Bases des algèbres des groupes quantiques en termes des mots de Lyndon

Compared to the original report submitted to the university, this version includes the following revisions: (1) a trivial paragraph intended for internal audience was deleted from the introduction; (2) MSC and PACS classifications were added; (3) keywords were added; (4) affiliation information was revised; and (5) small grammatical edits were made. There were no revisions affecting the scientific content of this work.We have reviewed some results on quantized shuffling, and in particular, the grading and structure of this algebra. In parallel, we have summarized certain details about classical shuffle algebras, including Lyndon words (primes) and the construction of bases of classical shuffle algebras in terms of Lyndon words. We have explained how to adapt this theory to the construction of bases of quantum group algebras in terms of Lyndon words. This method has a limited application to the specific case of the quantum group parameter being a root of unity, with the requirement that specialization to the root of unity is non-restricted. As an additional, applied part of this work, we have implemented a Wolfram Mathematica package with functions for quantum shuffle multiplication and constructions of bases in terms of Lyndon words.Nous avons examiné certains résultats sur le mélange quantifié, et en particulier la graduation et la structure de ce type d'algèbre. En parallèle, nous avons récapitulé certaines informations sur l'algèbre de mélange classique, y compris les mots de Lyndon (facteurs premiers) et la construction des bases des algèbres de mélange classiques en termes de mots de Lyndon. Nous avons expliqué comment adapter cette théorie à la construction des bases des algèbres des groupes quantiques en termes de mots de Lyndon. Cette méthode a une application limitée dans le cas spécifique où le paramètre du groupe quantique est une racine d'unité, avec la condition que la spécialisation à la racine de l'unité soit sans restriction. Comme partie supplémentaire et appliquée de ce travail, nous avons mis en œuvre un package Wolfram Mathematica avec des fonctions pour la multiplication de mélange quantique et des constructions de bases en termes de mots de Lyndon

Field Modeling, Symplectic Tracking, and Spin Decoherence for EDM and Muon g-2g\textrm{-}2 Lattices

While the first particle accelerators were electrostatic machines, and several electrostatic storage rings were subsequently commissioned and operated, electrostatic storage rings pose a number of challenges. Unlike motion in the magnetic field, where particle energy remains constant, particle energy generally changes in electrostatic elements. Conservation of energy in an electrostatic element is, in practice, only approximate, and it requires careful and accurate design, manufacturing, installation, and operational use. Electrostatic deflectors require relatively high electrostatic fields, tend to introduce nonlinear aberrations of all orders, and are more challenging to manufacture than homogeneous magnetic dipoles. Accordingly, magnetic storage rings are overwhelmingly prevalent.The search for electric dipole moments (EDMs) of fundamental particles is of key importance in the study of C and CP violations and their sources. C and CP violations are part of the Sakharov conditions that explain the matter–antimatter asymmetry in the universe. Determining the source of CP violations would provide valuable empirical insight for beyond-Standard-Model physics. EDMs of fundamental particles have not to this date been experimentally observed. The search for fundamental particle EDMs has narrowed the target search region; however, an EDM signal is yet to be discovered.In 2008, Brookhaven National Laboratory (BNL) had proposed the frozen spin (FS) concept for the search of a deuteron EDM. The FS concept envisions launching deuterons through a storage ring with combined electrostatic and magnetic fields. The electrostatic and magnetic fields are in a proportion that would, without an EDM, freeze the deuteron's spin along its momentum as the deuteron moves around the lattice. The radial electrostatic field would result in a torque on the spin vector, proportional to a deuteron EDM, rotating the spin vector out of the midplane.The principle of an anomalous magnetic dipole moment (MDM) measurement using a storage ring, shared by BNL's completed E821 Experiment and the ongoing E989 Experiment operated by Fermi National Accelerator Laboratory (FNAL), requires injecting muons into a magnetic ring at the so-called magic momentum. The magic momentum, as defined in this context, would freeze the muon's spin vector along its momentum if the anomalous MDM was zero. The spin precession in the horizontal plane relative to the momentum is proportional to the anomalous MDM.Storage rings for measurement of EDM and anomalous MDM present a new frontier in tracking code accuracy requirements. For accurate tracking of storage rings with electrostatic particle optical elements, it is necessary to model the fringe fields of such elements accurately, in particular, because not doing so provides a mechanism for energy conservation violation. However, the previous research on fringe fields tended to focus on magnetic rather than electrostatic particle optical elements. We will study and model the fringe fields of several electrostatic deflectors. Field falloffs of electrostatic deflectors are slower than exponential, and Enge functions are not suitable for accurate modeling of these falloffs. We will propose an alternative function to model field falloffs of electrostatic deflectors. We will use conformal mapping methods to obtain the main field of the Muon g-2 storage ring high voltage quadrupole, and we will calculate its fringe field and effective field boundary (EFB) using Fourier analysis.Furthermore, we will study tracking of storage rings with electrostatic elements using map methods. We will find that, for simultaneous symplecticity and energy conservation, it is only necessary to enforce symplecticity in COSY INFINITY. We will model and track several benchmark lattices – an electrostatic spherical deflector, a homogeneous magnetic dipole, and a proton EDM lattice – in COSY INFINITY and MSURK89, our in-house eighth order Runge–Kutta–Verner tracking code. Finally, we will investigate spin decoherence and systematic errors in FS and quasi-frozen spin (QFS) lattices. Spin decoherence effects are similar in FS and QFS lattices, and spin decoherence in said lattices often remains in the same range over time, indicating the feasibility of EDM measurement using FS and QFS lattices

