- #1
HomogenousCow
- 737
- 213
Beginning of page 303 of Schawrtz's QFT text (section 16.1.1), there's a part on the renormalization of the scalar propagator in ##\phi^3## theory to second order in ##g## the bare coupling. He writes (quote begins):
$$M(Q)=M^0 (Q)+M^1 (Q)=\frac{g^2}{Q^2}(1-\frac{1}{32 \pi^2} \frac{g^2}{Q^2}\text{ln}{ \frac{Q^2}{\Lambda^2}}+...) $$
Note that ##g## is not a number in ##\phi^3## theory but has dimensions of mass (...) Let us substitute for ##g## a new Q-dependent variable ##\tilde{g}^2=\frac{g^2}{Q^2}##, which is dimensionless. Then,
$$M(Q)=\tilde{g}^2-\frac{1}{32 \pi^2} \tilde{g}^4\text{ln}{ \frac{Q^2}{\Lambda^2}}+... $$
Then we can define a renormalized coupling ##\tilde{g}_R^2=M(Q_0)## (...)
It follows that ##\tilde{g}_R^2## is a formal power series in ##\tilde{g}##:
$$\tilde{g}_R^2=\tilde{g}^2-\frac{1}{32 \pi^2} \tilde{g}^4\text{ln}{ \frac{Q_0^2}{\Lambda^2}}+... $$
(...)
Substituting into ... produces a prediction fo rthe matrix element at the scale Q in terms of the matrix element at the scale ##Q_0##
$$M(Q)=\tilde{g}_R^2-\frac{1}{32 \pi^2} \tilde{g}_R^4\text{ln}{ \frac{Q^2}{Q_0^2}}+... $$
(End quote)
So I've got a real issue with this derivation, it looks like he's treating the ##\tilde{g}##s in the 2nd and 3rd equations as being the same when this is clearly not so. Shouldn't the two be related by
$$\tilde{g}^2(Q)=\frac{Q_0^2}{Q^2}\tilde{g}^2(Q_0)?$$ I'm very confused.
$$M(Q)=M^0 (Q)+M^1 (Q)=\frac{g^2}{Q^2}(1-\frac{1}{32 \pi^2} \frac{g^2}{Q^2}\text{ln}{ \frac{Q^2}{\Lambda^2}}+...) $$
Note that ##g## is not a number in ##\phi^3## theory but has dimensions of mass (...) Let us substitute for ##g## a new Q-dependent variable ##\tilde{g}^2=\frac{g^2}{Q^2}##, which is dimensionless. Then,
$$M(Q)=\tilde{g}^2-\frac{1}{32 \pi^2} \tilde{g}^4\text{ln}{ \frac{Q^2}{\Lambda^2}}+... $$
Then we can define a renormalized coupling ##\tilde{g}_R^2=M(Q_0)## (...)
It follows that ##\tilde{g}_R^2## is a formal power series in ##\tilde{g}##:
$$\tilde{g}_R^2=\tilde{g}^2-\frac{1}{32 \pi^2} \tilde{g}^4\text{ln}{ \frac{Q_0^2}{\Lambda^2}}+... $$
(...)
Substituting into ... produces a prediction fo rthe matrix element at the scale Q in terms of the matrix element at the scale ##Q_0##
$$M(Q)=\tilde{g}_R^2-\frac{1}{32 \pi^2} \tilde{g}_R^4\text{ln}{ \frac{Q^2}{Q_0^2}}+... $$
(End quote)
So I've got a real issue with this derivation, it looks like he's treating the ##\tilde{g}##s in the 2nd and 3rd equations as being the same when this is clearly not so. Shouldn't the two be related by
$$\tilde{g}^2(Q)=\frac{Q_0^2}{Q^2}\tilde{g}^2(Q_0)?$$ I'm very confused.