Jantzen coefficients and simplicity of generalized Verma modules

The main purpose of the paper is to establish new tools in the study of $\mathcal{O}^\mathfrak{p}$. We introduce the Jantzen coefficients of generalized Verma modules. It comes from the Jantzen's simplicity criteria for generalized Verma modules and has a deep relation with the structure of $\mathcal{O}^\mathfrak{p}$. We develop a reduction process to compute those coefficients. For which we need to consider generalized Verma modules induced from maximal parabolic subalgebras having maximal nontrivial singularity, so called basic generalized Verma modules. The classification of such modules is obtained in this paper. As the first application of our results, we give a refinement of Jantzen's simplicity criteria.Comment: 43 pages, 4 figure

On the non-thermal kappa-distributed electrons in planetary nebulae and HII regions: the kappa index and its correlations with other nebular properties

Recently, a suspicion arose that the free electrons in planetary nebulae (PNe) and HII regions might have non-thermal energy distributions. In this scenario, a kappa index is introduced to characterize the electron energy distributions, with smaller kappa values indicating larger deviations from Maxwell-Boltzmann distributions. Assuming that this is the case, we determine the kappa values for a sample of PNe and HII regions by comparing the intensities of [OIII] collisionally excited lines and the hydrogen Balmer jump. We find the average kappa indices of PNe and HII regions to be 27 and 32, respectively. Correlations between the resultant kappa values and various physical properties of the nebulae are examined to explore the potential origin of non-thermal electrons in photoionized gaseous nebulae. However, no positive result is obtained. Thus the current analysis does not lend to support to the idea that kappa-distributed electrons are present in PNe and HII regions.Comment: 23 pages, 6 figures, accepted for publication in Ap

H I Free-Bound Emission of Planetary Nebulae with Large Abundance Discrepancies: Two-Component Models versus Kappa-distributed electrons

The "abundance discrepancy" problem in the study of planetary nebulae (PNe), viz., the problem concerning systematically higher heavy-element abundances derived from optical recombination lines relative to those from collisionally excited lines, has been under discussion for decades, but no consensus on its solution has yet been reached. In this paper we investigate the hydrogen free-bound emission near the Balmer jump region of four PNe that are among those with the largest abundance discrepancies, aiming to examine two recently proposed solutions to this problem: two-component models and Kappa electron energy distributions. We find that the Balmer jump intensities and the spectrum slopes cannot be simultaneously matched by the theoretical calculations based upon single Maxwell-Boltzmann electron-energy distributions, whereas the fitting can be equally improved by introducing Kappa electron energy distributions or an additional Maxwell-Boltzmann component. We show that although H I free-bound emission alone cannot distinguish the two scenarios, it can provide important constraints on the electron energy distributions, especially for cold and low-Kappa plasmas.Comment: 23 pages, 10 figures, accepted for publication in Ap

Nuclear suppression of ϕ\phi meson yields with large pTp_T at the RHIC and the LHC

We calculate $\phi$ meson transverse momentum spectra in p+p collisions as well as their nuclear suppressions in central A+A collisions both at the RHIC and the LHC in LO and NLO with the QCD-improved parton model. We have included the parton energy loss effect in hot/dense QCD medium with the effectively medium-modified $\phi$ fragmentation functions in the higher-twist approach of jet quenching. The nuclear modification factors of $\phi$ meson in central Au+Au collisions at the RHIC and central Pb+Pb collisions at the LHC are provided, and a nice agreement of our numerical results at NLO with the ALICE measurement is observed. Predictions of yield ratios of neutral mesons such as $\phi/\pi^0$, $\phi/\eta$ and $\phi/\rho^0$ at large $p_T$ in relativistic heavy-ion collisions are also presented for the first time.Comment: 7 pages, 8 figure

η\eta meson production of high-energy nuclear collisions at NLO

The transverse momentum spectrum of $\eta$ meson in relativistic heavy-ion collisions is studied at the next-to-leading-order (NLO) within the perturbative QCD, where the jet quenching effect in the QGP is incorporated with the effectively medium-modified $\eta$ fragmentation functions using the higher-twist approach. We show that the theoretical simulations could give nice descriptions of PHENIX data on $\eta$ meson in both $\rm p+p$ and central $\rm Au+Au$ collisions at the RHIC, and also provide numerical predictions of $\eta$ spectra in central $\rm Pb+Pb$ collisions with $\sqrt{s_{NN}}=2.76$~TeV at the LHC. The ratios of $\eta/\pi^0$ in $\rm p+p$ and in central $\rm Au+Au$ collisions at $200$~GeV are found to overlap in a wide $p_T$ region, which matches well the measured ratio $\eta / \pi^0$ by PHENIX. We demonstrate that, at the asymptotic region when $p_{T} \rightarrow \infty$ the ratios of $\eta/\pi^{0}$ in both $\rm Au+Au$ and $\rm p+p$ are almost determined only by quark jets fragmentation and thus approach to the one in $e^{+} e^{-}$ scattering; in addition, the almost identical gluon (quark) contribution fractions to $\eta$ and to $\pi$ result in a rather moderate variation of $\eta/\pi^{0}$ distribution at intermediate and high $p_T$ region in $\rm A+A$ relative to that in $\rm p+p$; while a slightly higher $\eta/\pi^{0}$ at small $p_T$ in $\rm Au+Au$ can be observed due to larger suppression of gluon contribution fraction to $\pi^{0}$ as compared to the one to $\eta$. The theoretical prediction for $\eta / \pi^0$ at the LHC has also been presented.Comment: 7 pages, 8 figures, 2 typos corrected, revision for publicatio