Experimental review of the ϒ(1S, 2S, 3S) physics at e+e− colliders and the LHC

The three lowest-lying \mathrm{\Upsilon } states, i.e., \mathrm{\Upsilon }(1S), \mathrm{\Upsilon }(2S), and \mathrm{\Upsilon }(3S), composed of bb¯ pairs and below the BB ¯ threshold, provide a good platform for the researches of hadronic physics and physics beyond the Standard Model. They can be produced directly in e⁺e⁻ colliding experiments, such as CLEO, Babar, and Belle, with low continuum backgrounds. In these experiments, many measurements of the exclusive \mathrm{\Upsilon }(1S) and \mathrm{\Upsilon }(2S) decays into light hadrons, which shed light on the “80% rule” for the Okubo–Zweig–Iizuka suppressed decays in the bottomonium sector, were carried out. Meanwhile, many studies of the charmonium and bottomonium productions in \mathrm{\Upsilon }(1S, 2S, 3S) decays were performed, to distinguish different Quantum Chromodynamics (QCD) models. Besides, exotic states and new physics were also extensively explored in \mathrm{\Upsilon }(1S, 2S, 3S) decays at CLEO, BaBar, and Belle. The \mathrm{\Upsilon }(1S, 2S, 3S) states can also be produced in pp collisions and in collisions involving heavy ions. The precision measurements of their cross sections and polarizations at the large hadron collider (LHC), especially in the CMS, ATLAS, and LHCb experiments, help to understandΥproduction mechanisms in pp collisions. The observation of the sequentialΥsuppression in heavy ion collisions at CMS, LHCb, and ALICE is of great importance for verifying the quark–gluon plasma predicted by QCD. In this article, we review the experimental results on \mathrm{\Upsilon }(1S, 2S, 3S) at e⁺e⁻ colliders and the LHC, and summarize their prospects at Belle II and the LHC.     

Functionalization of multiwalled carbon nanotubes for reinforcing of poly(l-lactide-co--caprolactone) biodegradable copolymers

Up to now, synthetic polymers and biomacromolecules have been grafted or assembled onto the convex surface of carbon nanotubes (CNTs) via covalent bonds or chemisorptions. In this research, poly(l-lactide-co--caprolactone)-functionalized multiwalled carbon nanotubes (MWCNT-OH-g-PCLA)s are synthesized by in situ ring-opening copolymerization of l-lactide (LA) and -caprolactone (CL) using stannous octanoate and hydroxylated MWCNTs (MWCNT-OHs) as the initiating system. The pristine MWCNTs are modified to possess carboxyl groups and then hydroxyl groups. MWCNT-OHs are used as coinitiators to polymerize LA and CL by the surface-initiated ring-opening polymerization. The FT-IR spectra, SEM and TEM micrographs revealed that the PCLA grafted form the sidewall of MWCNTs strongly. The TGA analysis indicates that about 75 wt% of functionalized MWCNTs with PCLA belongs to grafted PCLA and the remaining 25 wt% to the initial MWCNT-OH.     

The H2+ molecular ion: Low-lying states

Matching for a wavefunction the WKB expansion at large distances and Taylor expansion at small distances leads to a compact, few-parametric uniform approximation found in Turbiner and Olivares-Pilon (2011). The ten low-lying eigenstates of H₂⁺ of the quantum numbers (n,m,Λ,±)$(n,m,\Lambda ,\pm )$  with n=m=0$n=m=0$ at Λ=0,1,2$\Lambda =0,1,2$, with n=1$n=1$, m=0$m=0$ and n=0$n=0$, m=1$m=1$ at Λ=0$\Lambda =0$ of both parities are explored for all interproton distances R$R$. For all these states this approximation provides the relative accuracy ?10⁻⁵$?10^{-5}$ (not less than 5 s.d.) locally, for any real coordinate x$x$ in eigenfunctions, when for total energy E(R)$E\left(R\right)$ it gives 10-11 s.d. for R∈[0,50]$R\in [0,50]$  a.u. Corrections to the approximation are evaluated in the specially-designed, convergent perturbation theory. Separation constants are found with not less than 8 s.d. The oscillator strength for the electric dipole transitions E1$E1$ is calculated with not less than 6 s.d. A dramatic dip in the E1$E1$ oscillator strength f_{1sσg−3pσu}$f_{1s{\sigma }_g-3p{\sigma }_u}$ at R∼R_eq$R\sim R_{eq}$ is observed. The magnetic dipole and electric quadrupole transitions are calculated for the first time with not less than 6 s.d. in oscillator strength. For two lowest states (0,0,0,±)$(0,0,0,\pm )$ (or, equivalently, 1sσ_g$1s{\sigma }_g$ and 2pσ_u$2p{\sigma }_u$ states) the potential curves are checked and confirmed in the Lagrange mesh method within 12 s.d. Based on them the Energy Gap between 1sσ_g$1s{\sigma }_g$ and 2pσ_u$2p{\sigma }_u$ potential curves is approximated with modified Pade Re^−R[Pade(8/7)](R)$R{e}^{-R}\left[Pade(8/7)\right]\left(R\right)$ with not less than 4-5 figures at R∈[0,40]$R\in [0,40]$ a.u. Sum of potential curves E_1sσg+E_2pσu$E_{1s{\sigma }_g}+E_{2p{\sigma }_u}$ is approximated by Pade 1/R[Pade(5/8)](R)$1/R\left[Pade(5/8)\right]\left(R\right)$ in R∈[0,40]$R\in [0,40]$ a.u. with not less than 3-4 figures.     

Equivalence of star products on a symplectic manifold; an introduction to Deligne''s ech cohomology classes

These notes grew out of the Quantisation Seminar 1997–1998 on Deligne's paper [P. Deligne, Déformations de l'algèbre des fonctions d'une variété symplectique: Comparison entre Fedosov et De Wilde, Lecomte, Selecta Math. (New Series) 1 (1995) 667–697] and the lecture of the first author in the Workshop on Quantisation and Momentum Maps at the University of Warwich in December 1997.We recall the definitions of the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manfold and show the properties of the relations between these classes by elementary methods based on ech cohomology.