The Standard Model (SM) of Elementary Particles has been extremely successful in correctly predicting and describing the properties of elementary particles studied at experimental facilities all around the world. The SM itself covers elementary particles such as quarks and electrons, for instance, and their anti-particles, which interact by means of force carriers like photons and gluons. All the visible matter surrounding us is made up of fundamental particles described by the SM.

The theory of strong interactions that describes how quarks are  bound together by the strong nuclear force, mediated by the exchange of gluons, is called Quantum ChromoDynamics (QCD). We are now able to make very precise predictions starting from the rather compact set of mathematical formulae defining QCD. For instance, the mass of the proton but also many other quantities [1] can now be predicted with sub-percent precision. Solving the equations of QCD to make these predictions requires large-scale simulations on high-performance computers such as the ones provided by DiRAC. It is really remarkable that we find such highly complicated predictions to be compatible with the experimental data – our formulae agree with nature!

However, over the past decades physicists have collected ample observational and experimental evidence for the limitations of the SM – it is now very clear that the SM does not cater for certain phenomena, like dark matter or neutrino masses. It is therefore our goal to improve SM predictions and to make them more precise. It is then expected that theory predictions and experiment or observation will start disagreeing. When this happens we will have discovered new physics. The way in which the discrepancies arise will help us understand the nature of the new physics.

QCD is only one of three theories included in the SM, the other theories being weak interactions and Quantum ElectroDynamics (QED). The force-carriers of QED are the photons, and they couple to electrically charged matter with a strength that is often much weaker than the strong coupling of gluons to quarks. The strong coupling is in fact so strong that it binds quarks into bound states called hadrons (e.g. the proton or the pion). Until recently we therefore simply neglected QED contributions in our simulations. However, with the results of our computer simulations of QCD now reaching sub-percent-level precision, QED contributions can no longer be neglected.

Before including QED contributions into the simulations a number of formal problems first had to be addressed, to which our collaboration contributed substantially [2]. For instance, our simulations are taking place in a small simulation volume  that is just large enough such that the existence of its boundary constitutes only a small systematic effect (finite-size effect). While gluons are confined to this small volume by the underlying physics, the massless photons aren’t. As a result they feel the presence of the boundary much more severely. We have now developed a thorough analytical understanding of these finite-size effects and have thereby developed ways to control them. There are however still a number of conceptual problems to be addressed in the future, like the treatment of infrared effects originating from the fact that photons are massless.

While QED contributions are often small relative to QCD their importance is huge. It is known, for instance, that while the mass of the proton is mainly determined by QCD, the contributions of QED to it, which are responsible for the tiny mass difference between proton and neutron, are of immense importance for our existence: If the interplay between QCD and QED was just a tiny bit different from what we find in nature, all matter that surrounds us might not even exist!

With our current DiRAC allocation, a team of researchers from the Universities of Edinburgh and Southampton, with international collaborators from Switzerland and the US, has set out to compute how mesons (which are hadrons containing two quarks) decay into charged leptons and neutrinos, while taking into account QED effects [3]. The computation in QCD is very well understood but the computation including QED effects has never before been achieved in simulations where all parameters have been tuned such that the result can readily be compared to experiment. The project was executed by a team of PhD students, postdocs and senior researchers and comprised a considerable amount of software development, code-performance improvement, development of novel algorithms and novel ideas in theoretical physics, and eventually sophisticated data analysis techniques. Our results are currently being prepared for publication. They confirm that with the precision now achieved in pure QCD calculations, QED effects can indeed  no longer be neglected. Our calculation therefore constitutes a milestone in precision physics towards unravelling a bit more the fundamental physics governing our universe.

Going forward, we are thinking about developing the methods enabling QCD+QED computations for more complicated processes, in this way increasing the range of high-precision tests of the SM. These efforts have to be seen in the context of world-wide efforts to increase precision at experimental facilities (e.g. at CERN in Switzerland, Fermilab in the USA, KEK in  Japan).

Figure: Feynman diagrams for the decay of charged kaons and pions decaying into a pair of charged and neutral leptons (here for the case of the muon). The green lines indicate how photons couple to the electric charge of the quarks and charged lepton, respectively.


  • [1] FLAG Review 2019: Flavour Lattice Averaging Group (FLAG), Y. Aoki et al., Eur.Phys.J.C 80 (2020) 2, 113, 1902.08191
  • [2] Relativistic, model-independent determination of electromagnetic finite-size effects beyond the point-like approximation, M. Di Carlo et al., 2109.05002; Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions, Z. Dvoudi et al, Phys.Rev.D 99 (2019) 3, 034510, 1810.05923
  • [3] Near-Physical Point Lattice Calculation of Isospin-Breaking Corrections to Kl2/pil2, A Yong et al., 2112.11823