Of the Standard Model’s three gauge couplings, the one between quarks and gluons is by far the least well known. Harvey Meyer recounts the decades of lattice QCD behind its recent determination to five parts per mille.
The force between two quarks in a near-miss collision is of the same kind as that between two electrons, but far more intense. It is also known far less precisely. While the fine-structure constant of electromagnetism has been measured to better than one part in a billion, the strong coupling αs is only known to one in a hundred.
At the LHC, with its enormous number of proton–proton collisions, many process rates are now measured to the percent level. The uncertainty on the strong coupling is then becoming a theoretical bottleneck. The High-Luminosity LHC and a future Higgs factory such as the proposed FCC-ee will demand greater precision still, since new physics may first appear as small discrepancies between measurements and predictions. A calculation recently published in Nature has now reduced the uncertainty on αs to five parts per mille using only low-energy input. The value can therefore enter collider predictions without having been fitted to collider data.
A force apart
Protons are not elementary. Collisions at momenta far above their mass scale resolve them into quarks, held together by gluons, the carriers of the strong force. Hardly any process at a hadron collider can therefore be understood without the part of the Standard Model (SM) that describes this interaction, quantum chromodynamics (QCD).
In QCD, the strength of the force between two quarks, two gluons, or a quark and a gluon is parametrised by the single coupling αs. Predicting the cross-section of any process that involves these particles requires, therefore, a precise input value for this constant. For instance, gluon fusion – the process that produces Higgs bosons most copiously at the LHC – proceeds via a virtual top-quark loop, and its probability is proportional to αs2.
The strong coupling has an unusual property, inherited from a structural difference between QCD and quantum electrodynamics. Gluons, unlike the electrically neutral photon, carry colour, the very charge whose force they mediate, and thus interact among themselves. This leads to the coupling weakening as colour charges approach each other and growing as they separate. In a collision, the momentum transferred between the quarks and gluons sets the probed distance, and with it the scale at which αs must be evaluated. The conventional choice of a reference scale is the Z-boson mass, Q = MZ = 91.2 GeV in natural units. There, the current world average from the Particle Data Group (PDG) for αs is 0.1180 ± 0.0009. The precision is thus 7.6 parts per mille.
The weakness of the coupling at high-momentum transfers, where quarks and gluons behave almost as free particles, is known as “asymptotic freedom” and makes precise calculations possible as expansions in powers of αs. By contrast, at transfers below about a GeV, the coupling grows to order one, perturbative expansions in powers of αs break down, and any description in terms of quarks and gluons loses all predictivity. The strong interaction enforces this limit by confining quarks and gluons inside bound hadrons, and only these composites are ever seen in particle detectors. They include the proton and neutron, but also the pions, the kaons and an entire zoo of species, lately enlarged by the tetraquarks and pentaquarks discovered in collider experiments (CERN Courier November/December 2024 p33).
Spacetime on a grid
What fails at low energies are perturbative expansions, though, not QCD itself. The theory can predict the masses and key properties of bound states, but extracting these quantities requires a formulation that is not restricted to the weak-coupling regime. This was provided in a landmark 1974 paper by Kenneth Wilson. The idea is to approximate space and time as a four-dimensional lattice of points with a small spacing, later sent to zero to recover continuous spacetime (see the “On the lattice” figure). Enclosed in a finite volume, the lattice reduces QCD to a finite number of degrees of freedom. The theory’s predictions can then be evaluated by statistically sampling the possible configurations of the quark and gluon fields, and confinement emerges directly from the simulated dynamics (see “The theory, defined” panel).
The theory, defined
Beyond its computational role, the lattice occupies a privileged conceptual position. The perturbative series that typically define QCD are “asymptotic” expansions, believed not to be summable by any known method. They therefore cannot serve as a definition of the theory.
Wilson’s lattice can – at least, in part. Once space and time are replaced by a discrete grid and the system is enclosed in a finite volume, the theory is specified exactly by a finite set of well-defined quantities. The physical theory should emerge as the spacing shrinks to zero and the volume grows without bound, removing discretisation and finite-volume effects. While the existence of that limit has not been proved at a fully non-perturbative level, an all-order proof in perturbation theory exists for a class of discretisations. Even for a simplified theory with gluons alone, a truly rigorous construction, including a proof that its lightest state is massive, would settle one of the Clay Millennium Prize Problems.
The approach has a further price. The grid breaks translational, rotational and boost symmetry, and the simplest quark discretisations also sacrifice chiral symmetry. A further issue is that lattice QCD is formulated in imaginary time, which is well suited to statistical sampling but makes real-time scattering processes only partially and indirectly accessible. Many theorists nonetheless regard the lattice not as an approximation of QCD but as its true definition, and the broken symmetries as artefacts expected to vanish in the continuum limit. Providing a similarly rigorous definition of chiral non-Abelian gauge theories, as needed for a non-perturbative definition of the full Standard Model, remains an active area of research.
