# β函數 (物理學)

${\displaystyle \beta (g)={\frac {\partial g}{\partial \log(\mu )}}~,}$

## 例子

### 量子电动力学

• ${\displaystyle \beta (e)={\frac {e^{3}}{12\pi ^{2}}}~,}$

• ${\displaystyle \beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}~,}$

### 量子色动力学

• ${\displaystyle \beta (g)=-\left(11-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,}$

• ${\displaystyle \beta (\alpha _{s})=-\left(11-{\frac {2n_{f}}{3}}\right){\frac {\alpha _{s}^{2}}{2\pi }}~,}$

### SU(N)非阿贝尔规范理論

${\displaystyle \beta (\alpha )=\mu ^{2}{\frac {\partial }{\partial \mu ^{2}}}{\frac {\alpha (\mu ^{2})}{4\pi }}=-\left[\beta _{0}\left({\frac {\alpha }{4\pi }}\right)^{2}+\beta _{1}\left({\frac {\alpha }{4\pi }}\right)^{3}+\beta _{2}\left({\frac {\alpha }{4\pi }}\right)^{4}+\cdots \right]}$

${\displaystyle \beta _{0}={\frac {11}{3}}C_{A}-{\frac {4}{3}}T_{F}n_{f}}$

${\displaystyle \beta _{1}={\frac {34}{3}}C_{A}^{2}-{\frac {20}{3}}C_{A}T_{F}n_{f}-4C_{F}T_{F}n_{f}}$

${\displaystyle \beta _{2}={\frac {2857}{54}}C_{A}^{3}-{\frac {1415}{27}}C_{A}^{2}T_{F}n_{f}+{\frac {158}{27}}C_{A}T_{F}^{2}n_{f}^{2}+{\frac {44}{9}}C_{F}T_{F}^{2}n_{f}^{2}-{\frac {205}{9}}C_{F}C_{A}T_{F}n_{f}+2C_{F}^{2}T_{F}n_{f}}$

## 參考資料

1. ^ 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.
2. ^ 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.
3. ^ The Nobel Prize in Physics 2004. NobelPrize.org. Nobel Media. [26 August 2011]. （原始内容存档于2018-06-29）.