QCD \Lambda Parameter: Definition & Its Role in Non-Perturbative QCD

In summary, the scale parameter \Lambda is needed in order to define the theory itself within the context of perturbation theory. However, non-perturbative QCD (such as confinement and hadron masses) should not depend on \Lambda. I am not aware of any systematic way of introducing \Lambda into QCD outside the context of perturbation theory and renormalization.
  • #1
fermi
76
5
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?
 
Physics news on Phys.org
  • #2
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
Any QFT with interactions always has a scale parameter [tex]\Lambda[/tex]. 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
xepma said:
Any QFT with interactions always has a scale parameter [tex]\Lambda[/tex]. 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...

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

The need for [tex]\Lambda[/tex] 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
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 occasion be similar to perturbative solutions in some other context, so that should be another hint that this behaviour should persist.
 

1. What is the QCD Lambda parameter?

The QCD Lambda parameter, also known as the QCD scale parameter, is a value that represents the energy scale at which the strong nuclear force, described by Quantum Chromodynamics (QCD), becomes strong enough to significantly affect the behavior of particles. It is an important parameter in non-perturbative QCD calculations.

2. How is the QCD Lambda parameter defined?

The QCD Lambda parameter is defined as the energy at which the strong coupling constant, which governs the strength of the strong force between quarks, becomes infinite. This energy scale is typically on the order of 200 MeV, or about 10-15 meters.

3. What is the role of the QCD Lambda parameter in non-perturbative QCD?

The QCD Lambda parameter plays a crucial role in non-perturbative QCD, which is the study of QCD at energies where perturbative methods do not work. This includes the study of hadrons, such as protons and neutrons, which are composed of quarks and are held together by the strong force. The QCD Lambda parameter is used in calculations to determine the masses and other properties of these particles.

4. How is the QCD Lambda parameter related to the Higgs boson?

The QCD Lambda parameter is not directly related to the Higgs boson, which is a particle that gives mass to other particles in the Standard Model of particle physics. However, the value of the QCD Lambda parameter affects the masses of particles, including the Higgs boson, through the process of hadronization, where quarks and gluons combine to form larger particles, such as protons and neutrons.

5. Can the QCD Lambda parameter be experimentally measured?

No, the QCD Lambda parameter cannot be directly measured in experiments. It is a theoretical parameter that is used in calculations to describe the behavior of the strong nuclear force. However, its value can be indirectly determined through measurements of other quantities, such as particle masses and scattering cross sections, which are affected by the QCD Lambda parameter.

Similar threads

  • Quantum Physics
Replies
5
Views
782
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
889
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
2K
  • Beyond the Standard Models
Replies
18
Views
3K
Replies
8
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
Back
Top