Erratum to: The flavor dependence of mϱ/fπ

A correction to this paper has been published: https://doi.org/10.1007/JHEP05(2019)197


Tree-level correlators and decay constant
The m /f π ratio was computed in the chiral-continuum limit in SU(3) gauge theory coupled to various numbers of fermions in the fundamental representation via non-perturbative lattice simulations [1,2]. None of the non-perturbative results are effected by this erratum.
The issue to be corrected here concerns the free theory which was used for illustration and comparison only. Clearly, in a free theory both m and f π first needs to be defined at finite fermion mass m and the chiral limit should be taken only for the ratio. Naturally, m = m π = 2m where m is the fermion mass. In [3] the result f π = √ 12/L was obtained from lattice simulations extrapolated to the continuum, in finite volume m π L = 1. The convention for the normalization of f π used in [3] was not specified and it turns out it corresponded to 130 M eV in QCD, which differs from our convention by a factor √ 2. In any case, from the finite ratio m /f π in finite volume m π L = 1, an incorrect conclusion was drawn in [1,2], namely that m /f π is volume independent and the value obtained in [3] holds in infinite volume too. Furthermore, m /f π was misquoted in [1, 2] by a factor √ 2, beyond the √ 2 difference in conventions. For completeness let us compute f π directly in the continuum both in finite and infinite volume at tree level in Euclidean signature. It is enough to consider N c = 1 and at the end restore the N c -dependence by f π → √ N c f π . The decay constant is defined from the large t behavior of the correlator at zero momentum, This normalization corresponds to f π = 92 M eV in QCD as in [1,2]. Now using the scalar and fermionic Green's functions, we obtain in Fourier space, hence we need d 3 x G 2 (x, t), which follows simply from (2), leading to the rather compact expression for t > 0,

JHEP06(2022)031
The last expression holds in infinite volume, but in finite L 3 volume the momentum integral simply needs to be replaced by a momentum sum. The fermion fields are assumed to be periodic in all spatial directions. Hence, in infinite volume, and positive time separation t > 0 we obtain, with the Bessel function K 2 . While in finite volume, mL fixed, where · · · refers to terms suppressed exponentially relative to the leading term e −2mt . It is clear from (6) that the amplitude vanishes for t → ∞ hence in infinite volume f π = 0 even at finite m. In finite volume (7) shows that the amplitude is finite for asymptotically large time separations and we get, using (1), which coincides with the continuum extrapolated result of [3] at the particular finite volume mL = 1/2 once it is multiplied by √ 3 since N c = 3 and also by √ 2 to take into account the normalization conventions (92 M eV vs 130 M eV ). As L → ∞ at finite m, clearly f π → 0, consistently with the analysis directly in infinite volume.
Hence the ratio m /f π is divergent in infinite volume at tree-level. The tree-level result is relevant at the upper end of the conformal window, N f = 11N c /2. Hence presumably the non-perturbative result m /f π = 7.85(14) with N c = 3 from [1, 2] valid for 2 ≤ N f ≤ 10 increases towards N f = 16.5 contrary to what was stated in [1,2].
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