Borel subalgebras of the Witt algebra W1W_1

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p>3$, and $\ggg$ the $p$-dimensional Witt algebra over $\mathbb{F}$. Let $\N$ be the nilpotent cone of $\ggg$. Explicit description of $\N$ is given, so that the conjugacy classes of Borel subalgebras of $\ggg$ under the automorphism group are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in $\ggg$. The representatives of conjugacy classes of Borel subalgebras, i.e., the so-called standard Borel subalgebras, are precisely given.Comment: 13 pages; All comments are welcome

Nilpotent commuting varieties of the Witt algebra

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimensional. Irreducible components of $\mathscr{C}(\mathscr{N})$ and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.Comment: 10 pages. Comments are welcom

Order Preservation for Path-Distribution Dependent SDEs

Sufficient and necessary conditions are presented for the order preservation of path-distribution dependent SDEs. Differently from the corresponding study of distribution independent SDEs, to investigate the necessity of order preservation for the present model we need to construct a family of probability spaces in terms of the ordered pair of initial distributions.Comment: 12 page

A global SU(3)/U(3)SU(3)/U(3) flavor symmetry analysis for B→PPB\to PP with η−η′\eta-\eta' Mixing

A large number of new experimental data on $B$ decay into two light pesudoscalar ($P$) mesons have been collected by the LHCb collaboration. Besides confirming information on $B_{u,d} \to PP$ decays obtained earlier by B-factories at KEK and SLAC, new information on $B_s\to PP$ and also more decay modes with $P$ being $\eta$ or $\eta'$ have been obtained. Using these new data, we perform a global fit for $B \to PP$ to determine decay amplitudes in the framework of $SU(3)/U(3)$ flavor symmetry. We find that $SU(3)$ flavor symmetry can explain data well. The annihilation amplitudes are found to be small as expected. Several CP violating relations predicted by $SU(3)$ flavor symmetry are in good agreement with data. Current available data can give constraints on the amplitudes which induce $P = \eta,\;\eta'$ decays in the framework of $U(3)$ flavor symmetry, and can also determine the $\eta-\eta'$ mixing angle $\theta$ with $\theta = (-18.4\pm1.2)^\circ$. Several $B \to PP$ decay modes which have not been measured are predicted with branching ratios accessible at the LHCb. These decays can provide further tests for the framework of $SU(3)/U(3)$ flavor symmetry for $B$ decays.Comment: RevTex, 19 pages. Added a "Note added" and a referenc

The remaining parts for the long-standing J/psi polarization puzzle

Based on the non-relativistic quantum chromodynamics factorization formalism, the polarization parameters $\lambda_{\theta\phi}$ and $\lambda_{\phi}$ of $J/\psi$ hadroproduction are analyzed in helicity frame and calculated at QCD next-to-leading order for the first time. For prompt $J/\psi$ production, we take into account the feeddown contributions from $\chi_{cJ}$ and $\psi(2S)$ decays. The theoretical predictions for the polarization parameters $\lambda_{\theta\phi}$ and $\lambda_{\phi}$ of $J/\psi$ are presented. With the theoretical results we have done the fit to the experimental measurements on yield and polarization for $J/\psi$ hadroproduction simultaneously, and found that the results are coincide with the experimental measurements at the LHC quite well.Comment: 5 pages, 4 figure

Finite-volume formalism in the 2→HI+HI22 \xrightarrow[]{H_I+H_I} 2 transition: an application to the lattice QCD calculation of double beta decays

We present the formalism for connecting a second-order electroweak $2\xrightarrow[]{H_I+H_I}2$ transition amplitudes in the finite volume (with two hadrons in the initial and final states) to the physical amplitudes in the infinite volume. Our study mainly focus on the case where the low-lying intermediate state consists of two scattering hadrons. As a side product we also reproduce the finite-volume formula for $2\xrightarrow[]{H_I}2$ transition, originally obtained by Brice\~no and Hansen. With the available finite-volume formalism, we further discuss how to treat with the finite-volume problem in the double beta decays $nn\to pp ee\bar{\nu}\bar{\nu}$ and $nn\to pp ee$.Comment: 18 page

