Dimensional regularization and renormalization scale.

In summary, there are different approaches to introducing a scale in order to keep the coupling constant dimensionless in the Lagrangian in QFT. While Ryder introduces a scale ##\mu##, Weinberg and Peskin and Schroeder choose not to do so. However, Weinberg avoids having dimensionful arguments in logarithms by directly calculating the renormalized values of certain quantities. This is not preferred by some, as it is clearer to calculate the regularized but unrenormalized values first and then introduce a scale to avoid dimensionful arguments in logarithms.
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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.
 
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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 [itex]Z_3[/itex] 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 [itex]d \rightarrow 4[/itex], 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 [itex]d=4[/itex], and that's where the renormalization scale enteres in dimensional regularization.
 

Related to Dimensional regularization and renormalization scale.

1. What is dimensional regularization?

Dimensional regularization is a mathematical technique used in quantum field theory to deal with divergent integrals. It involves analytically extending the number of spacetime dimensions from the usual four to a complex number, allowing for the cancellation of infinities in the calculations.

2. How does dimensional regularization work?

Dimensional regularization works by introducing a parameter, often denoted as D, which represents the number of spacetime dimensions. This allows for the evaluation of integrals in D dimensions, which can then be analytically continued to the physical dimension of four. The resulting expressions are finite and can be used to make predictions in quantum field theory.

3. What is the renormalization scale in dimensional regularization?

The renormalization scale in dimensional regularization is a parameter used to absorb the dependence of physical observables on the arbitrary scale introduced in the regularization process. It represents the energy scale at which the theory is being probed and can be chosen at will, but should be kept consistent throughout calculations to ensure accurate predictions.

4. Why is dimensional regularization preferred over other regularization methods?

Dimensional regularization is preferred over other regularization methods because it preserves the symmetries of the underlying physical theory, such as Lorentz invariance and gauge invariance. It also allows for the calculation of higher-order corrections in a systematic way, making it a powerful tool in quantum field theory calculations.

5. What are the limitations of dimensional regularization?

One limitation of dimensional regularization is that it cannot be used in all cases, particularly in theories with non-renormalizable interactions. It also requires careful handling of poles in the complex D-plane, which can lead to ambiguities in the final result. Additionally, it may not be as intuitive as other regularization methods, making it more challenging for non-experts to understand and use.

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