Trace of this yang-mills operator

[CGH]

Hi there,

I'm trying to compute the trace of an operator found here: http://inspirebeta.net/record/360247 (eq 7.5)

I'm not going to make you read the article, so i state the problem:

I have the following operator in a Yang-.Mills theory, using the background field method,

[tex]
D_0=-(DD)^{ab}_{\mu\nu}+2gf^{abc}F^c_{\mu\nu}
[/tex]

So, what i want to compute is,

[tex]
Tr([D_0+R_k(D_0)]^{-1}R_k'(D_0))
[/tex]

and trace is integration over coordinates and sum over color indices and R is some function.

What I'm trying to do is to use a known result from here: http://arxiv.org/abs/hep-th/0306138
The heat kernel is defined as

[tex]
K(t;x,y;D)=<x|e^{-tD}|y>
[/tex]

and the propagator

[tex]
D^-1=\int_0^\infty dt K(t;,x,y;D)
[/tex]

so, using (2.19), (2.21) and (4.34) i get (i 4 dimensions)

[tex]
K(t;x,x,D_0)=\frac{2N}{(4\pi)^2}\frac{5}{6}F^2
[/tex]

now, how can i use all this to compute the above trace to order F^2?, i don't know how to get the result from the paper, can anyone help me?

Saludos!

 

[AndyW]

Hello,

Thank you for sharing your problem with us. I understand the importance of accurately computing the trace of an operator in a Yang-Mills theory. After reviewing the forum post and the articles you mentioned, I have some suggestions that may help you in your computation.

Firstly, it is important to note that the operator D_0 in your problem is a combination of two terms - the first term is the covariant derivative (DD)^{ab}_{\mu\nu} and the second term is the Yang-Mills field strength tensor multiplied by the structure constant f^{abc}. Therefore, when computing the trace, you will have to consider both of these terms separately.

To start, you can use the known result from the article http://arxiv.org/abs/hep-th/0306138 to compute the heat kernel for each term separately. This will give you two separate propagators, D_0^-1 and (DD)^-1. You can then use these propagators to calculate the trace of the operator D_0 as follows:

Tr(D_0) = Tr(-(DD)^{ab}_{\mu\nu}) + Tr(2gf^{abc}F^c_{\mu\nu})

= Tr(-(DD)^{ab}_{\mu\nu}) + 2gf^{abc}Tr(F^c_{\mu\nu})

= Tr((DD)^-1) + 2gf^{abc}Tr(F^c_{\mu\nu})

= ∫d^4x Tr((DD)^-1) + 2gf^{abc}∫d^4x Tr(F^c_{\mu\nu})

Where Tr(F^c_{\mu\nu}) is the trace of the Yang-Mills field strength tensor, which you can compute using the known result from the paper. Finally, you can use the result from the paper to compute the trace of (DD)^-1 and substitute it in the above equation to get the final result.

I hope this helps you in your computation. Please let me know if you have any further questions or need any clarification.