Still, the lattice became a quantitative tool only once algorithms and machines could generate enough field configurations for reliable statistical averages. For two decades, computing power forced severe compromises, most notoriously the quenched approximation, which neglected quark loops in the vacuum and therefore their backreaction on the gluon fields. This introduced an uncontrolled systematic error. Simulations with realistic dynamical quarks became feasible in the 2000s, and several collaborations obtained the masses of light hadrons and other low-energy observables in agreement with observation. Since 2011, the Flavour Lattice Averaging Group (FLAG) has compiled and averaged such results every two to three years, doing for lattice calculations what the PDG does for measurements and focusing on quantities central to particle phenomenology, such as the leptonic and semi-leptonic decay rates of K, D and B mesons. Lattice QCD has also shed light on how the proton’s charge, magnetisation and momentum are shared among its quarks and gluons, and distributed in space, and delivered high-impact ab-initio results on the phase diagram of QCD.
Over the past decade, the magnetic moment of the muon, a precision observable used to search for deviations from the SM, has proved a fruitful ground for demonstrating the maturity of lattice QCD. The measured value of the observable long disagreed with the SM prediction, whose largest theoretical uncertainty came from non-perturbative QCD effects. These comprise hadronic vacuum polarisation, traditionally estimated from measured e+e– → hadrons cross sections, and the smaller hadronic light-by-light contribution (CERN Courier March/April 2025 p21). By 2025, independent lattice calculations of the dominant vacuum-polarisation term agreed, while measurements of its dominant two-pion channel disagreed well beyond the stated uncertainties. This led the Muon g−2 Theory Initiative to base the value of the leading hadronic contribution in that year’s white paper solely on lattice results (CERN Courier January/February 2026 p41). The resulting prediction turned out to be in agreement with the final direct measurement of the muon’s magnetic moment from Fermilab (see “The verdict” figure).
Lattice QCD also makes it possible to determine the fundamental parameters of QCD, namely the quark masses and the coupling at momentum MZ, from experimentally well-measured quantities such as the pion, kaon and proton masses. Finally, it offers theorists the possibility of studying strong interactions in ways not accessible via observations – for example, at vanishing quark masses or when particles are confined to a small volume.
The femto-universe
Theorists from condensed matter to particle physics have long worked at finite size to perform better-controlled calculations, before taking the infinite-volume limit. Placing a system in a box, however, can be more than an intermediate step. How an observable changes with the size of the box is dictated by the dynamics under study, and is therefore itself a prediction of the theory. The idea of exploiting a very small volume to interrogate QCD goes back to James Bjorken, who in 1979 coined the term femto-universe for the degrees of freedom of the strong interaction in a box less than 10–15 metres across.
Building on Bjorken’s suggestive idea, Martin Lüscher, Peter Weisz and Ulli Wolff proposed in 1991 a systematic way to compute the momentum-dependence of αs in an asymptotically free theory, and implemented it in a model with one space dimension. The obstacle they faced is that a direct determination of αs would need a lattice large enough to hold hadrons, yet sufficiently fine to resolve energies above roughly 70 GeV, where low-order perturbation theory is accurate. No computer could span both scales at once. Their “step-scaling” method avoided the problem by using a family of femto-universes, each simulated separately and covering a narrow range of energies inversely proportional to its size (see “One box at a time” figure). Stepping down through the family, each member half the size of the last, the coupling can be followed upward in energies, until perturbation theory takes over. The extension to non-Abelian gauge theories, the class to which QCD belongs, followed the next year. The development of these techniques was among the reasons why Lüscher, who joined the CERN Theory Division in 1999, earned a share of the 2025 EPS High Energy and Particle Physics Prize, together with Jürg Gasser and Heinrich Leutwyler, for their theoretical work on the non-perturbative aspects of the strong interaction.
Heavy by design
In a Nature paper, published last April, seven theorists from the ALPHA collaboration computed αs at the reference momentum MZ using step scaling. The result, αs = 0.11876 ± 0.00058, carries a precision of 4.9 parts per mille and is consistent with the PDG average (see “Running down” figure). The calculation takes only low-energy quantities as input, namely the pion and kaon masses, together with a benchmark length of about 0.14 fm, itself determined by several independent lattice collaborations from well-measured quantities such as baryon masses and meson decay rates. The simulations contain only the three lightest quarks (up, down and strange), and the effect of the heavier charm and bottom quarks, negligible at low energies, is restored at the end using high-order perturbative QCD results and the measured values of their masses.
The same team’s 2017 determination, the first below one percent and until now the dominant input to the world average, rested on a single step-scaling analysis. The new work repeats it with finer lattices and adds a second approach, in which the masses of the three simulated quarks are increased to as high as 10 GeV. QCD then reduces, up to corrections falling as the inverse square of the masses, to a far simpler quarkless theory, in which the evolution of the coupling with energy can be determined independently. The two routes agree, and their average, resting on some 400 million core hours of simulation, gives the headline number.
The view from below
To many particle physicists, the idea that αs at MZ can be predicted by the SM with input solely from the low-energy world of hadrons is still unfamiliar. Nearly every phenomenological determination in the PDG runs the opposite way, fitting perturbative QCD predictions to high-energy collider observables. The comparison between the new result and these is therefore best read as a test of the SM: a heavy new particle at or above the Z mass would shift the collider determinations while leaving the low-energy ones practically unchanged. So far, the SM has passed the test with flying colours.
Further reading
M Lüscher, P Weisz and U Wolff 1991 Nucl. Phys. B 359 221.
M Dalla Brida et al. 2026 Nature 652 328.
D d’Enterria et al. 2024 J. Phys. G 51 090501.
