Compton Amplitude and Nucleon Structure Functions via the Feynman-Hellmann theorem

Understanding the internal structure of hadrons from first principles remains one of the

foremost tasks in particle and nuclear physics. It is an active field of research with important

phenomenological implications in high-energy, nuclear and astroparticle physics. The

structure of hadrons relevant for deep-inelastic scattering are completely characterized by the

Compton amplitude (see LH figure). A direct calculation of the Compton amplitude within

the framework of lattice QCD [1] provides an opportunity to investigate the non-perturbative

effects at low scales which are less well-understood and might have significant implications

for global QCD analyses. In the RH figure [2] we show the lowest moment of the F2

structure function for the proton versus Q2. There is clearly an effect from higher twist terms

(terms proportional to1/Q2), following the experimental results.

Left panel: Deep inelastic scattering where a proton is broken up by a highly energetic electron

emitting a photon which strikes a quark or parton in the hadron. The measured cross section is the

square of this and is described by the Compton amplitude shown in the figure.

Right panel: The computed lowest moment of the F2 structure for the proton. The experimental

Cornwall-Norton results are shown as black stars. The fit is based on the twist expansion.

Several phenomenologically relevant quantities, e.g. low- and high-x regions of parton

distribution functions, power corrections, subtraction function, and generalised parton

distributions in the zero-skewness region to name a few, are accessible. Our long term

objective is improving particle and nuclear physics community’s overall understanding of

non-perturbative effects in hadron structure from first principles.

The simulations and analysis were performed on the DiRAC-3 Extreme Scaling service

(Edinburgh) and Data Intensive service (Cambridge).

[1] K. U. Can et al. [QCDSF/UKQCD/CSSM], Phys. Rev. D 102 (2020), 114505.

[2] M. Batelaan et al. [QCDSF/UKQCD/CSSM], Phys. Rev. D 107 (2023), 054503.

Extreme QCD: Quantifying the QCD Phase Diagram

Project: dp006

Science Highlights 2022

The FASTSUM collaboration uses DiRAC supercomputers to simulate the interaction of quarks, the

fundamental particles which make up protons, neutrons and other hadrons. The force which holds

quarks together inside these hadrons is Quantum ChromoDynamics, “QCD”. We are particularly

interested in the behavior of QCD as the temperature increases to billions, and even trillions of Kelvin.

These conditions existed in the first moments after the Big Bang, and are recreated on a much smaller

scale in heavy ion collision experiments in CERN (near Geneva) and the Brookhaven laboratory (near

New York).

The intriguing thing about QCD at these temperatures is that it undergoes a substantial change in

nature. At low temperatures, QCD is an extremely strong, attractive force and so it’s effectively

impossible to pull quarks apart, whereas at temperatures above the “confining” temperature Tc, it is

much weaker and the quarks are virtually free and the hadrons they once formed “melt”.

We study this effect by calculating the masses of protons and other hadrons and their “parity partners”,

which are like their mirror-image siblings. Understanding how these masses change with temperature

can give deep insight into the thermal nature of QCD and its symmetry structure.

Our most recent results are summarized in the plots below. On the left we show the temperature

variation of the masses of the D and D* mesons (which are made up of a charm and a light quark).

This shows that they become nearly degenerate at the deconfining temperature indicated by the vertical

red line. On the right we show the R parameter which measures the degeneracy of the positive and

negative parity states of particular baryons. Results are plotted for the N (nucleon, i.e. proton/neutron)

as well as three other baryons which contain strange quark(s). This shows that the two parity states

become near degenerate (corresponding to R→ 0) in the high temperature regime above the vertical


FLAMINGO — A large suite of state-of-the-art galaxy cluster simulations for emulations for high-precision cosmology

Leads: Ian McCarthy, Joop Schaye, Matthieu Schaller & Virgo II

The worldwide observational cosmology community is gearing up to receive a real deluge of new data from large survey campaigns currently mapping the sky. This data will allow, for the first time, to put tight constraints on some key properties of our standard cosmological model as well as measure the mass of the neutrino. However, to interpret this data, equally precise and accurate models of the universe must be available. Furthermore, model variations have to be designed so as to encompass the whole parameter range in which our Universe lies. 

The Virgo Consortium’s FLAMINGO project is designed to provide exactly the virtual twins of our Universe that will be used alongside these modern surveys. The project includes the largest cosmological simulation with gas ever run (shown above left) as well as the largest simulation ever run including neutrinos, one of the key parameters we hope to constrain. However, as argued, having one single simulation is not sufficient to encompass the real of plausible universes. FLAMINGO has been extended, using DiRAC time, to include many variations. In these,  we vary the neutrino masses and the background cosmology.

Besides targeting the largest ever simulations, the other feature that sets the project apart is the variations of the galaxy formation aspects. The details of how galaxies form, and especially how their central supermassive black hole interact with their environment are still too poorly understood for the requirement of the era of precision cosmology. In this project, we thus decided to vary wildly the model by scaling the effect of the black holes up and down by large fractions. This will be of crucial importance to teams attempting to marginalise over this effect in the data.  In the right hand figure we show the relative effects on the matter power spectrum due to these variations compated to a dark matter only universe. Our library of runs shows large multi-percent variations, which will be crucial to take into account when attempting to measure the key parameters of our Universe with sub-percent accuracy.

Quantum Electrodynamics meets Quantum Chromodynamics

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