When students are first told about the fundamental forces of the Universe, the strong nuclear force is described as the force binding protons and neutrons together inside nuclei. Later, they learn about quarks and gluons and the theory describing them, quantum chromodynamics (QCD). What they do not learn is how the theory of strong interactions (QCD) actually explains the strong interactions between nucleons. The reason, of course, is that nuclear forces are a long distance phenomenon and QCD is notoriously unyielding to analysis in this regime. In fact, after over 30 years since its discovery, QCD has provided very few clues regarding nuclear forces.

This is not to say that we don't know much about nuclear forces. In fact, we know a lot. If we look at nuclear systems with a resolution of no more than about 0.5 fermi (corresponding to energies below the threshold for pion production in nucleon-nucleon scattering), the complicated dance of quarks and gluons reduces to the much simpler motion of non-relativistic nucleons interacting through a  (possibly momentum dependent) potential. One can then just take the attitude that non-relativistic nucleons are the correct degrees of freedom in nuclei and learn about their interaction in an empirical way. This point of view was how few-nucleon physics developed even before the advent of QCD and was more recently systematized by the effective field theory approach. In its modern incarnations, the empirically well known proton-proton and neutron-proton phase shifts are very well fit by non-relativistic potentials ( AV18, CD-Bonn, Nijmegen s, ... ) and these potentials are used in the Schroedinger equation for systems containing several nucleons. The result is that the empirical potentials provide a good but not perfect match with what is found in Nature. The discrepancies, found empirically and predicted/postdicted by the power counting of effective theory, can be cast as the need for additional terms in the hamiltonian describing the physics occurring when more than two-nucleons occupy the same region of space-time. Examples include a three-nucleon force and two-nucleon-photon or two-nucleon-neutrino contact terms. These terms are, obviously, not constrained by nucleon-nucleon data. The problem is that the data that can constrain them are, frequently, very hard to measure. This problem is particularly acute in the extension of this approach to the study of hyperon interactions where the available data is  nowhere as complete as in the nucleon-nucleon case (and will never be).

We see then that, besides the conceptually important goal of a "first principles" understanding of nuclear binding, there are other, more "practical" reasons to hope for a QCD-based approach to nuclear interactions. Currently, the only general purpose approach to QCD in the non-perturbative regime is lattice QCD. QCD, like any field theory, can be defined by a functional integral over the fields describing gluons and quarks. This is nothing but the field theoretical version of Feynman's "sum over histories" in quantum mechanics. The basic idea of lattice QCD is to discretize space-time - so the functional integral becomes an integral in a space with a large but finite number of dimensions - and calculate the integral by Monte Carlo methods. Although straightforward in principle, this program is enormously difficult in practice, even when considering much simpler observables like stable hadron masses. In the last few years a number of advancements came together to make the promise of phenomenologically useful lattice calculations a reality. To the outsider, the most obvious progress is the increase of the sheer power of computers, be them special purpose machines (like QCDOC) or clusters of commodity pc's dedicated to lattice calculations. But equally important is the development in algorithms and innovative approaches to lattice calculations like the use of quark discretizations with exact chiral symmetry and, specially, the use of improved action discretizations. What is "improved" in these discretizations is the approach to the continuum limit: if the space-time lattice spacing is a, naive discretizations will have an error of order a Λ , where Λ is a typical QCD scale (say ~300 MeV). Improved actions will have smaller errors, scaling like (a Λ)². This improvement allows for precise results to come out even of rather coarse lattices (typical lattices nowadays have a of order of 0.1 fm). The saving in computer requirements is huge.

All these improvements help the lattice calculation of any observable. But the study of nuclear forces -- or hadron interactions in general -- has its own specific difficulties requiring specific solutions. Until about a year ago, the only study of nuclear forces in lattice QCD was a 1995 pioneer quenched calculation, and only recently dynamical, exploratory calculations started telling us what the real issues are. Let us consider two recent results of the NPLQCD (Nuclear Physics with Lattice QCD) collaboration.

