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## 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,

[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!

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!