- #1
pkwei99
- 1
- 0
Hi,
I'm very ashamed to not understand how even the simplest gluon amplitudes are conformally invariant. See eg http://arxiv.org/abs/hep-th/0312171 pages 11-12.
[tex]
M(1^-,2^-,3^+)=\delta(\sum_i \lambda_i\tilde{\lambda}_i)\frac{\langle12\rangle^4}{\langle12\rangle \langle 23\rangle\langle31\rangle}
[/tex]
the dilation operator is:
[tex]
D\sim \lambda\frac{\partial}{\partial \lambda}+\tilde{\lambda}\frac{\partial}{\partial \tilde{\lambda}}+2
[/tex]
First, I assume the dilation operator contains a sum over all particles. Next, Witten says the delta function carries weight -4 under D. Ok. Then he says that [tex]\langle 12\rangle^4[/tex] has weight 4. This I don't get. Doesn't it have weight 4 under just eg [tex]\lambda_1\frac{\partial}{\partial \lambda_1}[/tex]
So
[tex]D \langle12\rangle^4=\sum_i (\lambda\frac{\partial}{\partial \lambda}+\tilde{\lambda}\frac{\partial}{\partial \tilde{\lambda}}+2)\langle12\rangle^4=[(4+0+2)+(4+0+2)+(0+0+2) ] \langle12\rangle^4?=14\langle12\rangle^4[/tex]
Thanks for any help:)
I'm very ashamed to not understand how even the simplest gluon amplitudes are conformally invariant. See eg http://arxiv.org/abs/hep-th/0312171 pages 11-12.
[tex]
M(1^-,2^-,3^+)=\delta(\sum_i \lambda_i\tilde{\lambda}_i)\frac{\langle12\rangle^4}{\langle12\rangle \langle 23\rangle\langle31\rangle}
[/tex]
the dilation operator is:
[tex]
D\sim \lambda\frac{\partial}{\partial \lambda}+\tilde{\lambda}\frac{\partial}{\partial \tilde{\lambda}}+2
[/tex]
First, I assume the dilation operator contains a sum over all particles. Next, Witten says the delta function carries weight -4 under D. Ok. Then he says that [tex]\langle 12\rangle^4[/tex] has weight 4. This I don't get. Doesn't it have weight 4 under just eg [tex]\lambda_1\frac{\partial}{\partial \lambda_1}[/tex]
So
[tex]D \langle12\rangle^4=\sum_i (\lambda\frac{\partial}{\partial \lambda}+\tilde{\lambda}\frac{\partial}{\partial \tilde{\lambda}}+2)\langle12\rangle^4=[(4+0+2)+(4+0+2)+(0+0+2) ] \langle12\rangle^4?=14\langle12\rangle^4[/tex]
Thanks for any help:)