How strong is the strong interaction?

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. Ever smaller worlds Thomas Wright’s 1750 engraving of an array of bounded worlds. Physicists have now pinned down the strong coupling in a kindred family of tiny universes, each simulated on its own. Credit: Spencer Collection, The New York Public Library. “Plate XVII” The New York Public Library Digital Collections. 1750. https://digitalcollections.nypl.org/items/783751f0-c60a-012f-e5fc-58d385a7bc34. 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. On the lattice In lattice QCD, quark fields live on the sites of a grid and gluon fields on the links between them. A quark q and an antiquark q̅ pulled apart remain joined by a flux tube, whose energy grows steadily with the distance, and no quark is ever freed. Stretched far enough, the tube snaps and its energy materialises as new hadrons. Credit: D De Biasio 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.