In this thesis we compute masses, decay constants and gluonic matrix elements of the flavour-diagonal pseudoscalar mesons η and η' from lattice QCD.
To control all relevant systematic errors we employ Nf = 2 + 1 flavour simulations along two distinct quark mass trajectories leading to and including the physical point. The continuum extrapolation is guided by four lattice spacings. The ensembles were generated within the coordinated lattice simulations initiative, and we set their relative scales in this work.
We discuss noise reduction techniques for the efficient calculation of the disconnected contributions that are important building blocks of the relevant correlation functions of the η and η' system. The physical states are no flavour eigenstates, and hence sophisticated analysis methods are required to extract them from the data. We develop a matrix generalization of the effective mass
method which we employ in conjunction with additional techniques to determine masses and matrix elements. The physical point extrapolation employs next-to-leading order large-Nc chiral perturbation theory, and we determine all relevant low energy constants. For the first time also their renormalization scale dependence is taken into account, and this provides an important check of the range of validity of this effective field theory with implications on many existing phenomenological analyses. Our physical point results for the masses are in agreement with experimental values and read
Mη = 554.7(9.2) MeV and Mη' = 930(21) MeV. The determination of the four η and η' decay constants is the first from first principles and we obtain F8 = 115.0(2.8) MeV and θ8 = −25.8(2.3)◦ in the octet channel and F0 = 100.1(3.0) MeV and θ0 = −8.1(1.8)◦ for the singlet in the MS scheme at 2 GeV. These results are in excellent agreement with phenomenological
determinations and at a similar level of precision.
Finally, we connect these axialvector decay constants with pseudoscalar and gluonic matrix elements to test the axial Ward identities, and predict the anomalous matrix elements to be aη = <Ω|2ω|η> = 0.0170(10) GeV^3 and aη' = <Ω|2ω|η'> = 0.0381(84) GeV^3 at the physical point and µ = 2 GeV.

In dieser Dissertation werden Massen, Zerfallskontanten und gluonische Matrixelemente der Flavour-diagonalen pseudoskalaren Mesonen η und η' mittels Gitter-QCD berechnet.
Um alle relevanten systematischen Fehler miteinbeziehen zu können, werden Nf=2+1 Quarks entlang zweier Massentrajektorien simuliert, die beide zum physikalischen Punkt führen. Die Kontinuumsextrapolation wird durch vier Gitterabstände erleichtert. Die verwendeten Ensembles wurden in der "coordinated lattice simulations"-Initiative erzeugt und in dieser Arbeit wird deren relative Skala bestimmt.
Es werden Methoden zur Reduktion der statistischen Fehler diskutiert, die besonders für die Schätzung von sog. "disconnected" Beiträgen eine große Rolle spielen. Die Effektive-Massen-Methode wird für Matrizen generalisiert und angewandt um Massen und Matrixelemente zu bestimmen.
Zur Extrapolation zum physikalischen Punkt wird next-to-leading order large-Nc chirale Störungstheorie benutzt und alle Niederenergieparameter bestimmt. Zum ersten Mal wird auch die Skalenabhängigkeit mit einbezogen, was es ermöglicht, phänomenologische Näherungen zu überprüfen.