QCD

Simple Examples

1. Quark-Antiquark Interaction
   The basic Feyman rules for QCD can be discussed by considering the scattering between a quark and an anti-quark due to one-gluon exchange.
   To ease the discussion the flavors of the two quarks are assumed to be different. In this case, only a t-channel gluon-exchange has to be considered.
   
   The corresponding Feynman diagram is depicted in Applying the Feynman rules the transition amplitude is given by where δ^αβ guarantees conservation of colour when exchanging the gluon.
   
   The amplitude is like the electron-positron scattering amplitude in QED with the replacement: α -> C_Fα_s
   with C_F being an additional colour factor: Depending on the sign of C_F the colour force is attractive (C_F > 0) or repulsive (C_F < 0).
   
   There are two possibilities for C_F:
   
   1) C_F for colour octet structure
   
   As an example the scattering of a quark with colour R and of an anti-quark with colour anti-G is considered. Since colour charge is conserved the quark in the final state has the same colour as the one in the initial state. The same holds for the anti-quark. The explicite calculation yields:
   
   2) C_F for colour singlet structure
   
   In this case, the quark-anti-quark pair is in a singlet state: The calculation yields
   
   
   SUMMARY:
   
   The one-gluon exchange potential (at short distances!) reads It explains why mesons with net colour charge do not form bound system.
   For colour singlet mesons the potential is attractive at short distances.
   This consideration does not apply to long distances. Lattice QCD calculations are able to calculate the QCD potential at long distances showing that the potential for a colour neutral quark-anti-quark pair is also attractive at long distances!
   
   
2. Quark-Quark Interaction