Β函数 (物理学)在理论物理量子场论中β函数,β(g)描述的是在重正化群下,理论中耦合常数g随能量标度μ的变化,定义: β ( g ) = ∂ g ∂ log ( μ ) , {\displaystyle \beta (g)={\frac {\partial g}{\partial \log(\mu )}}~,} 目录 1 例子 1.1 量子电动力学 1.2 量子色动力学 1.3 SU(N)非阿贝尔规范理论 2 参考资料 例子 量子电动力学 量子电动力学 (QED)中β函数一圈图表示: β ( e ) = e 3 12 π 2 , {\displaystyle \beta (e)={\frac {e^{3}}{12\pi ^{2}}}~,} 或 β ( α ) = 2 α 2 3 π , {\displaystyle \beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}~,} 这里精细结构常数α = e2/4π . 量子色动力学 量子色动力学 (QCD)中β函数还与夸克的味数 n f {\displaystyle n_{f}} 有关其一圈图表示: β ( g ) = − ( 11 − 2 n f 3 ) g 3 16 π 2 , {\displaystyle \beta (g)=-\left(11-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,} 或 β ( α s ) = − ( 11 − 2 n f 3 ) α s 2 2 π , {\displaystyle \beta (\alpha _{s})=-\left(11-{\frac {2n_{f}}{3}}\right){\frac {\alpha _{s}^{2}}{2\pi }}~,} 这里 αs = g 2 4 π {\displaystyle {\frac {g^{2}}{4\pi }}} . 如果 nf ≤ 16则β函数为负数,理论存在渐近自由,这一现象在1973年,被弗朗克·韦尔切克和戴维·格娄斯[1],与休·波利策[2]两组人发现。他们三人在2004年因这项发现而获得了诺贝尔物理学奖[3]。SU(N)非阿贝尔规范理论 β ( α ) = μ 2 ∂ ∂ μ 2 α ( μ 2 ) 4 π = − [ β 0 ( α 4 π ) 2 + β 1 ( α 4 π ) 3 + β 2 ( α 4 π ) 4 + ⋯ ] {\displaystyle \beta (\alpha )=\mu ^{2}{\frac {\partial }{\partial \mu ^{2}}}{\frac {\alpha (\mu ^{2})}{4\pi }}=-\left[\beta _{0}\left({\frac {\alpha }{4\pi }}\right)^{2}+\beta _{1}\left({\frac {\alpha }{4\pi }}\right)^{3}+\beta _{2}\left({\frac {\alpha }{4\pi }}\right)^{4}+\cdots \right]} β 0 = 11 3 C A − 4 3 T F n f {\displaystyle \beta _{0}={\frac {11}{3}}C_{A}-{\frac {4}{3}}T_{F}n_{f}} β 1 = 34 3 C A 2 − 20 3 C A T F n f − 4 C F T F n f {\displaystyle \beta _{1}={\frac {34}{3}}C_{A}^{2}-{\frac {20}{3}}C_{A}T_{F}n_{f}-4C_{F}T_{F}n_{f}} β 2 = 2857 54 C A 3 − 1415 27 C A 2 T F n f + 158 27 C A T F 2 n f 2 + 44 9 C F T F 2 n f 2 − 205 9 C F C A T F n f + 2 C F 2 T F n f {\displaystyle \beta _{2}={\frac {2857}{54}}C_{A}^{3}-{\frac {1415}{27}}C_{A}^{2}T_{F}n_{f}+{\frac {158}{27}}C_{A}T_{F}^{2}n_{f}^{2}+{\frac {44}{9}}C_{F}T_{F}^{2}n_{f}^{2}-{\frac {205}{9}}C_{F}C_{A}T_{F}n_{f}+2C_{F}^{2}T_{F}n_{f}} 其中: T F = 1 2 , C F = N 2 − 1 2 N {\displaystyle T_{F}={\frac {1}{2}},C_{F}={\frac {N^{2}-1}{2N}}} 和 C A = N {\displaystyle C_{A}=N} 参考资料 ^ D.J. Gross, F. Wilczek. Ultraviolet behavior of non-abeilan gauge theories. Physical Review Letters. 1973, 30: 1343–1346. Bibcode:1973PhRvL..30.1343G. doi:10.1103/PhysRevLett.30.1343. ^ H.D. Politzer. Reliable perturbative results for strong interactions. Physical Review Letters. 1973, 30 (26): 1346–1349. Bibcode:1973PhRvL..30.1346P. doi:10.1103/PhysRevLett.30.1346. ^ The Nobel Prize in Physics 2004. NobelPrize.org. Nobel Media. [26 August 2011]. (原始内容存档于2018-06-29).