# Trace of this yang-mills operator

## Main Question or Discussion Point

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,

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

So, what i want to compute is,

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

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

$$K(t;x,y;D)=<x|e^{-tD}|y>$$

and the propagator

$$D^-1=\int_0^\infty dt K(t;,x,y;D)$$

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

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

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!