How Does Scalar QCD Beta Function Calculation Differ in Various Sources?

AI Thread Summary
The discussion centers on the calculation of the beta function in scalar quantum chromodynamics (QCD) and discrepancies between different sources. The initial calculations for the renormalization constants \(Z_1\), \(Z_2\), and \(Z_3\) were presented, leading to a beta function that differed from Srednicki's results. A participant pointed out that while \(Z_1\) and \(Z_2\) were correct, \(Z_3\) contained an error that needed correction to align with the expected beta function. The corrected expression for \(Z_3\) ultimately leads to the accurate beta function, confirming the importance of precise calculations in quantum field theory. The discussion highlights the nuances in deriving results in theoretical physics.
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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.
 
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Can someone help me?A:The answer is that your expressions for $Z_1$ and $Z_2$ are correct, but the expression for $Z_3$ is wrong. It should be$$Z_3 = 1 + \frac{g^2}{24\pi^2} \left[ 7 T(A) - T(R) \right] \frac1{\epsilon} + \cdots,$$which gives the correct result for the $\beta$-function.Edit: To see this in detail, write the renormalized Lagrangian as$$\mathcal L_R = Z_1\phi_i^* D_\mu D^\mu \phi_i + Z_2 \psi^\dagger D_\mu \gamma^\mu \psi + Z_3 F_{\mu\nu}F^{\mu\nu} + \mathcal O(g^3).$$As Srednicki points out, the contribution to the $\beta$-function from a term of the form $g^2F_{\mu\nu}F^{\mu\nu}$ is$$-g^3\frac{d\ln Z_3}{dg}\left(\frac{T(A)-T(R)}{24\pi^2}\right)\left(\frac{11T(A)-T(R)}{16\pi^2}\right).$$Using the expression for $Z_3$ you give, this evaluates to$$-\frac{17}{24}g^3\left(\frac{T(A)-T(R)}{24\pi^2}\right)\left(\frac{11T(A)-T(R)}{16\pi^2}\right),$$which is different from the correct answer. However, if we use the above expression for $Z_3$, then the same calculation gives$$-\frac{11}{16}g^3\left(\frac{T(A)-T(R)}{24\pi^2}\right)\left(\frac{11T(A)-T(R)}{16\pi^2}\right),$$which is the correct answer.
 
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