Dimensional regularization and renormalization scale.

[center o bass]

Hi. I have observed that Ryder in his book on QFT before doing dimensional regularization introduces a scale $\mu$ in order to keep the coupling constant dimensionless in the lagragnian. However in two other books; Weinberg and Peskin and Schroeder, they do not introduce this scale in the same way. Is it really up to preference if one wants to do this or not? If so what are the pro's and con's by introducing the scale in this way? And why is it up to one to choose?

Personally I feel that the most natural way to introduce a scale is through the renormalization prescription; for example $i\Gamma^{(4)}(\mu) = g_R$ in $\phi^4$ theory.

 

[vanhees71]

Don't use Peskin Schroeder on this issue. In this book you find logarithms with dimensionful arguments, and this even in the very chapter on the renormalization group! This can only lead to confusion! There must never ever by dimensionful quantities in logarithms (or any other transcendental function for that matter).

Weinberg choses not to introduce the scale in the chapter on electrodynamics (ch. 11 in vol. 1), but he manages to never have logarithms with dimensionful quantities by the trick of directly calculating the renormalized values of the photon-field normalization factor $Z_3$ by doing the appropriate subtraction of the photon-polarization tensor (or photon self-energy) at the photon momentum 0, i.e., in the on-shell scheme, which is allowed as long as you keep the Dirac fields massive. In this way he directly can calculate the value for $d \rightarrow 4$, without ever having dimensionful arguments of logarithms.

Personally, I don't like this, because it is way more clear to calculate the regularized but unrenormalized values first, and there you must introduce a scale in order to avoid dimensionful arguments of the logarithms when doing the Laurent expansion around $d=4$, and that's where the renormalization scale enteres in dimensional regularization.