QCD

QCD as a Non-Abelian Gauge Theory

  1. SU(3) Transformations
  2. Local SU(3)C Transformations
    Since there is evidence for Colour charge one may speculate if this is the 'charge' the strong interaction is coupling to.

    One starts with the Dirac equation for a free quark
    colourdiracequation
    where the wave function is the product of the ordinary Dirac wave function and a Colour wave function.

    Since there are three different Colour charges (red, green, blue) there are three different basis states which can be represented as
    colourbasisstates
    To construct the dynamical theory of strong interactions, called Quantum Chromo Dynamics, one requires invariance of the Dirac equation under local SU(3)C transformations:
    localsu3ctrafo
    Invariance under such a local phase transformation is possible if
    invarianceunderlocalsu3ctrafo
    The QCD four-potential is given by Gjμ and the QCD coupling constant is gs.

    Proof: To prove that the theory is invariant under local SU(3)C transformations
    invarianceofdiracequation
    when introducing the gauge fields Gjμ it is sufficient to show that
    sufficienttoshow
    The calculation yields:
    proof
    The second term is zero. The third term is zero as well since
    lemma



    The Lagrangian after the introduction of the gauge fields is given by:
    qcdlagrangian
    with the field strength tensor
    qcdfieldstrengthtensor


    CONSEQUENCES:

    1) GLUONS ARE CARRYING COLOUR
    The non-abelian character leads to the fact that the gauge bosons themselves are carrying colour charges.
    As consequence, gluons are self-coupling. This can be easily read off the QCD Lagrangian containing terms proportional to 'G3' and 'G4', respectively.
    gluonselfcouplings


    2) UNIVERSALITY OF THE COUPLINGS
    In contrast to the abelian gauge theory, QED, non-abelian gauge theories, as QCD, contain in the gauge transformation explicitely a term proportional to the coupling constant, gs.
    That is, if fermions are coupling to the gauge fields at all (in this case, the gluons) they are coupling all with the same strength! Otherwise, the theory would be inconsistent. This is not the case e.g. in QED where charged leptons and quarks have different electrical charges.

    This 'explains' also lepton universality in weak interactions. In turn, the fact that lepton universality is observed is already a strong argument that the underlying theory of weak interactions has to be a non-abelian gauge theory if it is a gauge theory at all!

  3. Quark-Gluon Coupling
  4. Colour Singlet, Nuclear Forces