The physics of heavy quarks with Lattice QCD

Quantum Chromodynamics (QCD) describes the interactions of quarks and gluons and predicts in principle the masses of their bound states (called hadrons). The calculation of hadron masses, however, can only be done by the numerical simulation of QCD on a lattice of space-time points. With U.S. collaborators, we have been calculating the masses of hadrons made up of the heavy charm (c) and bottom (b) quarks using Lattice QCD. Because these are heavy quarks, several times the proton mass, our results are much more accurate than those for light hadrons. Hadrons made of b and c quarks can be produced in high energy collisions at CERN and their masses measured. Comparing our results to experiment gives a good test of QCD. We can also extract from our results and the experiments an accurate value for the coupling constant, alpha_s, of QCD. This quantity is analogous to the fine structure constant, of electromagnetism, but far harder to obtain because it is much larger. It is this strong coupling between quarks and gluons in QCD which keeps the quarks and gluons bound into hadrons and unable to escape. The bound states of b quarks with up, down or strange quarks are very interesting because of the way that they decay via the weak interactions. Study of these decay modes theoretically and experimentally will tell us how the weak interaction violates some of Nature's symmetries. We are currently calculating the masses and decay constants of these hadrons as part of this programme. More technical details are given below.

Research at Glasgow

The approach of lattice QCD consists of discretising space-time onto a lattice with spacing a. The Lagrangian for QCD is discretised by replacing all derivatives of fields with finite differences of fields at lattice points, for example. A Monte Carlo integration of the (now finite) Feynman Path Integral can then carried out to calculate expectation values of appropriate operators. The integration is done by randomly generating gluon fields over the lattice (these are called configurations) and averaging over the expectation values of the operators on those configurations. For efficient integration we must use importance sampling, that is, those configurations with greatest weight in the integral are preferentially generated. Quark propagators are calculated and then put together in appropriate spin combinations to make hadron correlators which create and destroy hadrons with time separation, T, on the lattice. The correlators are fitted to an exponential form, exp(-MT), to extract a dimensionless mass for that hadron. Masses, M, measured in lattice units must be converted to masses, m, in GeV using M=ma. From the experimental value for one mass we can extract a and then convert all other masses to GeV. Heavy quarks initially posed a problem on the lattice because their mass in lattice units is bigger than 1, implying that they are sensitive to distances smaller than a, that cannot be simulated correctly on the lattice. However, the heavy quark mass itself is irrelevant to the physics of its bound states, which are non-relativistic systems with level splittings much smaller than the heavy quark mass. We can exploit this on the lattice by treating the heavy quark with a non-relativistic Lagrangian which allows for a very efficient extraction of quark propagators, which can then be put together into hadron correlators.

This gives a very accurate calculation of the spectrum of bound states, particularly for bound states of the b quark. Systematic errors from finite volume and finite lattice spacing can be kept small. Errors from the quenched approximation then become visible. Current calculations are being done on unquenched configurations to get results closer to the real world.

• Further reading on quarks and hadrons:
  New Scientist: Inside Science no. 63, World of Quarks, 10th July 1993.
• Staff contact: C. Davies