Mass of composite particles in a non abelian gauge theory.

In summary, non-Abelian Yang-Mills theories with massless fermions can produce massive composite particles, such as baryons. This is because at low energies, the coupling constant will logarithmically diverge and perturbation theory will break down, producing a new energy scale. In QCD, this scale is known as Lambda_QCD. So while the QCD Lagrangian may be scale invariant when the masses are set to zero, the quantum theory does not respect this symmetry and an energy scale is dynamically generated.
  • #1
rkrsnan
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Is it possible to produce massive composite particles from a non abelian gauge theory of massless fermions? I know that if the quarks were massless, the pions will be massless too (goldstone bosons). But what about baryons? Will they be also massless? If so, can we make a general statement that composite particles from any non abelian gauge theory of massless fermions will always be massless? Is there some mathematical proof for that?
 
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  • #2
Baryons would not be massless.
 
  • #3
Thanks for the reply. So I guess, if we have only massless up and down quarks and chromodynamics, then at low energy we will have massive protons and neutrons interacting via the exchange of massless pios. A theory with massless particles is scale invariant. But if we generate massive composite particles, then the low energy theory will not be scale invariant. Is there any contradiction in that?
 
  • #4
I'm also confused at how bound states of fermions would arise.

Consider the abelian case first. Look at the ground state energy of hydrogen or positronium. It is proportional to the reduced mass of the charged particles. So if the mass goes to zero, it looks like there would be no bound state.

This also makes sense dimensionally. Given the dimensionless coupling strength alpha, the only length scale in the problem is the mass of the particles.

Where does this logic fail? Is it at least correct for the Abelion case, and the non-abelion case has a loop-hole?

---

As an aside, if _all_ we had was fundamentally massless electrons, positrons, and photons ... at low energies, would the "effective" theory still give the electrons a mass? As the electron moves, the vacuum polarizes in its vicinity which must take energy. So would renormalization to low energies give the electron an effective mass anyway? Can a low energy theory actually have massless fermions?
 
  • #5
Non-Abelian Yang-Mills theories are "asymptotically free", which means that at large energies, the coupling constant is finite. However, this also means that as you run the renormalisation towards the infrared, the coupling constant will logarithmically diverge. Around when the coupling becomes unity, perturbation theory will break down, and that will produce a new energy scale.

Frank Wilczek has some nice papers about this, such as: http://arxiv.org/abs/hep-ph/0201222
 
  • #6
The scale is Lambda_QCD.

In QCD I can even make a bound state of entirely massless particles: glueballs.
 
  • #7
rkrsnan said:
A theory with massless particles is scale invariant.

Just to expand on what genneth Vanadium said, in case you're not familiar. It actually isn't true that a theory with only massless particles is necessarily scale invariant -- the obvious example being QCD. The QCD Lagrangian is scale invariant when the masses are set to zero, and therefore (as your intuition tells you) a classical field theory defined by this Lagrangian is scale invariant. However, the quantum theory does not actually respect this symmetry. An energy scale is dynamically generated, and we call such a scale Lambda_QCD.

You are correct that baryons couldn't have a non-zero mass in a scale invariant theory, but it turns out QCD with massless quarks isn't actually scale invariant, and there actually is an intrinsic length scale in the theory, even though it doesn't appear in the Lagrangian.
 

1. What is a composite particle?

A composite particle is a particle that is made up of smaller subparticles, such as quarks and gluons. These subparticles are bound together by strong nuclear forces.

2. What is a non abelian gauge theory?

A non abelian gauge theory is a mathematical framework used to describe the interactions between subatomic particles. It is based on the principles of symmetry and gauge invariance, and is used to study the strong nuclear force.

3. How is the mass of composite particles determined in a non abelian gauge theory?

In a non abelian gauge theory, the mass of composite particles is determined by the energy of the strong nuclear force that binds the subparticles together. This energy can be calculated using mathematical equations and techniques.

4. Can the mass of composite particles change in a non abelian gauge theory?

Yes, the mass of composite particles can change in a non abelian gauge theory. This can happen when the subparticles interact with other particles or when the energy of the strong nuclear force changes.

5. How does the concept of mass in a non abelian gauge theory differ from classical Newtonian mass?

The concept of mass in a non abelian gauge theory differs from classical Newtonian mass in that it is not a fixed, inherent property of a particle. Instead, it is a dynamic and relative quantity that is influenced by the interactions and energy of the subparticles within the composite particle.

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