Dynamically Running Mass of Light Quark and QCD Vacuum Condensates

Based on Dyson-Schwinger equations (DSEs) in “rainbow“ approximation, the dynamically running mass of light quark and QCD vacuum condensate are investigated. The structure of non-local quark vacuum condensate, the values of local vacuum condensate of quarks and quark-gluon mixture, and dynamical transition of quark mass from current quark to constituent quark are illustrated. At the same time, according to the knowledge and experience learned from an extensive study of the solutions of DSEs, a parameterized form of confining quark propagator is suggested for a practical use. The new parameterized form of quark propagator is analytic everywhere in the finite complex p2-plane and has no Lehmann representation. The predictions for p2-dependence of effective quark masses, Mf(p2), defined by the self-energy functions Af(p2) and Bf(p2), both from the numerical solutions of DSEs and from its parameterized form, are shown dynamically. Our conclusion is that all numerical results are consistent with empirical values used in QCD sum rules and lattice QCD calculations. For a qualitative study, the parameterized form is a sufficiently good approximation to confining quark propagator.     

QCD中等时方程的赝标介子解

本文讨论在背景场量子色动力学框架中的等时方程赝标介子解. 方程的积分核由微扰和非微扰贡献两部分组成．微拔部分是通常的单胶子交换图的贡献；非微扰部分的贡献来自于最低阶的夸克和胶子凝聚效应. 当输入合理的夸克质量参数和耦合常数时，可以得到与实验相符合的赝标介子谱．     

单胶子交换梯形近似下Bethe－Salpeter方程赝标解

将在动量空间具有积分形式的单胶子交换梯形近似下Ｂｅｔｈｅ－Ｓａｌｐｅｔｅｒ方程化为微分方程，求出该方程在四动量为零时的赝标解全部分量，其中第一分量为已知的Ｇｏｌｄｓｔｅｉｎ解．     

A Novel Series Solution to the Renormalization-Group Equation in QCD

Recently, the QCD renormalization-group (RG) equation at higher orders in MS-like renormalization schemes has been solved for the running coupling as a series expansion in powers of the exact two-loop-order coupling. In this work, we prove that the power series converge to all orders in perturbation theory. Solving the RG equation at higher orders, we determine the running coupling as an implicit function of the two-loop-order running coupling. Then we analyze the singularity structure of the higher-order coupling in the complex two-loop coupling plane. This enables us to calculate the radii of convergence of the series solutions at the three- and four-loop orders as a function of the number of quark flavours n _f . In parallel, we discuss in some detail the singularity structure of the coupling at the three- and four-loops in the complex-momentum squared plane for 0 ≤ n _f ≤ 16. The correspondence between the singularity structure of the running coupling in the complex-momentum squared plane and the convergence radius of the series solution is established. For sufficiently large n _f values, we find that the series converges for all values of the momentum-squared variable Q ² = −q ² > 0. For lower values of n _f , in the scheme, we determine the minimal value of the momentum-squared Q _min ² above which the series converges. We study properties of the non-power series corresponding to the presented power-series solution in the QCD analytic perturbation-theory approach of Shirkov and Solovtsov. The Euclidean and Minkowskian versions of the non-power series are found to be uniformly convergent over the whole ranges of the corresponding momentum-squared variables.     

Solution of the Schwinger-Dyson Equation Involving the Gluon condensate Kernel

The form of the quantum chromodynamic Schwinger-Dyson (SD) equation modified by the nonperturbative kernel associated with the gluon condensate is obtained. It is shown that the eolution for the renormalisation group invariant quark condensate from the SD equation is in agreement with that obtained from the consistent equation for quark condensate when the leading behavior of the dynamical quark mass is concerned^[1]. It is also found that the gluon condensate kernel makes the critical coupling constant above which the chiral symmetry breaks smaller and increases the value of the quark condensate.     

The Spectra of the Light 0++ Mesons in QCD Sum Rules

In this work, the fight J^PC = 0⁺⁺ mesons f₀(975), a₀(980) and S^*(980) are interpreted as the usual qq states, their masses are investigated in the framework of the QCD sum rules.     

Stability of Global Solution for the Relativistic Enskog Equation near Vacuum

The Cauchy problem of the relativistic Enskog equation with near-vacuum data is considered in this paper. Under the same assumption as that in Jiang (J. Stat. Phys. 127:805–812, 2007) for the relativistic Enskog equation, we obtain the uniform L ^∞-stability of the solution. What’s more important, is that for two new types of the scattering cross section σ, we give the global existence and L ¹(x,v)-stability for mild solution when the initial data lies in the space L ¹(x,v). As a corollary, we have a BV-type estimate. It is worth mentioning that the stability results in this paper can be applied to the case in Jiang (J. Stat. Phys. 127:805–812, 2007).     

The Importance of Boundary Conditions in Solving the Bound-State Bethe-Salpeter Equation

The characteristic of bound-state, Bethe-Salpeter equations that makes them so difficult to solve numerically can be overcome, in some if not many cases, by expanding solutions in terms of basis functions that obey the boundary conditions that are satisfied by the solutions. The utility of such basis functions is demonstrated by calculating the zero-energy, bound-state solutions of a spin-0 boson and a spin-&frac; fermion with unequal masses. The constituents interact via scalar electrodynamics and are described by the Bethe-Salpeter equation in the ladder approximation. Although the Bethe-Salpeter equation that is solved is separable in the zero-energy limit, the feature that typically prevents solutions from being obtained numerically is still present. A technique for calculating boundary conditions, which is readily generalized to other Bethe-Salpeter equations, is discussed in detail.Supported by a grant from the Ohio Supercomputer CenterReceived January 31, 2003; accepted April 4, 2003 Published online August 25, 2003     

O(4) Expansion of the Ladder Bethe-Salpeter Equation

The Bethe-Salpeter amplitude is expanded on a hyperspherical basis thereby reducing the original four-dimensional integral equation into an infinite set of coupled one-dimensional ones. It is shown that this representation offers a highly accurate method to determine numerically the bound state solutions. For generic cases only a few hyperspherical waves are needed to achieve convergence, both for the ground state as well as for radially or orbitally excited states. The wave function is reconstructed for several cases and in particular it is shown that it becomes independent of the relative time in the nonrelativistic regime. Received July 18, 1996; accepted in final form October 2, 1996     

非微扰QCD真空中夸克真空凝聚的结构(英文)

基于Dyson-Schwinger方程(DSEs)所确定的夸克传播子和算符成积展开(OPE),在彩虹近似下,预言了QCD真空中非定域夸克真空凝聚的结构。这种结构由夸克自能函数Af和Bf决定,通过数值求解DSEs就可以得到这些自能函数。但是,直接数值求解DSEs方程非常复杂,这里采用Roberts和Williams提出的参数化方法,用参数化的夸克传播函数σf v(p2)和σf s(p2)计算夸克自能函数。同时,也计算了定域的夸克真空凝聚值,夸克胶子混合的真空凝聚值,以及夸克的虚度。理论预言和计算结果均与标准QCD求和定则、格点QCD和瞬子模型的理论结果大致相符。和这些模型的结果相比,参数化方法得到的轻夸克(u,d,s)的定域真空凝聚偏大,这主要是由于模型依赖导致的。与u,d夸克相比,s夸克的真空凝聚比较大,这是因为s夸克自身质量较大的缘故。当然,Roberts-Williams参数化的夸克传播子只是一个经验公式,只能近似描述夸克的传播。     

Temperature Dependence of Two- and Three-Gluon Condensates in the Instanton Vacuum

The two-gluon and three-gluon condensates at finite temperature are calculated in the framework of the grand partition function for a weak-interacting instanton medium of a disordered phase. The interference parts of both chromomagnetic and chromoelectric two-gluon condensates are found to be small but still significant for a quantitative determination of the value of the condensates at very low temperature, and negligible at the temperature above 1.5Λ_MS, with respect to their non-interference ones. The interference parts of the threegluon condensates are found to be quite small even at very low temperature. With the QCD scale parameter fixed by the value of the Lorentz invariant two-gluon condensate at zero temperature, the value of the three-gluon condensate at zero temperature is found to be coincident with the phenomenological estimate. The values of the gluon condensates decrease when temperature increases, and are negligible when the temperature goes beyond 2.5Λ_MS in the case of the SU_C(3) pure gauge theory.     

