in table 16 (FeynHiggs pipelines), as well as exclusion contours from LUX [117] in solid red, PandaX [118] in solid blue, and LZ (projected) [119] in dashed black lines, for comparison. We now turn to a description of our results. Throughout our discussion, we will concentrate on the SOFTSUSY (magenta) and SPheno (cyan) pipelines. In the figures, we also plot the predictions from the Snowmass∗ and Snowmass† pipelines shown in solid green and solid black, respectively. Since the physics of these pipelines is very similar to the SOFTSUSY - micrOMEGAs pipeline shown by solid magenta shapes, we will not discuss them separately. 58 – 20 – JHEP05(2018)113 DarkSUSY gives Ωh2 = 0.120. We point out that these theoretical uncertainties exceed the current experimental uncertainty, which, here results in the hollow diamond lying within the Planck-allowed band, while the solid diamond does not. These discrepancies occur due to differences in the calculation of the effective cross section for each annihilation and coannihilation channel and the different relative weights of contributing final states in the coannihilation channels assigned by the calculators. We note that both micrOMEGAs and DarkSUSY give values for the coannihilation cross sections using an effective tree-level calculation. These values can be significantly altered if higher-order SUSY-QCD corrections are taken into account. After including loop diagrams containing a gluon, a gluino, a four-squark vertex, and incorporating gluon radiation, the authors of reference [120] have found a ∼ 20% discrepancy with the relic density computed by micrOMEGAs in the bino-stop coannihilation region. We will not consider these loop corrections further in this paper, but note that global ∼ 20% theoretical uncertainties are expected in all cases. The annihilation cross section of the bino-stop benchmark model in the current Universe is shown in the upper right panel of figure 2. There is remarkable agreement among the various pipelines. Since coannihilation channels are irrelevant in the present Universe, the sensitivity to the mass difference between the stop and the bino is absent. The annihilation proceeds mainly through the t-channel exchange of a stop, and the small difference in the stop mass reported by SOFTSUSY and SPheno does not affect this diagram as significantly. We find that both micrOMEGAs and DarkSUSY give similar values for the different channels in χ̃01 χ̃01 annihilation, the only difference being that DarkSUSY ascribes ∼ 5% contribution to the annihilation cross section from χ̃01 χ̃01 → gg final state, while micrOMEGAs does not return this channel. The SI scattering cross section is shown by the diamonds in the lower left panel of figure 2. For very heavy squarks, the leading scattering cross section is mediated by Higgs exchange, which is suppressed if the lightest neutralino is a pure higgsino or gauge state, as discussed in section 3. Thus, the combination of heavy squarks and pure bino eigenstate conspire to give suppressed scattering cross sections for the bino-stop coannihilation benchmark. The cross sections lie below the projected LZ limits, at approximately 10 −11 pb. There is a factor of ∼ 5 discrepancy between the SI scattering cross section yielded by SOFTSUSY - micrOMEGAs (solid magenta diamond) relative to SPheno-micrOMEGAs (solid cyan diamond). From tables 5, 6, and 7, we can see that while the higgsino fraction and the Higgs mass match to a high approximation for both SOFTSUSY and SPheno in the binostop coannihilation benchmark, there can be up to a 4 GeV difference in the masses of the squarks. While it is unlikely that this small variation in squark masses can account for the observed variation in the SI scattering cross section, we cannot pinpoint the exact source for the discrepancy. Comparing the dark matter calculators, we see that the solid and hollow magenta diamonds overlap entirely, meaning that after receiving the spectrum from SOFTSUSY, both micrOMEGAs and DarkSUSY computed the same scattering cross section. The matching is not quite as exact for the spectrum coming from SPheno (solid and hollow cyan diamonds), but the values are quite close. While DarkSUSY implements an effective Lagrangian in the heavy squark limit following [110] (see reference [67] and references therein for details), micrOMEGAs implements the full one-loop Lagrangian following reference [87]. 59 τ +τ − bb̄ Ωh2 ⟨σv⟩ [cm3/s] Planck Range mχ̃0 [GeV] σ SI [pb] 1 1 Pipeline Legend Snowmass† Black Snowmass∗ Green SOFTSUSY Magenta SPheno Cyan micrOMEGAs ⚫ DarkSUSY ○ PandaX 2016 LUX 2016 Benchmark Legend 𝐵̃ − 𝑡̃ ⧫ Pure ℎ̃ ★ 𝐵̃ − 𝑞̃ ⚫ 𝐴-funnel ⬟ Well Tempered 𝜒̃ ▸ LZ Projected mχ̃0 [GeV] 1 Figure 2. pMSSM results: the top left, top right, and bottom left panels show the neutralino relic density, annihilation cross section today, and the SI neutralino-nucleon elastic scattering cross section, respectively, all as functions of the dark matter mass, with a common legend in the lower right panel. For comparison, the Planck 3-sigma range for the dark matter abundance [115] is highlighted in red in the upper left panel, the limits on the annihilation cross section today from Fermi-LAT’s 6-year analysis of dwarf spheroidal galaxies [116] for annihilation to τ + τ − (red) and bb̄ (blue) are shown in the upper right panel, and exclusion limits from LUX (red) [117], PandaX (blue) [118], and LZ (projected; black) [119] are shown in the lower left panel. 4.2 Bino-squark coannihilation The physics for this benchmark is similar to that of the bino-stop case. Although there is good agreement in the neutralino spectrum, we see from table 6 that squarks can differ by up to 1.51% between SOFTSUSY and SPheno. The effect on the relic density can be seen in the upper left panel of figure 2. The solid magenta and cyan circles correspond to the relic density values computed by micrOMEGAs for spectra coming from SOFTSUSY and SPheno, respectively. While SOFTSUSY yields a value of Ωh2 = 0.087, SPheno yields a value of Ωh2 = 0.110. The difference can be attributed to the difference in squark masses. The fact that SOFTSUSY produces a lower value for the relic density than SPheno is due to the fact that squark masses from SOFTSUSY are lighter than those from SPheno, giving a stronger coannihilation effect. We also note that for a given spectrum generator, say SOFTSUSY, DarkSUSY gives a larger value of the relic density than micrOMEGAs. – 21 – JHEP05(2018)113 mχ̃0 [GeV] 60 4.3 Pure higgsino The results for the relic density, current annihilation cross section, and scattering cross section of the pure higgsino benchmark are shown by stars in the upper left, upper right, and lower left panels of figure 2, respectively. For the relic density there is good agreement between the different pipelines. The two spectrum calculators produce similar mass and higgsino fraction of the lightest neutralino for this benchmark, as is evident from table 5. Thus, the magenta and cyan stars for a fixed dark matter calculator overlap in the upper left panel of figure 2. There is some discrepancy between the results returned by micrOMEGAs and DarkSUSY due to differences in calculation of coannihilation processes between the neutral and charged higgsinos. Since coannihilation is unimportant in the current Universe, these discrepancies disappear and all stars align perfectly in the upper right panel of figure 2. For the SI scattering cross section, we notice that the rates for a pure higgsino are suppressed due to the small gaugino-higgsino mixing, similar to the case of a pure bino. As for the relic density, the main difference between pipelines comes from the choice of the dark matter calculator and whether an effective Lagrangian is used or the full one-loop Lagrangian is considered. – 22 – JHEP05(2018)113 The annihilation cross section in the current Universe is shown by the circles in the upper right panel of figure 2. It is evident that there is much more convergence of results among the various pipelines compared to the relic density computation. This can again be ascribed to the fact that coannihilation channels are absent in the current Universe and small differences in squark masses do not translate to large differences in the calculation of the t-channel squark exchange diagram. Comparing the diamonds (stop coannihilation) and circles (squark coannihilation) in the upper right panel of figure 2, we can see that the bino-squark coannihilation points exhibit greater spread in their current annihilation cross sections. This is due to the fact that the cross sections are driven by t-channel stop or squark exchange diagrams in the present Universe, and the different pipelines give a greater spread in the squark masses than they do the stop mass for the bino-stop benchmark. The SI scattering cross section is shown by the circles in the lower left panel of figure 2. As expected, the values are higher than for the bino-stop coannihilation case, due to the lighter squarks which make the leading contributions to the squark-exchange diagram. However, there is also greater disagreement between the different pipelines. Firstly, comparing the solid magenta circle with the solid cyan circle, we see that the cross section computed with the spectrum coming from SOFTSUSY is a factor ∼ 3 greater than that coming from SPheno. The same trend can be seen using DarkSUSY (comparing the hollow magenta circle with the hollow cyan circle), although the effect is smaller. For a given spectrum generator, it is clear that micrOMEGAs is giving larger values of the scattering cross section than DarkSUSY, since the solid circles are above the hollow ones. This is partly due to differences in the form factors used by the two calculators (see table 4). Using the form factors of DarkSUSY in micrOMEGAs, we find that the scattering cross sections reported by micrOMEGAs reduces by ∼ 30%, bringing the two calculators to greater agreement with each other. 61 4.4 Well-tempered neutralino The results for the relic density, current annihilation cross section, and SI scattering cross section of the well-tempered neutralino benchmark are shown by triangles in the upper left, upper right, and lower left panels of figure 2, respectively. On the other hand, for a given spectrum generator, there is almost no discrepancy in the relic abundance coming from the dark matter calculators. Thus, the solid and hollow triangles approximately overlap for a given color. What small discrepancy that is evident can be attributed to differences in the computation of the effective annihilation and coannihilation cross sections between the bino, neutral higgsino, and charged higgsinos, as well as the computation of the t-channel chargino exchange diagram. The annihilation cross section in the present Universe presents far fewer differences, since the coannihilation channels are absent. The triangles thus overlap in the upper right panel of figure 2. For the SI scattering cross section with nuclei, we see that the well-tempered benchmark is the only one of our pMSSM benchmarks that is constrained by current experiments, regardless of the pipeline adopted. This is because of the non-negligible higgsino fraction. The different higgsino fractions reported by SOFTSUSY and SPheno affect the relative positions of the magenta and cyan triangles in the lower left panel of figure 2. Since SOFTSUSY reports the larger higgsino fraction, the scattering cross section for magenta triangles are larger than those for the cyan triangles corresponding to the SPheno pipelines. For a given spectrum generator, say SOFTSUSY, there is some disagreement between the SI scattering cross sections computed by micrOMEGAs (solid magenta triangle) versus DarkSUSY (hollow magenta triangle). This can be attributed to the different form factors, as well as the way in which loop effects are incorporated. To check the effect of the different form factors, we used DarkSUSY’s form factors from table 4 in the micrOMEGAs code. For most points, doing so brings micrOMEGAs’s values for the scattering cross sections into better agreement with those of DarkSUSY. However, some differences remain, suggesting that other details of the calculation are also important. We also carried out the tree level calculation for SI scattering, following the discussion of appendix C in reference [121]7 and using the values in table 4. Surprisingly, we found O(10%) differences between our treelevel calculation and that of the codes’, implying substantial loop contributions that are different between the two codes. 7 We note that there are minor errors in Equation 204 of reference [121] related to the squark and Higgs masses. To correct these minor errors, we followed references [122] & [123] and references [124, 125], & [122]. – 23 – JHEP05(2018)113 We first discuss the relic density. The upper left panel of figure 2 shows that there are considerable differences between the various pipelines. The SPheno pipelines (cyan triangles) give a larger relic density than the SOFTSUSY pipelines (magenta triangles). This can be traced to the lower higgsino content of the lightest neutralino given by SPheno relative to SOFTSUSY. In fact, from table 5, the lightest neutralino is 30% higgsino when calculated by SPheno, but 36.4% higgsino when calculated by SOFTSUSY, although the masses agree to better than a percent. 