- #1

- 409

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## Main Question or Discussion Point

Hi,

In [tex] \phi^4[/tex] theory, Srednicki deduces the self energy (equation 31.5), as:

[tex] \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 [/tex]

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 [tex] \Pi(-m^2)=0 [/tex] and [tex] \Pi'(-m^2)=0 [/tex] (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) :

[tex] B=\frac{\lambda}{16\pi^2}\left[ \frac{1}{\epsilon}+\frac{1}{2}+\kappa_B+ln\left(\frac{\mu}{m^2}\right)\right] [/tex].

Thus

[tex] \Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_Bm^2-Ak^2 [/tex]

But now, imposing [tex] \Pi(-m^2)=0 [/tex] leads to:

[tex] A=\frac{\lambda}{16\pi^2}\kappa_B [/tex]

So,

[tex] \Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_B(m^2+k^2) [/tex]

Finally, imposing [tex] \Pi'(-m^2)=0 [/tex] leads to:

[tex] -\frac{\lambda}{16\pi^2}\kappa_B=0 [/tex]

and thus [tex] \Pi(k^2)=0 [/tex]

I really don't know what I'm doing that could be possibly wrong. Appreciate any help whatsoever, thanks.

In [tex] \phi^4[/tex] theory, Srednicki deduces the self energy (equation 31.5), as:

[tex] \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 [/tex]

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 [tex] \Pi(-m^2)=0 [/tex] and [tex] \Pi'(-m^2)=0 [/tex] (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) :

[tex] B=\frac{\lambda}{16\pi^2}\left[ \frac{1}{\epsilon}+\frac{1}{2}+\kappa_B+ln\left(\frac{\mu}{m^2}\right)\right] [/tex].

Thus

[tex] \Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_Bm^2-Ak^2 [/tex]

But now, imposing [tex] \Pi(-m^2)=0 [/tex] leads to:

[tex] A=\frac{\lambda}{16\pi^2}\kappa_B [/tex]

So,

[tex] \Pi(k^2)=-\frac{\lambda}{16\pi^2}\kappa_B(m^2+k^2) [/tex]

Finally, imposing [tex] \Pi'(-m^2)=0 [/tex] leads to:

[tex] -\frac{\lambda}{16\pi^2}\kappa_B=0 [/tex]

and thus [tex] \Pi(k^2)=0 [/tex]

I really don't know what I'm doing that could be possibly wrong. Appreciate any help whatsoever, thanks.