- #1

- 728

- 194

$$M(Q)=M^0 (Q)+M^1 (Q)=\frac{g^2}{Q^2}(1-\frac{1}{32 \pi^2} \frac{g^2}{Q^2}\text{ln}{ \frac{Q^2}{\Lambda^2}}+...) $$

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.