62 4.5 A-funnel ϵ ≡ ' ΓA mA (2 ∼ 5 × 10−6 . (4.1) Reference [75] compared various approximation schemes in the relic density calculation (such as Taylor expansion in v) to a full numerical computation for values of ϵ near this value. Depending on the approximation, the relic density can vary over several orders of magnitude. Even for a full numerical computation, the resonance region is sharp enough that a factor of ∼ 2 can easily appear, unless there is an exact matching of calculation. Finally, we note that there is a significant difference between the SOFTSUSY-FeynHiggsmicrOMEGAs and SOFTSUSY-SusyHD-micrOMEGAs pipelines for the relic density of the Afunnel benchmark point, as can be seen by comparing tables 12a and 12b. This discrepancy is due to the inclusion of the pseudoscalar width, as in eq. (3.1). FeynHiggs calculates the pseudoscalar width, which is read in by micrOMEGAs (but not DarkSUSY, as discussed below). However, SusyHD does not calculate the width, so any program further down the pipeline either takes the width approximation from the spectrum generator or calculates the width itself. SPheno does estimate a width with reasonable agreement between its width and that calculated by FeynHiggs, yielding decent agreement between the SPhenoFeynHiggs-micrOMEGAs and SPheno-SusyHD-micrOMEGAs pipelines. But SOFTSUSY does not report a width that can be used by micrOMEGAs, so for the SOFTSUSY-SusyHD-micrOMEGAs pipeline, micrOMEGAs calculates its own width, which differs by nearly a factor of 1.7 from that calculated by FeynHiggs. This leads to a discrepancy of nearly a factor of 2 in the relic abundances, as can be verified, for example, by evaluating eq. 41 of ref. [75]. We do not see a discrepancy in the relic densities of any of the DarkSUSY pipelines because DarkSUSY always calculates all relevant particle widths, since, as mentioned above, up to version 5.1.2 DarkSUSY does not read in SLHA decay blocks. The annihilation cross section today is given by an even sharper peak in ⟨σv⟩ centered at mA = 2mχ̃01 . From the upper right panel of figure 2, and comparing tables 12a and 14a, – 24 – JHEP05(2018)113 The results for the relic density, current annihilation cross section, and scattering cross section of the A-funnel neutralino benchmark are shown by pentagons in the upper left, upper right, and lower left panels of figure 2, respectively. We first discuss the relic density. From the upper left panel of figure 2 and table 12, it is evident that there is a large variation, of more than factor of two, among the different pipelines, though the variation is " 2 if one neglects the Snowmass and Snowmass* pipelines. We note that for a given spectrum generator, say SOFTSUSY, there is some difference in the calculation performed by micrOMEGAs (solid magenta pentagon) versus DarkSUSY (hollow magenta pentagon). This difference amounts to an uncertainty of 69% and 37% for SPheno and SOFTSUSY, respectively, both of which far exceed the experimental uncertainty in the measurement of the dark matter abundance. The resonance region is notoriously sensitive to the approximations used to compute the relic density, especially for sharp peaks. The sharpness parameter in our case (following the same notation as reference [75]) is 63 4.6 Summary: broad trends in the pMSSM analysis It is instructive to look back at our analysis and draw some broad conclusions. The dark matter models studied in this section are a well-known subset of pMSSM benchmarks that satisfy the observed dark matter relic density. These benchmarks have been used in numerous studies, typically relying on one of the pipelines described in our work. Furthermore, connections between supersymmetric model building and cosmology often concern regions of parameter space based around one of the (co)annihilation mechanisms that our benchmarks capture. It should be clear from our work that the theoretical uncertainty in the relic density calculation using standard pipelines far exceeds the experimental uncertainty. From the upper left panel of figure 2, it is apparent that only a minority of pipeline choices for any given benchmark actually fall within the red band that delineates the Planck range. For the coannihilation and funnel models, this spread is especially broad, with theoretical calculations yielding results that can vary by as much as 300%. The spread is somewhat lower for the well-tempered neutralino and pure higgsino benchmarks, but even in those cases it far exceeds the experimental uncertainty. We note also that there can be significant spread in the relic density due to updates to the software packages even for the same sequence of calculators. There are several underlying reasons for the discrepancies in the relic density and other dark matter observables among the pipeline choices: • Small variations in the spectrum. — Coannihilations and A-resonance models are critically dependent on the relative masses of the dark matter particle and other light supersymmetric states. Within a pMSSM framework, the spectrum generators can easily produce 1% - 2% variation in the low energy spectrum of squarks, stops, or the pseudoscalar Higgs, leading to a large variation of the relic density, evident in the upper left panel of figure 2. However, the annihilation cross section in the present Universe is far less dependent on the masses of other light states, since coannihilation – 25 – JHEP05(2018)113 we see that the values reported by the various pipelines are in agreement with what we expect from the upper left panel; that is, there is a factor of ∼ 2 spread in the annihilation cross sections, similar to the factor of ∼ 2 spread in the relic densities (with the largest annihilation cross section corresponding to the smallest relic abundance, and so forth). We note that the current annihilation cross sections in the upper right panel of figure 2 are a factor ∼ 5 reduced compared to their values in the early Universe, used to calculate the abundances in the upper left panel of figure 2. This happens due to the thermal broadening of the resonance region in the early Universe. The SI scattering cross sections reported in the lower left panel of figure 2 match very closely due to the close agreement in the relevant masses and higgsino fractions. For a given spectrum generator, say SOFTSUSY, there is some disagreement between the scattering cross section computed by micrOMEGAs (solid magenta pentagon) versus DarkSUSY (hollow magenta pentagon). As for other benchmarks, this can be attributed to the different form factors, as in table 4 and the way in which loop effects are incorporated. 64 channels become irrelevant. This is reflected in the much greater convergence among pipelines in the upper right panel of figure 2. • Composition of the lightest neutralino. — The well-tempered neutralino framework depends critically on the higgsino fraction of the dark matter for its relic density. We see from table 5 that the spectrum generators can vary by up to 20% in their calculation of the higgsino fraction relative to each other. Furthermore, the higgsino fraction plays a crucial role in the SI scattering cross section, most evident in the well-tempered benchmark. • Differences in form factors. — The form factors employed by micrOMEGAs and DarkSUSY are displayed in table 4. These differences affect the SI scattering cross sections reported in the lower left panel of figure 2 for a given spectrum calculator (discrepancies between solid and hollow points of the same shape and color). For a given spectrum, using the form factors of DarkSUSY in micrOMEGAs, we find a definitive shift in the SI cross sections, bringing them into closer agreement. • NLO effects in scattering cross section. — Even accounting for the differences in spectrum and form factors, we see that different dark matter calculators report different SI scattering cross sections, especially for very low cross sections. These differences are likely coming from the fact that DarkSUSY implements an effective Lagrangian in the heavy squark limit following reference [110] (see reference [67] and references therein for details) while micrOMEGAs implements the full one-loop Lagrangian following reference [87]. For example, the pure higgsino or pure bino benchmarks in the lower left panel of figure 2 show a lot of variation. The theory calculations for pure higgsino and wino scattering cross sections have only converged recently [97, 98]. The discrepancies among the pipelines is likely to become a pressing issue in the future, when experimental sensitivity reaches the relevant cross sections. 5 Results: GUT analysis In the previous sections, we have presented our analysis of several standard pMSSM benchmark scenarios. For electroweak-scale models like the pMSSM, the supersymmetric mass – 26 – JHEP05(2018)113 • LO and NLO calculation of coannihilation channels. — For a given spectrum, there is wide variation (∼ 50%) in the relic density reported by micrOMEGAs and DarkSUSY (discrepancies between solid and hollow points of the same shape and color in the upper left panel of figure 2), especially in scenarios where coannihilation channels become important. This stems from differences in the tree level computation implemented in these programs. Moreover, as we have pointed out, the incorporation of NLO SUSY-QCD will further change the relic density calculation, by up to as much as 20%. 65 Model CMSSM 1 CMSSM 2 NUHM A NUHM B CMSSM 1 CMSSM 2 NUHM A NUHM B mχ̃01 Table 8. Masses, in GeV, of the lightest and second-lightest neutralinos and the lighter chargino for the GUT benchmark points. The corresponding higgsino fraction is given in parentheses. spectrum is used as an input for the spectrum generators at low energies, and the final sparticle spectrum is used as an input for the dark matter calculators. Thus, there is very little running of the sparticle masses and the results reported by different spectrum generators are generally in good agreement. In this section, we will analyze dark matter benchmark points in the context of supersymmetric models with boundary conditions for soft terms specified at the GUT scale. In particular, we will study two models: CMSSM and NUHM. In this case, the effects of RG running performed by the two spectrum calculators are expected to become more important, and can substantially affect the dark matter observables. We begin this section by first discussing the sparticle spectra of the GUT benchmark models. The GUT models are defined in table 1. The low energy spectrum of neutralinos and squarks generated by SOFTSUSY and SPheno are shown in tables 8 and 9, and Higgs masses are shown in table 10. Masses presented in tables 10a & 10b are computed by FeynHiggs while those in tables 10c & 10d SusyHD. The higgsino mass parameter µ is shown in table 11. We also display in tables 8–11 the values obtained via the MasterCode† pipeline, as described in table 2. From table 8, we see that for the CMSSM points, the lightest neutralino is mainly bino, while for the NUHM points, it is mainly higgsino. The higgsino fraction calculated by the different spectrum generators has appreciable differences in the cases of the CMSSM 1 and NUHM B, which will significantly affect the dark matter observables reported for the two pipelines, as we shall see. Even in the case of the NUHM A point, the small difference in higgsino fraction (< 1% difference between SOFTSUSY and SPheno) will be important. There is good agreement between SOFTSUSY and SPheno for the mass of the dark matter in the CMSSM cases. However, there is a vast disagreement in the mass of dark matter – 27 – JHEP05(2018)113 CMSSM 1 CMSSM 2 NUHM A NUHM B 2 + N2 ) (N13 14 SOFTSUSY SPheno 936.10 (0.029) 938.00 (0.003) 386.74 (0.0008) 385.88 (0.0009) 654.14 (0.998) 1041.78 (0.992) 1084.49 (0.992) 1356.37 (0.822) 2 + N2 ) mχ̃02 (N23 24 1157.98 (0.970) 1147.50 (0.962) 1581.69 (0.901) – 736.77 (0.0039) 736.83 (0.0039) 691.54 (1.000) 658.50 (1.000) 1047.73 (1.000) 1112.63 (1.000) 1090.82 (1.000) 1374.98 (1.000) mχ̃+ 1 1154.62 1143.91 1581.15 – 736.93 737.02 689.76 656.75 1045.39 1110.48 1088.70 1372.18 MasterCode† 936.14 (0.022) – 687.09 (0.997) 1106.04 (0.989) 66 mũ,L Model CMSSM 1 CMSSM 2 NUHM A NUHM B 6777.40 – 6023.66 6123.45 CMSSM 1 CMSSM 2 NUHM A NUHM B 4694.85 – 4606.33 4688.72 mũ,R SOFTSUSY SPheno 6777.68 1966.32 6023.08 6122.97 md,L ˜ 6778.02 1967.77 6023.48 6123.36 mt̃,1 4696.36 1008.00 4608.33 4690.93 6755.77 1972.71 5967.12 6070.48 MasterCode† 6687.72 – 5759.01 5864.82 6756.15 1974.11 5967.60 6070.95 6677.77 – 5727.70 5834.10 4615.18 1005.62 4518.04 4602.46 5233.18 – 5302.46 5390.24 SOFTSUSY SPheno 6688.32 1900.87 5758.85 5864.75 md,R ˜ 6678.19 1893.87 5727.37 5833.89 mt̃,2 5183.89 1516.05 5240.45 5328.54 6669.38 1906.65 5709.44 5818.50 6659.78 1899.67 5679.26 5788.83 5226.83 1550.65 5235.73 5324.89 Table 9. Masses, in GeV, of the squarks for the GUT benchmark points. between SOFTSUSY and SPheno for both NUHM points. There are also large discrepancies in the mass of the second lightest neutralino and the charginos in all cases except CMSSM 2. These discrepancies all stem from differences in the value of µ, as is evident from table 11. From table 10, we see that there are also large discrepancies in the mass of the pseudoscalar Higgs A, which is critical for the dark matter relic density computation in the A-funnel region of parameter space. On the other hand, the squark spectrum agrees among different generators quite well, as is evident from table 9, although there can be variations of up to ∼ 2 % in the calculation of squark masses. Even these ∼ 2 % discrepancies will be important in the following analysis. We thus see that the largest discrepancies seem to occur in the calculation of the electroweak symmetry breaking (EWSB) sector between different spectrum generators. We now turn to a more detailed study of these differences: as a prelude to our investigation of dark matter observables, we discuss in detail the differences between the calculation of the higgsino mass parameter µ and the pseudoscalar Higgs mass mA by the different spectrum generators. Finally, we discuss the dark matter observables obtained from the different pipelines. 5.1 Comparison of EWSB sectors Here we explore the EWSB calculations performed by the different spectrum generators. The differences in the EWSB calculations are particularly exacerbated at large values of tan β and m0 , which is the regime where our GUT benchmark models CMSSM 1 and NUHM A/B lie. We discuss these issues in this section. – 28 – JHEP05(2018)113 CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 6777.06 – 6023.26 6123.06 67 mh Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 125.91 – 121.21 121.12 mH SOFTSUSY SPheno 123.40 122.45 123.21 123.27 123.49 122.32 123.25 123.30 MasterCode† 2117.34 – 3127.40 3201.72 SOFTSUSY SPheno 1469.30 1480.85 2732.38 2805.16 2593.21 1592.88 3256.13 3329.63 mA Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 2118.13 – 3127.86 3202.15 SOFTSUSY SPheno 1469.82 1481.30 2732.78 2805.52 2594.11 1593.36 3256.61 3330.07 MasterCode† 2118.65 – 3128.19 3202.48 mH ± SOFTSUSY SPheno 1479.90 1484.09 2733.24 2805.99 2586.37 1595.88 3255.75 3329.26 (b) mA and mH ± as computed via the FeynHiggs branch of the pipeline. mh Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 125.91 – 121.21 121.12 mH SOFTSUSY SPheno 123.19 121.27 123.02 123.02 123.06 121.16 123.37 123.38 MasterCode† 2117.34 – 3127.40 3201.72 SOFTSUSY SPheno 1469.83 1480.98 2732.80 2805.54 2594.15 1592.93 3256.64 3330.10 (c) mh and mH as computed via the SusyHD branch of the pipeline. mA Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 2118.13 – 3127.86 3202.15 SOFTSUSY SPheno 1469.82 1481.30 2732.79 2805.53 2594.12 1593.36 325.61 3330.08 MasterCode† 2118.65 – 3128.19 3202.48 mH ± SOFTSUSY SPheno 1472.30 1483.67 2734.10 2806.80 2595.71 1595.75 3257.76 3331.19 (d) mA and mH ± as computed via the SusyHD branches of the pipeline. Table 10. Masses, in GeV, of the CP-even Higgses (tables 10a and 10c) and the CP-odd and charged Higgses (tables 10b and 10d) for the GUT scale points. Masses presented in tables 10a & 10b are computed by FeynHiggs while those in tables 10c & 10d SusyHD. – 29 – JHEP05(2018)113 (a) mh and mH as computed via the FeynHiggs branch of the pipeline. 