Subatomic Physics

456 Strong Interactions
Figure 14.26: The potential V of
Eq. (14.59).
14.9 QCD at Low Energies
We expect the short-distance one-gluon exchange
force between the heavy quarks to
be primarily like a Coulomb force. The distance
dependence then is r−2 ,justasbetween
two fixed (heavy) electrical charges.
At large distances the confining force should
predominate. Good results in fitting the
spectrum of cc(J/ψ) and bb(Υ) systems
have been obtained with a potential of the
form
V = − αsk
+ Ar (14.59)
r
where k and A are constant coefficients.
This potential is illustrated in Fig. 14.26.
In Section 14.6 we discussed phenomenological approaches to extracting the nucleonnucleon
force. Ideally one would deduce this force from QCD, but, as we have
already mentioned, this is a complicated problem that has not been solved.
Chiral Perturbation QCD-inspired systematic methods have been introduced
at low energies. Use is made of the symmetries of QCD, particularly chirality (see
Section 11.7.) The left and right handed light quarks (up, down, and strange) are
decoupled from each other in the QCD Hamiltonian if their masses can be neglected;
it is the mass term which connects them. Since the masses are small, but not zero,
this symmetry is only approximate. At low energies a systematic expansion can
be carried out by constructing the most general Hamiltonian which incorporates all
terms with the symmetries of QCD, primarily chirality. For this reason, the method
is called chiral perturbation theory. (46) In addition, it is possible to carry out an
expansion in powers of p 2 /χ 2 ,wherep is the relative momentum of the nucleons
and χ is the chiral perturbation theory limit, of order ∼ 1 GeV, where the strong
fine structure constant become of order unity. (47)
46 S. Scherer, Introduction to Chiral Perturbation in Advances in Nuclear Physics 27, 277 (2003).
47 Nucleon-nucleon chiral potentials were developed in C. Ordoñez and U. VanKolck, Phys. Lett.
B291, 459 (1992); and brought to a fine point in D.R. Entem and R. Machleidt, Phys. Lett. B524,
93 (2002).