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Perhaps this is a very basic question.

In chapter 14 of Srednicki's QFT textbook (2007), $O(g^2)$ loop corrections to the propagator of $\phi^3$ theory is discussed. However, I don't know how to derive the Eq. 14.34 on page 103 (I can't present the full context as it would be too long). From \begin{equation} \Pi(k^2)=\frac{1}{2}\alpha\Gamma(-1+\epsilon/2)\int_0^1\,dxD(4\pi\mu^2/D)^{\epsilon/2}-Ak^2-Bm^2+O(\alpha^2), (14.32) \end{equation} and take the $\epsilon \rightarrow 0$ limit using \begin{equation} A^{\epsilon/2}=1+\frac{\epsilon}{2}ln(A)+O(\epsilon^2) \end{equation} and the following property of Gamma functions \begin{equation} \Gamma(-n+x)=\frac{(-1)^n}{n!}\left[\frac{1}{x}-\gamma+\sum_{k=1}^nk^{-1}+O(x)\right], \end{equation} he got \begin{equation} \Pi(k^2)=-\frac{1}{2}\alpha\left[\left(\frac{2}{\epsilon}+1\right)\left(\frac{1}{6}k^2+m^2\right)+\int_0^1\,dxDln\left(\frac{4\pi\mu^2}{e^\gamma D}\right)\right]-Ak^2-Bm^2+O(\alpha^2), \qquad (14.34) \end{equation} where $\alpha\equiv\frac{g^2}{(4\pi)^3}$.

I don't know how to do the integral correctly (although it seems elementary). Can anyone familiar with this part help me out?

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closed as off-topic by AccidentalFourierTransform, Qmechanic Jun 15 '18 at 3:31

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    $\begingroup$ What is $D$ ? Does it depends on $k$ and $x$ ? If it is only the integral over $x$ that you don't know how to do, it might be easier just to give it directly (or ask Mathematica)... $\endgroup$ – Adam Dec 30 '13 at 17:04
  • $\begingroup$ @Adam : $D= x(1-x)k^2 +m^2$, see link (14.14) $\endgroup$ – Trimok Dec 30 '13 at 19:20
  • $\begingroup$ Sorry that I forgot to give D. Trimok is correct. The author also used \begin{equation} \int_0^1\,dxD=\frac{1}{6}k^2+m^2, \end{equation} which is easy knowing $D=x(1-x)k^2+m^2$. $\endgroup$ – D-K Dec 31 '13 at 0:35
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I find it's not a problem if one simply omits the $O(\epsilon)$ terms since we are taking the $\epsilon \rightarrow 0$ limit. The author just didn't state it clearly.

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  • $\begingroup$ The term with the integral is $O(1)$, not $O(\epsilon)$ and it can not be omitted (without changing regularisation scheme). $\endgroup$ – fqq Aug 13 '14 at 22:50

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