Subtraction scheme invariance in QFT

Question

I'm currently reading Schwartz's QFT text and I'm confused on how observables are supposed to be independent of subtraction schemes. In the text it seems that the renormalized loop diagrams are different depending on the subtraction scheme chosen. So is it that the renormalized parameters have different meanings depending on the subtraction scheme and/or the regulator?

Answer 1

It's because we define, for instance, the mass to be the pole of the renormalized propagator. Namely in yukawa theory the propagator will be given by something like

$$G^R = \frac{1}{Z} \frac{i}{\not{p} - m_0 + \Sigma(\not{p})} $$

where $\Sigma_R$ is the sum over all 1PI insertions, and $m_R = \frac{1}{Z_m} m_0$ We can absorb the $Z, Z_m$ and plug in in for $m_0$ and redefine $\Sigma$ so that

$$ G^R = \frac{1}{1+\delta}\frac{i}{\not{p} - m_0 + \Sigma_2(\not{p})} \\ \boxed{G^R= \frac{i}{\not{p} - m_R + \Sigma_R(\not{p}, m_R, Z_m, Z)} } $$

Now the point is that you don't need your counter terms (or $Z_i$s) to be the same as somebody else's, and you don't have to agree on the $m_R$s either. Rather, all you need at the end of the day is to make sure that you have a pole at $\not{p} = m_P$. That is,

$$ \begin{cases} m_P - m_R + \Sigma_R(\not{p} = m_P) = 0 \\ 1 = \lim_{\not{p} \to m_P} \frac{\not{p} - m_P}{ \not{p} - m_R + \Sigma_R(\not{p}, m_R, Z_m, Z)} \end{cases} $$

where a stricter condition on $\Sigma_R$ can be found by using L'Hôpitals rule. Note that this is just the mathematical definition of a simple pole at $m_P$.

The condition that the mass $m_P$ gives a pole to the propagator is enforced in all schemes (it just looks different for different schemes). So no matter how we define $M_R$ or $\Sigma$, as long as we enforce the fact that the pole of the propagator is the physical mass then it doesn't matter how you get there.