GoKawiilAt the fundamental level, the strong nuclear force between nucleons arises from quantum chromodynamics (QCD), a quantum field theory formulated in terms of their ‘colour-charged’ elementary constituents, the quarks and gluons. The interaction between these constituents is characterized by being weak at very high energies and short distances, a phenomenon known as asymptotic freedom5,6, whereas, in contrast to the other forces, it is so strong at nuclear distances that thinking of quarks and gluons as individual particles makes no sense at all. We speak of ‘confinement’: fundamental quarks and gluons cannot be directly observed, but instead, only composite ‘colour-neutral’ states, such as protons, neutrons or π-mesons, are observed in experiments. This fact poses several challenges, including the fundamental question of how to determine the strength of the interaction between quarks and gluons at high energy.
The quark–gluon coupling, α x (μ), depends on the energy scale, μ, of the interaction and also on its detailed definition, summarized as the ‘scheme’, x. Owing to confinement, we cannot collide quarks with quarks or gluons and determine α x (μ), directly in experiments. Instead, phenomenological estimates of the strong coupling are obtained by examining different processes, such as electron–positron or proton–proton collisions at various energy scales. After decades of theoretical and experimental efforts to parameterize the effects of confinement and to identify observables in which these effects are minimized, significant uncertainties persist. In particular, in determining a world average of α x (μ), notably by the Particle Data Group (PDG), different categories still exhibit uncertainties in the range of 1.5–3% (compare ref. 3 and Fig. 5). In fact, in most cases, these are not simply due to the limited precision of the experimental data, but include significant systematic uncertainties originating from the lack of an analytic understanding of confinement. In this situation, we cannot profit much from having more experimental data in reducing the uncertainty in α x (μ).
The inaccuracy of α x (μ) limits the potential of current experiments that test the fundamental laws of nature4. Even when all phenomenology extractions of the strong coupling are combined, they lead to an error of about 1%. This uncertainty propagates, for example, into a 2–4% uncertainty in the rate of production of Higgs particles by gluon fusion7 or its decay into gluons8. Furthermore, reducing the current uncertainty in the strong coupling by a factor of 2 turns out to be crucial9 for finding out whether the vacuum of the Standard Model is stable10 and to constrain extensions of the Standard Model, which cure the possible instability11,12.
A first-principles, robust, free of modelling uncertainties determination of the strong coupling avoids the limitations of extractions from experimental data and will affect ongoing searches for new physics.
Here we provide such a determination. We analyse the scale dependence of the strong coupling, as described by its β-function:
$$\mu \frac{{\rm{d}}}{{\rm{d}}\mu }{\alpha }_{{\rm{x}}}(\mu )={\beta }_{{\rm{x}}}({\alpha }_{{\rm{x}}}(\mu )),$$ (1)
which has an expansion of the form \({\beta }_{{\rm{x}}}({\alpha }_{{\rm{x}}})=-{\beta }_{0}{\alpha }_{{\rm{x}}}^{2}-{\beta }_{1}{\alpha }_{{\rm{x}}}^{3}+{\rm{O}}({\alpha }_{{\rm{x}}}^{4})\), with leading positive coefficients β 0 , β 1 , which are independent of the scheme. This implies that α x (μ) runs with the scale μ, decreasing with increasing μ, with a leading behaviour proportional to 1/ln(μ/Λ x ), as shown in Fig. 1. This phenomenon, known as asymptotic freedom5,6, implies that perturbative series expansions in powers of the strong coupling become accurate at high energies, as shown by the β-function itself. The scheme independence of the leading coefficients, β 0 , β 1 , implies that the asymptotic scale dependence is universal, and the Λ-parameters of different schemes are simply related by exactly calculable constants. Conventionally, we use the modified minimal subtraction scheme13 (\(\overline{{\rm{M}}{\rm{S}}}\)) to quote the coupling \({\alpha }_{{\rm{s}}}\equiv {\alpha }_{\overline{{\rm{M}}{\rm{S}}}}\) and \({\Lambda }_{{\rm{Q}}{\rm{C}}{\rm{D}}}\equiv {\Lambda }_{\overline{{\rm{M}}{\rm{S}}}}\).
Fig. 1: Scale dependence of the strong coupling. Full size image The strong coupling for a wide range of energy scales, as determined from our result for Λ QCD , is represented by the red band. The data points show the experimental determinations from various processes with their uncertainties as quoted by the PDG3.
Knowledge of Λ QCD and the β-function is equivalent to knowing the coupling at any given scale μ. In the \(\overline{{\rm{M}}{\rm{S}}}\) scheme, the expansion coefficients of \({\beta }_{\overline{{\rm{M}}{\rm{S}}}}\) are known up to high order, including β 4 , that is, five-loop order14,15,16,17,18, so that the scale dependence of \({\alpha }_{\overline{{\rm{M}}S}}(\mu )\) can be accurately predicted down to μ of the order of 1 GeV.
In this paper, we determine Λ QCD with two independent, dedicated strategies replacing the modelling of confinement by numerical simulations of lattice QCD. Our α s -uncertainty of about 0.5% is due to the finite computational resources and not due to our limited theoretical understanding. (The total cost of our α s -dedicated simulations is 400 million core hours). Combined with \({\beta }_{\overline{{\rm{M}}{\rm{S}}}}\), our determination of the coupling can be compared with experimental estimates at various energies, as shown by the data points in Fig. 1. The new, precise result also provides opportunities for better understanding of how confinement manifests itself in different processes and for extracting more detailed information from the experimental data.
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