68 Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 1139.40 – 673.80 1090.70 µ SOFTSUSY 1129.36 1693.2 641.81 1070.04 SPheno 1552.31 1704.83 1019.52 1340.17 Table 11. Higgsino mass parameter, µ, after FeynHiggs. d 2 6 3 m = 3Xt − 6g22 |M2 |2 − g12 |M1 |2 + g12 S dt Hu 5 5 6 2 3 2 d 2 2 2 16π m = 3Xb + Xτ − 6g2 |M2 | − g1 |M1 |2 − g12 S . dt Hd 5 5 16π 2 In the above equations, 2 2/ 0 2 2 2 2 2 2 2 X(t,b,τ ) = 2 2y(t,b,τ ) 2 mH(u,d,d) + m(Q3 ,Q3 ,L3 ) + m(ū3 ,d¯3 ,ē3 ) + A(t,b,τ ) + , S = m2Hu − m2Hd + T r m2Q − m2L − 2mū2 + m2d̄ + mē2 , (5.1) (5.2) in standard notation. The higgsino mass parameter is given, in the large tan β limit, by µ2 ∼ m2Hd − m2Hu tan2 β tan2 β − 1 1 − m2Z , 2 (5.3) where all quantities are defined at mZ . The pseudoscalar Higgs mass is given at tree level by m2A = 2|µ|2 + m2Hu + m2Hd , (5.4) with all quantities defined at the scale MSU SY , and the quantities on the right hand side of eq. (5.4) related to those at mZ by radiative corrections. Obviously, the calculated value of mA depends on the computation of µ, m2Hu , and m2Hd . From eq. (5.1), eq. (5.3), and eq. (5.4), it is clear that the values of µ and mA reported by the programs will be greatly affected by their calculation of the top and bottom Yukawas yt and yb , as well as the calculation of squark and stop masses that enter into Xt and Xb . In table 9, there are variations of ∼ 2% in the stop masses between SOFTSUSY and SPheno. In fact, SOFTSUSY consistently reports higher values of the stop masses than SPheno across benchmark models. There are also variations in yt and yb between the programs, as studied in [45]. These factors result in vastly different values of µ and mA , especially for large values of m0 where the squark and Yukawa calculations differ substantially. Similarly, we find that τ̃ masses are reported ∼ 30% higher by SOFTSUSY than SPheno, following from a ∼ 20% difference in the calculation of the τ Yukawa couplings. In figures 3 and 4, we show the resulting variations the higgsino mass parameter, µ, and the pseudoscalar mass, mA , (left panels), as well as mh (right panels), each as functions of – 30 – JHEP05(2018)113 The RGE’s for the higgsino mass parameters in the MSSM are, following the notation of reference [126], m [GeV] mh [GeV] 69 m1/2 tan(β) A0 mt = = = = 2098.4 GeV 51.3 781.9 GeV 173.3 GeV m0 [TeV] Figure 3. We show the higgsino mass parameter, µ, and the pseudoscalar mass, mA , (left panel), as well as mh (right panel), each as functions of m0 , as computed by FeynHiggs with spectral input from SOFTSUSY (magenta) or SPheno (cyan) for the CMSSM. The CMSSM 1 benchmark is denoted with a solid grey line. Vertical dotted lines indicate the value of m0 above which µ becomes unphysical (µ2 < 0). Data near the benchmarks is presented in table 24. m [GeV] mh [GeV] m2Hd = = m1/2 = tan(β) = A0 = mt = A B A m0 [TeV] m2Hu 1.335 · 107 GeV 3200 GeV 39 3139.3 GeV 173.2 GeV B m0 [TeV] Figure 4. We show the higgsino mass parameter, µ, and the pseudoscalar mass, mA , (left panel), as well as mh (right panel), each as functions of m0 , as computed by FeynHiggs with spectral input from SOFTSUSY (magenta) or SPheno (cyan) for the NUHM. The NUHM A and B benchmarks are denoted by solid grey vertical lines, as labeled. m0 , as computed by FeynHiggs with spectral input from SOFTSUSY (magenta) and SPheno (cyan) near the CMSSM 1 and NUHM benchmarks. Our benchmark points are denoted with a solid grey line in each panel. Vertical dotted lines indicate the values of m0 above which µ becomes unphysical (µ2 < 0). Data near the benchmarks is presented in table 24. We now discuss some general features of the figures. From the left panel of figure 3, we see that for both generators, µ decreases as m0 increases. We can understand this as follows. From eq. (5.1), we see that increasing the scalar masses makes the RG running of both m2Hu and m2Hd steeper, decreasing their values at low scales. On the other hand, increasing the scalar masses within the CMSSM also increases the boundary values of m2Hu – 31 – JHEP05(2018)113 m0 [TeV] 70 – 32 – JHEP05(2018)113 and m2Hd at the GUT scale. The low-scale values of m2Hu and m2Hd are thus determined by these two competing effects. For our selection of m1/2 and tan β, we have checked that the cumulative effect is to decrease m2Hu with increasing m0 near the benchmark point, for both SOFTSUSY and SPheno. We have also verified this using the approximate relations for the renormalization group running given in reference [127]. On the other hand, we have found that near the benchmark, the effect of increasing m0 is to increase m2Hd . This is because while increasing m0 increases the slope of the renormalization group running from eq. (5.1), this increase is suppressed compared to the m2Hu case by the small value of the bottom Yukawa. The cumulative effect is that the increase in the boundary value of m2Hd for increasing m0 dominates, so m2Hd increases with increasing m0 . We see from eq. (5.3) that m2Hu and m2Hd contribute with opposite signs to µ. However, the dominant contribution is from m2Hu , since m2Hd is suppressed by the large value of tan β. Thus, following the behavior of m2Hu , µ too decreases with increasing m0 . The values of µ (solid curves) from SOFTSUSY and SPheno start to diverge radically for m0 > 2 TeV in the left panel of figure 3. This is the regime where differences in the squark masses and the top Yukawa calculated by the two spectrum generators start to become important in determining µ. From table 9, we see that SOFTSUSY produces heavier stop masses than SPheno for the same CMSSM model point. Thus, µ runs to smaller values faster in SOFTSUSY compared to SPheno. The values are µ = 1129 GeV for SOFTSUSY and µ = 1552 GeV for SPheno at the CMSSM 1 benchmark. From the right panel of figure 3, we see that for both programs the mass of the lightest CP-even Higgs increases with increasing m0 . This is expected, due to the usual loop corrections to the Higgs mass. We also note that SPheno reports a slightly larger Higgs mass than SOFTSUSY due to a combination of the low energy values of the stop mass and the trilinear coupling. Finally, we point out that the Higgs mass calculation is relatively robust to uncertainties in the EWSB calculations, since mh is sensitive only to mA (not µ independently) at tree level, which is reflected in the behavior of mh at very large m0 near where µ becomes unphysical. We note that the uncertainties in the EWSB calculations tend to cancel each other in the calculations of mh , while, in contrast, they do not cancel each other in the calculation of mH , which tracks mA quite closely. We now move on to a discussion of the pseudoscalar Higgs mass mA (dashed curves) in the left panel of figure 3. From the tree level relation for mA in eq. (5.4), we expect that the value of mA reported will depend on the relative magnitudes of µ, m2Hu , and m2Hd reported by the spectrum generators. The values of mA returned by SOFTSUSY decrease steadily with increasing m0 . However, the values of mA given by SPheno instead increase with increasing m0 . At the benchmark value of m0 , we have mA = 1469 GeV given by SOFTSUSY and mA = 2595 GeV given by SPheno. The NUHM benchmarks are somewhat different from the CMSSM case discussed above. From the left panel of figure 4, we see that with increasing m0 , the value of µ increases. This is due to m2Hu and m2Hd being fixed at the GUT scale, causing both masses at the low scale to now decrease with increasing m0 without any competing effects. With the steeper slope in the renormalization group equations, m2Hu runs to increasingly negative values faster than m2Hd . Between this and the fact that the large value of tan(β) suppresses 71 5.2 Dark matter observables In this section, we study the dark matter observables for the GUT benchmarks. For each of the CMSSM and NUHM benchmark models, we discuss the relic density, the annihilation cross section today, and the predicted scattering cross section, all of which are plotted in figure 5. As in figure 2, for comparison, the Planck 3-sigma range for the dark matter abundance [115] is highlighted in red in the upper left panel, the limits on the annihilation cross section today from Fermi-LAT’s 6-year analysis of dwarf spheroidal galaxies [116] for annihilation to τ + τ − (red) and bb̄ (blue) are shown in the upper right panel, and exclusion limits from LUX (red) [117], PandaX (blue) [118], and LZ (projected; black) [119] are shown in the lower left panel. Here, we introduce a new set of unique shapes to denote the various pipelines, but follow the same colour scheme as in figure 2 — black is used again for our implementation of the original pipeline (MasterCode† , rather than Snowmass† ).8 5.2.1 CMSSM benchmarks We first examine the relic density as plotted in the upper left panel of figure 5 and tabulated in table 20. For the CMSSM 1 point, the mass of the lightest neutralino from table 8 is mχ̃01 = 936.10 GeV from SOFTSUSY, and mχ̃01 = 938.0 GeV from SPheno. The dark matter is mostly bino in both cases. It is, however, the mass of the second lightest neutralino that differs radically between the two programs. For SOFTSUSY, we have mχ̃02 = 1147.50 GeV, while for SPheno, we have mχ̃02 = 1581.69 GeV. The second lightest neutralino is mostly higgsino. The huge discrepancy in masses is due to the very different values of µ reported by the two programs, as discussed above and shown in figure 3. The large difference in µ is also re8 There are no green markers to indicate an updated MasterCode pipeline, since the updated MasterCode pipeline is the same as our SOFTSUSY-FeynHiggs-micrOMEGAs pipeline. – 33 – JHEP05(2018)113 the m2Hd contribution to µ, the result is that µ increases with m0 . For the NUHM A point, the values are µ = 642 GeV from SOFTSUSY and µ = 1020 GeV from SPheno. For the NUHM B point, the values are µ = 1070 GeV from SOFTSUSY and µ = 1340 GeV from SPheno. The values of the pseudoscalar and lightest CP-even Higgs masses increase with increasing m0 for both spectrum generators, as can be seen from the left and right panel, respectively, of figure 4. Finally, the CMSSM 2 point, which is a τ̃ coannihilation model, is largely insensitive to uncertainties in the EWSB sector, with the exception of those associated with the τ Yukawa coupling. The dark matter physics of the CMSSM 2 benchmark primarily concerns the LSP, which is strongly bino-like with good agreement among the pipelines, and the lighter τ̃ , with masses of 538.29 GeV and 390.01 GeV from SOFTSUSY and SPheno, respectively. The large difference in the τ̃ masses comes primarily from the difference in the calculation of the τ Yukawa coupling, with SOFTSUSY reporting a value ∼ 20% larger than SPheno. Since the lightest neutralino mass for the CMSSM 2 is ∼ 386 GeV for both SOFTSUSY and SPheno, this means that only SPheno pipelines represent true coannihilation models, while SOFTSUSY pipelines do not coannihilate efficiently enough to achieve the correct relic abundance. 72 Planck Range mχ̃0 [GeV] 1 PandaX 2016 1 Pipeline Legend MasterCode† Black SOFTSUSY Magenta SPheno Cyan micrOMEGAs ⚫ DarkSUSY ○ LUX 2016 Benchmark Legend CMSSM 1 ▴ CMSSM 2 ⬥ NUHM A ◼ NUHM B ⬣ LZ Projected mχ̃0 [GeV] 1 Figure 5. GUT model results. The upper left, upper right, and lower left panels show the neutralino relic density, annihilation cross section today, and the SI neutralino-nucleon elastic scattering cross section, respectively, all as functions of the dark matter mass, with a common legend in the lower right panel. We note that relic density for some pipelines yields values far larger than those plotted here, and for this reason the CMSSM 1 SPheno points and the CMSSM 2 SOFTSUSY points do not appear in the upper left panel. For comparison, the Planck 3-sigma range for the dark matter abundance [115] is highlighted in red in the upper left panel, the limits on the annihilation cross section today from Fermi-LAT’s 6-year analysis of dwarf spheroidal galaxies [116] for annihilation to τ + τ − (red) and bb̄ (blue) are shown in the upper right panel, and exclusion limits from LUX (red) [117], PandaX (blue) [118], and LZ (projected; black) [119] are shown in the lower left panel. flected in the different higgsino fractions of the lightest neutralino. From SOFTSUSY, the higgsino fraction in χ̃01 is around ∼ 3%, while from SPheno, the higgsino fraction is only 0.3%. We note another large discrepancy among the results from the different pipelines is that the values of mA obtained from SOFTSUSY and SPheno are 1470 GeV and 2594 GeV, respectively. These differences in the spectra have a profound effect on the relic density. From the upper left panel of figure 5, we find that the SOFTSUSY-micrOMEGAs pipeline (solid magenta triangles) and SOFTSUSY-DarkSUSY pipeline (hollow magenta triangles) give values for the relic density Ωh2 = 0.037 and Ωh2 = 0.648, respectively, while the pipelines that involve SPheno as the spectrum calculator give relic densities that are Ωh2 ! 10, with the value returned from the DarkSUSY pipeline larger than that from the micrOMEGAs pipeline by a factor of 2. – 34 – JHEP05(2018)113 mχ̃0 [GeV] σ SI [pb] bb̄ ⟨σv⟩ [cm3/s] Ωh2 τ +τ − 73 9 The original MasterCode CMSSM best fit point from [30] was primarily an A-funnel point, as is our MasterCode† point. From updated pipelines, the value of mA is much farther from 2mχ̃0 such that the 1 A-funnel resonance does not have a significant impact. – 35 – JHEP05(2018)113 In fact, the CMSSM 1 benchmark model is not even a cosmologically-favored point (at least within a thermal history) if one uses SPheno as the spectrum calculator. The lightest neutralino in that case is an almost pure bino, far away from the A-resonance, and without any possible contributions from coannihilation channels. The pipelines involving SPheno give a relic density that is ! 2 orders of magnitude larger than those given by SOFTSUSY pipelines. For pipelines involving SOFTSUSY, the CMSSM 1 benchmark falls approximately into the category of well-tempered dark matter.9 With the SOFTSUSY spectrum, we see a factor of ∼ 20 difference in Ωh2 between micrOMEGAs (solid magenta triangles) and DarkSUSY (hollow magenta triangles), with neither giving a value within the limits of experimental uncertainty. As in the pMSSM, for well-tempered dark matter, DarkSUSY tends to give a relic density value that is larger than the one given by micrOMEGAs for the same spectrum. This could arise due to differences in the way the effective annihilation cross section between the dark matter and the neutral and charged higgsinos is computed by the two programs, as well as differences in the calculation of the t-channel chargino exchange diagram. While in the case of the pMSSM the difference was small, in the case of the GUT model benchmark the difference is enormous, possibly due to the proximity to the A-funnel region. With micrOMEGAs we obtain a dark matter candidate that annihilates too efficiently in the early Universe, while with DarkSUSY we obtain a candidate that does not annihilate efficiently enough. Regarding the relic density for the CMSSM 2 point, as mentioned above, only SPheno pipelines represent true coannihilation models, while SOFTSUSY pipelines do not coannihilate efficiently enough to achieve the correct relic abundance. Thus, the relic abundance from SOFTSUSY pipelines appears at large values of Ωh2 beyond the range shown in the upper left panel of figure 5. We now turn to the annihilation cross section in the current Universe for the CMSSM benchmarks. The results are plotted in the upper right panel of figure 5 and tabulated in table 21. We see that the enormous difference in the relic density computation between spectra coming from SOFTSUSY and SPheno continues to persist in the computation of the current annihilation cross section for the CMSSM 1. For this benchmark we see the largest discrepancies in the calucation of the annihilation cross section today, nearly three orders of magnitude. Since the charged higgsinos are much heavier in the spectrum generated by SPheno, the t−channel chargino exchange diagram is suppressed in this case. This leads to a much smaller annihilation cross section, ∼ 10−28 cm3 s−1 . For a given spectrum, the difference between micrOMEGAs and DarkSUSY persists, with DarkSUSY giving smaller annihilation cross section as before. For the CMSSM 2 benchmark, neutralino-stau coannihilations play no role in the annihilation today, so we see reasonably good agreement among the pipelines, albeit with a very low annihilation cross section. Finally, we consider the neutralino-nucleon scattering cross sections, which are presented in the lower left panel of figure 5 (as per-nucleon scattering cross sections) and 74 5.2.2 NUHM benchmarks As discussed in section 2.1, since the original MasterCode NUHM best fit point has a nearly pure higgsino LSP and is therefore very sensitive to the details of the spectrum, our NUHM A and B benchmarks were chosen with the requirements that a valid relic density would be achieved by NUHM A via the SPheno pipelines