Grand Unified Theories based on Dynamical Symmetry Breaking

本课题在1994-1995两个学年的研究成果如下：1.提出了一种新的拉格朗日方法来定义Nambu-Jona-Lasinio（NJL）模型，它具有由费米子动力学高阶导数项截断的内禀紫外截止，并保持了所有的外规范不变性.讨论了高阶导数系统的量子化方法、电流算符和手征反常问题.基于重整化群方法，分析了规范-Higgs-Yukawa系统是否是非平凡的，即当紫外截止达到无穷大时，它是否能给出一个定义良好的相互作用理论.我们证明了它在下列条件下是非平凡的：i）存在渐近自由规范相互作用，它的渐近自由不应该如此迅速地接近; ii）Yukawa耦合应该小于或等于一个上界; iii）标量四次耦合不是一个独立的参数，而是由Yukawa和规范耦合常数唯一地确定的。在此基础上，证明了规范NJL模型在四维空间是可重正化的.阐明了在背景场方法中如何从规范不变的有效作用量得到物理S矩阵的过程。我们给出了一个精确的证明，证明了它给出的S-矩阵与通常的非规范不变（BRS不变）有效作用量得到的S-矩阵相同.我们发现了一个新的关系式1+u=Z_1/Z_3=Z_<vertex>/Z_<wave-function>，其中左边是1+u = 0给出色禁闭发生的充分条件的量，右边是重整化因子之比Z_<vertex>/Z_<wave-function>，它是普适的（即，独立于物质），

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),