The most recent experimental data for all measured production and decay channels of the bottomoniumlike states Z b ð10610Þ and Z b ð10650Þ are analyzed simultaneously using solutions of the Lippmann-Schwinger equations which respect constraints from unitarity and analyticity. The interaction potential in the open-bottom channels B ðÃÞBà þ c:c: contains short-range interactions as well as a one-pion exchange. It is found that the long-range interaction does not affect the line shapes as long as only S waves are considered. Meanwhile, the line shapes can be visibly modified once D waves, mediated by the strong tensor forces from the pion exchange potentials, are included. However, in the fit they get balanced largely by a momentum dependent contact term that appears to be needed also to render the results for the line shapes independent of the cutoff. The resulting line shapes are found to be insensitive to various higherorder interactions included to verify stability of the results. Both Z b states are found to be described by the poles located on the unphysical Riemann sheets in the vicinity of the corresponding thresholds. In particular, the Z b ð10610Þ state is associated with a virtual state residing just below the BB à =BB à threshold while the Z b ð10650Þ state most likely is a shallow state located just above the B ÃBà threshold.

We propose a practical parametrisation for the line shapes of near-threshold states compatible with all requirements of unitarity and analyticity. The coupled-channel system underlying the proposed parametrisation includes bare poles and an arbitrary number of elastic and inelastic channels treated fully nonperturbatively. The resulting formulae are general enough to be used for a simultaneous analysis of the data in all available production and decay channels of the (system of) state(s) under consideration for a quite wide class of reactions. As an example, we fit the experimental data currently available for several decay channels for the charged Zstates in the spectrum of bottomonia and find a good overall description of the data. We find the present data to be consistent with the Z b (10610) as a virtual state and with the Z b (10650) as a resonance, both residing very close to the BB * and B * B * threshold, respectively.

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The Hubbard model arises naturally when electron-electron interactions are added to the tightbinding descriptions of many condensed matter systems. For instance, the two-dimensional Hubbard model on the honeycomb lattice is central to the ab initio description of the electronic structure of carbon nanomaterials, such as graphene. Such low-dimensional Hubbard models are advantageously studied with Markov chain Monte Carlo methods, such as Hybrid Monte Carlo (HMC). HMC is the standard algorithm of the lattice gauge theory community, as it is well suited to theories of dynamical fermions. As HMC performs continuous, global updates of the lattice degrees of freedom, it provides superior scaling with system size relative to local updating methods. A potential drawback of HMC is its susceptibility to ergodicity problems due to so-called exceptional configurations, for which the fermion operator cannot be inverted. Recently, ergodicity problems were found in some formulations of HMC simulations of the Hubbard model. Here, we address this issue directly and clarify under what conditions ergodicity is maintained or violated in HMC simulations of the Hubbard model. We study different lattice formulations of the fermion operator and provide explicit, representative calculations for small systems, often comparing to exact results. We show that a fermion operator can be found which is both computationally convenient and free of ergodicity problems. CONTENTS

Numerical simulations are playing an increasingly important role in modern science. In this work it is suggested to use a numerical study of the famous perihelion motion of the planet Mercury (one of the prime observables supporting Einsteins General Relativity) as a test case to teach numerical simulations to high school students. The paper includes details about the development of the code as well as a discussion of the visualization of the results. In addition a method is discussed that allows one to estimate the size of the effect as well as the uncertainty of the approach a priori. At the same time this enables the students to double check the results found numerically. The course is structured into a basic block and two further refinements which aim at more advanced students.