Since we do not yet understand the internal structure of an isolated hadron from first principles, it is clear that we cannot presently have a fundamental, quantitative theory of the interactions between hadrons. In particular, there is currently no fundamental understanding of the very foundation of nuclear physics, the nucleon-nucleon interaction.
Historically, the interactions between hadrons have been described
in terms of meson exchange with phenomenologically determined
parameters. More recently, the techniques of effective field theory
have been used to try to understand some aspects of hadronic
interactions solely in terms of the pion field, but this approach is
intrinsically limited to low momentum and energy. Hence, as in the
case of hadron structure, there is an increasing appreciation of the
need to understand hadronic interactions in terms of the underlying
field theory, QCD, and recognition that lattice QCD and computing
resources have advanced to the point that one can hope to obtain
genuine insight from ab initio lattice calculations.
Understanding the strong interaction in multi-hadron systems from lattice QCD is a notoriously difficult problem. Multi-hadron states involve the computation of a four-point function and are relatively massive, and therefore the corresponding correlation functions quickly vanish into noise at increasing temporal separations. Furthermore, multi-hadron systems are large, and therefore the volume of the finite lattice needs to be correspondingly larger than that used in hadron spectroscopy. Finally, the use of a Euclidean lattice obscures the extraction of the phase of the full scattering matrix. Despite these difficulties, the problem is fundamental and compelling.
Historically, there have been two approaches to this problem within lattice QCD. The first aimed at extracting certain quantitative parameters of the hadron-hadron interaction by direct lattice simulation. Lüscher exploited finite-size effects to extract a discrete set of s-wave scattering lengths. The method was thoroughly tested within an O(4)-symmetric φ4-model and scattering lengths have been computed within QCD for pions, and for nucleons. The I=2 π-π system has been explored by extracting a residual interaction potential from which the scattering phase shifts were computed and compared with experiment. Whereas the extraction of the s-wave scattering lengths of the I=2 π-π interaction has been encouraging, the investigation of the N-N system is more problematical. Here the scattering lengths are of the order of ~10fm, rather than less than ~1fm as is the case for the π-π interaction. Therefore, while scattering lengths for the N-N interaction were found to be considerably larger than those for the π-π and π-N interaction, this approach is limited by our present inability to simulate on lattice sizes of the order of 10-20fm, and at physical values of the pseudoscalar mass.
The second approach is motivated by the realization that important insight into the nucleon-nucleon interaction can be gleaned by studying much simpler systems. The interactions between two heavy-light mesons with static heavy quarks exhibit most of the salient features of the nucleon-nucleon system, including quark exchange, flavor exchange and color polarization. Lattice calculations already indicate that, at large distances, flavor-exchange processes can be interpreted in terms of π or ρ exchange.
The especially attractive feature of heavy-light systems is that the heavy quarks admit the definition of a relative coordinate, and thereby a local adiabatic potential. Exploratory studies have been made of this potential, and evidence for nuclear binding sought. The investigation of this class of potentials is important and more feasible, because the large scattering lengths for the N-N system come from a short-range potential. Furthermore, by exploring the potential, we can discover the relative importance of gluon and meson exchange contributions at various distances.
The principle aim, then, in this approach, is an understanding of the underlying QCD dynamics describing hadron-hadron interactions, rather than a detailed quantitative description of the N-N interaction itself. Thus studying interactions of systems containing a single heavy quark, which, as we have noted, exhibit most of the desired features of the true nucleon-nucleon interaction, enable us to formulate an adiabatic potential. Long-range goals for investigations of hadrons containing heavy quarks include the following:
A calculation of the interaction between heavy-light mesons, using static heavy quarks, over a wide range of separations, and at a variety of lattice spacings. Such a calculation will enable the relative importance of quark-exchange and gluonic contributions to the potential to be assessed, and the spin- and isospin-dependence of the interaction to be explored.
A comparison of this interaction with quark-model results. Such a comparison for this simplified system would provide invaluable insight into the application of quark-model calculations to the real nucleon-nucleon system. Furthermore, we can make direct comparisons with meson-exchange interactions.
An extension of the calculation to the interaction between baryons, each containing a single heavy quark, where both the spin and flavor structure are more involved.
In addition to employing the formalism for extracting a potential the calculations shall utilize many of the same techniques developed for other projects: the use of anisotropic lattices to enable correlation matrices to be extracted at shorter time separations, the calculation of all-to-all propagators using random-source or other methods, the study of matrices of correlators to extract excited-state masses, Bayesian methods for spectral analysis, and the exploitation of developments in chiral fermion formulations to enable the calculations to be performed at sufficiently light quark masses for the pion cloud to emerge. We believe that the study of interactions of hadrons containing heavy quarks will provide physical insight into the nature of the nucleon-nucleon force, and is an essential precursor to the application of lattice QCD to multi-hadron systems.
An extension to the case of a finite but large heavy-quark mass mQ, to study how nonlocalities modify the local potential.
Finally, we conclude by noting that the issues we are addressing
in this part of our program are intimately connected with those of
other sections. For example, the question of the existence of the H
dibaryon, and other exotic, multi-quark states, may be more usefully
addressed by exploring the potential between two heavy-light hadrons,
rather than by a direct study of the spectroscopy of a six-quark
system. Furthermore, as a long-term prospect, the measurement of
two-, three- and four-point meson correlation functions
may enable the extraction of the
parameters of effective chiral Lagrangians.