Abstract

For the effective potential in the leading logarithmic approximation, we construct a renormalization group equation that holds for arbitrary scalar field theories, including nonrenormalizable ones, in four dimensions. This equation reduces to the usual renormalization group equation with a one-loop beta-function in the renormalizable case. The solution of this equation sums up the leading logarithmic contributions in the field in all orders of the perturbation theory. This is a nonlinear second-order partial differential equation in general, but it can be reduced to an ordinary one in some cases. In specific examples, we propose a numerical solution of this equation and construct the effective potential in the leading logarithmic approximation. We consider two examples as an illustration: a power-law potential and a cosmological potential of the \(\tan^2\phi\) type. The obtained equation in physically interesting cases opens up the possibility of studying the properties of the effective potential, the presence of additional minima, spontaneous symmetry breaking, stability of the ground state, etc.

中文翻译：

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对不可重整化理论中有效势的量子修正

摘要

对于领先对数近似中的有效势，我们构造了一个重​​整化群方程，该方程适用于四个维度的任意标量场论，包括不可重整化场论。在可重整化的情况下，该方程简化为具有单循环 beta 函数的常见重整化群方程。该方程的解总结了微扰理论所有阶中该领域的主要对数贡献。一般而言，这是一个非线性二阶偏微分方程，但在某些情况下可以简化为普通方程。在具体例子中，我们提出了该方程的数值解，并构造了领先对数近似中的有效势。我们考虑两个例子作为说明：幂律势和\(\tan^2\phi\)类型的宇宙势。在物理有趣的情况下获得的方程为研究有效势的性质、额外最小值的存在、自发对称性破缺、基态稳定性等提供了可能性。