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earth2
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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: