DESY, Zeuthen (Germany)
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Quarks and gluons form protons and neutrons and thus most of the matter. The strength with which they interact is called the strong coupling. It is one of the fundamental parameters of Nature, but not that well known. Researchers used simulations on a space-time lattices to determine the coupling with good overall precision. The experimental inputs are the masses of pi-mesons and K-mesons as well as their decay rates into leptons (such as electrons), neutrinos and photons. Many simulations and their subsequent analysis were necessary in order to extrapolate to the required space-time continuum in all steps.
Decades of research have lead to the Standard Model of particle physics. It is a theory which describes the structure of matter at length scales below the diameters of nuclei to an astonishing level of precision. Equivalently it successfully predicts particle decays and scattering cross sections of high-energy processes up to the energies reached at the Large Hadron Collider (LHC) at CERN in Geneva. Indeed, we also believe that physics at larger length scales, i.e., nuclear and atomic physics, would emerge from the fundamental equations of the Standard Model, if we were able to solve them directly.
A very attractive feature of the Standard Model is that it has very few free parameters. These are – as far as we presently know – fundamental parameters of Nature. Their precise determination is thus an important part of particle physics and physics in general. It is also essential in order to put the Standard Model to tests of ever increasing precision. Such tests are especially motivated by observations that go beyond the physics described by the Standard Model, such as the existence of dark matter or the degree of asymmetry between matter and anti-mattter in the universe. Thus, despite its tremendous success, the Standard Model must be incomplete! In the quest for a more complete theory, precision tests of the Standard Model complement direct searches for dark matter candidates and other effects of “new” physics at the LHC and other experiments.
Both the determination of the fundamental parameters and the precision tests of the theory require precision experiments on the one hand and a precise solution of the theory (as a function of the fundamental constants) on the other hand. For many processes, the theory can be accurately solved as a series expansion in the couplings of the theory. An exception are processes that are dominated or affected by the ”strong” interactions part of the theory. This part is called Quantum Chromo Dynamics (QCD). Its most important feature is that it describes the structure of the smallest nucleus, the proton, in terms of constituents called quarks. These are bound inside the proton by the strong force, which is mediated by the exchange of quanta called gluons. The analogous phenomenon on the atomic level is the binding of the electrons by the exchange of photons with the nucleus. However, while the Coulomb force between electron and nucleus falls off rapidly with the distance (and is relatively weak altogether, characterized by a small fine structure constant α=1/137), the force between quarks remains strong at arbitrarily large distances and leads to the phenomenon of confinement: quarks are always bound. They do not exist as true particles by themselves. But how do we know that confinement is indeed a property of the theory? It is only due to its “simulations” on a space-time grid on super-computers such as SuperMUC and JUQUEEN. We put “simulations" in quotation marks, since these are not simulations of how particles move in space-time, but rather represent stochastic solutions of Feynman’s path integral, which provides the quantization of the fundamental fields, the quarks and the gluons. The stochastic solution of the path integral is possible, independently of any series (i.e., perturbative) expansions. It thus provides non-perturbative predictions of the theory. The stochastic evaluation of the path integral on a grid is called a lattice QCD simulation.
Despite its strong interactions, one may define αs, the analogue of the fine structure constant in QCD. This is the coupling of the theory, and a perturbative treatment means the series expansion in this coupling. A simple, physically appealing definition of the coupling is the force between static (infinitely heavy) quarks multiplied by the square of the distance. The aforementioned property of confinement means that at distances around a proton radius this coupling is much larger than one; hence, a series expansion makes no sense. However, at smaller distances also the QCD force becomes Coulomb-like, and the distance-dependent (“running") coupling, αs, becomes weaker and weaker. Perturbation theory then predicts that αs vanishes at small distances, r, as -1/log(rΛ). The constant Λ characterizes the coupling uniquely and thus corresponds to the fundamental intrinsic energy scale of the theory. Once it is known, perturbative predictions, valid at short distances or, equivalently, high energies become parameter-free. This is important for tests of QCD at the LHC providing the necessary high energies.
Results and Methods
A precision determination of the Λ-parameter is a challenge. A physical observable has to be evaluated which simultaneously has three properties: 1) it is a short distance quantity; 2) it can be obtained with high precision and 3) its perturbative expansion is known to reasonably high order. Lattice QCD, once its free parameters are determined from low-energy experimental data, can provide such observables, but in addition to the above, care has to be taken that 4) lattice spacings are small compared to the physical short distance involved. Our collaboration has developed a systematic strategy to cope with all challenges, in particular with 4). After applications to simplified theories, we have by now obtained a very precise result in the three-flavor theory with u, d, and s quarks. It can be connected perturbatively to the physical five-flavor number . We thereby determine Λ in a controlled fashion, from experimental input at the lowest energy: masses and decay constants of Pion and Kaon.
Our strategy (Figure 1) to connect large distances L=1/μ (μ: energy) and small distances involves couplings defined deliberately in a finite small volume, where very small grid spacings can be simulated. Two couplings with complementary properties at larger and smaller distances are employed. Their μ-dependence, and the connection between them, are both computed non-perturbatively by simulations of lattice QCD and extrapolations to vanishing grid spacing a. The two couplings are denoted ”Gradient flow" and ”Schrödinger Functional” in Figure 1. In the weak coupling region, the non-perturbative results are compared to the perturbative expression in terms of Λ, and that fundamental parameter is determined. Here the precision of perturbation theory is at the % level, because a coupling of αs=0.1 is reached and the unknown perturbative corrections are proportional to its square.
In our SuperMUC project, we deal with the connection of the coupling to the low-energy, non-perturbative scales of the theory. This is numerically most challenging, because now physically large volumes have to be simulated and different grid spacings need to be considered in order to take a continuum limit by extrapolation. Also this part is split in two individual steps, which can both be carried out separately and again with the most suitable resolutions. In the previous report we mentioned the determination of an intermediate reference scale, μref*, in units of the experimentally accessible decay constants of Pion and Kaon. By our computations, this scale is now related to the lowest scales, μhad, reached by the Gradient flow running coupling in the second step. It thus consists in the determination of μref*/ μhad. With very fine lattice spacings a, the continuum limit extrapolation of that ratio could again be carried out with confidence, see Figure 2.
Result / Conclusions
The last step, Figure 2, has been combined with the previous results to obtain 
which compares very well, but is more precise than the Particle Data Group (PDG) world average of phenomenological determinations of the strong coupling at the Z-boson mass from scattering experiments,
This agreement provides a very good test of the correctness of QCD as the theory of the strong interactions. Here it is of particular importance that our result is based entirely on experimental results for hadrons, i.e., QCD properties governed by confinement. In contrast, the PDG world average uses scattering data at relatively large energy, selected such that confinement can be hoped to lead only to small corrections to the perturbative expressions. Both analyses lead to the same coupling. Of course, the higher precision in our determination is particularly valuable for future predictions of QCD effects at particle physics colliders.
ALPHA Collaboration: Mattia Bruno, Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, Stefan Schaefer, Hubert Simma, Stefan Sint
Humboldt Universität zu Berlin, Universität Wuppertal, Trinity College Dublin, CERN, University Milan Bicocca, Brookhaven National Laboratory
 M. Dalla Brida, Phys. Rev. Lett. 117 (2015) 182001
 M. Bruno et al, J. High E. Phys., 1502 (2015) 043
 M. Bruno et al, Phys. Rev. Lett. 119 (2017) 102001
Prof. Dr. Rainer Sommer
Deutsches Elektronen-Synchrotron (DESY)
Platanenallee 6, D-15738 Zeuthen/Germany
e-mail: rainer.sommer [@] desy.de
NOTE: This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2018":
LRZ project ID: pr84mi, JSC project ID: hde09