Michael A. Schmidt

Distilling Free-Form Natural Laws from Experimental Data

For centuries, scientists have attempted to identify and document analytical laws that underlie physical phenomena in nature. Despite the prevalence of computing power, the process of finding natural laws and their corresponding equations has resisted automation. A key challenge to finding analytic relations automatically is defining algorithmically what makes a correlation in observed data important and insightful. We propose a principle for the identification of nontriviality. We demonstrated this approach by automatically searching motion-tracking data captured from various physical systems, ranging from simple harmonic oscillators to chaotic double-pendula. Without any prior knowledge about physics, kinematics, or geometry, the algorithm discovered Hamiltonians, Lagrangians, and other laws of geometric and momentum conservation. The discovery rate accelerated as laws found for simpler systems were used to bootstrap explanations for more complex systems, gradually uncovering the “alphabet” used to describe those systems.

Running neutrino mass parameters in see-saw scenarios

We systematically analyze quantum corrections in see-saw scenarios, including effects from above as well as below the see-saw scales. We derive approximate renormalization group equations for neutrino masses, lepton mixings and CP phases, yielding an analytic understanding and a simple estimate of the size of the effects. Even for hierarchical masses, they often exceed the precision of future experiments. Furthermore, we provide a software package allowing for a convenient numerical renormalization group analysis, with heavy singlets being integrated out successively at their mass thresholds. We also discuss applications to model building and related topics.

Real-time three-dimensional echocardiography for measurement of left ventricular volumes

Left ventricular (LV) volumes are important prognostic indexes in patients with heart disease. Although several methods can evaluate LV volumes, most have important intrinsic limitations. Real-time 3-dimensional echocardiography (RT3D echo) is a novel technique capable of instantaneous acquisition of volumetric images. The purpose of this study was to validate LV volume calculations with RT3D echo and to determine their usefulness in cardiac patients. To this end, 4 normal subjects and 21 cardiac patients underwent magnetic resonance imaging (MRI) and RT3D echo on the same day. A strong correlation was found between LV volumes calculated with MRI and with RT3D echo (r = 0.91; y = 20.1 + 0.71x; SEE 28 ml). LV volumes obtained with MRI were greater than those obtained with RT3D echo (126 +/- 83 vs 110 +/- 65 ml; p = 0.002), probably due to the fact that heart rate during MRI acquisition was lower than that during RT3D echo examination (62 +/- 11 vs 79 +/- 16 beats/min; p = 0.0001). Analysis of intra- and interobserver variability showed strong indexes of agreement in the measurement of LV volumes with RT3D echo. Thus, LV volume measurements with RT3D echo are accurate and reproducible. This technique expands the use of ultrasound for the noninvasive evaluation of cardiac patients and provides a new tool for the investigational study of cardiovascular disease.

CP and discrete flavour symmetries

A bstractWe give a consistent definition of generalised CP transformations in the context of discrete flavour symmetries. Non-trivial consistency conditions imply that every generalised CP transformation can be interpreted as a representation of an automorphism of the discrete group. This allows us to give consistent generalised CP transformations of popular flavour groups. We are able to clear up issues concerning recent claims about geometrical CP violation in models based on T′, clarify the origin of ”calculable phases” in Δ(27) and explain why apparently CP violating scalar potentials of A4 result in a CP conserving ground state.

The Scale-Invariant NMSSM and the 126 GeV Higgs Boson

A bstractThe recent LHC discovery of a Higgs-like resonance at 126 GeV suggests that the minimal supersymmetric standard model must be modified in order to preserve naturalness. A simple extension is to include a singlet superfield and consider the scale-invariant NMSSM, whose renormalizable superpotential contains no dimensionful parameters. This extension not only solves the μ-problem, but can easily accommodate a 126 GeV Higgs. We study the naturalness of the scale-invariant NMSSM taking into account the recent constraints from LHC searches, flavor physics and electroweak precision tests. We show that TeV-scale stop masses are still allowed in much of the parameter space with 5% tuning for a low messenger scale of 20 TeV, split families (with third-generation sleptons decoupled) and Higgs-singlet coupling λ of order one. For larger values of the Higgs-singlet coupling, which can relieve the tuning in the Higgs VEV, an additional tuning in the Higgs mass limits increasing the (lightest) stop mass beyond 1.2 TeV, the gluino mass above 3 TeV, and electroweak charginos and neutralinos beyond 400 GeV for a combined tuning better than 5%. This implies that the natural region of parameter space for the scale-invariant NMSSM will be fully explored at the 14 TeV LHC.

