Grand Unified Theories based on Dynamical Symmetry Breaking

本课题1994-1995两学年的研究成果如下：提出了一种新的拉格朗日方法来定义nambujna - lasinio （NJL）模型，该模型具有高导数费米子动力学项的本征紫外截止，并保持所有外部规范不变性。讨论了这种高导数系统的量化方法、当前算子和手性异常问题。基于重整化群方法，分析了量规-希格斯-汤川系统的非平凡性；也就是说，当紫外截止到无穷大时，它是否能给出一个定义良好的相互作用理论。我们在以下条件下证明了它的非平凡性：i)存在渐近自由规范相互作用，其接近渐近自由的速度不应如此之快；ii)汤川耦合应小于或等于上界；iii)标量四次耦合不是一个独立的参数，而是由汤川和规范耦合常数唯一决定的。在此基础上，证明了测量后的NJL模型在四个维度上是可重整的。阐述了如何用标准不变的背景场有效作用法得到物理s矩阵的过程。给出了与一般非规范不变（BRS不变）有效作用得到的s -矩阵相同的精确证明。我们发现了一个新的关系1+u=Z_1/Z_3=Z_<顶点>/Z_<波函数>，其中左手边是已知的1+u= 0给出发生颜色约束的充分条件的量，右手边表示重整化因子的比值，Z_<顶点>/Z_<波函数>，这是普适的（即与物质无关）。

The research results of our project for two academic years in 1994-1995 are as follows :1. A new lagrangian way of defining the Nambu-Jona-Lasinio (NJL) model is proposed, which possesses an intrinsic ultraviolet cutoff by the higher derivative fermion kinetic term and maintains all the external gauge invariances. There we also discussed the quantization method of such a higher-derivative system, current operators and chiral anomaly problems.2. Based on the renormalization group method, the gauge-Higgs-Yukawa system was analyzed whether it is nontrivial or not ; namely, whether it can give a well-defined interacting theory when the ultraviolet cutoff goes to infinity. We proved that it is nontrivial under the following conditions : i) there exists an asymptotically free gauge interaction, whose approach to the asymptotic freedom should not be so rapid, ii) the Yukawa coupling should be smaller than or equal to an upperbound, and iii) the scalar quartic coupling is not an independent parameter but is uniquely determined by the Yukawa and gauge coupling constants. Based on the result, the gauged NJL model is proved to be renormalizable in four dimensions.3. The procedure was clarified how one can obtain the physical S-matrix from the gauge-invariant effective action in the background field (BF) method. We gave a precise proof that it gives the same S-matrix as the one obtained by the usual non-gauge-invariant (BRS invariant) effective action.4. We found a new relation 1+u=Z_1/Z_3=Z_<vertex>/Z_<wave-function>, where the left-hand side is the quantuty that 1+u = 0 is known to give a sufficient condition for the color confinement to occur, and the righthand side stands for the ratio of the renormalization factors, Z_<vertex>/Z_<wave-function>, which is universal (i.e., matter-independent),