- #1

earth2

- 86

- 0

Hey,

I've been reading a bit in the QFT book by Pierre Ramond about loops and i stumbled about some derivations in appendix B that I can't follow.

First Ramond derives a formula for integration of momenta in arbitrary dimensions

[tex]\int \frac{d^Nl}{(l^2+2p\cdot l+b^2)^A}=\pi^{N/2}\frac{\Gamma(A-N/2)}{\Gamma(A)}\frac{1}{(b^2-p^2)^{A-N/2}}[/tex]

This is fine with me. What I don't get is the following. He says if we differentiate this formula with respect to [tex]p^\mu[/tex] we'd get

[tex]\int \frac{d^Nl \quad l_\mu}{(l^2+2p\cdot l+b^2)^A}=\pi^{N/2}\frac{\Gamma(A-N/2)}{\Gamma(A)}\frac{-p_\mu}{(b^2-p^2)^{A-N/2}}[/tex]

I don't see how that comes about. If I differentiate the first formula wrt p I get

[tex]\int \frac{d^Nl \quad (-A)2l_\mu}{(l^2+2p\cdot l+b^2)^{A+1}}=\pi^{N/2}\frac{\Gamma(A-N/2)}{\Gamma(A)}\frac{-p_\mu(-A+N/2)}{(b^2-p^2)^{A-N/2+1}}[/tex]

I haven't found another derivation of this formula, nor do I see where I go wrong...

Can anyone help my with this?

Thanks

earth2

I've been reading a bit in the QFT book by Pierre Ramond about loops and i stumbled about some derivations in appendix B that I can't follow.

First Ramond derives a formula for integration of momenta in arbitrary dimensions

[tex]\int \frac{d^Nl}{(l^2+2p\cdot l+b^2)^A}=\pi^{N/2}\frac{\Gamma(A-N/2)}{\Gamma(A)}\frac{1}{(b^2-p^2)^{A-N/2}}[/tex]

This is fine with me. What I don't get is the following. He says if we differentiate this formula with respect to [tex]p^\mu[/tex] we'd get

[tex]\int \frac{d^Nl \quad l_\mu}{(l^2+2p\cdot l+b^2)^A}=\pi^{N/2}\frac{\Gamma(A-N/2)}{\Gamma(A)}\frac{-p_\mu}{(b^2-p^2)^{A-N/2}}[/tex]

I don't see how that comes about. If I differentiate the first formula wrt p I get

[tex]\int \frac{d^Nl \quad (-A)2l_\mu}{(l^2+2p\cdot l+b^2)^{A+1}}=\pi^{N/2}\frac{\Gamma(A-N/2)}{\Gamma(A)}\frac{-p_\mu(-A+N/2)}{(b^2-p^2)^{A-N/2+1}}[/tex]

I haven't found another derivation of this formula, nor do I see where I go wrong...

Can anyone help my with this?

Thanks

earth2

Last edited: