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σfv (p2)和σfs (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。
Based on the Dyson-Schwinger Equations (DSEs) with the rainbow truncation, and Op-erator Product Expansion, the structure of nonlocal quark vacuum condensate in QCD, described by quark self-energy functions Af and Bf given usually by the solutions of the DSEs of quark propagator, is predicted numerically. We also calculate the local quark vacuum condensate, quark-gluon mixed local vacuum condensate, and quark virtuality. The self-energy functions Af and Bf are given by the parameterized quark propagator functions σfv (p2) and σfs (p2) of Roberts and Williams, instead of the numerical solutions of the DSEs. Our calculated results are in reasonable agreement with those of QCD sum rules, Lattice QCD calculations, and instanton model predictions, although the result-ing local quark vacuum condensate for light quarks, u, d, s, are a little bit larger than those of the above theoretical predictions. We think the differences are caused by model dependence. The larger of strange quark vacuum condensate than u, d quark is due to the s quark mass which is more larger than u, d quark masses. Of course, the Roberts-Williams parameterized quark propagator is an empirical formulism, which approximately describes quark propagation.
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