Fomenko–Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra ${S(\mathfrak{g})}$ over a Lie algebra ${\mathfrak{g}}$ has the structure of a Poisson algebra. Assume ${\mathfrak{g}}$ is complex semisimple. Then results of Fomenko–Mischenko (translation of invariants) and Tarasov construct a polynomial subalgebra ${{\mathcal {H}} = {\mathbb C}[q_1,\ldots,q_b]}$ of ${S(\mathfrak{g})}$ which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of ${\mathfrak{g}}$ . Let G be the adjoint group of ${\mathfrak{g}}$ and let ? = rank ${\mathfrak{g}}$ . Using the Killing form, identify ${\mathfrak{g}}$ with its dual so that any G-orbit O in ${\mathfrak{g}}$ has the structure (KKS) of a symplectic manifold and ${S(\mathfrak{g})}$ can be identified with the affine algebra of ${\mathfrak{g}}$ . An element ${x\in \mathfrak{g}}$ will be called strongly regular if ${\{({\rm d}q_i)_x\},\,i=1,\ldots,b}$ , are linearly independent. Then the set ${\mathfrak{g}^{\rm{sreg}}}$ of all strongly regular elements is Zariski open and dense in ${\mathfrak{g}}$ and also ${\mathfrak{g}^{\rm{sreg}}\subset \mathfrak{g}^{\rm{ reg}}}$ where ${\mathfrak{g}^{\rm{reg}}}$ is the set of all regular elements in ${\mathfrak{g}}$ . A Hessenberg variety is the b-dimensional affine plane in ${\mathfrak{g}}$ , obtained by translating a Borel subalgebra by a suitable principal nilpotent element. Such a variety was introduced in Kostant (Am J Math 85:327–404, 1963). Defining Hess to be a particular Hessenberg variety, Tarasov has shown that ${{\rm{Hess}}\subset \mathfrak{g}^{\rm{sreg}}}$ . Let R be the set of all regular G-orbits in ${\mathfrak{g}}$ . Thus if ${O\in R}$ , then O is a symplectic manifold of dimension 2n where n = b ? ?. For any ${O\in R}$ let ${O^{\rm{sreg}} = \mathfrak{g}^{\rm{sreg}} \cap O}$ . One shows that O ^sreg is Zariski open and dense in O so that O ^sreg is again a symplectic manifold of dimension 2n. For any ${O\in R}$ let ${{\rm{Hess}}(O) = {\rm{Hess}}\cap O}$ . One proves that Hess(O) is a Lagrangian submanifold of O ^sreg and that $${\rm{Hess}} = \sqcup_{O\in R}{\rm{Hess}}(O).$$ The main result of this paper is to show that there exists simultaneously over all ${O\in R}$ , an explicit polarization (i.e., a “fibration” by Lagrangian submanifolds) of O ^sreg which makes O ^sreg simulate, in some sense, the cotangent bundle of Hess(O).     

Theoretical Investigation into Production of Double-Λ Hypernuclei from Stopped Ξ − on 6Li

We have investigated theoretically a feasible nuclear reaction to produce light double-Λ hypernuclei by choosing a suitable target. In the reaction from stopped Ξ ^? on ⁶Li target light doubly-strange nuclei, ${^5_{\Lambda\Lambda}{\rm H}}$ and ${^6_{\Lambda\Lambda}{\rm He}}$ , are produced: we have calculated the formation ratio of ${^5_{\Lambda\Lambda}{\rm H}}$ to ${^6_{\Lambda\Lambda}{\rm He}}$ for Ξ ^? absorptions from 2S, 2P and 3D orbitals of Ξ ^?–⁶Li atom by assuming a d?α cluster model for ⁶Li. From this cluster model the d?α relative wave functions has a node due to Pauli exclusion among nucleons belonging to d and α clusters. Two kinds of d?α wave functions, namely 1s relative wave function with a phenomenological one-range Gaussian (ORG) potential and that of an orthogonality-condition model (OCM) are used. It is found that the probability of ${^5_{\Lambda\Lambda}{\rm H}}$ formation is larger than that of ${^6_{\Lambda\Lambda}{\rm He}}$ for all absorption orbitals: in the case of the major 3D absorption their ratio is 1.08 for ORG and 1.96 for OCM. The dominant low momentum component of the d?α relative wave function favors the ${^5_{\Lambda\Lambda}{\rm H}}$ formation with a low Q value compared to the ${^6_{\Lambda\Lambda}{\rm He}}$ formation with a high Q value. We have also calculated momentum distributions of emitted particles, d and n, displaying continuum spectra for single-Λ hypernuclei, ${^4_{\Lambda}{\rm H}}$ and ${^5_{\Lambda}{\rm He}}$ , and line spectra for the ${^5_{\Lambda\Lambda}{\rm H}}$ and ${^6_{\Lambda\Lambda}{\rm He}}$ nuclei. Thus, our present theoretical analysis would be a significant contribution to experiments in the strangeness ?2 sector of hypernuclear physics.     

