A Monte-Carlo simulation of the equilibrium beam polarization in ultra-high energy electron (positron) storage rings

With the recently emerging global interest in building a next generation of circular electron-positron colliders to study the properties of the Higgs boson, and other important topics in particle physics at ultra-high beam energies, it is also important to pursue the possibility of implementing polarized beams at this energy scale. It is therefore necessary to set up simulation tools to evaluate the beam polarization at these ultra-high beam energies. In this paper, a Monte-Carlo simulation of the equilibrium beam polarization based on the Polymorphic Tracking Code(PTC) (Schmidt et al., 2002 [1]) is described. The simulations are for a model storage ring with parameters similar to those of proposed circular colliders in this energy range, and they are compared with the suggestion (Derbenev et al., 1978 [2]) that there are different regimes for the spin dynamics underlying the polarization of a beam in the presence of synchrotron radiation at ultra-high beam energies. In particular, it has been suggested that the so-called "correlated" crossing of spin resonances during synchrotron oscillations at current energies, evolves into "uncorrelated" crossing of spin resonances at ultra-high energies.Comment: submitted to and accepted by Nucl. Instrum. Meth.

A New and Unifying Approach to Spin Dynamics and Beam Polarization in Storage Rings

With this paper we extend our studies [1] on polarized beams by distilling tools from the theory of principal bundles. Four major theorems are presented, one which ties invariant fields with the notion of normal form, one which allows one to compare different invariant fields, and two that relate the existence of invariant fields to the existence of certain invariant sets and relations between them. We then apply the theory to the dynamics of spin-1/2 and spin-1 particles and their density matrices describing statistically the particle-spin content of bunches. Our approach thus unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics. This unifying aspect of our approach relates the examples elegantly and uncovers relations between the various underlying dynamical systems in a transparent way

An Informal Summary of a New Formalism for Classifying Spin-Orbit Systems Using Tools Distilled from the Theory of Bundles

We give an informal summary of ongoing work which uses tools distilled from the theory of fibre bundles to classify and connect invariant fields associated with spin motion in storage rings. We mention four major theorems. One ties invariant fields with the notion of normal form, the second allows comparison of different invariant fields and the two others tie the existence of invariant fields to the existence of certain invariant sets. We explain how the theorems apply to the spin dynamics of spin-$1/2$ and spin-$1$ particles. Our approach elegantly unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics and suggests an avenue for addressing the question of the existence of the invariant spin field.Comment: Based on a presentation at Spin2014, The 21st International Symposium on Spin Physics, Beijing, China, October 2014. To be published in the International Journal of Modern Physics, Conference Serie

A detailed and unified treatment of spin-orbit systems using tools distilled from the theory of bundles

We return to our study \cite{BEH} of invariant spin fields and spin tunes for polarized beams in storage rings but in contrast to the continuous-time treatment in \cite{BEH}, we now employ a discrete-time formalism, beginning with the $\rm{Poincar\acute{e}}$ maps of the continuous time formalism. We then substantially extend our toolset and generalize the notions of invariant spin field and invariant frame field. We revisit some old theorems and prove several theorems believed to be new. In particular we study two transformation rules, one of them known and the other new, where the former turns out to be an $SO(3)$-gauge transformation rule. We then apply the theory to the dynamics of spin-$1/2$ and spin-$1$ particle bunches and their density matrix functions, describing semiclassically the particle-spin content of bunches. Our approach thus unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics. This unifying aspect of our approach relates the examples elegantly and uncovers relations between the various underlying dynamical systems in a transparent way. As in \cite{BEH}, the particle motion is integrable but we now allow for nonlinear particle motion on each torus. Since this work is inspired by notions from the theory of bundles, we also provide insight into the underlying bundle-theoretic aspects of the well-established concepts of invariant spin field, spin tune and invariant frame field. Since we neglect, as is usual, the Stern-Gerlach force, the underlying principal bundle is of product formso that we can present the theory in a fashion which does not use bundle theory. Nevertheless we occasionally mention the bundle-theoretic meaningof our concepts and we also mention the similarities with the geometrical approach to Yang-Mills Theory

Accurate and efficient spin integration for particle accelerators

Accurate spin tracking is a valuable tool for understanding spin dynamics in particle accelerators and can help improve the performance of an accelerator. In this paper, we present a detailed discussion of the integrators in the spin tracking code gpuSpinTrack. We have implemented orbital integrators based on drift-kick, bend-kick, and matrix-kick splits. On top of the orbital integrators, we have implemented various integrators for the spin motion. These integrators use quaternions and Romberg quadratures to accelerate both the computation and the convergence of spin rotations. We evaluate their performance and accuracy in quantitative detail for individual elements as well as for the entire RHIC lattice. We exploit the inherently data-parallel nature of spin tracking to accelerate our algorithms on graphics processing units.Comment: 43 pages, 17 figure

Probing for Binding Regions of the FtsZ Protein Surface through Site-Directed Insertions: Discovery of Fully Functional FtsZ-Fluorescent Proteins

FtsZ, a bacterial tubulin homologue, is a cytoskeletal protein that assembles into protofilaments that are one subunit thick. These protofilaments assemble further to form a “Z ring” at the center of prokaryotic cells. The Z ring generates a constriction force on the inner membrane and also serves as a scaffold to recruit cell wall remodeling proteins for complete cell division in vivo. One model of the Z ring proposes that protofilaments associate via lateral bonds to form ribbons; however, lateral bonds are still only hypothetical. To explore potential lateral bonding sites, we probed the surface of Escherichia coli FtsZ by inserting either small peptides or whole fluorescent proteins (FPs). Among the four lateral surfaces on FtsZ protofilaments, we obtained inserts on the front and back surfaces that were functional for cell division. We concluded that these faces are not sites of essential interactions. Inserts at two sites, G124 and R174, located on the left and right surfaces, completely blocked function, and these sites were identified as possible sites for essential lateral interactions. However, the insert at R174 did not interfere with association of protofilaments into sheets and bundles in vitro. Another goal was to find a location within FtsZ that supported insertion of FP reporter proteins while allowing the FtsZ-FPs to function as the sole source of FtsZ. We discovered one internal site, G55-Q56, where several different FPs could be inserted without impairing function. These FtsZ-FPs may provide advances for imaging Z-ring structure by superresolution techniques. IMPORTANCE One model for the Z-ring structure proposes that protofilaments are assembled into ribbons by lateral bonds between FtsZ subunits. Our study excluded the involvement of the front and back faces of the protofilament in essential interactions in vivo but pointed to two potential lateral bond sites, on the right and left sides. We also identified an FtsZ loop where various fluorescent proteins could be inserted without blocking function; these FtsZ-FPs functioned as the sole source of FtsZ. This advance provides improved tools for all fluorescence imaging of the Z ring and may be especially important for superresolution imaging