- #1

- 76

- 5

## Main Question or Discussion Point

I have a simple sounding question about QCD and its scale parameter [tex]\Lambda[/tex]:

The general consensus amongst field theorists is that QCD is not a well defined theory by its Lagrangian and the quantization procedure alone. One needs an additional parameter [tex]\Lambda[/tex], which sets the scale of renormalization. The coupling constant becomes strong roughly at the scale of [tex]\Lambda[/tex]. While the QCD renormalization is a well defined and self consistent prescription, I find the need for [tex]\Lambda[/tex] in order to define the theory itself a bit disturbing. [tex]\Lambda[/tex] should only be needed to define perturbative QCD, where renormalization is needed. However, the non-perturbative QCD (such as confinement and hadron masses) should not depend on [tex]\Lambda[/tex]. In fact, I am not aware of any systematic way of introducing [tex]\Lambda[/tex] into QCD outside the context of perturbation theory and renormalization.

One may argue that the usual renormalization prescription is not merely perturbative. The renormalization group flows and the Callan-Symanzik equation may apply to non-perturbative interactions. Unfortunately, I have never seen a rigorous derivation of these totally and completely outside the context of the perturbation theory. (I think even the non-perturbative lattice calculations use [tex]\Lambda[/tex]) If this is not so, please refer me to a reference (thanks in advance.) On the other hand, if you believe this statement to be true, then we should go back to the first paragraph above and ask how we can define QCD without the [tex]\Lambda[/tex] parameter. Alternatively, can one prove that it is still necessary to introduce the parameter [tex]\Lambda[/tex], with no reference whatsoever to the perturbation theory or to renormalization?

The general consensus amongst field theorists is that QCD is not a well defined theory by its Lagrangian and the quantization procedure alone. One needs an additional parameter [tex]\Lambda[/tex], which sets the scale of renormalization. The coupling constant becomes strong roughly at the scale of [tex]\Lambda[/tex]. While the QCD renormalization is a well defined and self consistent prescription, I find the need for [tex]\Lambda[/tex] in order to define the theory itself a bit disturbing. [tex]\Lambda[/tex] should only be needed to define perturbative QCD, where renormalization is needed. However, the non-perturbative QCD (such as confinement and hadron masses) should not depend on [tex]\Lambda[/tex]. In fact, I am not aware of any systematic way of introducing [tex]\Lambda[/tex] into QCD outside the context of perturbation theory and renormalization.

One may argue that the usual renormalization prescription is not merely perturbative. The renormalization group flows and the Callan-Symanzik equation may apply to non-perturbative interactions. Unfortunately, I have never seen a rigorous derivation of these totally and completely outside the context of the perturbation theory. (I think even the non-perturbative lattice calculations use [tex]\Lambda[/tex]) If this is not so, please refer me to a reference (thanks in advance.) On the other hand, if you believe this statement to be true, then we should go back to the first paragraph above and ask how we can define QCD without the [tex]\Lambda[/tex] parameter. Alternatively, can one prove that it is still necessary to introduce the parameter [tex]\Lambda[/tex], with no reference whatsoever to the perturbation theory or to renormalization?