Lattice QCD with QED Corrections

In recent years, lattice calculations in pure quantum chromodynamics have advanced significantly, to the point where they are now performed to relatively high accuracy. It is no longer reasonable to assume that effects of electromagnetism can be ignored. Due to this, there is now interest in looking into QED corrections to QCD calculations.

One area where these corrections will be of interest is when looking into the effects of electromagnetism on particle masses, particularly considering the splittings within isospin multiplets, such as the mass difference between the neutron and the proton. It is due to this mass difference that protons are observed to be stable in nature, such as in the centre of the hydrogen atom, whereas neutrons are rarely seen free in nature. Neutrons are more massive than protons, and decay into protons with a lifetime of a few minutes. If this mass difference were changed, say with the proton being heavier than the neutron, then hydrogen atoms would not be stable, and the universe would look very different to how we observe it today.

Another area of interest is the decay of pi mesons, where the decay rate depends on the parameter f pi which is a measure of how often the quarks within the pi meson interact with one another. This has been calculated in pure QCD, but it would be useful to also consider the effects of electromagnetism on f pi, as the electric charges of the quarks will influence their interactions.

There are some challenges to consider when working with QED and QCD simultaneously on the lattice. Ideally, the lattice spacing, alpha, should be small, while the lattice size, L , should be large. It is clear that both for a smaller lattice spacing and a larger lattice size we require more computing power, and so calculations become more costly. In pure QCD, we do not deal with massless particles, and the least massive particles considered are the pions. The finite size effects in the case of pure QCD depend upon the value of m pi L, where mpi is the mass of the pion. When this value is large, the finite size effects are small. When we begin to consider QED corrections, however, we have photons to consider, and so these effects are much larger for a lattice of the same size and with the same spacing. As well as these effects, the addition of soft photons also brings the issue of infrared divergences, which are not present in the pure QCD case.

The first two projects will involve looking into f pi and the issues involved with the addition of massless photons from the QED corrections. There are many other QED corrections which will be considered in future.