Solutions of the Bethe-Salpeter Equation in Minkowski Space: A Comparative Study

The Bethe–Salpeter equation for a two-scalar, S-wave bound system, interacting through a massive scalar, is investigated within the ladder approximation. By assuming a Nakanishi integral representation of the Bethe–Salpeter amplitude, one can deduce new integral equations that can be solved and quantitatively studied, overcoming the analytic difficulties of the Minkowski space. Finally, it is shown that the Light-front distributions of the valence state, directly obtained from the Bethe–Salpeter amplitude, open an effective window for studying the two-body dynamics.     

Photoproduction of Vector Meson ø off Deuteron in QCD Inspired Model

Based on the quark-gluon contents of nucleon and strongly believing that the force mediators, Pomeron and its counterpart in the conventional approach of Regge theory, for high energy diffractive process would be the tensor glueball and Odderonrespectively, we discuss photo-production of vector meson ø off the deuteron at energy less than 3 GeV in the QCD inspired model in which the quark gluon degrees of freedom and glueball, Odderon exchange are taken into account. A calculation isperformed for γ+D → ø+D, and the theoretical predictions of the differential cross section dσ ^γD/dt, are presented and compared with available experimental data. Our QCD inspired model reproduces data quite well in the whole range of the experimental measurements up to |t|≌ 0.4~GeV. Our results can be used toextract γn →øn data, which cannot be measured in experiment.     

Vacuum Condensates from the Modification of Global Color Symmetry Model

Based on the quark propagator in the instanton dilute liquid approximation, we have determined the quark condensate , the mixed quark gluon condensate g,μνσ^μνq) and the four-quark condensate at the mean-field level in the framework of global color symmetry model. The numerical calculation shows that our result is compatible with the range obtained within other nonperturbative approaches. In particular, we have found that even at the mean-field level the naive vacuum saturation approximation is not a good approximation when we consider nonlocal four-quark condensate.     

The Spin Symmetry of Heavy Baryons in the Framework of the Bethe-Salpeter Equation

We study the baryons containing a heavy quark in the framework of Bethe-Salpeter (BS) equation. The most general forms of the BS wavefunctions are given. In the heavy-quark limit we simplify the BS equations and we show clearly that the spin symmetry exists in heavy baryon states.     

An Equivalent Reduction of the Bethe-Salpeter Equation for Multi-Fermion Bound States

By means of the renormalized Dirac spinor wave function and through. the introduction of an effective interaction operator, the exact ~ethe-saypetere quation for multi-fermion bound states has been reduced to an equivalent Pauli-Schrodinger equation. As a result, the specific form of the latter equation in the static approximation has directly been given as well. In comparison of the effective interaction operator appearing in the Pauli-Schrodinger equation with the corresponding S-matrix, a substantial difference between both interactions acting in the bound state and the scattering state emerges which is important to determine an interaction potential in the bound state.     

重离子碰撞中的轻矢量介子光生过程(英文)

计算了相对论重离子碰撞中轻矢量介子(ρ,ω和φ)的光生过程。基于窄-宽近似,严格推导了相对论重离子碰撞中ρ,ω和φ半弹性和非弹性光生过程的遍举和单举横截面。从数值结果可以看出对于p-p碰撞,光生过程的贡献是不重要的,但是对于pT2.5 GeV Au-Au碰撞和pT3 GeV Pb-Pb碰撞,轻矢量介子光生过程的贡献是明显的。