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Nuclear masses calculated from scratch

An exhaustive calculation of proton and neutron masses vindicates the Standard Model.

Atoms are much heavier than the fundamental particles from which they are made. European scientists have now shown that this oddity can be fully understood using the conventional Standard Model of particle physics.

Zoltán Fodor of the University of Wuppertal in Germany and his co-workers have carried out an awesomely difficult computation to deduce the masses of the proton and the neutron — the so-called baryon particles that constitute atomic nuclei — from theory alone1.

This drawing shows the supercomputers the researchers used to calculate the masses of protons and neutrons. Credit: Forschungszentrum Jülich/Seitenplan

Baryons are themselves made up of quarks, bound together by the 'strong' nuclear force produced by particles called gluons that zip between quarks. Before now, no one had demonstrated conclusively that the Standard Model, which explains how baryons are built from quarks, can accurately predict baryons' masses — which are well known from experiments.

That's largely because of the teeming nature of the subatomic world, in which 'empty' space is no such thing. According to quantum theory, a vacuum is filled with virtual particles that pop in and out of existence in pairs, corresponding to particles and their antimatter equivalents.

All at sea

These virtual quarks form a kind of pervasive 'sea' in which those that make up the actual baryons are bathed. "The calculations are complicated because of the need to include the effect of the sea quarks," says Christine Davies, a particle physicist at Glasgow University in Scotland who was part of a team that made some of the first accurate calculations of these particle masses in 2003.

But worst of all, the calculations needed to describe this morass don't 'converge'. For atoms and molecules interacting in gases and liquids, most of the interaction energy comes from direct interplay between pairs of particles, a little comes from triplets of particles, a tiny bit from groups of four and so on. Quarks don't have this handy tailing off as the number of interacting particles multiplies – they are said to be 'strongly coupled' – so all the possibilities need to be taken into account.

The theory for describing quark-gluon interactions is called quantum chromodynamics (QCD). The strong coupling of these interactions means that there's no exact way of making calculations using pen and paper. Instead, the equations must be solved numerically on a computer.

The usual way of doing this, devised 20 years ago, is to divide up four-dimensional time and space into a series of discrete cells, like those of a two-dimensional chess board, and then compute how each cell is affected by all the others. This gridding is essentially the same trick that scientists use in other complicated computations such as global weather and climate models. In nuclear physics it is called lattice QCD.

Virtually accurate

But even with these simplifications, the calculation is immense. So until the mid-1990s, it was common for lattice QCD calculations of the masses of hadrons (all particles composed of quarks, including baryons) simply to ignore all the virtual quark-antiquark pairs in the vacuum. That gave only rough estimates.

Five years ago, physicists hit on a way to include some of these pairs without overloading the computation. This stimulated calculations such as those of Davies's team that produced mass values for protons and neutrons close to those observed. But all such efforts have been forced to leave out some aspects of the problem, and so were not exhaustive tests of the Standard Model's accuracy.

Now Fodor and his colleagues have cracked the problem of how to include all the ingredients and the results of their calculations match experimental values extremely closely.

The researchers' success is down to no magic ingredient, but a combination of doggedness and judicious planning. It has required "a huge amount of computer time", says team member Stefan Dürr of the John von Neumann Institute for Computing in Zeuthen, Germany. But he adds that making the calculation tractable also involved a careful choice of how to describe the key physical processes involved.

Collider question

The results verify that only a small part of the baryon masses comes from the masses of the quarks themselves. The remaining mass comes from the energy that the quarks carry by virtue of being bound together or confined within a hadron.

This puts some perspective on the aims of the new Large Hadron Collider particle accelerator at CERN, the European Laboratory for Particle Physics in Geneva. The popular story is that the Large Hadron Collider will explore the 'origins of mass' by probing how quarks and other fundamental particles obtain their masses, thought to happen through the mediation of a particle called the Higgs boson.

In fact, the new work confirms that the mass of the stuff around us is due only in very small part to the masses of quarks themselves. Most of it comes from the way they interact inside baryons. "Ninety-nine per cent of the mass of the proton and neutron, and therefore the visible Universe, is QCD binding energy," says Davies. "The Higgs then just explains the 1% of it that comes from quark masses." All the same, she adds, where that last per cent comes from is still "a very important fundamental question".

Credit: Forschungszentrum Jülich/Seitenplan


  1. Dürr, S. et al. Science 322, 1224-1227 (2008).

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Ball, P. Nuclear masses calculated from scratch. Nature (2008).

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