and by NUHM B via the SOFTSUSY pipelines. From table 8, the mass of the lightest neutralino obtained by SOFTSUSY is far smaller than that obtained by SPheno for both NUHM A and B. The dark matter is mostly higgsino in all cases, and the radically different masses for the LSP (and other light-inos) are due to the very different values of µ reported by the two programs, as discussed previously in section 5.1. The analyses of the dark matter observables for both NUHM benchmarks follow the trends of the pure higgsino case discussed previously for the pMSSM. For both benchmarks, we expect that the relic density for pipelines involving SOFTSUSY should be lower than that given by pipelines involving SPheno, due to the smaller higgsino mass in the former case. This expectation is borne out in the upper left panel of figure 5. For a given spectrum, the relic densities computed by micrOMEGAs and DarkSUSY match quite well, as evidenced by the fact that solid and hollow NUHM markers more or less overlap. For the annihilation cross section in the current Universe, we expect that the lower higgsino mass of the SOFTSUSY pipelines should correspond to a larger annihilation cross section than the SPheno pipelines. This is borne out by the relative positions of the magenta and cyan NUHM markers in the upper right panel of figure 5, though the difference is less pronounced for micrOMEGAs pipelines for the NUHM B benchmark. The scattering cross sections with nuclei are shown in the lower left panel of figure 5. We first note that the SI scattering cross sections for higgsino dark matter in this case are within a factor of a few of ∼ 10−9 pb, which is much larger than the cross section for the – 36 – JHEP05(2018)113 tabulated in table 22 and 23, where the subtables are organized by proton then neutron scattering cross sections, first for the spin-independent then for spin-dependent scattering. From the lower left panel of figure 5, we see that the scattering cross sections for the CMSSM 1 spectrum coming from SOFTSUSY are much larger than those for the corresponding spectrum coming from SPheno for both DarkSUSY and micrOMEGAs. In fact, the SOFTSUSY model is already being constrained by current experiments. This is due to the larger higgsino content of the dark matter in the SOFTSUSY case, which leads to an enhancement of the Higgs exchange diagram. Moreover, as was observed in the pMSSM cases, there are discrepancies between the scattering cross sections reported by micrOMEGAs versus DarkSUSY, even for the same spectrum, from the differences in form factors employed by each code (see table 4) and the way in which loop effects are incorporated. For the CMSSM 2, micrOMEGAs gives the same SI scattering cross section no matter which spectrum generator is employed, while DarkSUSY yields somewhat larger SI scattering cross sections that do depend somewhat on the details of the spectrum that differ. Since the LSP for the CMSSM 2 benchmark is nearly pure bino and the SI cross sections are strongly suppressed, differences in the SI scattering cross sections from the SPheno-DarkSUSY versus SOFTSUSY-DarkSUSY pipelines likely come from loop corrections. 75 pure higgsino case in the pMSSM analysis. This can be attributed to the fact that the dark matter in the NUHM benchmarks is less pure higgsino than in the “pure higgsino” pMSSM benchmark, as can be seen by comparing tables 8 and 5. The more pure the LSP, the smaller the SI scattering cross section. We see also that DarkSUSY gives larger scattering cross sections than micrOMEGAs, consistent with results discussed above. As before, we can attribute this to the difference in form factors between the two programs. 6 Conclusions – 37 – JHEP05(2018)113 In this paper, we have performed comparative studies of the physics of supersymmetric dark matter calculated with a sequence of spectrum generators (SOFTSUSY and SPheno), Higgs sector calculators (FeynHiggs and SusyHD) and to dark matter observable calculators (micrOMEGAs and DarkSUSY). We placed our study in the context of several SUSY benchmark models that are interesting in light of LHC Run-1 and null results from recent dark matter searches as studied previously by [26, 30, 54]. We have compared calculations for the sparticle spectra, the Higgs sectors, and the dark matter observables for each benchmark, and we have incorporated the various generators and calculators into comprehensive pipelines to study not only the effects of the choice of an individual package, but also all downstream effects of those choices on subsequent calculations. This study was conducted in two parts. In the first part, we investigated a set of pMSSM models from ref. [54]. The dark matter scenarios we considered were coannihilation (bino-stop and bino-squark), A-funnel, well-tempered neutralinos, and pure higgsinos. We discovered that the spectrum generators can differ by up to 1 - 2 % in their predicted masses for the stop and the first two generations of squarks, and by up to 20% in the gauge composition of the lightest neutralino, for a given pMSSM model. As for the dark matter observables, differences of up to a factor of ∼ 3 − 5 in the relic density and current annihilation cross section, and up to a factor of ∼ 10 in the predicted scattering cross section, were found to exist between the different pipelines. These discrepancies are already pressing in the case of the relic abundance of dark matter, for which the uncertainty in the experimental value is small compared to the theoretical uncertainty in pMSSM predictions. For the annihilation cross section today and the SI scattering cross section, the discrepancies will become important if/when future dark matter indirect and direct detection experimental sensitivities reach the predicted levels. In the second part of our study, we considered four interesting benchmark models defined at the GUT scale — two CMSSM points and two NUHM points. For GUT-scale models, we found that discrepancies among the various pipelines are often amplified by the renormalization group running. For our CMSSM and NUHM benchmarks, we found that the spectrum generators can give low energy values of the higgsino mass parameter µ and the pseudoscalar Higgs mass mA that differ by up to 150% - 200% (though the differences can be much greater at larger m0 ). This can lead to large differences in the annihilation and scattering cross sections computed by the dark matter calculators. As a community, we have made tremendous progress in predicting signals of the particle nature of dark matter, made possible by pioneering work in theory and computation and 76 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass† Snowmass* 0.116 0.119 0.116 0.112 0.120 0.106 0.118 0.100 0.131 0.091 Ωh2 micrOMEGAs SOFTSUSY SPheno 0.094 0.035 0.115 0.116 0.087 0.110 0.078 0.064 0.088 0.127 DarkSUSY SOFTSUSY SPheno 0.120 0.045 0.109 0.110 0.123 0.156 0.107 0.108 0.091 0.128 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass† Snowmass* 0.116 0.119 0.116 0.112 0.120 0.106 0.118 0.100 0.131 0.091 Ωh2 micrOMEGAs SOFTSUSY SPheno 0.094 0.035 0.115 0.116 0.087 0.110 0.135 0.069 0.088 0.127 DarkSUSY SOFTSUSY SPheno 0.120 0.045 0.109 0.110 0.123 0.156 0.107 0.108 0.092 0.128 (b) Ωh2 as computed via the SusyHD branch of the pipeline. Table 12. pMSSM dark matter relic density: the relic density as computed by the various pipelines. Table 12a’s pipelines make use of FeynHiggs, while table 12b uses SusyHD. Here “Snowmass∗ refers to the updated version of the Snowmass pipeline and “Snowmass† ” refers to our incarnation of the Snowmass pipeline with micrOMEGAs 2.4.5 (rather than v2.4; see table 2). Values for the FeynHiggs pipelines (table 12a) are plotted in the upper left panel of figure 2 and percent differences for both pipelines may be found in table 13. closely related to increasing experimental sophistication. Though no definitive signals have yet emerged, ongoing attention to the theoretical calculations related to MSSM neutralino dark matter, and dark matter observables in general, is now more important than ever as experiments begin to probe the canonical SUSY WIMP parameter space. Acknowledgments We thank Paolo Gondolo, Werner Porod, and Alexander Pukhov for clarifications related to the implementation of calculations in DarkSUSY, SPheno, and micrOMEGAs, respectively, and Keith Olive for information on the MasterCode best fit points. PS would also like to thank Nordita, the Nordic Institute for Theoretical Physics, for their hospitality. The work of PS is supported in part by NSF Grant No. PHY-1720282. A Data tables for pMSSM models In this appendix, we collect data tables for the pMSSM section of our paper. – 38 – JHEP05(2018)113 (a) Ωh2 as computed via the FeynHiggs branch of the pipeline. 77 B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass† Snowmass∗ −23.40% −3.48% −33.33% −43.59% −36.36% −12.77% −2.61% −14.94% −67.95% −3.41% DarkSUSY SOFTSUSY SPheno −27.87% 52.33% 4.80% 4.63% −41.76% −78.94% −37.56% −38.02% −3.97% −46.00% (a) Percent differences in Ωh2 as computed via the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass† Snowmass* −23.40% −3.48% −33.33% 17.04% −36.36% −12.77% −2.61% −14.94% 2.96% −3.41% Ωh2 micrOMEGAs SOFTSUSY SPheno 0% 62.77% 0% −0.87% 0% −26.44% 0% 48.89% 0% −44.32% DarkSUSY SOFTSUSY SPheno −27.82% 52.36% 4.83% 4.63% −41.76% −78.96% 20.61% 20.35% −4.19% −45.39% (b) Percent differences in Ωh2 as computed via the SusyHD Branch of the pipeline. Table 13. pMSSM dark matter relic density: percent differences Ωh2 , relative to the SOFTSUSYmicrOMEGAs pipelines. – 39 – JHEP05(2018)113 Model Ωh2 micrOMEGAs SOFTSUSY SPheno 0% 62.77% 0% −0.87% 0% −26.44% 0% 17.95% 0% −44.32% 78 B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 4.3 · 10−30 1.0 · 10−26 1.4 · 10−29 3.8 · 10−27 1.9 · 10−26 3.90 · 10−30 9.94 · 10−27 5.37 · 10−30 3.23 · 10−27 2.41 · 10−26 < σv > micrOMEGAs SOFTSUSY SPheno 3.91 · 10−30 4.33 · 10−30 1.02 · 10−26 1.01 · 10−26 6.73 · 10−30 4.17 · 10−30 5.06 · 10−27 6.26 · 10−27 2.50 · 10−26 1.68 · 10−26 DarkSUSY SOFTSUSY SPheno 4.00 · 10−30 4.34 · 10−30 1.02 · 10−26 1.03 · 10−26 1.45 · 10−29 1.36 · 10−29 3.91 · 10−27 3.88 · 10−27 2.44 · 10−26 1.70 · 10−26 (a) ⟨σv⟩ as computed via the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 4.3 · 10−30 1.0 · 10−26 1.4 · 10−29 3.8 · 10−27 1.9 · 10−26 3.90 · 10−30 9.94 · 10−27 5.37 · 10−30 3.23 · 10−27 2.41 · 10−26 < σv > micrOMEGAs SOFTSUSY SPheno 3.91 · 10−30 4.33 · 10−30 1.02 · 10−26 1.01 · 10−26 6.73 · 10−30 4.17 · 10−30 5.04 · 10−27 6.23 · 10−27 2.50 · 10−26 1.68 · 10−26 DarkSUSY SOFTSUSY SPheno 4.00 · 10−30 4.34 · 10−30 1.02 · 10−26 1.03 · 10−26 1.45 · 10−29 1.36 · 10−29 3.88 · 10−27 3.85 · 10−27 2.43 · 10−26 1.70 · 10−26 (b) ⟨σv⟩ as computed via the SusyHD branch of the pipeline. Table 14. pMSSM dark matter annihilation cross section: dark matter annihilation cross section today, in cm3 s−1 , for pMSSM benchmarks as computed by the various pipelines. Table 14a displays results from the pipelines using FeynHiggs while table 14b’s pipelines use SusyHD. Values for the FeynHiggs pipelines (table 14a) are plotted in the upper right panel of figure 2 and the percent differences are given in table 15. – 40 – JHEP05(2018)113 Model 79 B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass∗ −9.97% 1.96% −108.02% 24.90% 24.00% 0.26% 2.55% 20.21% 36.17% 3.60% < σv > micrOMEGAS SOFTSUSY SPheno 0% −10.74% 0% 0.98% 0% 38.04% 0% −23.72% 0% 32.80% DarkSUSY SOFTSUSY SPheno −2.24% −11.02% −0.08% −0.81% −114.98% −101.90% 22.77% 23.28% 2.56% 31.81% (a) Percent differences for ⟨σv⟩ as computed via the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −9.97% 1.96% −108.02% 24.60% 24.00% 0.26% 2.55% 20.21% 35.91% 3.60% < σv > micrOMEGAs SOFTSUSY SPheno 0% −10.74% 0% 0.98% 0% 38.04% 0% −23.61% 0% 32.80% DarkSUSY SOFTSUSY SPheno −2.27% −11.04% −0.17% −0.92% −115.01% −101.91% 23.09% 23.67% 2.62% 31.86% (b) Percent differences for ⟨σv⟩ as computed via the SusyHD branch of the pipeline. Table 15. pMSSM dark matter annihilation cross section: percent differences for ⟨σv⟩, relative to the SOFTSUSY-micrOMEGAs pipelines. – 41 – JHEP05(2018)113 Model 80 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 1.5 · 10−11 1.7 · 10−10 1.6 · 10−11 3.1 · 10−11 4.3 · 10−8 7.88 · 10−12 4.90 · 10−11 5.10 · 10−10 1.48 · 10−11 1.64 · 10−8 σpSI micrOMEGAs SOFTSUSY SPheno 6.74 · 10−12 1.05 · 10−11 5.10 · 10−11 5.09 · 10−11 5.07 · 10−10 1.82 · 10−10 1.55 · 10−11 1.39 · 10−11 1.75 · 10−08 1.24 · 10−08 DarkSUSY SOFTSUSY SPheno 6.30 · 10−12 6.90 · 10−12 1.09 · 10−10 1.09 · 10−10 4.59 · 10−11 4.20 · 10−11 2.36 · 10−11 2.24 · 10−11 2.64 · 10−08 2.34 · 10−08 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 1.5 · 10−11 1.8 · 10−10 3.6 · 10−11 3.1 · 10−11 4.5 · 10−8 7.98 · 10−12 5.01 · 10−11 1.49 · 10−10 1.49 · 10−11 1.73 · 10−8 σnSI micrOMEGAs SOFTSUSY SPheno 6.84 · 10−12 1.06 · 10−11 5.21 · 10−11 5.19 · 10−11 1.51 · 10−10 4.39 · 10−11 1.57 · 10−11 1.41 · 10−11 1.84 · 10−08 1.30 · 10−08 DarkSUSY SOFTSUSY SPheno 6.42 · 10−12 7.02 · 10−12 1.11 · 10−10 1.10 · 10−10 4.71 · 10−11 4.31 · 10−11 2.39 · 10−11 2.26 · 10−11 2.73 · 10−08 2.41 · 10−08 (b) σnSI as computed in the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 1.5 · 10−9 2.5 · 10−8 4.6 · 10−5 1.9 · 10−8 3.1 · 10−4 1.37 · 10−9 3.70 · 10−8 4.62 · 10−5 1.76 · 10−8 3.24 · 10−4 σpSD micrOMEGAs SOFTSUSY SPheno 1.43 · 10−09 1.43 · 10−09 3.80 · 10−08 2.35 · 10−08 4.58 · 10−05 3.06 · 10−05 1.83 · 10−08 1.82 · 10−08 3.33 · 10−04 2.86 · 10−04 DarkSUSY SOFTSUSY SPheno 1.40 · 10−09 1.40 · 10−09 3.37 · 10−08 2.09 · 10−08 1.00 · 10−07 9.88 · 10−08 1.99 · 10−08 1.97 · 10−08 2.96 · 10−04 2.54 · 10−04 (c) σpSD as computed in the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 2.6 · 10−9 1.9 · 10−8 1.1 · 10−5 2.6 · 10−8 2.4 · 10−4 2.44 · 10−9 2.83 · 10−8 1.06 · 10−5 2.45 · 10−8 2.50 · 10−4 σnSD micrOMEGAs SOFTSUSY SPheno 2.51 · 10−09 2.50 · 10−09 2.91 · 10−08 1.80 · 10−08 1.05 · 10−05 6.90 · 10−06 2.52 · 10−08 2.48 · 10−08 2.57 · 10−04 2.21 · 10−04 DarkSUSY SOFTSUSY SPheno 1.81 · 10−09 1.81 · 10−09 2.22 · 10−08 1.37 · 10−08 1.81 · 10−08 1.83 · 10−08 1.75 · 10−08 1.73 · 10−08 1.96 · 10−04 1.68 · 10−04 (d) σnSD as computed in the FeynHiggs branch of the pipeline. Table 16. pMSSM dark matter scattering cross section: spin independent neutralino-nucleon elastic scattering cross sections, in pb, as computed by the various FeynHiggs pipelines. The spinindependent per-nucleon average for Xe is plotted in the lower left panel of figure 2 and percent differences are given in table 18. – 42 – JHEP05(2018)113 (a) σpSI as computed in the FeynHiggs branch of the pipeline. 