Symmetry of Generalized Randall-Sundrum Model and Distribution of 3-Branes in Six-Dimensional Spacetime

A generalization from the usual $5$-dimensional two-brane Randall-Sundrum (RS) model to a $6$-dimensional multi-brane RS model is presented. The extra dimensions are extended from one to two; correspondingly the single-variable warp function is generalized to be a double-variable function, to represent the two extra dimensions. In the analysis of the Einstein equation we have two remarkable discoveries. One is that, when branes are absent, the cosmological parameter distributed in the two extra dimensions acts as a function describing a family of circles. These circles are not artificially added ones but stem from the equations of motion, while their radii are inversely proportional to the square root of the cosmological parameter. The other discovery is that, on any circle, there symmetrically distribute four branes. Their tensions, $V_1 \sim V_4$, satisfy a particular relationship $V_1=V_3=-V_2=-V_4=3M^4$, where $M$ is the $6$-dimensional fundamental scale of the RS model.Comment: 12 pages. 1 figur

Light-Neutrino Exchange and Long-Distance Contributions to 0ν2β0\nu2\beta Decays: An Exploratory Study on ππ→ee\pi\pi\to ee

We present an exploratory lattice QCD calculation of the neutrinoless double beta decay $\pi\pi\to ee$. Under the mechanism of light-neutrino exchange, the decay amplitude involves significant long-distance contributions. The calculation reported here, with pion masses $m_\pi=420$ and 140 MeV, demonstrates that the decay amplitude can be computed from first principles using lattice methods. At unphysical and physical pion masses, we obtain that amplitudes are $24\%$ and $9\%$ smaller than the predication from leading order chiral perturbation theory. Our findings provide the lattice QCD inputs and constraints for effective field theory. A follow-on calculation with fully controlled systematic errors will be possible with adequate computational resources.Comment: 6 pages, 3 figures. V2: version accepted by PRL; minor changes compared to v

Cavity QED implementation of the multi-qubit refined Deutsch-Jozsa algorithm

We theoretically study the realization of a multi-qubit refined Deutsch-Jozsa (DJ) algorithm using resonant interaction of many Rydberg atoms with a single-mode microwave cavity, in which the multi-qubit controlled phase gates could be accomplished efficiently. We show how to achieve a multi-qubit refined DJ algorithm in high fidelity, even in the case of a weak cavity decay and some imperfection. We argue that the required operations in our scheme are almost within the present experimental possibilities

A New Interpretation of Three-Dimensional Particle Geometry: M-A-V-L

This study provides a new interpretation of 3D particle geometry that unravels the 'interrelation' of the four geometry parameters, i.e., morphology M, surface area A, volume V, and size L, for which a new formula, M = A/V$\times$L/6, is introduced to translate the 3D particle morphology as a function of surface area, volume, and size. The A/V$\times$L of a sphere is invariably 6, which is placed in the denominator of the formula, and therefore M indicates a relative morphological irregularity compared to the sphere. The minimum possible value of M is clearly one, and M may range approximately to three for coarse-grained mineral particles. Furthermore, the proposed formula, M = A/V$\times$L/6, enables to graphically preserve the four parameters' relations when plotting the geometry parameter distributions. This study demonstrates the approach with two plot spaces that represent (i) L vs. M and (ii) A/V vs. V, where A/V works as the messenger between these two spaces as A/V = M/L$\times$6. Therefore, this approach helps comprehensively address the four-dimensional aspects of the 3D particle geometry and better understand the parameters' combined influence on the mechanical behavior of granular materials. Keywords: 3D particle geometry; Morphology; Surface area; Volume; Size;Comment: 36 pages, 13 figures; Transportation Geotechnics (2020) https://www.sciencedirect.com/science/article/abs/pii/S221439121930298