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

• fenyutanchan
In summary, Srednicki shows that the $beta$-function is $\beta(g) = - \frac{g^3}{16\pi^2} \left[ \frac{11}3 T(A) - \frac13 T(R) \right] + \mathcal O(g^5)$, while you show that the $beta$-function is $\beta(g) = - \frac{g^3}{96\pi^2} \left[ 19 T(A) - 2 T(R) \right] + \mathcal O(g^5)$.
fenyutanchan
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.

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.

## 1. What is the Beta Function for Scalar QCD?

The Beta Function for Scalar QCD is a mathematical function that describes how the coupling constant of the strong nuclear force changes with the energy scale at which it is measured. It is an important tool for understanding the behavior of quarks and gluons, the fundamental particles that make up protons and neutrons.

## 2. How is the Beta Function calculated?

The Beta Function is calculated using a mathematical technique called perturbation theory, which involves breaking down a complex problem into simpler parts. In the case of Scalar QCD, the Beta Function is calculated by considering the interactions between quarks and gluons at different energy scales.

## 3. What is the significance of the Beta Function for Scalar QCD?

The Beta Function is significant because it allows us to understand how the strength of the strong nuclear force changes with energy scale. This is important for predicting the behavior of particles at different energy levels, such as in high-energy particle collisions.

## 4. How does the Beta Function relate to the running of the strong coupling constant?

The Beta Function is directly related to the running of the strong coupling constant, which describes how the strength of the strong nuclear force changes with distance. The Beta Function tells us how the value of the coupling constant changes as we move to higher or lower energy scales.

## 5. Are there any applications of the Beta Function for Scalar QCD?

Yes, the Beta Function has many applications in particle physics and cosmology. It is used to study the behavior of quarks and gluons in high-energy collisions, as well as in the early universe. It is also an important tool for developing theories of quantum gravity and understanding the fundamental laws of nature.

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