81 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 1.5 · 10−11 1.7 · 10−10 1.6 · 10−11 3.1 · 10−11 4.3 · 10−08 7.88 · 10−12 4.90 · 10−11 5.10 · 10−10 1.48 · 10−11 1.64 · 10−08 σpSI micrOMEGAs SOFTSUSY SPheno 7.18 · 10−12 1.11 · 10−11 5.50 · 10−11 5.50 · 10−11 5.02 · 10−10 1.78 · 10−10 1.60 · 10−11 1.44 · 10−11 1.83 · 10−08 1.30 · 10−08 DarkSUSY SOFTSUSY SPheno 6.96 · 10−12 7.63 · 10−12 1.19 · 10−10 1.19 · 10−10 4.80 · 10−11 4.42 · 10−11 2.46 · 10−11 2.45 · 10−11 2.77 · 10−08 2.45 · 10−08 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 1.5 · 10−11 1.8 · 10−10 3.6 · 10−11 3.1 · 10−11 4.5 · 10−8 7.98 · 10−12 5.01 · 10−11 1.49 · 10−10 1.49 · 10−11 1.73 · 10−8 σnSI micrOMEGAs SOFTSUSY SPheno 7.30 · 10−12 1.13 · 10−11 5.63 · 10−11 5.62 · 10−11 1.48 · 10−10 4.22 · 10−11 1.62 · 10−11 1.46 · 10−11 1.92 · 10−08 1.37 · 10−08 DarkSUSY SOFTSUSY SPheno 7.09 · 10−12 7.77 · 10−12 1.21 · 10−10 1.21 · 10−10 4.92 · 10−11 4.54 · 10−11 2.49 · 10−11 2.38 · 10−11 2.86 · 10−08 2.53 · 10−08 (b) σnSI as computed in the SusyHD branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 1.5 · 10−09 2.5 · 10−08 4.6 · 10−05 1.9 · 10−08 3.1 · 10−04 1.37 · 10−09 3.70 · 10−08 4.62 · 10−05 1.76 · 10−08 3.24 · 10−04 σpSD micrOMEGAs SOFTSUSY SPheno 1.43 · 10−09 1.43 · 10−09 3.80 · 10−08 2.35 · 10−08 4.58 · 10−05 3.06 · 10−05 1.83 · 10−08 1.82 · 10−08 3.33 · 10−04 2.86 · 10−04 DarkSUSY SOFTSUSY SPheno 1.40 · 10−09 1.40 · 10−09 3.37 · 10−08 2.09 · 10−08 1.00 · 10−07 9.88 · 10−08 1.99 · 10−08 1.99 · 10−08 2.96 · 10−04 2.54 · 10−04 (c) σpSD as computed in the SusyHD branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* 2.6 · 10−9 1.9 · 10−8 1.1 · 10−5 2.6 · 10−8 2.4 · 10−4 2.44 · 10−9 2.83 · 10−8 1.06 · 10−5 2.45 · 10−8 2.50 · 10−4 σnSD micrOMEGAs SOFTSUSY SPheno 2.51 · 10−09 2.50 · 10−09 2.91 · 10−08 1.80 · 10−08 1.05 · 10−05 6.90 · 10−06 2.52 · 10−08 2.48 · 10−08 2.57 · 10−04 2.21 · 10−04 DarkSUSY SOFTSUSY SPheno 1.81 · 10−09 1.81 · 10−09 2.22 · 10−08 1.37 · 10−08 1.81 · 10−08 1.83 · 10−08 1.75 · 10−08 1.73 · 10−08 1.96 · 10−04 1.68 · 10−04 (d) σnSD as computed in the SusyHD branch of the pipeline. Table 17. pMSSM dark matter scattering cross section: spin independent neutralino-nucleon elastic scattering cross sections, in pb, where pipelines not labeled as “Snowmass” are computed by the various SusyHD pipelines. Percent differences for the values presented here may be found in table 19. – 43 – JHEP05(2018)113 (a) σpSI as computed in the SusyHD branch of the pipeline. 82 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −122.55% −233.33% 96.84% −100.00% −145.71% −16.91% 3.92% −0.59% 4.52% 6.29% σpSI micrOMEGAs SOFTSUSY SPheno 0% −55.79% 0% 0.20% 0% 64.10% 0% 10.32% 0% 29.14% DarkSUSY SOFTSUSY SPheno 6.48% −2.30% −113.28% −112.81% 90.95% 91.71% −52.50% −44.64% −51.08% −33.51% Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −119.30% −245.49% 76.16% −97.45% −144.57% −16.67% 3.84% 1.32% 5.10% 5.98% σnSI micrOMEGAs SOFTSUSY SPheno 0% −54.97% 0% 0.38% 0% 70.93% 0% 10.19% 0% 29.35% DarkSUSY SOFTSUSY SPheno 6.17% −2.62% −112.25% −111.77% 68.83% 71.45% −52.03% −44.18% −48.20% −30.99% (b) Percent differences for σnSI as computed in the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −4.90% 34.21% −0.44% −3.83% 6.91% 4.20% 2.63% −0.87% 3.83% 2.70% σpSD micrOMEGAs SOFTSUSY SPheno 0% < 0.01% 0% 38.16% 0% 33.19% 0% 0.55% 0% 14.11% DarkSUSY SOFTSUSY SPheno 1.94% 1.92% 11.27% 45.11% 99.78% 99.78% −8.50% −7.42% 11.16% 23.65% (c) Percent differences for σpSD as computed in the FeynHiggs branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −3.59% 34.71% −4.76% −3.17% 6.61% 2.79% 2.75% −0.95% 2.78% 2.72% σnSD micrOMEGAs SOFTSUSY SPheno 0% 0.40% 0% 38.14% 0% 34.29% 0% 1.59% 0% 14.01% DarkSUSY SOFTSUSY SPheno 27.76% 27.81% 23.84% 52.87% 99.83% 99.83% 30.45% 31.24% 23.88% 34.51% (d) Percent differences for σnSD as computed in the FeynHiggs branch of the pipeline. Table 18. pMSSM dark matter scattering cross section: percent difference of the values found in table 16 (pipelines with FeynHiggs) relative to the SOFTSUSY-micrOMEGAs pipelines. – 44 – JHEP05(2018)113 (a) Percent differences for σpSI as computed in the FeynHiggs branch of the pipeline. 83 Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −108.91% −209.09% 96.81% −93.75% −134.97% −9.75% 10.91% −1.59% 7.50% 10.38% σpSI micrOMEGAs SOFTSUSY SPheno 0% −54.60% 0% < 0.01% 0% 64.54% 0% 10.00% 0% 28.96% DarkSUSY SOFTSUSY SPheno 3.12% −6.26% −116.44% −116.38% 90.45% 91.19% −53.60% −47.03% −51.36% −33.86% Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −105.48% −219.72% 75.68% −91.36% −134.38% −9.32% 11.01% −0.68% 8.02% 9.90% σnSI micrOMEGAs SOFTSUSY SPheno 0% −54.79% 0% 0.18% 0% 71.49% 0% 9.88% 0% 28.65% DarkSUSY SOFTSUSY SPheno 2.92% −6.47% −115.21% −115.14% 66.75% 69.34% −53.42% −46.85% −48.75% −31.58% (b) Percent differences for σnSI as computed in the SusyHD branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −4.90% 34.21% −0.44% −3.83% 6.91% 4.20% 2.63% −0.87% 3.83% 2.70% σpSD micrOMEGAs SOFTSUSY SPheno 0% < 0.01% 0% 38.16% 0% 33.19% 0% 0.55% 0% 14.11% DarkSUSY SOFTSUSY SPheno 1.94% 1.92% 11.27% 45.11% 99.78% 99.78% −8.50% −7.42% 11.16% 23.65% (c) Percent differences for σpSD as computed in the SusyHD branch of the pipeline. Model B̃ − t̃ Coann. Pure h̃ B̃ − q̃ Coann. A funnel Well Tempered χ̃ Snowmass Snowmass* −3.59% 34.71% −4.76% −3.17% 6.61% 2.79% 2.75% −0.95% 2.78% 2.72% σnSD micrOMEGAs SOFTSUSY SPheno 0% 0.40% 0% 38.14% 0% 34.29% 0% 1.59% 0% 14.01% DarkSUSY SOFTSUSY SPheno 27.76% 27.81% 23.84% 52.87% 99.83% 99.83% 30.45% 31.24% 23.88% 34.51% (d) Percent differences for σnSD as computed in the SusyHD branch of the pipeline. Table 19. pMSSM dark matter scattering cross section: percent difference of the values found in table 17 (pipelines with SusyHD) relative to the SOFTSUSY-micrOMEGAs pipelines. – 45 – JHEP05(2018)113 (a) Percent differences for σpSI as computed in the SusyHD branch of the pipeline. 84 Model MasterCode† CMSSM 1 CMSSM 2 NUHM A NUHM B 0.191 – 0.051 0.131 Ωh2 micrOMEGAs SOFTSUSY SPheno 0.037 14.9 3.58 0.113 0.046 0.115 0.125 0.207 DarkSUSY SOFTSUSY SPheno 0.648 28.1 3.63 0.111 0.045 0.111 0.118 0.200 (a) Ωh2 as computed via the FeynHiggs branch of the pipeline. MasterCode† CMSSM 1 CMSSM 2 NUHM A NUHM B 0.191 – 0.051 0.131 Ωh2 micrOMEGAs SOFTSUSY SPheno 0.036 14.9 3.58 0.113 0.046 0.115 0.125 0.208 DarkSUSY SOFTSUSY SPheno 0.637 28.1 3.63 0.110 0.045 0.111 0.118 0.200 (b) Ωh2 as computed via the SusyHD branch of the pipeline. Table 20. GUT model dark matter relic density: dark matter relic density as computed by the various pipelines. Table 20a’s pipelines make use of FeynHiggs, while table 20b uses SusyHD. The values from table 20a are plotted in the upper left panel of figure 5. B Data tables for GUT models In this appendix, we collect data tables for the GUT model analysis of our paper. – 46 – JHEP05(2018)113 Model 85 CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 4.63 · 10−27 – 2.21 · 10−26 8.96 · 10−27 < σv > micrOMEGAs SOFTSUSY SPheno 7.56 · 10−26 1.03 · 10−28 7.89 · 10−29 1.17 · 10−28 2.49 · 10−26 1.03 · 10−26 9.49 · 10−27 8.60 · 10−27 DarkSUSY SOFTSUSY SPheno 3.85 · 10−27 4.97 · 10−29 7.46 · 10−29 1.06 · 10−28 2.46 · 10−26 1.01 · 10−26 9.51 · 10−27 7.00 · 10−27 (a) < σv > today as computed in the FeynHiggs pipelines. Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 4.63 · 10−27 – 2.21 · 10−26 8.96 · 10−27 < σv > micrOMEGAs SOFTSUSY SPheno 7.81 · 10−26 1.03 · 10−28 7.89 · 10−29 1.17 · 10−28 2.49 · 10−26 1.03 · 10−26 9.49 · 10−27 8.59 · 10−27 DarkSUSY SOFTSUSY SPheno 3.93 · 10−27 4.97 · 10−29 7.46 · 10−29 1.06 · 10−28 2.46 · 10−26 1.01 · 10−26 9.53 · 10−27 6.99 · 10−27 (b) < σv > today as computed in the SusyHD pipelines. Table 21. GUT model dark matter annihilation cross section: dark matter annihilation cross section today, in cm3 s−1 , for GUT models as computed by the various pipelines. Table 21a shows the results from the pipelines that make use of FeynHiggs, while table 21b shows those that use SusyHD. The values in table 21a are plotted in the upper right panel of figure 5. – 47 – JHEP05(2018)113 Model 86 Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 8.76 · 10−10 – 4.20 · 10−10 1.26 · 10−9 σpSI micrOMEGAs SOFTSUSY SPheno 3.78 · 10−09 7.48 · 10−11 1.28 · 10−11 1.17 · 10−11 4.00 · 10−10 9.54 · 10−10 9.38 · 10−10 8.90 · 10−09 DarkSUSY SOFTSUSY SPheno 2.92 · 10−09 1.25 · 10−10 3.04 · 10−11 2.25 · 10−11 8.09 · 10−10 1.71 · 10−09 2.12 · 10−09 1.52 · 10−08 Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 9.02 · 10−10 – 4.20 · 10−10 1.26 · 10−9 σnSI micrOMEGAs SOFTSUSY SPheno 4.01 · 10−09 7.69 · 10−11 1.34 · 10−11 1.33 · 10−11 4.11 · 10−10 9.75 · 10−10 9.61 · 10−10 9.10 · 10−09 DarkSUSY SOFTSUSY SPheno 2.99 · 10−09 1.28 · 10−10 3.15 · 10−11 2.32 · 10−11 8.26 · 10−10 1.74 · 10−09 2.16 · 10−09 1.55 · 10−08 (b) σnSI as computed in the FeynHiggs branch of the pipeline. Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 7.49 · 10−7 – 7.25 · 10−7 8.38 · 10−7 σpSD micrOMEGAs SOFTSUSY SPheno 1.07 · 10−06 5.63 · 10−08 1.75 · 10−9 2.15 · 10−9 −07 7.79 · 10 7.75 · 10−07 7.54 · 10−07 4.20 · 10−06 DarkSUSY SOFTSUSY SPheno 9.48 · 10−07 5.03 · 10−08 2.77 · 10−9 3.23 · 10−9 −07 6.91 · 10 6.87 · 10−07 6.69 · 10−07 3.73 · 10−06 (c) σpSD as computed in the FeynHiggs branch of the pipeline. Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 5.80 · 10−7 – 5.55 · 10−7 6.41 · 10−7 σnSD micrOMEGAs SOFTSUSY SPheno 8.24 · 10−07 4.51 · 10−08 9.65 · 10−9 1.04 · 10−8 5.96 · 10−07 5.93 · 10−07 5.77 · 10−07 3.22 · 10−06 DarkSUSY SOFTSUSY SPheno 6.27 · 10−07 3.42 · 10−08 6.56 · 10−9 7.08 · 10−9 4.54 · 10−07 4.52 · 10−07 4.40 · 10−07 2.45 · 10−06 (d) σnSD as computed in the FeynHiggs branch of the pipeline. Table 22. GUT model dark matter scattering cross section: spin independent neutralino-nucleon elastic scattering cross sections, in pb, as computed by the various FeynHiggs branches of the pipelines. The per nucleon average for Xe is plotted in the lower left panel of figure 5 using the FeynHiggs pipelines’ values (tables 22a & 22b). – 48 – JHEP05(2018)113 (a) σpSI as computed in the FeynHiggs branch of the pipeline. 87 Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 8.76 · 10−10 – 4.20 · 10−10 1.26 · 10−09 σpSI micrOMEGAs SOFTSUSY SPheno 3.73 · 10−09 7.63 · 10−11 1.14 · 10−11 1.34 · 10−11 3.65 · 10−10 9.67 · 10−10 8.59 · 10−10 9.03 · 10−09 DarkSUSY SOFTSUSY SPheno 2.85 · 10−09 1.28 · 10−10 2.62 · 10−11 2.38 · 10−11 7.16 · 10−10 1.74 · 10−09 1.87 · 10−09 1.55 · 10−08 Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 9.02 · 10−10 – 4.20 · 10−10 1.26 · 10−09 σnSI micrOMEGAs SOFTSUSY SPheno 3.95 · 10−09 7.85 · 10−11 1.20 · 10−11 1.40 · 10−11 3.74 · 10−10 9.88 · 10−10 8.76 · 10−10 9.24 · 10−09 DarkSUSY SOFTSUSY SPheno 2.92 · 10−09 1.31 · 10−10 2.71 · 10−11 2.45 · 10−11 7.29 · 10−10 1.77 · 10−09 1.91 · 10−09 1.58 · 10−08 (b) σnSI as computed in the SusyHD branch of the pipeline. Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 7.49 · 10−07 – 7.25 · 10−07 8.38 · 10−07 σpSD micrOMEGAs SOFTSUSY SPheno 1.07 · 10−06 5.63 · 10−08 1.27 · 10−09 2.15 · 10−09 7.79 · 10−07 7.75 · 10−07 7.54 · 10−07 4.20 · 10−06 DarkSUSY SOFTSUSY SPheno 9.48 · 10−07 5.03 · 10−08 2.77 · 10−09 3.23 · 10−09 6.91 · 10−07 6.87 · 10−07 6.69 · 10−07 3.73 · 10−06 (c) σnSD as computed in the SusyHD branch of the pipeline. Model CMSSM 1 CMSSM 2 NUHM A NUHM B MasterCode† 5.80 · 10−07 – 5.55 · 10−07 6.41 · 10−07 σnSD micrOMEGAs SOFTSUSY SPheno 8.24 · 10−07 4.51 · 10−08 9.65 · 10−09 1.04 · 10−08 5.96 · 10−07 5.93 · 10−07 5.77 · 10−07 3.22 · 10−06 DarkSUSY SOFTSUSY SPheno 6.27 · 10−07 3.42 · 10−08 6.56 · 10−09 7.08 · 10−09 4.54 · 10−07 4.52 · 10−07 4.40 · 10−07 2.45 · 10−06 (d) σnSD as computed in the SusyHD branch of the pipeline. Table 23. GUT model dark matter scattering cross section: spin independent neutralino-nucleon elastic scattering cross sections, in pb, as computed by the various SusyHD branches of the pipelines (except for the MasterCode† pipeline, which uses FeynHiggs.). – 49 – JHEP05(2018)113 (a) σpSI as computed in the SusyHD branch of the pipeline. 88 mχ [GeV] 937.25 937.06 936.82 936.50 936.10 936.09 935.54 934.80 933.77 932.27 2 + N2 N13 14 0.0140 0.0163 0.0194 0.0235 0.0289 0.0292 0.0374 0.0498 0.0698 0.1042 SOFTSUSY mA [GeV] µ [GeV] 1489.6 1214.8 1485.1 1194.0 1480.2 1172.7 1475.1 1150.9 1469.8 1129.4 1469.6 1128.5 1463.8 1105.5 1457.8 1081.9 1451.4 1057.6 1444.6 1032.6 < σv > [cm3 /s] 3.02 · 10−26 3.66 · 10−26 4.54 · 10−26 5.80 · 10−26 7.56 · 10−26 7.65 · 10−26 1.05 · 10−25 1.52 · 10−25 2.30 · 10−25 3.66 · 10−25 Ωh2 0.090 0.075 0.061 0.048 0.037 0.037 0.027 0.019 0.013 0.008 σpSI [pb] 1.41 · 10−9 1.74 · 10−9 2.20 · 10−9 2.85 · 10−9 3.78 · 10−9 3.83 · 10−9 5.35 · 10−9 7.85 · 10−9 1.22 · 10−8 1.99 · 10−8 (a) CMSSM points in the neighborhood of the CMSSM 1 benchmark point (here denoted by the star) as computed via SOFTSUSY-FeynHiggs. m0 [GeV] 5450 5500 5550 5600 ⋆ 5648 5650 5700 5750 5800 5850 mχ [GeV] 937.50 937.63 937.75 937.87 937.98 937.99 938.10 938.22 938.33 938.43 2 + N2 N13 14 0.0023 0.0024 0.0024 0.0025 0.0026 0.0026 0.0027 0.0028 0.0029 0.0030 SPheno mA [GeV] µ [GeV] 2542.5 1589.6 2554.3 1579.2 2566.2 1568.5 2578.1 1557.6 2589.4 1547.0 2589.9 1546.6 2601.7 1535.4 2613.5 1524.0 2625.3 1512.4 2638.7 1500.6 < σv > [cm3 /s] 1.14 · 10−28 1.11 · 10−28 1.09 · 10−28 1.07 · 10−28 1.06 · 10−28 1.06 · 10−28 1.05 · 10−28 1.04 · 10−28 1.04 · 10−28 1.04 · 10−28 Ωh2 13.6 13.9 14.2 14.4 14.6 14.6 14.8 14.9 15.0 15.0 σpSI [pb] 6.66 · 10−11 6.88 · 10−11 7.11 · 10−11 7.37 · 10−11 7.63 · 10−11 7.64 · 10−11 7.93 · 10−11 8.25 · 10−11 8.60 · 10−11 8.98 · 10−11 (b) CMSSM points in the neighborhood of the CMSSM 1 benchmark point (here denoted by the star) as computed via SPheno-FeynHiggs. Table 24. CMSSM EWSB sector: the variation of µ, mA , and dark matter observables with universal scalar mass m0 in the neighborhood of the CMSSM 1 benchmark point (denoted by the star), i.e. m1/2 = 2098.41 GeV, A0 = 781.89 GeV, tan(β) = 51.28, µ > 0, mt = 173.30 GeV. These results are also plotted in figure 3. The dark matter observables are computed by micrOMEGAs for both table 24a and table 24b. – 50 – JHEP05(2018)113 m0 [GeV] 5450 5500 5550 5600 ⋆ 5648 5650 5700 5750 5800 5850 89 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE]. [3] CMS collaboration, Search for new phenomena with the MT2 variable in the all-hadronic √ final state produced in proton-proton collisions at s = 13 TeV, Eur. Phys. J. C 77 (2017) 710 [arXiv:1705.04650] [INSPIRE]. [4] ATLAS collaboration, Search for top squarks in final √ states with one isolated lepton, jets and missing transverse momentum using 36.1 fb−1 of 13 TeV pp collision data with the ATLAS detector, ATLAS-CONF-2017-037, CERN, Geneva Switzerland, (2017). [5] Particle Data Group collaboration, C. Patrignani et al., Review of particle physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE]. [6] H. Baer, V. Barger and A. Mustafayev, Implications of a 125 GeV Higgs scalar for LHC SUSY and neutralino dark matter searches, Phys. Rev. D 85 (2012) 075010 [arXiv:1112.3017] [INSPIRE]. [7] J.L. Feng, K.T. Matchev and D. Sanford, Focus point supersymmetry redux, Phys. Rev. D 85 (2012) 075007 [arXiv:1112.3021] [INSPIRE]. [8] S. Heinemeyer, O. Stal and G. Weiglein, Interpreting the LHC Higgs search results in the MSSM, Phys. Lett. B 710 (2012) 201 [arXiv:1112.3026] [INSPIRE]. [9] A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi and J. Quevillon, Implications of a 125 GeV Higgs for supersymmetric models, Phys. Lett. B 708 (2012) 162 [arXiv:1112.3028] [INSPIRE]. [10] P. Draper, P. Meade, M. Reece and D. Shih, Implications of a 125 GeV Higgs for the MSSM and low-scale SUSY breaking, Phys. Rev. D 85 (2012) 095007 [arXiv:1112.3068] [INSPIRE]. [11] O. Buchmueller et al., Higgs and supersymmetry, Eur. Phys. J. C 72 (2012) 2020 [arXiv:1112.3564] [INSPIRE]. [12] M. Kadastik, K. Kannike, A. Racioppi and M. Raidal, Implications of the 125 GeV Higgs boson for scalar dark matter and for the CMSSM phenomenology, JHEP 05 (2012) 061 [arXiv:1112.3647] [INSPIRE]. [13] C. Strege, G. Bertone, D.G. Cerdeno, M. Fornasa, R. Ruiz de Austri and R. Trotta, Updated global fits of the CMSSM including the latest LHC SUSY and Higgs searches and XENON100 data, JCAP 03 (2012) 030 [arXiv:1112.4192] [INSPIRE]. [14] L. Aparicio, D.G. Cerdeno and L.E. Ibáñez, A 119–125 GeV Higgs from a string derived slice of the CMSSM, JHEP 04 (2012) 126 [arXiv:1202.0822] [INSPIRE]. [15] J. Ellis and K.A. Olive, Revisiting the Higgs mass and dark matter in the CMSSM, Eur. Phys. J. C 72 (2012) 2005 [arXiv:1202.3262] [INSPIRE]. – 51 – JHEP05(2018)113 [2] CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE]. 