Gluon Scattering - Colored Feynman Rules for Yang Mills Theory

In summary, the conversation discusses the generators and algebra of SU(N) and the normalization of the generators. It is then shown that the factor of 2 in the calculation of f^{a b c} is correct. The author also states that the factor of 2 should be present in the calculation of f^{a b e} f^{c d e}, but the reason for this is not clear. The conversation ends with a question about the relationship between <1 2>^4 and p_1 \cdot p_2.
  • #1
maverick280857
1,789
4
Hi,

I'm reading Appendix 1 of Section N2 (Gluon Scattering) in "Quantum Field Theory in a Nutshell" by Anthony Zee. The generators for SU(N) have the usual algebra

[tex][T^a, T^b] = i \epsilon^{a b c}T^c[/tex]

Suppose we adopt the following normalization

[tex]\text{tr}(T^a T^b) = \frac{1}{2}\delta^{a b}[/tex]

Then, we have

[tex]\text{tr}([T^a, T^b]T^c) = \text{tr}(i f^{a b e} T^{e} T^{c}) = i f^{a b e}\text{tr}(T^e T^c) = \frac{i f^{a b e} \delta^{e c}}{2} = \frac{i f ^{a b c}}{2}[/tex]

so that

[tex]f^{a b c} = - 2 i \text{tr}([T^a, T^b]T^c)[/tex]

Also,

[tex]\text{tr}([T^a, T^b][T^c, T^d]) = \text{tr}(i f^{a b \alpha} T^{\alpha} i f ^{c d \beta} T^{\beta}) = -f^{a b \alpha} f^{c d \beta}\text{tr}(T^\alpha T^\beta) = -\frac{1}{2}f^{a b e} f^{c d e}[/tex]

which implies that

[tex]f^{a b e} f^{c d e} = -2 \text{tr}([T^a, T^b][T^c, T^d])[/tex]

However, the author explicitly states that

[tex]f^{a b e} f^{c d e} = -4 \text{tr}([T^a, T^b][T^c, T^d])[/tex]

I don't get this additional factor of 2. What am I missing? Is there an error somewhere?

Thanks in advance!
 
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  • #2
Right on, the factor should be 2, not 4. For example check it for a = c = 1, b = d = 2. You get

f123f123 = 2 tr(T3T3)

The LHS is 1 and the trace of (T3)2 is indeed 1/2, so your form is correct.
 
  • #3
Thanks Bill.

How is

[tex]\frac{\langle 1 2\rangle^4}{\langle 1 2\rangle \langle 2 3\rangle \langle 3 4\rangle \langle 4 1\rangle} = \frac{p_1 \cdot p_2}{p_2 \cdot p_3}[/tex]?

For a given set of momenta, doesn't one have to find out what <1|2>, <3|4> etc are explicitly? How does this simple relationship arise? I think I missed it somehow. I can't get the LHS to equal the RHS. Any ideas?

Normally, papers/books seem to derive it up to the LHS..
 

1. What is gluon scattering?

Gluon scattering is a quantum process in which gluons, the particles that mediate the strong nuclear force, interact with each other. This process is described by the colored Feynman rules for Yang-Mills theory.

2. What is the significance of colored Feynman rules in gluon scattering?

Colored Feynman rules are a set of mathematical rules that are used to calculate the probabilities of different gluon scattering processes. They take into account the color charges of the gluons and the interaction between them.

3. How do colored Feynman rules differ from regular Feynman rules?

Colored Feynman rules are specific to Yang-Mills theory, which describes the strong nuclear force. Regular Feynman rules are used in other quantum field theories, such as quantum electrodynamics (QED).

4. What is Yang-Mills theory?

Yang-Mills theory is a quantum field theory that describes the interactions between particles that carry a color charge, such as gluons and quarks. It is an important part of the Standard Model of particle physics.

5. How are colored Feynman rules used in practical applications?

Colored Feynman rules are used in theoretical calculations to predict the outcomes of gluon scattering experiments. They are also used in computer simulations to model the behavior of particles in high-energy collisions, such as those at the Large Hadron Collider.

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