- #1
Hao
- 93
- 0
In Eq 11.72 in the QFT text by Peskin, the following equality is stated:
[tex]i\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\log(k_{E}^{2}+m^{2})=-i\frac{\partial}{\partial\alpha}\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}[/tex]
This suggests that
[tex]\log(k_{E}^{2}+m^{2})=-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}[/tex]
However, I can't see how this identity follows. Differentiating the right hand side gives
[tex]-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}=\frac{\alpha}{(k_{E}^{2}+m^{2})^{\alpha+1}}|_{\alpha=0}\rightarrow\frac{0}{(k_{E}^{2}+m^{2})^{1}}[/tex]
Any help would be greatly appreciated.
[tex]i\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\log(k_{E}^{2}+m^{2})=-i\frac{\partial}{\partial\alpha}\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}[/tex]
This suggests that
[tex]\log(k_{E}^{2}+m^{2})=-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}[/tex]
However, I can't see how this identity follows. Differentiating the right hand side gives
[tex]-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}=\frac{\alpha}{(k_{E}^{2}+m^{2})^{\alpha+1}}|_{\alpha=0}\rightarrow\frac{0}{(k_{E}^{2}+m^{2})^{1}}[/tex]
Any help would be greatly appreciated.