- #1
CGH
- 7
- 0
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!