Identification of \mathrm{\ensuremath J^{\pi} = 19/2^+} and \mathrm{\ensuremath 23/2^+} isomeric states in 127Sb

The nucleus $\ensuremath {\rm ^{127}Sb}$ , which is on the neutron-rich periphery of the $\ensuremath \beta$ -stability region, has been populated in complex nuclear reactions involving deep-inelastic and fusion-fission processes with $\ensuremath {\rm {}^{136}Xe}$ beams incident on thick targets. The previously known isomer at 2325 keV in $\ensuremath {\rm {}^{127}Sb}$ has been assigned spin and parity $\ensuremath 23/2^+$ , based on the measured $\ensuremath \gamma$ - $\ensuremath \gamma$ angular correlations and total internal conversion coefficients. The half-life has been determined to be 234(12) ns, somewhat longer than the value reported previously. The 2194 keV state has been assigned $\ensuremath J^{\pi} = 19/2^+$ and identified as an isomer with $\ensuremath T_{1/2} = 14(1) {\rm ns}$ , decaying by two $\ensuremath E2$ branches. The observed level energies and transition strengths are compared with the predictions of a shell model calculation. Two $\ensuremath 15/2^+$ states have been identified close in energy, and their properties are discussed in terms of mixing between vibrational and three-quasiparticle configurations.     

Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .     

Structure of neutron-rich nuclei around the N = 126 closed shell; the yrast structure of 205Au126 up to spin-parity I^{\pi} = (19/2+)

Heavy neutron-rich nuclei have been populated through the relativistic fragmentation of a $\ensuremath ^{208}_{\ 82}{\rm Pb}$ beam at $\ensuremath E/A = 1$ GeV on a $\ensuremath 2.5 {\rm g/cm^2}$ thick Be target. The synthesised nuclei were selected and identified in-flight using the fragment separator at GSI. Approximately 300 ns after production, the selected nuclei were implanted in an $\ensuremath \sim 8$ mm thick perspex stopper, positioned at the centre of the RISING $\ensuremath \gamma$ -ray detector spectrometer array. A previously unreported isomer with a half-life $\ensuremath T_{1/2} = 163(5)$ ns has been observed in the N = 126 closed-shell nucleus $\ensuremath ^{205}_{\ 79}{\rm Au}$ . Through $ \gamma$ -ray singles and $ \gamma$ - $ \gamma$ coincidence analysis a level scheme was established. The comparison with a shell model calculation tentatively identifies the spin-parity of the excited states, including the isomer itself, which is found to be $\ensuremath I^{\pi} = (19/2^+)$ .     

Finite W-Superalgebras and Truncated Super Yangians

Let ${Y_{m|n}^{\ell}}$ be the super Yangian of general linear Lie superalgebra for ${\mathfrak{gl}_{m|n}}$ . Let ${e \in \mathfrak{gl}_{m\ell|n\ell}}$ be a “rectangular” nilpotent element and ${\mathcal{W}_e}$ be the finite W-superalgebra associated to e. We show that ${Y_{m|n}^{\ell}}$ is isomorphic to ${\mathcal{W}_e}$ .     