The first, a warm up problem with its own physics interest, is the ππ scattering in the I=2 channel. Since lattice QCD calculations are done in imaginary time, scattering amplitudes are not directly accessible. The way NPLQCD chose to extract scattering information was the use of the "Luscher method". It is based on the observation that while scattering amplitudes cannot be extracted from euclidean correlators, energy levels can. For instance, if π(t,r) is some operator with pion quantum numbers, |n> are energy eigenstates and |π> a one pion at rest state we have

Thus we can extract the mass of a particle from the large time behavior of an euclidean correlator. Similarly, we can extract from the two-particle correlator the energy levels of a two-particle state. The bulk of the energy of a two-hadron state is given by the sum of the two hadron masses. At finite volume though there is a small component proportional to their interaction: the energy levels go up a bit for repulsive interactions and down a bit for attractive interactions. There is in fact a formula (known as Luscher formula) that relates the shift in the energy level ΔE to the phase shift describing scattering at energy ΔE. For boxes with size L much larger than the scattering length a the Luscher formula reduces to 

which is easy to derive using perturbation theory. The applicability of the method is not restricted by a/L << 1, the only condition is that L be much larger than the range of the force between the particles. In particular, the nuclear s-wave scattering lengths that are fairly large (5-20 fm) can be studied in boxes just a few fermi wide. In a box of size L ~ 2.5 fm the typical values of  ΔE is only about 20 MeV and very precise measurements of the energy are then necessary for a good determination of ΔE and, consequently, of the phase shifts.  Recently a  high statistics calculation reported an error of only a few percent on the scattering length. Since the computer power needed for the simulation of light quarks is large, the values of the quark masses used were larger than the realistic ones, corresponding to pion masses in the 300-500 MeV range. An extrapolation down to realistic quark masses is then necessary for a comparison to experiments. The use of chiral perturbation theory is fundamental for that since it provides approximate but model independent expressions for the dependence of various observables as a function of the quark mass. In fact, the precision of this calculation is high enough that special variations of chiral perturbation theory including discretization effects are used. The result of this extrapolation is shown in the blue band in the figure together with the lattice QCD points (in black) and the empirical point in green. Notice  that the precision of the extrapolated point is higher than any individual point at higher quark masses. This is due to the fact that the scattering length at zero quark mass, where the pion is a Goldstone bosons, must vanish. Due to this exact result, the "extrapolation" is actually an interpolation between lattice results and the exact answer at m=0..

Several factors make the lattice study of nucleon-nucleon interactions more difficult than the ππ case:

1. Until recently, lattice calculations almost invariably were done using the quenched calculation. This means that out of the QCD picture of, for instance, a proton -- three valence quarks surrounded by many gluons and sea quark-anti-quark pairs -- only the valence quarks and gluons are kept. Dropping the sea quarks makes it the action local and reduces the computational cost of the calculation by many orders of magnitude. For certain observables and for large quark masses, the quenching approximation is not a huge effect (~10%). Unfortunately, nucleon-nucleon forces are strongly changed by quenching and the much more expensive calculations with dynamical sea quarks are essential.

2. An effective field theory for nucleon interactions, in the mold of chiral perturbation theory for pions, exists and has been the subject of intense scrutiny in the last 10 years. It is, however, much less effective than it's pionic counterpart. In particular, it suggests that the s-wave scattering lengths in both spin channels of nucleon-nucleon scattering will diverge for values of the quark masses lying between the physical values and the values used up to here. Clearly, the chiral extrapolation will be a delicate point for this observable (phase shifts at generic values of momenta do not suffer from this fine tuning problem and are easier to extrapolate).

3. Perhaps the most serious problem is that the signal-to-statistical noise ratio in baryon-baryon correlators decrease exponentially with (euclidean) time. The rate of decrease equals 2M-3m (M is the nucleon mass and m the pion mass) so it is large and it grows as more realistic pion masses are used. Thus, the statistical noise renders the information about energy levels residing at large t useless.

Due mainly to the problem 3) above, the only dynamical calculation up to now was only able to put bounds on the values of NN scattering lengths, of the order of 1 fm. This already contains some non trivial information. It shows, for instance, that the values of the scattering lengths are indeed fine tuned and modest changes in the quark masses are enough to make them relax to more natural values. Also, using the knowledge we have about the nuclear effective theories (and the experimental value of a at realistic quark masses) we can perform the rough extrapolation down in pion masses shown in the figure below.

 

Further progress of the nuclear physics on the lattice program will require a continued effort along two prongs:

1. Community support: modern lattice calculations resemble large experiments in some respects. They involve innovations in hardware, software as well as smoothly running facilities. This can only be accomplished with the involvement of top experts in different areas of computer technology.

2. Innovative ideas: new ideas on how to tackle the problems unearthed by the pioneering calculations are needed, brute computer power can't solve everything. These problems are directly linked to the physics of nuclear interactions and experts in this area -- and not only lattice experts -- are the most likely to make progress. The use of chiral perturbation theory/nuclear effective theory in the chiral extrapolations above is one example of profitable interaction between lattice and nuclear community. We can expect that, in the future, experts on both  nuclear force/few-nucleon physics and lattice practitioners will become more aware of each others fields and able to contribute to the success of the program.