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?


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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – AccidentalFourierTransform, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 2
    $\begingroup$ 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

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.

  • $\begingroup$ 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

Not the answer you're looking for? Browse other questions tagged or ask your own question.