Abstract
We compute both sides of the Witten-Veneziano formula using lattice techniques. For the one side we perform dedicated quenched simulations and use the spectral projector method to determine the topological susceptibility in the pure Yang-Mills theory. The other side we determine in lattice QCD with $N_f=2+1+1$ dynamical Wilson twisted mass fermions including for the first time also the flavour singlet decay constant. The Witten-Veneziano formula represents a leading order expression in the framework of chiral perturbation theory and we also employ leading order chiral perturbation theory to relate the flavor singlet decay constant to the relevant decay constant parameters in the quark flavor basis and flavor non-singlet decay constants. After taking the continuum and the SU$(2)$ chiral limits we compare both sides and find good agreement within uncertainties.
Highlights
We compute both sides of the Witten-Veneziano formula using lattice techniques
The other side we determine in lattice QCD with Nf = 2 + 1 + 1 dynamical Wilson twisted mass fermions including for the first time the flavour singlet decay constant
Moving away from the chiral limit leads to corrections to the formula which are linear in the quadratic meson masses f02 4Nf
Summary
We compute both sides of the Witten-Veneziano formula using lattice techniques. The question remains whether Nc = 3 in QCD is large enough in practice to sufficiently suppress corrections to this formula to higher order in 1/Nc. An alternative way to derive the Witten-Veneziano formula is to expand the anomalous flavor-singlet Ward-Takahashi identities of the theory order by order in u = Nf /Nc around u = 0 [2, 4]. We remark that for lattice QCD it is possible to obtain an unambiguous, theoretical sound implementation of the Witten-Veneziano formula through the study of anomalous flavor-singlet Ward-Takahashi identities in the limit u → 0 [5, 6]. Moving away from the chiral limit leads to corrections to the formula which are linear in the quadratic meson masses f02 4Nf
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