Radiative symmetry breaking of the minimal left-right symmetric model

Under the assumption of classical conformal invariance, we study the Coleman-Weinberg symmetry breaking mechanism in the minimal left-right symmetric model. This model is attractive as it provides a natural framework for small neutrino masses and the restoration of parity as a good symmetry of nature. We find that, in a large fraction of the parameter space, the parity symmetry is maximally broken by quantum corrections in the Coleman-Weinberg potential, which are a consequence of the conformal anomaly. As the left-right symmetry breaking scale is connected to the Planck scale through the logarithmic running of the dimensionless couplings of the scalar potential, a large separation of the two scales can be dynamically generated. The symmetry breaking dynamics of the model was studied using a renormalization group analysis. Electroweak symmetry breaking is triggered by the breakdown of left-right symmetry, and the left-right breaking scale is therefore expected in the few-TeV range. The phenomenological implications of the symmetry breaking mechanism are discussed.

Testing atmospheric mixing sum rules at precision neutrino facilities

We study the prospects for testing classes of atmospheric mixing sum rules at precision neutrino facilities. Such sum rules, which correlate the atmospheric mixing angle �23 with the recently measured reactor angle �13 and the cosine of the oscillation phase �, are predicted by a variety of semi-direct models based on discrete family symmetry, classified in terms of finite von Dyck groups. We perform a detailed simulation of the performance of the next generation of oscillation experiments, including the wide band superbeam and low-energy neutrino factory proposals, and compare their discriminating power for testing atmospheric mixing sum rules.

Symbolic Regression of Implicit Equations

Traditional Symbolic Regression applications are a form of supervised learning, where a label y is provided for every \(\vec{x}\) and an explicit symbolic relationship of the form \(y = f(\vec{x})\) is sought. This chapter explores the use of symbolic regression to perform unsupervised learning by searching for implicit relationships of the form \(f(\vec{x}, y) = 0\). Implicit relationships are more general and more expressive than explicit equations in that they can also represent closed surfaces, as well as continuous and discontinuous multi-dimensional manifolds. However, searching these types of equations is particularly challenging because an error metric is difficult to define. We studied several direct and indirect techniques, and present a successful method based on implicit derivatives. Our experiments identified implicit relationships found in a variety of datasets, such as equations of circles, elliptic curves, spheres, equations of motion, and energy manifolds.

Recipes and Ingredients for Neutrino Mass at Loop Level

A bstractThe large hierarchy between the neutrino mass scale and that of the other fermions seems to be unnatural from a theoretical point of view. Various strategies have been devised in order to generate naturally small values of neutrino masses. One of these techniques is neutrino mass generation at the loop level which requires a mechanism, e.g., a symmetry, to forbid the lower order contributions. Here, we study in detail the conditions on this type of symmetries. We put special emphasis on the discrete Zn symmetries as a simple example but our results can be also extended to more general groups. We find that regardless of the details of the symmetry, in certain cases the existence of a lower order contribution to neutrino masses can be determined by the topology of the diagrams with a given number of loops. We discuss the lepton flavor violating rare decays as well as (g − 2)μ in this class of models, which generically appear at the one loop level. Typically the imposed symmetry has important implications for dark matter, with the possibility of stabilizing one or even multiple dark matter candidates.

AMEND : a model explaining neutrino masses and dark matter testable at the LHC and MEG.

Despite being very successful in explaining the wide range of precision experimental results obtained so far, the Standard Model (SM) of elementary particles fails to address two of the greatest observations of the recent decades: tiny but nonzero neutrino masses and the well-known problem of missing mass in the Universe. Typically the new models beyond the SM explain only one of these observations. Instead, in the present article, we take the view that they both point towards the same new extension of the Standard Model. The new particles introduced are responsible simultaneously for neutrino masses and for the dark matter of the Universe. The stability of dark matter and the smallness of neutrino masses are guaranteed by a U(1) global symmetry, broken to a remnant