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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Banks–Zaks fixed point ： ウィキペディア英語版 Banks–Zaks fixed point In quantum chromodynamics (and also ''N'' = 1 superquantum chromodynamics) with massless flavors, if the number of flavors, ''N''f, is sufficiently small (i.e. small enough to guarantee asymptotic freedom, depending on the number of colors), the theory can flow to an interacting conformal fixed point of the renormalization group. If the value of the coupling at that point is less than one (''i.e.'' one can perform perturbation theory in weak coupling), then the fixed point is called a Banks–Zaks fixed point. The existence of the fixed point was first reported by William E. Caswell in 1974, and later used by Banks and Zaks in their analysis of the phase structure of vector-like gauge theories with massless fermions. For this reason one also justifiably finds references to a Caswell-Banks–Zaks fixed point. More specifically, suppose that we find that the beta function of a theory up to two loops has the form : $\backslash beta(g)\; =\; -b\_0\; g^3\; +\; b\_1\; g^5\; +\; \backslash mathcal(g^7)\; \backslash ,$ where $b\_0$ and $b\_1$ are positive constants. Then there exists a value $g=g\_\backslash ast$ such that $\backslash beta(g\_\backslash ast)\; =0$: : $g\_\backslash ast^2\; =\; \backslash frac.$ If we can arrange $b\_0$ to be smaller than $b\_1$, then we have . It follows that when the theory flows to the IR it is a conformal, weakly coupled theory with coupling $g\_\backslash ast$. For the case of a non-Abelian gauge theory with gauge group $SU(N\_c)$ and Dirac fermions in the fundamental representation of the gauge group for the flavored particles we have :$b\_0\; =\; \backslash frac\backslash frac(11N\_c-2N\_f)\; \backslash ;\backslash ;\backslash ;\backslash ;\; \backslash text\backslash ;\backslash ;\backslash ;\backslash ;\; b\_1\; =\; -\backslash frac\backslash left(\backslash fracN\_c^2\; -\; \backslash fracN\_f\backslash left(2\; \backslash frac\; +\; \backslash fracN\_c\; \backslash right)\; \backslash right)$ where $N\_c$ is the number of colors and $N\_f$ the number of flavors. Then $N\_f$ should lie just below $\backslash tfracN\_c$ in order for the Banks–Zaks fixed point to appear. Note that this fixed point only occurs if, in addition to the previous requirement on $N\_f$ (which guarantees asymptotic freedom), :$\backslash fracN\_c>N\_f>\backslash frac$ where the lower bound comes from requiring $b\_1>0$. This way $b\_1$ remains positive while $-b\_0$ is still negative (see first equation in article) and one can solve $\backslash beta\; (g)\; =\; 0$ with real solutions for $g$. ==See also== *Beta function 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Banks–Zaks fixed point」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.