Symmetry of Solutions for a Fractional System

We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for continuous $f, g$ and $\alpha, \beta \in (0,2)$. Under some natural assumptions on $f$ and $g$, by applying the \emph{method of moving planes} directly to the system, we obtain symmetry on non-negative solutions without any decay assumption on the solutions at infinity

Co-evolution of Extreme Star Formation and Quasar: hints from {\it Herschel} and the Sloan Digital Sky Survey

Using the public data from the Herschel wide field surveys, we study the far-infrared properties of optical-selected quasars from the Sloan Digital Sky Survey. Within the common area of $\sim 172~deg^2$, we have identified the far-infrared counterparts for 354 quasars, among which 134 are highly secure detections in the Herschel $250~\mu m$ band (signal-to-noise ratios $\geq5$). This sample is the largest far-infrared quasar sample of its kind, and spans a wide redshift range of $0.14{\leq}z\leq 4.7$. Their far-infrared spectral energy distributions, which are due to the cold dust components within the host galaxies, are consistent with being heated by active star formation. In most cases ($\gtrsim80$\%), their total infrared luminosities as inferred from only their far-infrared emissions ($L_{IR}^{(cd)}$) already exceed $10^{12}~L_{\odot}$, and thus these objects qualify as ultra-luminous infrared galaxies. There is no correlation between $L_{IR}^{(cd)}$ and the absolute magnitudes, the black hole masses or the X-ray luminosities of the quasars, which further support that their far-infrared emissions are not due to their active galactic nuclei. A large fraction of these objects ($\gtrsim50\text{--}60\%$) have star formation rates $\gtrsim 300~M_{\odot}yr^{-1}$. Such extreme starbursts among optical quasars, however, is only a few per cent. This fraction varies with redshift, and peaks at around $z\approx2$. Among the entire sample, 136 objects have secure estimates of their cold-dust temperatures ($T$), and we find that there is a dramatic increasing trend of $T$ with increasing $L_{IR}^{(cd)}$. We interpret this trend as the envelope of the general distribution of infrared galaxies on the ($T$, $L_{IR}^{(cd)}$) plane.Comment: 26 pages, 20 figures, Accepted for publication in Ap

On the Limited Sizes of Dusty Starbursting Regions at High Redshifts

Using the far-infrared data obtained by the Herschel Space Observatory, we study the relation between the infrared luminosity (L_IR) and the dust temperature (T) of dusty starbursting galaxies at high redshifts (high-z). We focus on the total infrared luminosity from the cold-dust component (L_IR^(cd)), whose emission can be described by a modified black body (MBB) of a single temperature (T_mbb). An object on the (L_IR^(cd), T_mbb) plane can be explained by the equivalent of the Stefan-Boltzmann law for a MBB with an effective radius of R_eff. We show that R_eff is a good measure of the combined size of the dusty starbursting regions (DSBRs) of the host galaxy. In at least one case where the individual DSBRs are well resolved through strong gravitational lensing, R_eff is consistent with the direct size measurement. We show that the observed L_IR-T relation is simply due to the limited R_eff (<~ 2 kpc). The small R_eff values also agree with the compact sizes of the DSBRs seen in the local universe. However, previous interferometric observations to resolve high-z dusty starbursting galaxies often quote much larger sizes. This inconsistency can be reconciled by the blending effect when considering that the current interferometry might still not be of sufficient resolution. From R_eff we infer the lower limits to the volume densities of the star formation rate ("minSFR3D") in the DSBRs, and find that the $L_{IR}$-$T$ relation outlines a boundary on the (L_IR^(cd), T) plane, below which is the "zone of avoidance" in terms of minSFR3D.Comment: Submitted to ApJ

Determine Arbitrary Feynman Integrals by Vacuum Integrals

By introducing an auxiliary parameter, we find a new representation for Feynman integrals, which defines a Feynman integral by analytical continuation of a series containing only vacuum integrals. The new representation therefore conceptually translates the problem of computing Feynman integrals to the problem of performing analytical continuations. As an application of the new representation, we use it to construct a novel reduction method for multi-loop Feynman integrals, which is expected to be more efficient than known integration-by-parts reduction method. Using the new method, we successfully reduced all complicated two-loop integrals in $gg\to HH$ process and $gg\to ggg$ process.Comment: Version published in PRD (Rapid Communication

Comprehensive description of J/ψJ/\psi production in proton-proton collisions at collider energies