Search for the Optimal Spin Decoherence Effect in a QFS Lattice

Measurement of electric dipole moment (EDM) in a storage ring requires the spin decoherence in a particle bunch to be less than 1 rad in 1000 s, which corresponds to about 1 billion turns. The quasi-frozen spin (QFS) method* has been proposed for deuteron EDM search. In a QFS lattice, spin direction turn in magnetic bend sections is compensated by spin direction turn in electrostatic bend sections, and thus the spin direction at a point in the lattice is approximately constant. We consider a QFS lattice with an RF cavity and seven families of sextupoles. In COSY Infinity, calculations were done using transfer maps of the 7th order, with symplectic tracking using the Extended Poincaré (EXPO) generating function and the most accurate COSY Infinity fringe field mode. We have optimized the sextupole strengths to minimize the spin decoherence. Using these sextupole strengths, we have done spin tracking of the lattice and analyzed the growth of spin decoherence as a function of the number of turns. Within their scope, our results indicate the feasibility of the QFS method

Investigation of Lattice for Deuteron EDM Ring

The quasi-frozen spin (QFS) concept of a storage ring for deuteron EDM measurement is based on the fact that the anomalous magnetic moment has a small negative value. Due to this fact, the rotation of spin in two parts of ring with the magnetic and electric fields relative to the momentum can compensate each other. In contrast to the concept of frozen spin we have the freedom to choose the ring parameters and also greatly simplified lattice. We consider two possible options for the lattice based on QFS concept and compare them with the frozen spin lattice proposed by BNL. In the first QFS option, we use completely separate electric and magnetic parts that form a structure. In the second option, we suggest using only two magnetic arcs with two straight sections having the straight elements with magnetic and electric fields. The straight elements have a horizontal electric field of 120 kV/cm and a vertical magnetic field of 80 mT. They provide the compensation for the spin rotation in the arc and at the same time allow having straight electric plates without the higher orders field. This scheme could be tested in the COSY ring at FZ Jülich to prove the quasi-frozen spin concept

Quasi-Frozen Spin Concept of Deuteron Storage Ring as an Instrument to Search for the Electric Dipole Moment

One of the possible arguments for the breaking of CP invariance is the existence of non-vanishing electric dipole moments (EDM) of elementary particles. Currently, the Jülich Electric Dipole Moment Investigation (JEDI) collaboration works under the conceptual design of the ring specifically for search of the deuteron electrical dipole moment (dEDM). The proposed Quasi-Frozen Spin concept differs from the Frozen Spin concept in that the spin of the reference particle is alternately deflected by a few degrees in different directions relative to momentum in the electric and magnetic parts of the ring. The QFS concept will allow using the existing COSY ring as pilot facility. The paper presents conceptual approach to ring design based on results of a study of spin decoherence and systematic errors, as well as the sensitivity estimation of the method to the determination of EDM

Systematic Errors Investigation in Frozen and Quasi-Frozen Spin Lattices of Deuteron EDM Ring

The search for the electric dipole moment (EDM) in the storage ring raises two questions: how to create conditions for maximum growth of the total EDM signal of all particles in bunch, and how to differentiate the EDM signal from the induced magnetic dipole moment (MDM) signal. The T-BMT equation distinctly addresses each issue. Because the EDM signal is proportional to the projection of the spin on the direction of the momentum, it is desirable to freeze the spin direction of all particles in a bunch along momentum. It can be successfully implemented in the Quasi Frozen (QFS) and Frozen (FS) Spin structures. However, in case of magnet misalignments, the induced MDM signal may arise in the same plane as the EDM signal and thereby prevent its registration. In this paper, we analyze the effect of errors together with the spin-tune decoherence of all particles in the bunch for FS and QFS options