90 [16] H. Baer, V. Barger and A. Mustafayev, Neutralino dark matter in mSUGRA/CMSSM with a 125 GeV light Higgs scalar, JHEP 05 (2012) 091 [arXiv:1202.4038] [INSPIRE]. [17] P. Bechtle et al., Constrained supersymmetry after two years of LHC data: a global view with Fittino, JHEP 06 (2012) 098 [arXiv:1204.4199] [INSPIRE]. [18] C. Balázs, A. Buckley, D. Carter, B. Farmer and M. White, Should we still believe in constrained supersymmetry?, Eur. Phys. J. C 73 (2013) 2563 [arXiv:1205.1568] [INSPIRE]. [19] D. Ghosh, M. Guchait, S. Raychaudhuri and D. Sengupta, How constrained is the cMSSM?, Phys. Rev. D 86 (2012) 055007 [arXiv:1205.2283] [INSPIRE]. [21] O. Buchmueller et al., The CMSSM and NUHM1 in light of 7 TeV LHC, Bs → µ+ µ− and XENON100 data, Eur. Phys. J. C 72 (2012) 2243 [arXiv:1207.7315] [INSPIRE]. [22] K. Kowalska, S. Munir, L. Roszkowski, E.M. Sessolo, S. Trojanowski and Y.-L.S. Tsai, Constrained next-to-minimal supersymmetric Standard Model with a 126 GeV Higgs boson: a global analysis, Phys. Rev. D 87 (2013) 115010 [arXiv:1211.1693] [INSPIRE]. [23] C. Strege, G. Bertone, F. Feroz, M. Fornasa, R. Ruiz de Austri and R. Trotta, Global fits of the CMSSM and NUHM including the LHC Higgs discovery and new XENON100 constraints, JCAP 04 (2013) 013 [arXiv:1212.2636] [INSPIRE]. [24] M.E. Cabrera, J.A. Casas and R. Ruiz de Austri, The health of SUSY after the Higgs discovery and the XENON100 data, JHEP 07 (2013) 182 [arXiv:1212.4821] [INSPIRE]. [25] P. Draper, J.L. Feng, P. Kant, S. Profumo and D. Sanford, Dark matter detection in focus point supersymmetry, Phys. Rev. D 88 (2013) 015025 [arXiv:1304.1159] [INSPIRE]. [26] T. Cohen and J.G. Wacker, Here be dragons: the unexplored continents of the CMSSM, JHEP 09 (2013) 061 [arXiv:1305.2914] [INSPIRE]. [27] S. Henrot-Versillé et al., Constraining supersymmetry using the relic density and the Higgs boson, Phys. Rev. D 89 (2014) 055017 [arXiv:1309.6958] [INSPIRE]. [28] P. Bechtle et al., Constrained supersymmetry after the Higgs boson discovery: a global analysis with Fittino, PoS(EPS-HEP 2013)313 [arXiv:1310.3045] [INSPIRE]. [29] O. Buchmueller et al., Implications of improved Higgs mass calculations for supersymmetric models, Eur. Phys. J. C 74 (2014) 2809 [arXiv:1312.5233] [INSPIRE]. [30] O. Buchmueller et al., The CMSSM and NUHM1 after LHC run 1, Eur. Phys. J. C 74 (2014) 2922 [arXiv:1312.5250] [INSPIRE]. [31] L. Roszkowski, E.M. Sessolo and A.J. Williams, What next for the CMSSM and the NUHM: improved prospects for superpartner and dark matter detection, JHEP 08 (2014) 067 [arXiv:1405.4289] [INSPIRE]. [32] O. Buchmueller et al., Collider interplay for supersymmetry, Higgs and dark matter, Eur. Phys. J. C 75 (2015) 469 [Erratum ibid. C 76 (2016) 190] [arXiv:1505.04702] [INSPIRE]. [33] E.A. Bagnaschi et al., Supersymmetric dark matter after LHC run 1, Eur. Phys. J. C 75 (2015) 500 [arXiv:1508.01173] [INSPIRE]. [34] B. Dutta et al., Probing squeezed bino-slepton spectra with the Large Hadron Collider, Phys. Rev. D 96 (2017) 075037 [arXiv:1706.05339] [INSPIRE]. – 52 – JHEP05(2018)113 [20] A. Fowlie et al., The CMSSM favoring new territories: the impact of new LHC limits and a 125 GeV Higgs, Phys. Rev. D 86 (2012) 075010 [arXiv:1206.0264] [INSPIRE]. 91 [35] H. Baer, V. Barger, H. Serce and K. Sinha, Higgs and superparticle mass predictions from the landscape, JHEP 03 (2018) 002 [arXiv:1712.01399] [INSPIRE]. [36] F.E. Paige, S.D. Protopopescu, H. Baer and X. Tata, ISAJET 7.69: a Monte Carlo event generator for pp, p̄p and e+ e− reactions, hep-ph/0312045 [INSPIRE]. [37] B.C. Allanach, SOFTSUSY: a program for calculating supersymmetric spectra, Comput. Phys. Commun. 143 (2002) 305 [hep-ph/0104145] [INSPIRE]. [39] A. Djouadi, J.-L. Kneur and G. Moultaka, SuSpect: a Fortran code for the supersymmetric and Higgs particle spectrum in the MSSM, Comput. Phys. Commun. 176 (2007) 426 [hep-ph/0211331] [INSPIRE]. [40] T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, High-precision predictions for the light CP-even Higgs boson mass of the minimal supersymmetric Standard Model, Phys. Rev. Lett. 112 (2014) 141801 [arXiv:1312.4937] [INSPIRE]. [41] J. Pardo Vega and G. Villadoro, SusyHD: Higgs mass determination in supersymmetry, JHEP 07 (2015) 159 [arXiv:1504.05200] [INSPIRE]. [42] G. Bélanger, F. Boudjema, A. Pukhov and A. Semenov, MicrOMEGAs4.1: two dark matter candidates, Comput. Phys. Commun. 192 (2015) 322 [arXiv:1407.6129] [INSPIRE]. [43] P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E.A. Baltz, DarkSUSY: computing supersymmetric dark matter properties numerically, JCAP 07 (2004) 008 [astro-ph/0406204] [INSPIRE]. [44] B. Allanach, S. Kraml and W. Porod, Comparison of SUSY mass spectrum calculations, in Supersymmetry and unification of fundamental interactions. Proceedings, 10th International Conference, SUSY’02, Hamburg Germany, 17–23 June 2002, pg. 904 [hep-ph/0207314] [INSPIRE]. [45] B.C. Allanach, S. Kraml and W. Porod, Theoretical uncertainties in sparticle mass predictions from computational tools, JHEP 03 (2003) 016 [hep-ph/0302102] [INSPIRE]. [46] G. Bélanger, S. Kraml and A. Pukhov, Comparison of SUSY spectrum calculations and impact on the relic density constraints from WMAP, Phys. Rev. D 72 (2005) 015003 [hep-ph/0502079] [INSPIRE]. [47] B.C. Allanach, A. Djouadi, J.L. Kneur, W. Porod and P. Slavich, Precise determination of the neutral Higgs boson masses in the MSSM, JHEP 09 (2004) 044 [hep-ph/0406166] [INSPIRE]. [48] E. Bagnaschi, G.F. Giudice, P. Slavich and A. Strumia, Higgs mass and unnatural supersymmetry, JHEP 09 (2014) 092 [arXiv:1407.4081] [INSPIRE]. [49] P. Draper and H. Rzehak, A review of Higgs mass calculations in supersymmetric models, Phys. Rept. 619 (2016) 1 [arXiv:1601.01890] [INSPIRE]. [50] R. Catena and B. Schwabe, Form factors for dark matter capture by the sun in effective theories, JCAP 04 (2015) 042 [arXiv:1501.03729] [INSPIRE]. – 53 – JHEP05(2018)113 [38] W. Porod, SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e− colliders, Comput. Phys. Commun. 153 (2003) 275 [hep-ph/0301101] [INSPIRE]. 92 [51] J. Harz, B. Herrmann, M. Klasen, K. Kovarik and P. Steppeler, Precise prediction of the dark matter relic density within the MSSM, PoS(EPS-HEP2015)410 [arXiv:1510.06295] [INSPIRE]. [52] B.C. Allanach, G. Bélanger, F. Boudjema and A. Pukhov, Requirements on collider data to match the precision of WMAP on supersymmetric dark matter, JHEP 12 (2004) 020 [hep-ph/0410091] [INSPIRE]. [53] M.E. Gomez, G. Lazarides and C. Pallis, Yukawa quasi-unification, Nucl. Phys. B 638 (2002) 165 [hep-ph/0203131] [INSPIRE]. [55] pMSSM benchmarks for Snowmass 2013 webpage, http://www.slac.stanford.edu/ aismail/snowmass/index.html, accessed July 2017. [56] W. Porod and F. Staub, SPheno 3.1: extensions including flavour, CP-phases and models beyond the MSSM, Comput. Phys. Commun. 183 (2012) 2458 [arXiv:1104.1573] [INSPIRE]. [57] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, The Higgs boson masses and mixings of the complex MSSM in the Feynman-diagrammatic approach, JHEP 02 (2007) 047 [hep-ph/0611326] [INSPIRE]. [58] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Towards high precision predictions for the MSSM Higgs sector, Eur. Phys. J. C 28 (2003) 133 [hep-ph/0212020] [INSPIRE]. [59] S. Heinemeyer, W. Hollik and G. Weiglein, The masses of the neutral CP-even Higgs bosons in the MSSM: accurate analysis at the two loop level, Eur. Phys. J. C 9 (1999) 343 [hep-ph/9812472] [INSPIRE]. [60] S. Heinemeyer, W. Hollik and G. Weiglein, FeynHiggs: a program for the calculation of the masses of the neutral CP even Higgs bosons in the MSSM, Comput. Phys. Commun. 124 (2000) 76 [hep-ph/9812320] [INSPIRE]. [61] G. Bélanger, F. Boudjema, A. Pukhov and A. Semenov, MicrOMEGAs: a program for calculating the relic density in the MSSM, Comput. Phys. Commun. 149 (2002) 103 [hep-ph/0112278] [INSPIRE]. [62] G. Bélanger, F. Boudjema, A. Pukhov and A. Semenov, MicrOMEGAs: version 1.3, Comput. Phys. Commun. 174 (2006) 577 [hep-ph/0405253] [INSPIRE]. [63] G. Bélanger et al., Indirect search for dark matter with MicrOMEGAs2.4, Comput. Phys. Commun. 182 (2011) 842 [arXiv:1004.1092] [INSPIRE]. [64] P. Gondolo and G. Gelmini, Cosmic abundances of stable particles: improved analysis, Nucl. Phys. B 360 (1991) 145 [INSPIRE]. [65] J. Edsjo and P. Gondolo, Neutralino relic density including coannihilations, Phys. Rev. D 56 (1997) 1879 [hep-ph/9704361] [INSPIRE]. [66] J. Edsjo, M. Schelke, P. Ullio and P. Gondolo, Accurate relic densities with neutralino, chargino and sfermion coannihilations in mSUGRA, JCAP 04 (2003) 001 [hep-ph/0301106] [INSPIRE]. – 54 – JHEP05(2018)113 [54] M.W. Cahill-Rowley, J.L. Hewett, A. Ismail, M.E. Peskin and T.G. Rizzo, pMSSM benchmark models for Snowmass 2013, in Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013), Minneapolis MN U.S.A., 29 July–6 August 2013 [arXiv:1305.2419] [INSPIRE]. 93 [67] L. Bergstrom and P. Gondolo, Limits on direct detection of neutralino dark matter from b → sγ decays, Astropart. Phys. 5 (1996) 263 [hep-ph/9510252] [INSPIRE]. [68] B.C. Allanach et al., SUSY Les Houches accord 2, Comput. Phys. Commun. 180 (2009) 8 [arXiv:0801.0045] [INSPIRE]. [69] E. Bagnaschi et al., Likelihood analysis of the minimal AMSB model, Eur. Phys. J. C 77 (2017) 268 [arXiv:1612.05210] [INSPIRE]. [71] B.C. Allanach, A. Bednyakov and R. Ruiz de Austri, Higher order corrections and unification in the minimal supersymmetric Standard Model: SOFTSUSY3.5, Comput. Phys. Commun. 189 (2015) 192 [arXiv:1407.6130] [INSPIRE]. [72] D.M. Pierce, J.A. Bagger, K.T. Matchev and R.-J. Zhang, Precision corrections in the minimal supersymmetric Standard Model, Nucl. Phys. B 491 (1997) 3 [hep-ph/9606211] [INSPIRE]. [73] L.V. Avdeev and M. Yu. Kalmykov, Pole masses of quarks in dimensional reduction, Nucl. Phys. B 502 (1997) 419 [hep-ph/9701308] [INSPIRE]. [74] A. Bednyakov, A. Onishchenko, V. Velizhanin and O. Veretin, Two loop O(αs2 ) MSSM corrections to the pole masses of heavy quarks, Eur. Phys. J. C 29 (2003) 87 [hep-ph/0210258] [INSPIRE]. [75] K. Griest and D. Seckel, Three exceptions in the calculation of relic abundances, Phys. Rev. D 43 (1991) 3191 [INSPIRE]. [76] K.A. Olive, D.N. Schramm and G. Steigman, Limits on new superweakly interacting particles from primordial nucleosynthesis, Nucl. Phys. B 180 (1981) 497 [INSPIRE]. [77] M. Srednicki, R. Watkins and K.A. Olive, Calculations of relic densities in the early universe, Nucl. Phys. B 310 (1988) 693 [INSPIRE]. [78] M. Hindmarsh and O. Philipsen, WIMP dark matter and the QCD equation of state, Phys. Rev. D 71 (2005) 087302 [hep-ph/0501232] [INSPIRE]. [79] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes in C: the art of scientific computing, second edition, Cambridge University Press, New York NY U.S.A., (1992). [80] G. Hairer, Solving ordinary differential equations II, Springer, Berlin Heidelberg Germany, (2010). [81] J.R. Ellis, T. Falk and K.A. Olive, Neutralino-stau coannihilation and the cosmological upper limit on the mass of the lightest supersymmetric particle, Phys. Lett. B 444 (1998) 367 [hep-ph/9810360] [INSPIRE]. [82] J.R. Ellis, T. Falk, K.A. Olive and M. Srednicki, Calculations of neutralino-stau coannihilation channels and the cosmologically relevant region of MSSM parameter space, Astropart. Phys. 13 (2000) 181 [Erratum ibid. 15 (2001) 413] [hep-ph/9905481] [INSPIRE]. – 55 – JHEP05(2018)113 [70] B.C. Allanach, S.P. Martin, D.G. Robertson and R. Ruiz de Austri, The inclusion of two-loop SUSYQCD corrections to gluino and squark pole masses in the minimal and next-to-minimal supersymmetric Standard Model: SOFTSUSY3.7, Comput. Phys. Commun. 219 (2017) 339 [arXiv:1601.06657] [INSPIRE]. 94 [83] G. Bélanger, F. Boudjema, A. Pukhov and A. Semenov, MicrOMEGAs3 : a program for calculating dark matter observables, Comput. Phys. Commun. 185 (2014) 960 [arXiv:1305.0237] [INSPIRE]. [84] A. Pukhov, CalcHEP 2.3: MSSM, structure functions, event generation, batchs and generation of matrix elements for other packages, hep-ph/0412191 [INSPIRE]. [85] A.C. Hearn, REDUCE 3.5, Rand publication CP78, Rand Corporation, Santa Monica CA U.S.A., (1993). [86] J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE]. [88] J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Non-perturbative effect on thermal relic abundance of dark matter, Phys. Lett. B 646 (2007) 34 [hep-ph/0610249] [INSPIRE]. [89] A. Hryczuk, R. Iengo and P. Ullio, Relic densities including Sommerfeld enhancements in the MSSM, JHEP 03 (2011) 069 [arXiv:1010.2172] [INSPIRE]. [90] A. Hryczuk and R. Iengo, The one-loop and Sommerfeld electroweak corrections to the wino dark matter annihilation, JHEP 01 (2012) 163 [Erratum ibid. 06 (2012) 137] [arXiv:1111.2916] [INSPIRE]. [91] A. Hryczuk, I. Cholis, R. Iengo, M. Tavakoli and P. Ullio, Indirect detection analysis: wino dark matter case study, JCAP 07 (2014) 031 [arXiv:1401.6212] [INSPIRE]. [92] L. Roszkowski, E.M. Sessolo and S. Trojanowski, WIMP dark matter candidates and searches — current issues and future prospects, arXiv:1707.06277 [INSPIRE]. [93] J. Bramante, N. Desai, P. Fox, A. Martin, B. Ostdiek and T. Plehn, Towards the final word on neutralino dark matter, Phys. Rev. D 93 (2016) 063525 [arXiv:1510.03460] [INSPIRE]. [94] S. Mizuta and M. Yamaguchi, Coannihilation effects and relic abundance of Higgsino dominant LSP(s), Phys. Lett. B 298 (1993) 120 [hep-ph/9208251] [INSPIRE]. [95] G. Kane, K. Sinha and S. Watson, Cosmological moduli and the post-inflationary universe: a critical review, Int. J. Mod. Phys. D 24 (2015) 1530022 [arXiv:1502.07746] [INSPIRE]. [96] B. Dutta, L. Leblond and K. Sinha, Mirage in the sky: non-thermal dark matter, gravitino problem and cosmic ray anomalies, Phys. Rev. D 80 (2009) 035014 [arXiv:0904.3773] [INSPIRE]. [97] C. Cheung, L.J. Hall, D. Pinner and J.T. Ruderman, Prospects and blind spots for neutralino dark matter, JHEP 05 (2013) 100 [arXiv:1211.4873] [INSPIRE]. [98] J. Hisano, S. Matsumoto, M.M. Nojiri and O. Saito, Direct detection of the wino and higgsino-like neutralino dark matters at one-loop level, Phys. Rev. D 71 (2005) 015007 [hep-ph/0407168] [INSPIRE]. [99] B. Dutta, T. Kamon, N. Kolev, K. Sinha, K. Wang and S. Wu, Top squark searches using dilepton invariant mass distributions and bino-higgsino dark matter at the LHC, Phys. Rev. D 87 (2013) 095007 [arXiv:1302.3231] [INSPIRE]. [100] M. Badziak, M. Olechowski and P. Szczerbiak, Is well-tempered neutralino in MSSM still alive after 2016 LUX results?, Phys. Lett. B 770 (2017) 226 [arXiv:1701.05869] [INSPIRE]. – 56 – JHEP05(2018)113 [87] M. Drees and M. Nojiri, Neutralino-nucleon scattering revisited, Phys. Rev. D 48 (1993) 3483 [hep-ph/9307208] [INSPIRE]. 