Leptonic decay-constant ratio ${f}_{K}/{f}_{\ensuremath{\pi}}$ from lattice QCD using $2+1$ clover-improved fermion flavors with 2-HEX smearing

Leptonic decay-constant ratio $f_{K} / f_{π}$ from lattice QCD using $2 + 1$ clover-improved fermion flavors with 2-HEX smearing

Stephan Dürr, Zoltán Fodor, Christian Hoelbling, Stefan Krieg, Laurent Lellouch, Thomas Lippert, Thomas Rae, Andreas Schäfer, Enno E. Scholz, Kálmán K. Szabó, and Lukas Varnhorst

Phys. Rev. D 95, 054513 – Published 31 March 2017

Abstract

We present a calculation of the leptonic decay-constant ratio $f_{K} / f_{π}$ in $2 + 1$ flavor QCD. Our data set includes five lattice spacings and pion masses reaching down below the physical one. Special emphasis is placed on a careful study of all systematic uncertainties, especially the continuum extrapolation. Our result is perfectly compatible with the first-row unitarity constraint of the Standard Model.

Received 2 February 2016

DOI:https://doi.org/10.1103/PhysRevD.95.054513

Physics Subject Headings (PhySH)

Lattice QCD Leptonic, semileptonic & radiative decays

Particles & Fields

Authors & Affiliations

Stephan Dürr^1,2,*, Zoltán Fodor^1,2,3, Christian Hoelbling¹, Stefan Krieg^1,2, Laurent Lellouch⁴, Thomas Lippert^1,2, Thomas Rae¹, Andreas Schäfer⁵, Enno E. Scholz^5,†, Kálmán K. Szabó^1,2, and Lukas Varnhorst¹

¹University of Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany
²Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52425 Jülich, Germany
³Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary
⁴CNRS, Aix Marseille U., U. de Toulon, CPT, UMR 7332, F-13288 Marseille, France
⁵University of Regensburg, Universitätsstraße 31, D-93053 Regensburg, Germany

^*durr@itp.unibe.ch
^†enno.scholz@physik.uni-regensburg.de

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Vol. 95, Iss. 5 — 1 March 2017

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Figure 1
Scatter plot of $(2 M_{K}^{2} - M_{π}^{2})$ versus $M_{π}^{2}$ for all ensembles. Here the lattice scales from Table 1 have been used to convert the masses from lattice units into GeV. The physical mass point, as defined in Eq. (2), is indicated by a circle.
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Figure 2
Scatter plots of $M_{K}$ (left panel) and $(M_{π} L)$ (right panel) versus $M_{π}$ . Also shown are the bounds for the fit ranges used in our analysis (see Sec. 4). Here the lattice scales from Table 1 have been used to convert the masses from lattice units into GeV.
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Figure 3
Scatter plot of the $p$ value of the fit versus the physical-world ratio $f_{K} / f_{π}$ from each one of the 1368 single fits used in the global analysis. The symbols distinguish fits using the ChPT or the polynomial form for the interpolation in the quark masses. Additionally, for the results using the polynomial form, the color indicates whether the 3-, 4-, or 6-parameter polynomial form has been used. For better readability no error bars are shown for the points in this figure. The vertical solid and dashed lines and grey band show our final result from Eq. (16) and its statistical and total uncertainty, respectively.
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Figure 4
Example of a fit with SU(3)-ChPT mass dependence, $a^{2}$ discretization, full FV, and mass-independent $a^{- 1}$ . Included are ensembles with $M_{π} \leq 300 MeV$ , $M_{K} \leq 600 MeV$ , $(M_{π} L) \geq 3.85$ , and $β \geq 3.50$ . The left panel compares the value of $f_{K} / f_{π}$ from the fit and measured on the ensemble for each data point (labeled from 0 to 46, ordered as in Table 3). The right panel shows the dependence of the fitted $f_{K} / f_{π}$ on the squared pion mass $M_{π}^{2}$ at the physical strange-quark mass and in the infinite volume and continuum limit. The data points shown in this panel have been shifted according to these values/limits using the fit results. This particular fit resulted in $χ^{2} = 7.6$ with $n_{d . o . f .} = 6$ , giving a $p$ value of 0.27.
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Figure 5
Example of a fit with 4-parameter polynomial mass dependence, $a^{2}$ discretization, full FV, and mass-independent $a^{- 1}$ . Included are ensembles with $β \geq 3.61$ . The left panel compares the value of $f_{K} / f_{π}$ from the fit and measured on the ensemble for each data point (labeled from 0 to 46, ordered as in Table 3). The right panel shows the dependence of the fitted $f_{K} / f_{π}$ on the squared pion mass $M_{π}^{2}$ at the physical strange-quark mass and in the infinite volume and continuum limit. The data points shown in this panel have been shifted according to these values/limits using the fit results. This particular fit resulted in $χ^{2} = 72.1$ with $n_{d . o . f .} = 18$ , giving a $p$ value of $10^{- 8} \approx 0$ .
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Figure 6
Overview of the results from the global analysis with different fit types (left) and different fit ranges (right). Shown are the value and the statistical and combined statistical and systematic errors, see Table 2 for details. The vertical black solid and dashed lines and the grey shaded band show the central value, the statistical and combined error, respectively, of our final result for $f_{K} / f_{π}$ in $2 + 1$ flavor QCD as given in Eq. (16).
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Physical Review D

covering particles, fields, gravitation, and cosmology

Leptonic decay-constant ratio $f_{K} / f_{π}$ from lattice QCD using $2 + 1$ clover-improved fermion flavors with 2-HEX smearing

Stephan Dürr, Zoltán Fodor, Christian Hoelbling, Stefan Krieg, Laurent Lellouch, Thomas Lippert, Thomas Rae, Andreas Schäfer, Enno E. Scholz, Kálmán K. Szabó, and Lukas Varnhorst

Phys. Rev. D 95, 054513 – Published 31 March 2017

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Physical Review D

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Leptonic decay-constant ratio fK/fπ from lattice QCD using 2+1 clover-improved fermion flavors with 2-HEX smearing

Stephan Dürr, Zoltán Fodor, Christian Hoelbling, Stefan Krieg, Laurent Lellouch, Thomas Lippert, Thomas Rae, Andreas Schäfer, Enno E. Scholz, Kálmán K. Szabó, and Lukas Varnhorst

Phys. Rev. D 95, 054513 – Published 31 March 2017

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Leptonic decay-constant ratio $f_{K} / f_{π}$ from lattice QCD using $2 + 1$ clover-improved fermion flavors with 2-HEX smearing