Non-abelian gauge fields 3-vertex and 4-vertex?

In summary, the conversation discusses the computation of the vertex factor for 3-boson and 4-boson fields in Feynman diagrams. The process involves using the gauge-covariant derivatives, non-Abelian Faraday tensor, and classical QCD Lagrangian. The Feynman rules can be determined from the Lagrangian and the resulting vertex factor can be obtained by contracting the four gluon fields in the interaction Lagrangian.
  • #1
moss
49
2
I want to understand the 'vertex factor' of 3-bosons field and 4-bosons field but get confused.
(I know the lagrangian and have computed the interaction vertices already) only need to understand the vertex factor.

In other words, I want to learn how 3-boson vertex and 4-boson vertex are computed in Feynman diagrams.

It is, for example, given in Peskin&Schrioeder on page 507 but I don't get it as to what is permuted there, the color indices a,b,c or the lorentz indices and the momentum?

any help/hint is much appricaited !
Thanks.
 
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  • #2
Ok, let's see how to sort this out. The gauge-covariant derivatives, acting on the quark fields read
$$\mathrm{D}_{\mu}=\partial_{\mu} - \mathrm{i} g t^a A_{\mu}^a,$$
where ##t^a=\lambda^a/2## are the generators of the su(3) Lie algebra in its fundamental representation.

The non-Abelian Faraday tensor is given by
$$F_{\mu \nu}^a=\partial_{\mu} A_{\nu}^a-\partial_{\nu} A_{\mu}^a + g f^{abc} A_{\mu}^b A_{\nu}^c.$$
Since SU(3) is a semisimple Lie group, the structure constants, defined by
$$[t^a,t^b]=\mathrm{i} f^{abc} t^c$$
can be made totally antisymmetric by a clever choice of the Lie algebra's basis, and this is the case for our choice of this basis ##t^a##.

The classical QCD Lagrangian reads
$$\mathrm{L}=-\frac{1}{4} F_{\mu \nu}^a F^{a \mu \nu} + \overline{\psi}(\mathrm{i} \mathrm{D}_{\mu} \gamma^{\mu} -M) \psi,$$
where ##\psi## consists of 6 quarks with three colors each. The ##t^a## contained in ##\mathrm{D}_{\mu}## act on the color indices. ##M## is a diagonal real matrix in flavor space.

When quantizing the theory, you have to introduce Faddeev-Popov ghosts, but these do not affect the quark-gluon and the three- and four-gluon vertices (at least not for the usually used gauges). Thus you can read off the Feynman rules immideately from the Lagrangian. The three- and four-gluon vertices come from the first term in the Lagrangian. You have to multiply out the terms. The four-gluon contribution to ##\mathrm{L}## reads
$$\mathrm{L}_{4g}=-\frac{1}{4} f^{eab} f^{ecd} A_{\mu}^a A_{\nu}^b A_{\rho}^{c}A_{\sigma}^{d} g^{\mu \rho} g^{\nu \sigma}.$$
Now to read off the Feynman rule for the corresponding vertex, you have to symmetrize/anti-symmetrize the coefficient in front of the four fields according to the symmetry of the objects. The ##f^{abc}## are totally antisymmetric, and this gives the 12 expressions with the corresponding signs in the Feynman rule given in Fig. 16.1.

In a more graphical way you can think of the vertex as the contraction of the four gluon fields in the interaction Lagrangian written above with the external-field points with the external lines amputated. Then you can collect all the combinations by just building up this Feynman diagram. Note that Peskin/Schroeder uses the usual convention to give the sum of all these contractions. This you have to take into account when using the Feynman rules and determining the symmetry factors of the diagram.
 

1. What is a non-abelian gauge field?

A non-abelian gauge field is a type of field in physics that describes the interactions between particles. It is a generalization of the concept of an abelian gauge field, which only considers interactions between particles that have a single charge. Non-abelian gauge fields allow for interactions between particles with multiple charges, making them more complex but also more powerful in describing the behavior of particles.

2. What is a 3-vertex in non-abelian gauge fields?

A 3-vertex in non-abelian gauge fields refers to a specific type of interaction between three particles. In this interaction, one particle emits a gauge boson, or force carrier particle, which is then absorbed by one of the other two particles. This process is important in understanding the behavior of particles in non-abelian gauge fields and is often represented graphically as a three-point vertex.

3. What is a 4-vertex in non-abelian gauge fields?

Similar to the 3-vertex, a 4-vertex in non-abelian gauge fields refers to an interaction between four particles. In this case, two particles emit and absorb gauge bosons, creating a more complex interaction. This type of vertex is important in understanding how particles interact in non-abelian gauge fields and is often represented graphically as a four-point vertex.

4. How do non-abelian gauge fields differ from abelian gauge fields?

Non-abelian gauge fields differ from abelian gauge fields in several ways. One of the main differences is that non-abelian gauge fields allow for interactions between particles with multiple charges, while abelian gauge fields only consider interactions with a single charge. Additionally, non-abelian gauge fields have more complex mathematical structures and require more advanced techniques to study and understand.

5. What are some real-world applications of non-abelian gauge fields?

Non-abelian gauge fields have a wide range of applications in physics, particularly in the study of subatomic particles and their interactions. For example, they are essential in understanding the strong nuclear force, which holds together the protons and neutrons in an atom's nucleus. Non-abelian gauge fields also play a role in theories such as the Standard Model, which describes the fundamental particles and forces in the universe.

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