95 [101] N. Arkani-Hamed, A. Delgado and G.F. Giudice, The well-tempered neutralino, Nucl. Phys. B 741 (2006) 108 [hep-ph/0601041] [INSPIRE]. [102] A. Arbey, M. Battaglia, A. Djouadi and F. Mahmoudi, The Higgs sector of the phenomenological MSSM in the light of the Higgs boson discovery, JHEP 09 (2012) 107 [arXiv:1207.1348] [INSPIRE]. [103] A. Djouadi, L. Maiani, A. Polosa, J. Quevillon and V. Riquer, Fully covering the MSSM Higgs sector at the LHC, JHEP 06 (2015) 168 [arXiv:1502.05653] [INSPIRE]. [104] G. Lee and C.E.M. Wagner, Higgs bosons in heavy supersymmetry with an intermediate mA , Phys. Rev. D 92 (2015) 075032 [arXiv:1508.00576] [INSPIRE]. [106] A. Anandakrishnan and K. Sinha, Viability of thermal well-tempered dark matter in SUSY GUTs, Phys. Rev. D 89 (2014) 055015 [arXiv:1310.7579] [INSPIRE]. [107] P. Huang and C.E.M. Wagner, Blind spots for neutralino dark matter in the MSSM with an intermediate mA , Phys. Rev. D 90 (2014) 015018 [arXiv:1404.0392] [INSPIRE]. [108] D. Hooper, C. Kelso, P. Sandick and W. Xue, Closing supersymmetric resonance regions with direct detection experiments, Phys. Rev. D 88 (2013) 015010 [arXiv:1304.2417] [INSPIRE]. [109] A.L. Erickcek, K. Sinha and S. Watson, Bringing isolated dark matter out of isolation: late-time reheating and indirect detection, Phys. Rev. D 94 (2016) 063502 [arXiv:1510.04291] [INSPIRE]. [110] K. Griest, Cross-sections, relic abundance and detection rates for neutralino dark matter, Phys. Rev. D 38 (1988) 2357 [Erratum ibid. D 39 (1989) 3802] [INSPIRE]. [111] M. Cirelli, N. Fornengo and A. Strumia, Minimal dark matter, Nucl. Phys. B 753 (2006) 178 [hep-ph/0512090] [INSPIRE]. [112] FeynHiggs: history of the code webpage, http://wwwth.mpp.mpg.de/members/heinemey/feynhiggs/history.html, accessed November 2017. [113] H. Bahl and W. Hollik, Precise prediction for the light MSSM Higgs boson mass combining effective field theory and fixed-order calculations, Eur. Phys. J. C 76 (2016) 499 [arXiv:1608.01880] [INSPIRE]. [114] H. Bahl, S. Heinemeyer, W. Hollik and G. Weiglein, Reconciling EFT and hybrid calculations of the light MSSM Higgs-boson mass, Eur. Phys. J. C 78 (2018) 57 [arXiv:1706.00346] [INSPIRE]. [115] Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE]. [116] Fermi-LAT collaboration, M. Ackermann et al., Searching for dark matter annihilation from milky way dwarf spheroidal galaxies with six years of Fermi Large Area Telescope data, Phys. Rev. Lett. 115 (2015) 231301 [arXiv:1503.02641] [INSPIRE]. [117] LUX collaboration, D.S. Akerib et al., Results from a search for dark matter in the complete LUX exposure, Phys. Rev. Lett. 118 (2017) 021303 [arXiv:1608.07648] [INSPIRE]. – 57 – JHEP05(2018)113 [105] A. Anandakrishnan, B. Shakya and K. Sinha, Dark matter at the pseudoscalar Higgs resonance in the phenomenological MSSM and SUSY GUTs, Phys. Rev. D 91 (2015) 035029 [arXiv:1410.0356] [INSPIRE]. 96 [118] PandaX-II collaboration, A. Tan et al., Dark matter results from first 98.7 days of data from the PandaX-II experiment, Phys. Rev. Lett. 117 (2016) 121303 [arXiv:1607.07400] [INSPIRE]. [119] LZ collaboration, D.S. Akerib et al., LUX-ZEPLIN (LZ) conceptual design report, arXiv:1509.02910 [INSPIRE]. [120] J. Harz, B. Herrmann, M. Klasen, K. Kovarik and Q.L. Boulc’h, Neutralino-stop coannihilation into electroweak gauge and Higgs bosons at one loop, Phys. Rev. D 87 (2013) 054031 [arXiv:1212.5241] [INSPIRE]. [122] T. Falk, A. Ferstl and K.A. Olive, Variations of the neutralino elastic cross-section with CP-violating phases, Astropart. Phys. 13 (2000) 301 [hep-ph/9908311] [INSPIRE]. [123] J.R. Ellis, A. Ferstl and K.A. Olive, Reevaluation of the elastic scattering of supersymmetric dark matter, Phys. Lett. B 481 (2000) 304 [hep-ph/0001005] [INSPIRE]. [124] G. Jungman, M. Kamionkowski and K. Griest, Supersymmetric dark matter, Phys. Rept. 267 (1996) 195 [hep-ph/9506380] [INSPIRE]. [125] T. Falk, A. Ferstl and K.A. Olive, New contributions to neutralino elastic cross-sections from CP-violating phases in the MSSM, Phys. Rev. D 59 (1999) 055009 [Erratum ibid. D 60 (1999) 119904] [hep-ph/9806413] [INSPIRE]. [126] S.P. Martin, A supersymmetry primer, Adv. Ser. Direct. High Energy Phys. 18 (1998) 1 [Adv. Ser. Direct. High Energy Phys. 21 (2010) 1] [hep-ph/9709356] [INSPIRE]. [127] M. Drees and S.P. Martin, Implications of SUSY model building, in Electroweak symmetry breaking and new physics at the TeV scale, T.L. Barklow et al. eds., World Scientific, Singapore, (1997), pg. 146 [hep-ph/9504324] [INSPIRE]. – 58 – JHEP05(2018)113 [121] G. Bertone, D. Hooper and J. Silk, Particle dark matter: evidence, candidates and constraints, Phys. Rept. 405 (2005) 279 [hep-ph/0404175] [INSPIRE]. CHAPTER 3 FUTURE WORK 3.1 Uncertainties in Flavor Physics Precision cosmology and dark matter searches are not the only source of constraints now being placed on SUSY. The LHC’s Run 2 results have ruled out the simplest models of SUSY in the low mass corners of the sparticle spectrum. Furthermore, new experiments, such as Femilab’s Muon g − 2 Experiment, offer yet more windows to new physics through precision SM tests [274]. As such, we must again ask if we understand the uncertainties present in the predictions of SUSY. In the previous chapter, we tackled this question primarily from the perspective of astrophysical observables centered around the hunt for dark matter. In this chapter, we will turn our attention to a different avenue in the search for new physics, and ask this question of flavor physics observables that would be affected by new SUSY particles. As discussed in Section 1.3.3, the symmetry breaking processes introduced to complete the MSSM were motivated, in part, by a need to eliminate substantial CP -violating terms. As a result, what CP violation that is manifested in SUSY will result in FCNCs that will be highly constrained by the SM and present a cleanly identifiable signal. We will discuss three prominent signatures of SUSY in flavor physics: flavor change in the the decay of B mesons (b → sγ), leptonic decays of B mesons (Bs0 → µ+ µ− ), and the calculation of the anomalous magnetic moment of the muon (aµ ). 3.1.1 3.1.1.1 Physics of the Observables Radiative B 0 Decays: b → sγ CP violation of the weak interaction leads to flavor changing charged currents in the SM. These processes lead to, most commonly, changes in quark flavor within the same generation. For example, the decay of neutron to a proton is one such process where a d quark decays to a u quark while radiating a W − boson. Furthermore, the more exotic intergenerational 98 couplings, such as between a dc̄ or sū can also occur, but much more rarely. In all cases, the couplings follow a pattern of connecting up-type to down-type quarks; couplings between 2 down-type quarks, such as a bs coupling, does not occur at tree level. Instead, b → s decays happen at the loop level through tW − loops, as in Figure 3.1, left. Consequently, the observation of b → sγ at experiments is a strong test of any new physics, including SUSY, and has been widely studied as such (e.g., see References [275– 280]). SUSY can effect b → sγ through loops, including top-Higgs loops (Figure 3.1, right), squark-chargino, squark-gluino, and squark-neutralino loops [281]. However, the squark-neutralino and squark-gluino loops are only a nominal contribution while the squarkchargino loops will either add constructively or destructively with W -Higgs loops. This leaves the top-Higgs loops as the most promising channel through which to observe deviations from the SM [281, 282]. Experiments such as CLEO [283], BaBar [284], and Belle [285] have put strong constraints on b → sγ through observations of radiative B 0 meson decay. Combining these results, the Heavy Flavor Averaging Group has found [286] found the global experimental value of the branching fraction to be Bexp (B 0 → Xs γ) = (3.32 ± 0.15) · 10−4 , while next-to-next-to-leading order SM calculations [287, 288] predict that BSM (B 0 → Xs γ) = (3.36 ± 0.23) · 10−4 , leaving little room for SUSY and thus providing a strong test of its phenomenology. When close to the weak scale, SUSY loop contributions can contribute to b → sγ on the order of the SM contributions, leaving large swaths of the cosmologically favored parameter space already ruled out. Higher mass scales for both the gauge and scalar masses – the direction W± b H± t s γ b t s γ Figure 3.1: Radiative Contributions to b → sγ. Shown here are contributing diagrams of an SM top-W Boson (tW − ) loop and a SUSY top-Higgs (tH − ) loop. 99 in which the search for SUSY is headed – result in much smaller SUSY loop corrections, thus evading constraints [282]. 3.1.1.2 Neutral Bs Meson Decay: Bs0 → µ+ µ− The decay of the meson Bs0 is rare in the SM thanks to the fact that it arises only at the loop level and is helicity suppressed.1 The main contributions are from box diagrams involving W bosons and penguin diagrams [289], such as the top two diagrams in Figure 3.2. In the SM, the contributions add to a branching ratio [289] of BSM (Bs0 → µ+ µ− ) = (3.65 ± 0.23) · 10−9 . This compares to the LHCb experiment’s measured value [290] of −9 , BLHCb (Bs0 → µ+ µ− ) = (3.0 ± 0.6+0.3 −0.2 ) · 10 which places the SM into good agreement with experiment, and leaves very little room in which to fit BSM physics. 1 Bs0 is a spin-0 particle, which requires that the µ+ -µ− pair be produced with opposite helicities. s µ+ s W+ µ+ c Z 0, γ Bs0 νµ u, c, t W− b Bs0 W± c µ− µ− b s Bs0 h, H 0 , A0 u, c, t b µ+ H− H+ s Bs0 χ̃± µ− b µ+ ũ, c̃, t̃ h, H 0 , A0 µ− Figure 3.2: Contributions to the Leptonic Decay of Bs0 . The top register shows the W -box (right) and penguin (left) diagram contributions to Bs0 → µ+ µ− from the SM. The bottom register depicts the diagrams contributing to Bs0 → µ+ µ− from SUSY, including a Higgs penguin diagram (left) and a contribution from self-energy corrections to the external quark lines (right). 100 The inclusion of extra Higgs doublets, as added to SUSY per the discussion in Section 1.3.2, allows extra contributions that can exacerbate the small difference between experiment and theory, allowing for strong constraints to be placed on such models [289, 291, 292]. These new contributions arise generically from the exchange of Higgses, taking on the same form as the SM W -box and penguin diagrams, as in, for example, the bottom left diagram in Figure 3.2. Here, the helicity suppression is lifted by a factor that goes as tan4 (β) due to the Higgs Yukawa coupling to the final state muons [289], making the high tan(β) regime very important. However, for large Higgs masses these enhancements can again become suppressed [293]. Furthermore, in the case of SUSY, the most important of the new diagrams are ones that arise from chargino and Higgs loops that provide corrections to the external quarks’ self energies that go as tan6 (β) [292, 294]. For example, see the bottom right diagram in Figure 3.2. 3.1.1.3 The Muon Anomalous Magnetic Moment, aµ The anomalous magnetic moment of the muon, aµ = (gµ − 2)/2, is the higher-order radiative vertex correction to the muon [295], as illustrated schematically in Figure 3.3, left. The precision with which this process is measured makes this a compelling indicator of new physics [274], and further precision measurements can help distinguish SUSY from other models that predict heavy partners to the known SM particles [296]. In fact, measurements of aµ allow for probing of the mSUGRA parameters sign(µ) and tan(β), making experiments like the Muon g − 2 Experiment complimentary to colliders, which remain insensitive to µ− µ− µ− χ̃± γ µ̃ γ ν̃ χ̃± µ− µ− γ χ̃0 µ̃ µ− Figure 3.3: Radiative Vertex Corrections to the Muon. Such corrections to the muon’s magnetic moment arise through the replacement of the interaction vertex with a loop (left). SUSY particles further modify the anomalous magnetic moment through running in additional loops; two such loops are shown here: a chargino-sneutrino loop (χ̃± -ν̃; middle) and a smuon-neutralino loop (µ̃-χ̃0 ; right). 101 them [297]. The current limits on aµ are informed by the E821 Experiment [298] at Brookhaven National Laboratory, setting the experimental world average as −10 aexp . µ = 11 659 209.1(5.4)(3.3) · 10 Whereas the predicted SM value [294] of −10 aSM , µ = 11 659 180.3(0.1)(4.2)(2.6) · 10 results in a difference of [294] ∆aµ = 28.8(6.3)(4.9) · 10−10 . As one can see, the difference between theory and experiment leaves much room for new physics. The SUSY contribution depends strongly on tan(β) but is also highly suppressed by the mass scale of SUSY [293], meaning contributions to aµ from the high mass regions now under scrutiny do little to remedy the difference between theory and experiment. In the right diagram of Figure 3.3, the muon vertex is replaced by a χ̃01 -µ̃ loop, which is important in models that exhibit strong smuon coannihilation. However, while this diagram can give very large contributions to aµ (depending on the masses of µ̃R and µ̃L ) [299], it is typically subdominant with respect to chargino-snuetrino loop diagrams, shown in the middle diagram in Figure 3.3. This is true whenever there is degeneracy between the sparticle masses, and especially in mSUGRA models [300]. 3.1.2 Comparison Framework As was the case with dark matter observables, the calculations of the flavor physics observables are sophisticated endeavors. As such, many calculators exist to help achieve predictions from theory. Of those available that are still being maintained, we consider the popular codes SuperIso [301, 302], GM2Calc [303], SusyBSG [304], and ISAJET’s ISATOOLS [305, 306] alongside DarkSUSY [307, 308] and micrOMEGAs [309–311]. While the latter two are predominantly dark matter calculators, we include them now as they both contain calculations of flavor physics observables which are often utilized within dark matter studies. Not all of the codes include calculations of b → sγ, Bs0 → µ+ µ− , and aµ ; a summary of which observables are calculated by which codes is presented in Table 3.1. It is worth 102 Table 3.1: Flavor Observables Computed by Each Computational Tool. A Xdenotes inclusion in a computational package. Note that we use original routines in our comparison, rather than observables reported via calls to other programs (as is the case for DarkSUSY v6 which includes SuperIso v1.3). Calculator Observable b → sγ Bs0 → µ+ µ− aµ SuperIso X X X GM2Calc - - X SusyBSG X - - micrOMEGAs X X X DarkSUSY X - X ISATOOLS X X X noting that the major update to DarkSUSY v6 [308] has been implemented since our work on uncertainties in dark matter calculations, which used DarkSUSY v5.1.2. This updated version is the one we are using for the study of uncertainties in flavor physics observables, and while, through a call to SuperIso (v1.3), DarkSUSY now reports all three flavor physics observables, we consider the observables for which DarkSUSY includes original code (i.e., b → sγ and aµ ). Additionally, we also expand the range of spectrum calculators, adding: SuSpect [312], FlexibleSUSY [313], and ISAJET [305, 306]. While we use the current versions of the calculators as of May 2019, we use SuSpect v2.43, as the v3 overhaul to transition the code from Fortran to C++ is only in its alpha release. As before, we also include FeynHiggs [314–320] as an intermediate step between the spectrum generators and the flavor physics calculators. Since our previous study indicated agreement between FeynHiggs and SusyHD, we do not consider the latter here. The inclusion of FlexibleSUSY and SuperIso is particularly important, since these two calculators will allow us to mimic the pipeline utilized by the GAMBIT Collaboration [321– 323]. GAMBIT is a dedicated framework that ties together several of the codes above for the purpose of providing likelihood analyses of the parameter spaces of models of new physics, including SUSY. To this end, the collaboration has released a number of best-fit benchmark models within the MSSM based on scans over the SUSY parameter space and fits to constraints from dark matter searches, electroweak precision measurements, flavor 103 physics measurements, and collider searches. These results include 5 pMSSM7 benchmark points [324] and 4 benchmark points from each of the CMSSM, NUHM1, and NUHM2 frameworks [325]. These 17 new points are added to the 9 points previously studies for a total of 26 points to be used to study the uncertainties across 26 pipelines (to our best understanding, ISATOOLS cannot easily be used separately from ISAJET). Work on the uncertainties in flavor physics observables is in progress as of May 2019. 