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Scalar fields and skyrmions

  1. Oct 31, 2007 #1
    I'm lazy so I'm going to start bringing my questions here.

    Correct me if I'm wrong, but isn't it true that baryons only enter the standard model through this subtle topological effect. Now this is where I'm at. I kind of got the Goldstone boson concept, maybe someone could better explain that to me too. Anyways, say you have scalar fields. Now, these scalar fields have a vacuum expectation value. To my understanding this is where the pion and pion's mass comes from? Now due to some topological effect "skyrmions" baryons also enter.
    Let me paraphrase a line in Weinberg vol II I don't get "the fields may furnish a reducible rather than irreducible representation of G (some group before its symmetry is broken) as for instance became the case when we introduced the nucleon fields in the previous section.

    So to me this means that baryons in some sense come from from the fact that the fields furnish a reducible rather than irreducible representation of group G.

    Where G for example could be SO(4), Spin(4) or whatever is isomorphic to SU(3)XSU(3)

    Could someone flesh these ideas out for me?
  2. jcsd
  3. Nov 6, 2007 #2
    When a global symmetry breaks, you expect some scalar particles to be massless (which may or may not be composites of fundamental fields). When an approximate global symmetry breaks you get light scalar particles (pseudo-Goldstones). When the symmetry group breaks from G to K, the massless or light bosons form a representation of the coset space G/K (elements of G that aren't elements of K). This represents the statement: "There is a (pseudo)Goldstone boson for each broken symmetry". You can look for topologically non-trivial field configurations by looking at fields as being a mapping from the space they inhabit to a representation space of some group. For example, you can look at the bosons forming a rep of G/K, and find topologically non-trivial configurations called skyrmions as described by Weinberg in Vol 2. In the physical case you mentioned, if you consider chiral symmetry to be an (approximate or exact) symmetry that's broken, the resulting (light or massless) bosons can be called "nucleons". Pions are vector bosons that mediate interaction of the "nucleons". This model is an effective field theory, and the particular framework is a type of "sigma model". We know that, in a technical sense, quarks and gluons compose nucleons and pions, though this view becomes very complicated and subtle at lower energies where the nucleons and pions are formed. For that reason people sometimes turn to effective sigma models, but their representation of the physical situation is approximate.
    The statement about reducible vs irreducible representations is not directly relevant. Usually we label particle states with irreducible representations of groups (that commute with the Hamiltonian, so that they are "good" quantum numbers). But we're allowed, and it may be convenient, to talk about a field that happens to form a reducible representation with respect to some symmetry group. That's all Weinberg is pointing out.
  4. Nov 6, 2007 #3
    I take it you actually mean "pseudoscalar" particles... i.e. the pion.

    You mean "fermions" instead of "bosons" in this last sentence.

    Pions are pseudoscalar bosons, I would pick the rho meson for the primary vector boson field in the inter-nucleon potential. Just remember that other more massive I=1 and I=0 spin-0 and spin-1 particles can participate, just suppressed by their higher masses.
  5. Nov 7, 2007 #4
    I kind of get what your saying, but let me phrase my question as simply as possible. How do baryons enter the standard model
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