# QCD lambda parameter

1. Apr 12, 2009

### fermi

I have a simple sounding question about QCD and its scale parameter $$\Lambda$$:

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 $$\Lambda$$, which sets the scale of renormalization. The coupling constant becomes strong roughly at the scale of $$\Lambda$$. While the QCD renormalization is a well defined and self consistent prescription, I find the need for $$\Lambda$$ in order to define the theory itself a bit disturbing. $$\Lambda$$ 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 $$\Lambda$$. In fact, I am not aware of any systematic way of introducing $$\Lambda$$ 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 $$\Lambda$$) 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 $$\Lambda$$ parameter. Alternatively, can one prove that it is still necessary to introduce the parameter $$\Lambda$$, with no reference whatsoever to the perturbation theory or to renormalization?

2. Apr 16, 2009

### fermi

Moderators:

Please consider moving this question to the Forum for Nuclear and Particle Physics. I posted it here originally, because this is the forum for Quantum Physics and Field Theory. But it appears that questions and discussions about QFT attract little attention here. My chances of getting answer from the particle physicists might be better.

Thank you

3. Apr 16, 2009

### xepma

Any QFT with interactions always has a scale parameter $$\Lambda$$. This is due to the need for renormalization. If you would naively start from some action S, in which one integrates over all momenta, one will always encounter divergences due to loop diagrams. Renormalization resolves this and introduces the concept of a scale parameter.

This is not some artifact of the theory - you can measure this scale dependence as well. Just look in what way the coupling constant depends on the energy scale. Or in the case of QCD, how the QCD becomes a weakly coupling theory at high energies (asympotitic freedom). The coupling constant doesn't just come into a play for perturbation theory; it also plays a role in non-perturbative objects. The simplest way to see this is that non-perturbative objects also carry energy, can interact etc, all of which depend on the original coupling constant.

So just to be clear: renormalization is a concept that goes beyond perturbation theory. Every interacting QFT needs to be renormalized, or else the theory is ill-defined. But since perturbation theory is usually the easiest, first calculational approach to any QFT it's not completely suprising you always see these two concepts go hand in hand.

4. Apr 16, 2009

### fermi

But that's perturbation theory: What's a "loop diagram" if defined outside the perturbation theory?

The need for $$\Lambda$$ is clear within the perturbation theory. The divergent integrals will force the issue. But for non-perturbative analysis, there are no loops (that I can undestand) and no divergent integrals to renormalize. Can you give me an example of a non-perturbative loop calculation?

5. Apr 17, 2009

### Haelfix

Lambda will show up for as you say for the Callan-Symanzik and renormalization group flow equations. The question is should you be surprised that nonperturbative effects still carry this dependance?

I think the answer is probably no (but no one really understands this fully either). Not just b/c you started from a manifestly perturbative framework in both those cases. Even for classical fluid dynamics you will typically find solutions or part solutions for some simplified model that involves behaviour that scales with some factor of energy. In the renormalization flow language you are moving between orbits around tentative fixed points, so its really no surprise that closed form solutions break down as you move far away from their regimes of validity. Ok, we haven't probed the full extent of the nonperturbative behaviour either, but still the hint remains.

Dualities also teach us that in many instances full nonperturbative solutions can on occassion be similar to perturbative solutions in some other context, so that should be another hint that this behaviour should persist.