Calculating Beta Function for Scalar QCD Theory

In summary, the conversation is about calculating the beta function for scalar QCD theory and the need to derive Feynman rules and diagrams. The Lagrangian for scalar QCD is discussed, specifically the replacement of partial derivatives with covariant derivatives and the addition of the gauge field Lagrangian. A recommended resource for further understanding is Srednicki's textbook on scalar electrodynamics.
  • #1
abrata
1
0
Hi all,

I am currently trying to calculate the beta function for scalar QCD theory (one loop for general su(n)).

I therefore need to calculate the Feynman rules in order to apply them to the one loop diagrams. Unfortunately I am getting very confused with what the Lagrangian for scalar QCD should be. If anyone knows of some clear examples of this Lagrangian and possibly the derivation of the corresponding Feynman rules and diagrams I would be very appreciated.

Many thanks
Abrata
 
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  • #2
First start with the Lagrangian for a scalar field with an internal SU(N) symmetry:

[tex](\partial_\mu \phi^a)^\dagger (\partial^\mu \phi^a) - m^2 \phi^{a \dagger} \phi^a - \frac{\lambda}{4}(\phi^{a\dagger} \phi^a)^2[/tex]

Then replace the partial derivatives with covariant derivatives:

[tex](D_\mu \phi^a)^\dagger (D^\mu \phi^a) - m^2 \phi^{a \dagger} \phi^a - \frac{\lambda}{4}(\phi^{a\dagger} \phi^a)^2[/tex]

where ##D_\mu \phi^a = \partial_\mu \phi^a - i g T^{a b} \phi^b##. Add in the gauge field Lagrangian and you have the Lagrangian for scalar QCD.

Srednicki's textbook has some chapters on scalar electrodynamics, which might help you.
 

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

The Beta Function for Scalar QCD Theory is a mathematical function that describes the behavior of the coupling constant in the theory. It is used to calculate the running of the coupling constant at different energy scales.

2. How is the Beta Function calculated?

The Beta Function is calculated using perturbative methods in quantum field theory. This involves using Feynman diagrams and performing calculations to determine the behavior of the coupling constant at different energy scales.

3. What is the significance of the Beta Function in Scalar QCD Theory?

The Beta Function is an important tool in understanding the behavior of the coupling constant in Scalar QCD Theory. It allows us to make predictions about the theory at different energy scales and test its validity through experimental observations.

4. How does the Beta Function change with different parameters in Scalar QCD Theory?

The Beta Function is dependent on the parameters of Scalar QCD Theory, such as the number of colors and flavors of particles. These parameters affect the behavior of the coupling constant and can lead to different predictions for the theory at different energy scales.

5. Can the Beta Function be used in other theories besides Scalar QCD Theory?

Yes, the Beta Function is a general concept in quantum field theory and can be applied to other theories besides Scalar QCD. It is commonly used in theories such as QED and the Standard Model to understand the behavior of the coupling constant at different energy scales.

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