# β函數 (物理學)

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

## 例子

### 量子电动力学

• $\beta(e)=\frac{e^3}{12\pi^2}~,$

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

### 量子色动力学

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

• $\beta(\alpha_s)=-\left(11-\frac{2n_f}{3}\right)\frac{\alpha_s^2}{2\pi}~,$

### SU(N)非阿贝尔规范理論

$\beta(\alpha) = \mu^2 \frac{\partial}{\partial \mu^2} \frac{\alpha(\mu^2)}{4\pi} = - \left[ \beta_0 \left( \frac{\alpha}{4\pi} \right)^2 + \beta_1 \left( \frac{\alpha}{4\pi} \right)^3 + \beta_2 \left( \frac{\alpha}{4\pi} \right)^4+ \cdots\right]$

$\beta_0 = \frac{11}{3}C_A - \frac{4}{3}T_F n_f$

$\beta_1 = \frac{34}{3}C_A^2 - \frac{20}{3}C_A T_F n_f - 4C_F T_F n_f$

$\beta_2 = \frac{2857}{54}C_A^3 - \frac{1415}{27}C_A^2 T_F n_f + \frac{158}{27}C_A T_F^2 n_f^2 + \frac{44}{9}C_F T_F^2 n_f^2 - \frac{205}{9}C_F C_A T_F n_f +2 C_F^2 T_F n_f$

## 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].