 #1
Markus Kahn
 112
 14
 Homework Statement:

Consider $\phi^4$ theory described by
$$\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^2 +\frac{1}{2}m^2\phi^2  \frac{\lambda}{4!}\phi^4.$$
We want to renormalize the theory. Assume that the renormalization for the 1PI 4point function has been done, resulting in a shift of $\lambda$, i.e.
$$\lambda_{r e n}=\lambda+\frac{\lambda^{2}}{32 \pi^{2}} \int_{0}^{1} d x\left[3\left(\frac{2}{\varepsilon}\gamma_{E}+\ln (4 \pi)\right)\ln \left(m^{2}4 m^{2} x(1x)\right)2 \ln \left(m^{2}\right)\right].$$
We are left with renormalizing the 2point function. Show that the renormalized mass ##m_{ren}## is given by
$$m_{r e n}^{2} = m^2 + \frac{\lambda}{2(4 \pi)^{\frac{d}{2}}} \frac{\Gamma\left(1\frac{d}{2}\right)}{\left(m^{2}\right)^{1d / 2}}.$$
 Relevant Equations:
 All given above.
Before I start, let me say that I have looked into textbooks and I know this is a standard problem, but I just can't get the result right...
My attempt goes as follows:
$$I(p^2)_{p^2=m^2}=0 \quad\text{and}\quad \frac{d}{dp^2}I(p^2)_{p^2m^2}=0.$$ Assuming this is true (I need to think about this a bit more...), how is one supposed to find ##m_{ren}## and ##Z##?
My attempt goes as follows:
 We notice that the amplitude of this diagram is given by $$\begin{align*}K_2(p) &= \frac{i(i m)^{2}(i)^{4}}{\left(p^{2}m^{2}+i \epsilon\right)^{2}} \frac{1}{2} \int \frac{i \mathrm{~d} \ell^{4}}{(2 \pi)^{4}} \frac{1}{\ell^{2}m^{2}+i \epsilon} \frac{1}{(p\ell)^{2}m^{2}+i \epsilon}\\&=: \frac{i(i m)^{2}(i)^{4}}{\left(p^{2}m^{2}+i \epsilon\right)^{2}}i I\left(p^{2}\right).\end{align*}$$
 We then remember that the 2point function can be expanded as shown in this diagram. The amplitude of this diagram is then given by $$ \begin{align*}K(p) &= \sum_i K_i(p)= \frac{1}{p^{2}m^{2}+i \epsilon} \sum_{j=0}^{\infty}\left(\frac{m^{2} I\left(p^{2}\right)}{p^{2}m^{2}+i \epsilon}\right)^{j}+\ldots \\&= \frac{1}{p^{2}m^{2}m^2I(p^2) +i \epsilon}+ \ldots.\end{align*}$$ If I'm not mistaken, this is essentially what one finds in Peskins book in eq. (10.27).
 What we now need is a renormalization condition. This was provided as a hint to the exercise and is given by $$K(p) = \frac{i Z }{p^2m_{ren}^2}+(\text{terms regular at }p^2=m^2).$$ Note that ##Z## is defined by ##\sqrt{Z}\phi_{ren}=\phi##.
$$I(p^2)_{p^2=m^2}=0 \quad\text{and}\quad \frac{d}{dp^2}I(p^2)_{p^2m^2}=0.$$ Assuming this is true (I need to think about this a bit more...), how is one supposed to find ##m_{ren}## and ##Z##?
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