- #1
LAHLH
- 405
- 2
Hi,
In \phi^4 theory, Srednicki deduces the self energy (equation 31.5), as:
\Pi(k^2)=\frac{\lambda}{2(4\pi)^2)} \left[ \frac{2}{\epsilon}+1+ln\left(\frac{\mu}{m^2}\right)\right]m^2-Ak^2-Bm^2
I understand how he gets to this just fine, but now I'm trying to impose the usual on-shel (OS) renormalization scheme, which I believe means \Pi(-m^2)=0 and \Pi'(-m^2)=0 (Which are due to needing the exact prop to have poles and residues in correspondance with the Lehman-Callen form of it).
I'm having some issues doing this and I really don't understand why. Firstly I set (to remove the mu depedence and the 1/epsilon infinity) :
B=\frac{\lambda}{16\pi^2}\left[ \frac{1}{\epsilon}+\frac{1}{2}+\kappa_B+ln\left(\frac{\mu}{m^2}\right)\right].
Thus
\Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_Bm^2-Ak^2
But now, imposing \Pi(-m^2)=0 leads to:
A=\frac{\lambda}{16\pi^2}\kappa_B
So,
\Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_B(m^2+k^2)
Finally, imposing \Pi'(-m^2)=0 leads to:
-\frac{\lambda}{16\pi^2}\kappa_B=0
and thus \Pi(k^2)=0
I really don't know what I'm doing that could be possibly wrong. Appreciate any help whatsoever, thanks.
In \phi^4 theory, Srednicki deduces the self energy (equation 31.5), as:
\Pi(k^2)=\frac{\lambda}{2(4\pi)^2)} \left[ \frac{2}{\epsilon}+1+ln\left(\frac{\mu}{m^2}\right)\right]m^2-Ak^2-Bm^2
I understand how he gets to this just fine, but now I'm trying to impose the usual on-shel (OS) renormalization scheme, which I believe means \Pi(-m^2)=0 and \Pi'(-m^2)=0 (Which are due to needing the exact prop to have poles and residues in correspondance with the Lehman-Callen form of it).
I'm having some issues doing this and I really don't understand why. Firstly I set (to remove the mu depedence and the 1/epsilon infinity) :
B=\frac{\lambda}{16\pi^2}\left[ \frac{1}{\epsilon}+\frac{1}{2}+\kappa_B+ln\left(\frac{\mu}{m^2}\right)\right].
Thus
\Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_Bm^2-Ak^2
But now, imposing \Pi(-m^2)=0 leads to:
A=\frac{\lambda}{16\pi^2}\kappa_B
So,
\Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_B(m^2+k^2)
Finally, imposing \Pi'(-m^2)=0 leads to:
-\frac{\lambda}{16\pi^2}\kappa_B=0
and thus \Pi(k^2)=0
I really don't know what I'm doing that could be possibly wrong. Appreciate any help whatsoever, thanks.