We employ a small $x$ Color Glass Condensate (CGC)+ Non-Relativistic QCD (NRQCD) formalism to compute $J/\psi$ production at low $p_\perp$ in proton-proton collisions at collider energies. Very good agreement is obtained for total cross-sections, rapidity distributions and low momentum $p_\perp$ distributions. Similar agreement is obtained for $\psi^\prime$ production. We observe an overlap region in $p_\perp$ where our results match smoothly to those obtained in a next-to-leading order (NLO) collinearly factorized NRQCD formalism. The relative contribution of color singlet and color octet contributions can be quantified in the CGC+NRQCD framework, with the former contributing approximately $10\%$ of the total cross-section.Comment: 6 pages, 3 figures, typos corrected, final version accepted by Physical Review Letter

New factorization theory for heavy quarkonium production and decay

The widely used nonrelativistic QCD (NRQCD) factorization theory now encounters some notable difficulties in describing quarkonium production. This may be due to the inadequate treatment of soft hadrons emitted in the hadronization process, which causes bad convergence of velocity expansion in NRQCD. In this paper, starting from QCD we propose a rigorously defined factorization approach, soft gluon factorization (SGF), to better deal with the effects of soft hadrons. After a careful velocity expansion, the SGF can be as simple as the NRQCD factorization in phenomenological studies, but has a much better convergence. The SGF may provide a new insight to understand the mechanisms of quarkonium production and decay.Comment: 12 pages, 3 figures, version published in PR

Symmetry and nonexistence of positive solutions for fractional systems

This paper is devoted to study the nonexistence results of positive solutions for the following fractional H$\acute{e}$non system \begin{eqnarray*}\left\{ \begin{array}{lll} &(-\triangle)^{\alpha/2}u=|x|^av^p,~~~&x\in R^n, &(-\triangle)^{\alpha/2}v=|x|^bu^q,~~~ &x\in R^n, &u\geq0, v\geq 0, \end{array} \right. \end{eqnarray*} where $0<\alpha<2$, $0<p,q<\infty$, $a$, $b$ $\geq0$, $n\geq2$. Using a direct method of moving planes, we prove non-existence of positive solution in the subcritical case

Extracting Parton Distribution Functions from Lattice QCD Calculations

Parton distribution functions (PDFs) are nonperturbative quantities describing the relation between a hadron and quarks and gluons within it. We propose to extract PDFs from QCD global analysis of "data" generated by lattice QCD calculations of good "lattice cross sections", which are basically single-hadron matrix elements that are lattice QCD calculable and perturbative QCD factorizable into the PDFs. To demonstrate the existence of good "lattice cross sections", we take quasi-quark distribution introduced by Ji [1] as a case study to show that it could be factorized into the PDFs to all orders in perturbation theory if it can be multiplicatively renormalized. We calculate the factorized coefficients at the next-to-leading order in $\alpha_s$.Comment: references updated, version published at PR

Hom-Nijienhuis operator and TT*-extension of Hom-Lie Superalgebras

In this paper, we study hom-Lie superalgebras. We give the definition of hom-Nijienhuis operators of regualr hom-Lie superalgebras and show that the deformation generated by a hom-Nijienhuis operator is trivial. Moreover, we introduce the definition of $T^*$-extensions of Hom-Lie superalgebras and show that $T^*$-extensions preserve many properties such as nilpotency, solvability and decomposition in some sense. We also investigate the equivalence of $T^*$-extensions.Comment: arXiv admin note: text overlap with arXiv:1005.0140 by other author

Theory for quarkonium: from NRQCD factorization to soft gluon factorization

We demonstrate that the recently proposed soft gluon factorization (SGF) is equivalent to the nonrelativistic QCD (NRQCD) factorization for heavy quarkonium production or decay, which means that for any given process these two factorization theories are either both valid or both violated. We use two methods to achieve this conclusion. In the first method, we apply the two factorization theories to the physical process $J/\psi \to e^+e^-$. Our explicit calculation shows that both SGF and NRQCD can correctly reproduce low energy physics of full QCD, and thus the two factorizations are equivalent. In the second method, by using equations of motion we successfully deduce SGF from NRQCD effective field theory. By identifying SGF with NRQCD factorization, we establish relations between the two factorization theories and prove the generalized Gremm-Kapustin relations as a by product. Comparing with the NRQCD factorization, the advantage of SGF is that it resums the series of relativistic corrections originated from kinematic effects to all powers, which gives rise to a better convergence in relativistic expansion.Comment: 15 pages, 1 figur