Show How Theta Term in QCD Lagrangian is a Total Derivative

In summary, the conversation discusses how to show that the theta term in the QCD Lagrangian can be written as a total derivative. The use of the wedge product and the Levi-Civita symbol is suggested, but the derivation is unclear. The suggestion to use the definition of the dual and relabel the indices is also mentioned. Finally, a possible derivation from a paper is cited. The expert summarizer provides a step-by-step explanation for deriving the term as a total derivative.
  • #1
Kara386
208
2
I'm trying to show that the theta term in the QCD Lagrangian, ##\alpha G^a_{\mu\nu} \widetilde{G^a_{\mu\nu}}##, can be written as a total derivative, where
##\begin{equation} G^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu}G^a_{\mu}-gf_{bca}G^b_{\mu}G^c_{\nu} \end{equation} ##
##\widetilde{G^a_{\mu\nu}} = \frac{1}{2} \epsilon^{\mu\nu\lambda\rho}G^a_{\lambda\rho} ##
And ##\epsilon## is the Levi-Civita symbol.

The thing is, plenty of papers and books say that this is a total derivative, and give absolutely no indication of how to show that. Well, it's been suggested I should use the wedge product, but I've never encountered it and I thought brute force should work fine. So I expand ##G^a_{\lambda\rho}## using the first equation, multiply the two and expand the brackets:
##\frac{1}{2} \epsilon^{\mu\nu\lambda\rho} (\partial_{\lambda}G^a_{\rho}\partial_{\mu}G^a_{\nu}-\partial_{\lambda}G^a_{\rho}\partial_{\nu}G^a_{\mu}-\partial_{\lambda}G^a_{\rho}gf_{bca}G^b_{\mu}G^c_{\nu}-\partial_{rho}G^a_{\lambda}\partial_{\mu}G^a_{\nu}+\partial_{\rho}G^a_{\lambda}\partial_{\nu}G^a_{\mu}+\partial_{\rho}G^a_{\lambda}gf_{bca}G^b_{\mu}G^c_{\nu}-gf_{bca}G^b_{\lambda}G^c_{\rho}\partial_{\mu}G^a_{\nu}+gf_{bca}G^b_{\lambda}G^c_{\rho}\partial_{\nu}G^a_{\mu}+g^2f^2_{bca}G^b_{\lambda}G^c_{\rho}G^b_{\mu}G^c_{\nu})##
So that looks a mess, but because the Levi-Civita symbol is antisymmetric that equals
##\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}(4\partial_{\lambda}G^a_{\rho}\partial_{\mu}G^a_{\nu}+2\partial_{\rho}G^a_{\lambda}gf_{bca}G^b_{\mu}G^c_{\nu}+2gf_{bca}G^b_{\lambda}G^c_{\rho}\partial_{\nu}G^a_{\mu}+g^2f^2_{bca}G^b_{\lambda}G^c_{\rho}G^b_{\mu}G^c_{\nu})##
What it's meant to be, somehow, is
##\partial_{\mu}\left(\alpha \epsilon^{\mu\nu\lambda\rho}G^a_{\nu}(G^a_{\lambda\rho} +\frac{1}{3}g f_{bca}G^b_{\lambda}G^c_{\rho}) \right)##
But I really, really can't see how you'd get there! I'm also a little confused about why the indices ##\mu## and ##\nu## are different in the definition of the dual, but the ##a## is kept the same - don't I need to relabel b,c,a?

I'd massively appreciate any pointers, I've been stuck on this for quite a while!
 
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  • #3
Maybe this derivation appears in Srednicki's textbook or its solution manual.
 
  • #4
for those terms with no g, you can use ∂A∂B = ∂(A∂B) - A∂∂B,then you can find the second term εμναβ∂∂ should be zero.
for terms with g, you need ∂(ABC) = ∂ABC+A∂BC+AB∂C, and combining them with εμναβ fabc, you will see the right hand three terms are equivalent with each other, that's why you get a 1/3 before AAA.
as for the last term with g2, you will find it is zero after you use the property of εμναβ and fabcfaef.
 

1. What is the Theta term in the QCD Lagrangian?

The Theta term is a term in the Quantum Chromodynamics (QCD) Lagrangian that describes the interactions of quarks and gluons, the fundamental particles that make up protons and neutrons. It is a total derivative term that arises due to the topological properties of QCD.

2. Why is the Theta term important in QCD?

The Theta term is important because it plays a crucial role in understanding the dynamics of strong interactions between quarks and gluons. It also helps explain the properties of hadrons, such as their mass and spin, and provides insights into the nature of confinement and asymptotic freedom in QCD.

3. How is the Theta term related to the strong CP problem?

The Theta term is closely related to the strong CP problem, which is a puzzle in particle physics that asks why the strong nuclear force conserves parity and charge conjugation, but not CP (the combination of charge conjugation and parity). The Theta term is a possible source of CP violation in QCD.

4. Can the Theta term be experimentally measured?

Unfortunately, the Theta term cannot be directly measured in experiments due to its total derivative nature. However, its effects can be indirectly measured through experiments that study CP violation and the properties of hadrons.

5. How has the understanding of the Theta term evolved over time?

The Theta term has been a topic of ongoing research and debate in particle physics. Initially, it was thought to be a fundamental parameter of QCD, but the discovery of asymptotic freedom in the 1970s led to the realization that it is a total derivative term. Currently, the Theta term is still an area of active research, with scientists trying to understand its role in CP violation and the strong CP problem.

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