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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:)

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# Conformal invariance of gluon amplitudes

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