Antihydrogen (hydrogen) atom formation

In this paper we show how the ATHENA data samples on the antihydrogen ( ${\bar{\rm H}}$ ) formation in very different conditions provide useful information on the two different possible mechanisms: the 3-body reaction ( $\bar{p}+{e^+}+{e^+}\rightarrow {\bar{\rm H}}+ e^+$ ) and the 2-body reaction ( $\bar{p}+{\rm e^+}\rightarrow {\bar{\rm H}}+{h\nu}$ ).     

Few-Body Aspects of Hypernuclear Physics

Recent development in the study of the structure of light Λ and double Λ hypernuclei is reviewed from the view point of few-body problems and interactions between the constituent particles. In the study the present author and collaborators employed Gaussian expansion method for few-body calculations; the method has been applied to many kinds of few-body systems in the fields of nuclear physics and exotic atomic/molecular physics. We reviewed the following subjects studied using the method: (1) Precise three- and four-body calculations of ${^7_{\Lambda}{\rm He}}$ , ${^7_{\Lambda}{\rm Li}}$ , ${^7_{\Lambda}{\rm Be}}$ , ${^8_{\Lambda}{\rm Li}}$ , ${^8_{\Lambda}{\rm Be}}$ , ${^9_{\Lambda}{\rm Be}}$ , ${^{10}_{\Lambda}{\rm Be}}$ , ${^{10}_{\Lambda}{\rm B}}$ and ${^{13}_{\Lambda}{\rm C}}$ provide important information on the spin structure of the underlying Λ N interaction by comparing the calculated results with the recent experimental data by γ-ray hypernuclear spectroscopy. (2) The Λ-Σ coupling effect was investigated in ${^4_{\Lambda}{\rm H}}$ and ${^4_{\Lambda}{\rm He}}$ on the basis of the N?+?N?+?N?+?Λ (Σ) four-body model. (3) A systematic study of double-Λ hypernuclei and the Λ Λ interaction, based on the NAGARA event data ( ${^6_{\Lambda\Lambda}{\rm He}}$ ), was performed within the α +?x?+?Λ +?Λ cluster model (x = n, p, d, t,³He and α) and α +?α +?n?+?Λ +?Λ cluster model, (4) The Demachi-Yanagi event was interpreted as observation of the 2⁺ state of ${^{10}_{\Lambda \Lambda}{\rm Be}}$ , (5) The Hida event was interpreted as observation of the ground state of ${^{11}_{\Lambda \Lambda}{\rm Be}}$ .     

Physics beyond the Standard Model through b → s μ ?+? μ ? transition

A comprehensive study of the impact of new-physics operators with different Lorentz structures on decays involving the b ?? s ?? ^?+? ?? ^? transition is performed. The effects of new vector?Caxial vector (VA), scalar?Cpseudoscalar (SP) and tensor (T) interactions on the differential branching ratios, forward?Cbackward asymmetries (A _FB??s), and direct CP asymmetries of ${\bar B}_{\rm s}^0 \to \mu^+ \mu^-$ , ${\bar B}_{\rm d}^0 \to$ $ X_{\rm s} \mu^+ \mu^-$ , ${\bar B}_{\rm s}^0 \to \mu^+ \mu^- \gamma$ , ${\bar B}_{\rm d}^0 \to {\bar K} \mu^+ \mu^-$ , and ${\bar B}_{\rm d}^0\to {\bar{K}^*} \mu^+ \mu^-$ are examined. In ${\bar B}_{\rm d}^0\to {\bar{K}^*} \mu^+ \mu^-$ , we also explore the longitudinal polarization fraction f _L and the angular asymmetries $A_{\rm T}^{(2)}$ and A _LT, the direct CP asymmetries in them, as well as the triple-product CP asymmetries $A_{\rm T}^{\rm (im)}$ and $A^{\rm (im)}_{\rm LT}$ . While the new VA operators can significantly enhance most of the observables beyond the Standard Model predictions, the SP and T operators can do this only for A _FB in ${\bar B}_{\rm d}^0 \to {\bar K} \mu^+ \mu^-$ .     

