- #1

fenyutanchan

- 1

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- Homework Statement
- Compute the beta function for $g$ in Yang–Mills theory with a complex scalar field in the representation $R$ of the gauge group.

- Relevant Equations
- Lagrangian:

$$

\mathcal L = \frac12 Z_3 A^{a\mu} \left( g_{\mu\nu} \Box - \partial_\mu \partial_\nu + \frac1{2\xi} A^{a\mu} \partial_\mu \partial_\nu \right) A^{a\nu} - Z_{3g} g f^{abc} A^{a\mu} A^{b\nu} \partial_\mu A^c_\nu - \frac14 Z_{4g} g^2 f^{abe} f^{cde} A^{a\mu} A^{b\nu} A^c_\mu A^d_\nu - Z_{2^\prime} \left( \partial^\mu \bar C^a \right) \partial_\mu C^a + Z_{1^\prime g} f^{abc} A^c_\mu \left( \partial^\mu \bar C^a \right) C^b - Z_2 \left( \partial^\mu \varphi_i^\dagger \right) \partial_\mu \varphi_i - Z_m m^2 \varphi_i^\dagger \varphi + i Z_1 g A^a_\mu \left( T_R^a \right)_{ij} \left[ \varphi_i^\dagger \left( \partial^\mu \varphi_j \right) - \left( \partial^\mu \varphi_i^\dagger \right) \varphi_j \right] - Z_4 g^2 A^a_\mu A^{b\mu} \varphi_j^\dagger \left( T_R^a \right)_{jk} \left( T_R^b \right)_{ki} \varphi_i - \frac14 Z_\lambda \lambda \left( \varphi_i^\dagger \varphi_i \right)^2.

$$

I have calculated $Z$s as

$$

\begin{aligned}

Z_1 & = 1 + \frac{3g^2}{16\pi^2} \left[ 2 C(R) - \frac12 T(A) \right] \frac1{\epsilon} + \cdots, \\

Z_2 &= 1 + \frac{3g^2}{8\pi^2} C(R) \frac1{\epsilon} + \cdots, \\

Z_3 &= 1 + \frac{g^2}{24\pi^2} \left[ 5 T(A) - T(R) \right] \frac1{\epsilon} + \cdots.

\end{aligned}

$$

It shows that the $beta$-function is

$$

\beta(g) = - \frac{g^3}{96\pi^2} \left[ 19 T(A) - 2 T(R) \right] + \mathcal O(g^5).

$$

However, Srednicki shows that the $beta$-function is

$$

\beta(g) = - \frac{g^3}{16\pi^2} \left[ \frac{11}3 T(A) - \frac13 T(R) \right] + \mathcal O(g^5).

$$

I think that I have make a mistake in $Z_1$, but I cannot find it and fix it.

$$

\begin{aligned}

Z_1 & = 1 + \frac{3g^2}{16\pi^2} \left[ 2 C(R) - \frac12 T(A) \right] \frac1{\epsilon} + \cdots, \\

Z_2 &= 1 + \frac{3g^2}{8\pi^2} C(R) \frac1{\epsilon} + \cdots, \\

Z_3 &= 1 + \frac{g^2}{24\pi^2} \left[ 5 T(A) - T(R) \right] \frac1{\epsilon} + \cdots.

\end{aligned}

$$

It shows that the $beta$-function is

$$

\beta(g) = - \frac{g^3}{96\pi^2} \left[ 19 T(A) - 2 T(R) \right] + \mathcal O(g^5).

$$

However, Srednicki shows that the $beta$-function is

$$

\beta(g) = - \frac{g^3}{16\pi^2} \left[ \frac{11}3 T(A) - \frac13 T(R) \right] + \mathcal O(g^5).

$$

I think that I have make a mistake in $Z_1$, but I cannot find it and fix it.