3.2 Uncertainties from the Computational Environment While we have focused on the physics underlying the tools used to do these calculations, that is only one possible source of discrepant predictions for observable quantities. Also extremely relevant are the architecture on which the calculations are done, the compilers with which we chose to compile the code, and the compilation flags utilized to guide the compilation. On-going work is being done to address these points. To evaluate the consequences of these choices, we use the “fuzzer” FLiT [326–328]. This is a program that compiles and executes C++ code, but does so in a way that loops over the different choices of compilation flags for each compiler and tracks differences in results between flag and compiler choices. The computational packages we are testing are highly sophisticated programs, many of which specify multiple options for compilation that can potentially affect their physics results. Encouragingly, initial testing suggests that SOFTSUSY’s routines are fairly robust to compilation choices. The goal is to expand this to other calculators, including not just C++ based packages but also C and Fortran based ones as well. Of particular interest are micrOMEGAs and SuSpect which offer the same code in multiple languages (micrOMEGAs uses C and Fortran while SuSpect is switching to C++ from Fortran in its next update). With a collaboration with the School of Computing, we in the process of are collecting these results and report them to the authors of the relevant codes so they may continue to improve these incredibly important computational tools. 104 3.3 References [274] D. W. Hertzog, J. P. Miller, E. de Rafael, B. Lee Roberts, and D. Stockinger, (2007), arXiv:0705.4617 [hep-ph]. [275] S. Bertolini, F. Borzumati, A. Masiero, and G. Ridolfi, Nucl. Phys. B353, 591 (1991). [276] L. Wolfenstein and Y. L. Wu, Phys. Rev. Lett. 73, 2809 (1994), arXiv:hep-ph/9410253 [hep-ph]. [277] G. M. Asatrian and A. Ioannisian, Phys. Rev. D54, 5642 (1996), arXiv:hepph/9603318 [hep-ph]. [278] S. Baek and P. Ko, Phys. Rev. Lett. 83, 488 (1999), arXiv:hep-ph/9812229 [hep-ph]. [279] A. L. Kagan and M. Neubert, Phys. Rev. D58, 094012 (1998), arXiv:hep-ph/9803368 [hep-ph]. [280] K. Kiers, A. Soni, and G.-H. Wu, Phys. Rev. D62, 116004 (2000), arXiv:hepph/0006280 [hep-ph]. [281] H. Baer, V. Barger, P. Huang, D. Mickelson, A. Mustafayev, and X. Tata, Phys. Rev. D87, 115028 (2013), arXiv:1212.2655 [hep-ph]. [282] H. Baer and M. Brhlik, Phys. Rev. D55, 3201 (1997), arXiv:hep-ph/9610224 [hep-ph]. [283] S. Chen et al. (CLEO), Phys. Rev. Lett. 87, 251807 (2001), arXiv:hep-ex/0108032 [hep-ex]. [284] J. P. Lees et al. (BaBar), Phys. Rev. D86, 112008 (2012), arXiv:1207.5772 [hep-ex]. [285] T. Saito et al. (Belle), Phys. Rev. D91, 052004 (2015), arXiv:1411.7198 [hep-ex]. [286] Y. Amhis et al. (HFLAV), Eur. Phys. J. C77, 895 (2017), arXiv:1612.07233 [hep-ex]. [287] M. Misiak et al., Phys. Rev. Lett. 114, 221801 (2015), arXiv:1503.01789 [hep-ph]. [288] M. Czakon, P. Fiedler, T. Huber, M. Misiak, T. Schutzmeier, and M. Steinhauser, JHEP 04, 168, arXiv:1503.01791 [hep-ph]. [289] C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou, and M. Steinhauser, Phys. Rev. Lett. 112, 101801 (2014), arXiv:1311.0903 [hep-ph]. [290] R. Aaij et al. (LHCb), Phys. Rev. Lett. 118, 191801 (2017), arXiv:1703.05747 [hep-ex]. [291] F. Beaujean, C. Bobeth, and D. van Dyk, Eur. Phys. J. C74, 2897 (2014), [Erratum: Eur. Phys. J.C74,3179(2014)], arXiv:1310.2478 [hep-ph]. [292] C. Bobeth, T. Ewerth, F. Kruger, and J. Urban, Phys. Rev. D64, 074014 (2001), arXiv:hep-ph/0104284 [hep-ph]. [293] H. Baer and X. Tata, Weak scale supersymmetry: From superfields to scattering events (Cambridge University Press, 2006). [294] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, 030001 (2018). 105 [295] M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory (AddisonWesley, Reading, USA, 1995). [296] J. M. Smillie and B. R. Webber, JHEP 10, 069, arXiv:hep-ph/0507170 [hep-ph]. [297] R. Lafaye, T. Plehn, and D. Zerwas, (2004), arXiv:hep-ph/0404282 [hep-ph]. [298] G. W. Bennett et al. (Muon g-2), Phys. Rev. D73, 072003 (2006), arXiv:hepex/0602035 [hep-ex]. [299] T. Moroi, Phys. Rev. D53, 6565 (1996), [Erratum: Phys. Rev.D56,4424(1997)], arXiv:hep-ph/9512396 [hep-ph]. [300] U. Chattopadhyay and P. Nath, Phys. Rev. D53, 1648 (1996), arXiv:hep-ph/9507386 [hep-ph]. [301] F. Mahmoudi, Comput. Phys. Commun. 178, 745 (2008), arXiv:0710.2067 [hep-ph]. [302] F. Mahmoudi, Comput. Phys. Commun. 180, 1579 (2009), arXiv:0808.3144 [hep-ph]. [303] P. Athron, M. Bach, H. G. Fargnoli, C. Gnendiger, R. Greifenhagen, J.-h. Park, S. Paehr, D. Stckinger, H. Stckinger-Kim, and A. Voigt, Eur. Phys. J. C76, 62 (2016), arXiv:1510.08071 [hep-ph]. [304] G. Degrassi, P. Gambino, and P. Slavich, Comput. Phys. Commun. 179, 759 (2008), arXiv:0712.3265 [hep-ph]. [305] H. Baer, F. E. Paige, S. D. Protopopescu, and X. Tata, in Workshop on Physics at Current Accelerators and the Supercollider Argonne, Illinois, June 2-5, 1993 (1993) pp. 0703–720, arXiv:hep-ph/9305342 [hep-ph]. [306] F. E. Paige, S. D. Protopopescu, H. Baer, and X. Tata, (2003), arXiv:hep-ph/0312045 [hep-ph]. [307] P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke, and E. A. Baltz, JCAP 0407, 008, arXiv:astro-ph/0406204 [astro-ph]. [308] T. Bringmann, J. Edsj, P. Gondolo, P. Ullio, and L. Bergstrm, JCAP 1807 (07), 033, arXiv:1802.03399 [hep-ph]. [309] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 149, 103 (2002), arXiv:hep-ph/0112278 [hep-ph]. [310] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 174, 577 (2006), arXiv:hep-ph/0405253 [hep-ph]. [311] G. Blanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, Comput. Phys. Commun. 231, 173 (2018), arXiv:1801.03509 [hep-ph]. [312] A. Djouadi, J.-L. Kneur, and G. Moultaka, Comput. Phys. Commun. 176, 426 (2007), arXiv:hep-ph/0211331 [hep-ph]. [313] P. Athron, M. Bach, D. Harries, T. Kwasnitza, J.-h. Park, D. Stckinger, A. Voigt, and J. Ziebell, Comput. Phys. Commun. 230, 145 (2018), arXiv:1710.03760 [hep-ph]. 106 [314] S. Heinemeyer, W. Hollik, and G. Weiglein, Comput. Phys. Commun. 124, 76 (2000), arXiv:hep-ph/9812320 [hep-ph]. [315] S. Heinemeyer, W. Hollik, and G. Weiglein, Eur. Phys. J. C9, 343 (1999), arXiv:hepph/9812472 [hep-ph]. [316] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich, and G. Weiglein, Eur. Phys. J. C28, 133 (2003), arXiv:hep-ph/0212020 [hep-ph]. [317] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein, JHEP 02, 047, arXiv:hep-ph/0611326 [hep-ph]. [318] T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein, Phys. Rev. Lett. 112, 141801 (2014), arXiv:1312.4937 [hep-ph]. [319] H. Bahl and W. Hollik, Eur. Phys. J. C76, 499 (2016), arXiv:1608.01880 [hep-ph]. [320] H. Bahl, S. Heinemeyer, W. Hollik, and G. Weiglein, Eur. Phys. J. C78, 57 (2018), arXiv:1706.00346 [hep-ph]. [321] P. Athron et al. (GAMBIT Models Workgroup), Eur. Phys. J. C78, 22 (2018), arXiv:1705.07936 [hep-ph]. [322] F. U. Bernlochner et al. (The GAMBIT Flavour Workgroup), Eur. Phys. J. C77, 786 (2017), arXiv:1705.07933 [hep-ph]. [323] P. Athron et al. (GAMBIT), Eur. Phys. J. C77, 784 (2017), [Addendum: Eur. Phys. J.C78,no.2,98(2018)], arXiv:1705.07908 [hep-ph]. [324] P. Athron et al. (GAMBIT), Eur. Phys. J. C77, 879 (2017), arXiv:1705.07917 [hepph]. [325] P. Athron et al. (GAMBIT), Eur. Phys. J. C77, 824 (2017), arXiv:1705.07935 [hepph]. [326] G. Sawaya, M. Bentley, I. Briggs, G. Gopalakrishnan, and D. H. Ahn, in 2017 IEEE International Symposium on Workload Characterization (IISWC) (IEEE, 2017) pp. 229–238. [327] M. Bentley, I. Briggs, G. Gopalakrishnan, D. H. Ahn, I. Laguna, G. L. Lee, and H. E. Jones, arXiv preprint arXiv:1811.05618 (2018). [328] M. Bentley, I. Briggs, G. Gopalakrishnan, D. H. Ahn, I. Laguna, G. L. Lee, and H. E. Jones, in The 28th International Symposium on High-Performance Parallel and Distributed Computing (HPDC’19) (ACM, 2019). CHAPTER 4 CONCLUSIONS The fact that new physics has not been conclusively observed at the LHC has ruled out large swaths of the supersymmetric parameter space. Yet, many models that relax potentially overly stringent assumptions or that reside in higher mass corners of the theory remain untouched for now. The focus we have taken in this dissertation is to determine whether or not we are ready to vet these remaining areas of SUSY. That is, by comparing results from the different computational packages used by the community, we can determine the theoretical uncertainty in predictions by evaluating the degree to which concordance exists between the various computational tools. To this end, we’ve presented results from several of the popular computational packages in the high mass regimes of the CMSSM, NUHM, and pMSSM frameworks. To do so, we have studied specific benchmark models designed to be in agreement with experimental constraints on SUSY. We find that there is still much work to do to reduce the theoretical uncertainties in the calculations of SUSY observables. In the pMSSM, which has the least amount of RGE running, we found small differences in the sparticle spectrum, as expected. These amounted to only a 1-2% difference in the stop, sdown, and sup quark masses. However, larger differences were apparent in the neutralino and chargino compositions; for example, we found that the well-tempered point exhibited a 20% difference in the higgsino admixture of the lightest neutralino. In both cases, these differences resulted in significant discrepancies – as much as a factor of 5 – in the value of the dark matter relic density, which is sensitive to the t-channel exchange of squarks (in bino coannihilation points) or charginos (in well-tempered points). These small differences were also enough to shift a model’s position in the parameter space, substantially altering expectations for models in highly sensitive funnel regions. While uncertainties in the mass spectrum affected the calculation of dark matter observables, they compounded sources of uncertainties already present in those calculations. 108 We found that even when the mass spectra agree, the dark matter calculators still disagree in their calculations of the relic density, as they handle the neutral and charged higgsinos differently for coannihilation processes. Moreover, whether or not they utilized an effective or a full Lagrangian at the loop level noticeably impacted results. Furthermore, as expected, calculations of the annihilation rate today were also dependent on what processes were included in the code. For instance, DarkSUSY does not include final state radiation when calculating the annihilation rate today as the process is dominant only in the early universe (and DarkSUSY does include it for the determination of the relic density). While the process is subdominant, according to micrOMEGAs such processes account for about 12% of the annihilation rate today for the B̃-q̃ point. On the other hand, micrOMEGAs does not include gluon-gluon final states in its accounting of the annihilation rate today, an annihilation channel that DarkSUSY counts as 66% of the processes for the same B̃-q̃ point. It’s also important to note which programs do which calculations when incorporating them into a pipeline. This was quite relevant to the relic density calculation while in the A-funnel, as not all of the spectrum and Higgs calculators calculate and report the pseudoscalar Higgs’s decay width (SPheno and FeynHiggs both report decay widths). Further complicating the matter is that the acceptance of this parameter as input is also not universal (DarkSUSY never uses the value reported whereas micrOMEGAs does). In GUT models, the input parameters are defined at the high scale, requiring significant running of the RGEs and allowing for significant variation to occur in the determination of the sparticle spectrum. Consequently, we studied the details of the electroweak symmetry breaking sector for the CMSSM and NUHM benchmarks, finding notable variation between the two spectrum generators. The two codes significantly diverge in their treatment of the Higgs mass parameter, µ, above approximately 2 TeV, due to the differences in reported squark masses and Yukawa couplings which become important at high m0 . In the CMSSM, unlike the NUHM, the two Higgs doublet masses are not fixed at the high scale. This allows the calculators to profoundly disagree and we consequentially find opposite trends for the pseudoscalar Higgs mass. Qualitatively, the RGE equations for m2Hu and m2Hd suggest a decrease in mA for the value of tan(β) we are studying. However, while SOFTSUSY exhibits this trend, SPheno does not – mA increases with m0 , in contrast to the behvior in SPheno. 109 The dark matter observables showed all the same trends in the case of GUT models as in the pMSSM analysis. However, being defined at the high scale, the significant running of the RGEs makes more room for disagreements to occur between the spectrum calculators, which propagates to the dark matter calculators. Indeed, substantial differences were found in the calculation of the sparticle spectrum, particularly in µ which can differ by up to 50%. This affected of the masses of the lightest and next-to-lightest neutralinos as well as the charginos. In the case of the NUHM, the result was a greater rate of Higgs exchange, which increased the uncertainty in the scattering cross-section. Depending on one’s choice of calculator, the sparticle mass differences for the CMSSM points were great enough to fundamentally change the point’s phenomenology. For the CMSSM 2 point, coannihilation is much more suppressed in SOFTSUSY relative to SPheno, making it no longer a τ̃ coannihilation point due to the differences in the τ̃ masses. Meanwhile, for the CMSSM 1, the dramatic difference in µ resulted in a reduction of an order of magnitude in the higgsino composition (down to ∼ 0.3% in SPheno) that completely shifted the point out of the funnel region, changing it into a nominally well-tempered point. Consequently, the LSP is no longer a cosmologically favored candidate for dark matter within a thermal history, as the relic abundance now spans a range of several orders of magnitude – far surpassing the precisely determined range of uncertainty set by experiment. SUSY presents a wide range of signatures relevant for both collider and astrophysical experiments. Neutrino and flavor physics phenomenology represent further windows into SUSY, and the question we pose here remains open for those as well: are the theoretical calculations in good enough agreement that the uncertainty for these new regimes is consistent with the precision set by experiments? So far, the answer is, “no”. For less constrained models and higher mass regimes, work still needs to be done to bring the various calculations into better agreement so that scientists can make reliable predictions that our colleagues at the next generation of experiments can confidently test. This is true of dark matter phenomenology and may be borne out on other experimental fronts as well. If nothing else, we certainly expect a wide uncertainty in, say, flavor physics observables simply due to the uncertainties in the spectrum generation that we’ve uncovered herein. As of now, how these uncertainties further propagate is unknown, and that is a future chapter in SUSY phenomenology.