Coupling constant in dispersive model

The average of the moments for event shapes in e ^?+? e??→hadrons within the context of next-to-leading order (NLO) perturbative QCD prediction in dispersive model is studied. Moments used in this article are $\langle {1-T}\rangle$ , $\langle \rho\rangle$ , $\langle {B_{\rm T}}\rangle$ and $\langle {B_{\rm W} }\rangle$ . We extract α _s, the coupling constant in perturbative theory and α ₀ in the non-perturbative theory using the dispersive model. By fitting the experimental data, the values of $\alpha_{\rm s} ({M_{\rm Z^0} })=0.1171\pm 0.00229$ and $\alpha_0 \left( {\mu_{\rm I} =2\,{\rm GeV}} \right)=0.5068\pm 0.0440$ are found. Our results are consistent with the above model. Our results are also consistent with those obtained from other experiments at different energies. All these features are explained in this paper.     

Study of Light Hypernuclei by the (e,e′K +) Reaction

We have been performing Λ hypernuclear spectroscopic experiments by the (e,e′K ⁺) reaction since 2000 at Thomas Jefferson National Accelerator Facility (JLab). The (e,e′K ⁺) experiment can achieve a few 100 keV (FWHM) energy resolution compared to a few MeV (FWHM) by the (K ^?, π ^?) and (π ⁺, K ⁺) experiments. Therefore, more precise Λ hypernuclear structures can be investigated by the (e,e′K ⁺) experiment. ${^{7}_{\Lambda}{\rm He}}$ , ${^{9}_{\Lambda}{\rm Li}}$ , ${^{10}_{\Lambda}{\rm Be}}$ , ${^{12}_{\Lambda}{\rm B}}$ , ${^{28}_{\Lambda}{\rm Al}}$ , and ${^{52}_{\Lambda}{\rm V}}$ were measured in the experiment at JLab Hall-C. In addition, ${^{9}_{\Lambda}{\rm Li}}$ , ${^{12}_{\Lambda}{\rm B}}$ , and ${^{16}_{\Lambda}{\rm N}}$ were measured in the experiment at JLab Hall-A.     

Analysis of the \frac{1} {2}^ - and \frac{3} {2}^ - heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the $\frac{1} {2}^ -$ and $\frac{3} {2}^ -$ heavy and doubly heavy baryon states $\Sigma _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi '_Q \left( {\frac{1} {2}^ - } \right)$ , $\Omega _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Omega _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Sigma _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Omega _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ and $\Omega _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ by subtracting the contributions from the corresponding $\frac{1} {2}^ +$ and $\frac{3} {2}^ +$ heavy and doubly heavy baryon states with the QCD sum rules in a systematic way, and make reasonable predictions for their masses.     

A DFG-based cavity ring-down spectrometer for trace gas sensing in the mid-infrared

Continuing studies into an all-diode laser-based 3.3 μm difference frequency generation cavity ring-down spectroscopy system are presented. Light from a 1,560 nm diode laser, amplified by an erbium-doped fibre amplifier, was mixed with 1,064 nm diode laser radiation in a bulk periodically poled lithium niobate crystal to generate 16 μW of mid-IR light at 3,346 nm with a conversion efficiency of $0.05\,\%\,{\text{W}}^{-1}\,{\text{cm}}^{-1}$ . This radiation was coupled into a 77 cm long linear cavity with average mirror reflectivities of 0.9996, and a measured baseline ring-down time of $6.07\pm 0.03\,\upmu{\rm s}$ . The potential of such a spectrometer was illustrated by investigating the $P(3)$ transition in the fundamental $\nu_{3}(F_{2})$ band of ${\text{CH}}_4$ both in a 7.5 ppmv calibrated mixture of ${\text{CH}}_4$ in air and in breath samples from methane and non-methane producers under conditions where the minimum detectable absorption coefficient ( $\alpha_{\rm min}$ ) was $2.8 \times 10^{-8}\,{\rm cm}^{-1}$ over 6 s using a ring-down time acquisition rate of 20 Hz. Allan variance measurements indicated an optimum $\alpha_{\rm min}$ of $2.9\times 10^{-9}\,{\rm cm}^{-1}$ over 44 s.     

Muonic Anti-hydrogen {(\bar{\rm H}_{\mu}}) Formation in Low-energy Three-body Reactions

A few-body type computation is performed for a three-charge-particle collision with participation of a slow antiproton ${\bar{\rm{p}}}$ and a muonic muonium atom (true muonium), i.e. a bound state of two muons ${(\mu^{+}\mu^{-})}$ in its ground state. The total cross section of the following reaction ${\bar{\rm p}+(\mu^{+}\mu^{-}) \rightarrow \bar{\rm{H}}_{\mu} + \mu^{-}}$ , where muonic anti-hydrogen ${\bar{\rm{H}}_{\mu}=(\bar{\rm p}\mu^{+})}$ is a bound state of an antiproton and positive muon, is computed in the framework of a set of coupled two-component Faddeev-Hahn-type equation. A better known negative muon transfer low energy three-body reaction: ${{\rm t}^{+} + ({\rm d}^{+}\mu^{-})\rightarrow ({\rm t}^{+}\mu^{-}) + {\rm d}^{+}}$ is also computed as a test system. Here, t⁺ is triton and d⁺ is deuterium.     

Joint System Quantum Descriptions Arising from Local Quantumness

We study the entropy flux in the stationary state of a finite one-dimensional sample ${\mathcal{S}}$ connected at its left and right ends to two infinitely extended reservoirs ${\mathcal{R}_{l/r}}$ at distinct (inverse) temperatures ${\beta_{l/r}}$ and chemical potentials ${\mu_{l/r}}$ . The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator ${h_{\mathcal{S}, L}= - \Delta + v}$ . The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system ${\mathcal{R}_l + \mathcal{S} + \mathcal{R}_r}$ in terms of scattering data. We study the behaviour of this steady state entropy flux in the limit ${L \to \infty}$ and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator ${h_\mathcal{S}}$ .     

Optimal Paths for Symmetric Actions in the Unitary Group

Given a positive and unitarily invariant Lagrangian ${\mathcal{L}}$ defined in the algebra of matrices, and a fixed time interval ${[0,t_0]\subset\mathbb R}$ , we study the action defined in the Lie group of ${n\times n}$ unitary matrices ${\mathcal{U}(n)}$ by $$\mathcal{S}(\alpha)=\int_0^{t_0} \mathcal{L}(\dot\alpha(t))\,dt, $$ where ${\alpha:[0,t_0]\to\mathcal{U}(n)}$ is a rectifiable curve. We prove that the one-parameter subgroups of ${\mathcal{U}(n)}$ are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if ${\mathcal{L}}$ is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in ${\mathcal{U}(n)}$ as well as angular metrics in the Grassmann manifold.     

Haag Duality and the Distal Split Property for Cones in the Toric Code

We prove that Haag duality holds for cones in the toric code model. That is, for a cone ??, the algebra ${\mathcal{R}_{\Lambda}}$ of observables localized in ?? and the algebra ${\mathcal{R}_{\Lambda^c}}$ of observables localized in the complement ?? ^c generate each other??s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if ${\Lambda_1 \subset \Lambda_2}$ are two cones whose boundaries are well separated, there is a Type I factor ${\mathcal{N}}$ such that ${\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}$ . We demonstrate this by explicitly constructing ${\mathcal{N}}$ .     

The Charmonium-Molecule Hybrid Structure of the X(3872)

In order to understand the structure of the X(3872) the effects of the ${{\rm c\overline{c}}}$ charmonium core state coupling to the ${D^0\overline{D}^{*0}}$ and D ⁺ D ^*? molecule states are studied. The obtained structure of the X(3872) is about 9 % of ${{\rm c}\overline{{\rm c}}}$ charmonium, 75 % of the isoscalar ${D\overline{D}}$ molecule and 16 % of the isovector ${D\overline{D}}$ molecule which explains observed properties of the X(3872) well.