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arxiv: 2512.19294 · v2 · submitted 2025-12-22 · ✦ hep-lat · hep-ph

f_K/f_(π) in iso-symmetric QCD and the CKM matrix unitarity

Alessandro Conigli , Julien Frison , Alejandro S\'aez This is my paper

Pith reviewed 2026-05-16 20:52 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords lattice QCDf_K/f_piCKM matrixisospin-symmetric QCDV_usV_udkaon decay constantunitarity
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The pith

Lattice QCD computes the kaon-to-pion decay constant ratio in the isospin-symmetric limit to determine |V_us|/|V_ud| and test first-row CKM unitarity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates the ratio of kaon to pion decay constants using lattice simulations of QCD with two light and one strange quark, but with the up and down quark masses set equal. This ratio enters the extraction of the Cabibbo-Kobayashi-Maskawa matrix elements V_us and V_ud from experimental decay rates. The authors combine a unitary Wilson action with a mixed-action setup to tighten control over how results approach the physical continuum limit. They then add corrections for strong isospin breaking and electromagnetic effects to check whether the first row of the CKM matrix sums to one.

Core claim

In the isospin-symmetric limit of QCD with N_f=2+1, the ratio f_K/f_π is extracted from lattice correlation functions on ensembles generated with a combination of Wilson unitary and mixed actions, yielding a value that, after inclusion of strong isospin-breaking and QED corrections, supports a determination of |V_us|/|V_ud| and a consistency check of first-row CKM unitarity.

What carries the argument

The ratio f_K/f_π extracted from two-point correlation functions in the isospin-symmetric limit of N_f=2+1 QCD, with continuum extrapolation controlled by combining two lattice regularizations.

If this is right

  • The computed ratio supplies a direct lattice input for extracting |V_us| from kaon leptonic decays.
  • After adding isospin-breaking and QED corrections the result enters a test of whether |V_ud|^2 + |V_us|^2 + |V_ub|^2 equals unity.
  • The dual-regularization approach reduces the dominant systematic uncertainty in the continuum limit for this quantity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A confirmed deviation from unitarity at the current precision level would indicate physics beyond the Standard Model in the weak sector.
  • The same combined-action strategy could be applied to other decay constants or form factors to improve continuum control.
  • Higher-statistics runs or inclusion of dynamical charm quarks would further shrink the error on |V_us|/|V_ud|.

Load-bearing premise

The combination of the Wilson unitary and mixed-action setups fully removes discretization effects so the continuum extrapolation can be performed without significant residual bias.

What would settle it

An independent experimental or phenomenological determination of |V_us|/|V_ud| from tau decays or hyperon decays lying outside the uncertainty range obtained here would falsify the central result.

Figures

Figures reproduced from arXiv: 2512.19294 by Alejandro S\'aez, Alessandro Conigli, Julien Frison.

Figure 1
Figure 1. Figure 1: Measured values of ρ2, ρ4, defined in Eq. (2.2) and Eq. (2.3), for the set of CLS ensembles employed in this work, following the tr(Mq) = 2ml + ms ≈ const. chiral trajectory. Empty points correspond to the Wilson unitary setup, while filled ones correspond to the Wtm mixed action. The difference in ρ2 and ρ4 between the Wilson and Wtm mixed action for any given ensemble is due to different cutoff effects i… view at source ↗
Figure 2
Figure 2. Figure 2: RX for X = mπ, fπ, fK for the set of ensembles considered in this work in the Wilson unitary setup, as given by Eq. (3.1) (blue crosses), together with the expected dependence on mπL for each of these ensembles (grey dashed lines). It can be seen that the finite-volume correction is of a few per mille at most, well below the statistical precision of the lattice data (the red horizontal line shows a represe… view at source ↗
Figure 3
Figure 3. Figure 3: Model exploration for the chiral-continuum extrapolation of the ratio [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Contribution of the different lattice ensembles to the statistical error squared in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left column: chiral-continuum extrapolation employing Eq. (4.12), using µ = fπ in the definition of the chiral logarithms, and adding cutoff effects according to Eq. (4.11) with Γi = 0, removing mπ > 360 MeV. Right column: chiral-continuum extrapolation employing Eq. (4.5), using µ = fπ, and adding cutoff effects according to Eq. (4.11) with Γi = 0, removing the coarsest lattice spacing β = 3.40. In the tw… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of our result for fK± /fπ± with strong isospin-breaking corrections, as given in Eq. (5.7), to other group’s results entering the FLAG average for Nf = 2 + 1 [8]. CLQCD 23 refers to [31], QCDSF/UKQCD 16 to [32], BMW 16 to [33], RBC/UKQCD 14B to [34], MILC 10 to [35], BMW 10 to [36] and HPQCD/UKQCD 07 to [37]. The point ALPHA 25 corresponds to our previous determination of fK in [1] based on an a… view at source ↗
read the original abstract

We present lattice results for $f_K/f_{\pi}$ in the iso-symmetric limit of pure QCD (isoQCD) with $N_f=2+1$ flavours, along with a determination of $|V_{us}|/|V_{ud}|$ and a study on the unitarity of the first row of the Cabibbo-Kobayashi-Maskawa (CKM) matrix after introducing strong isospin-breaking and QED effects. The results obtained are based on a combination of a Wilson unitary action and the mixed-action setup introduced in arXiv:2309.14154, arXiv:2510.20450. The combination of the two regularisations enables a more precise control over the continuum-limit extrapolation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper computes the ratio f_K/f_π in the iso-symmetric limit of N_f=2+1 QCD using lattice simulations that combine a Wilson unitary action with the mixed-action setup from prior works (arXiv:2309.14154, arXiv:2510.20450). It then incorporates strong isospin-breaking and QED corrections to extract |V_us|/|V_ud| and test unitarity of the first row of the CKM matrix, claiming improved control over the continuum extrapolation from the dual regularizations.

Significance. If the central results hold, the work supplies a high-precision lattice input for |V_us|/|V_ud| that directly informs the CKM unitarity test, a key probe for beyond-Standard-Model physics. The explicit use of two regularizations to tighten the continuum limit is a methodological strength that, when accompanied by quantitative error budgets, can reduce systematic uncertainties relative to single-action studies.

minor comments (3)
  1. [Abstract] The abstract states the goals and setup but supplies no numerical values, error bars, or fit details; adding a single representative result (e.g., the final f_K/f_π value with uncertainty) would allow immediate assessment of precision.
  2. [Continuum extrapolation] In the continuum-extrapolation section, the claim that the combination of regularizations 'enables a more precise control' should be supported by an explicit comparison of extrapolation errors or by showing the joint fit quality (e.g., via a table of fit parameters and χ²/dof for single- vs. dual-action extrapolations).
  3. [Results tables] Ensure all tables reporting final results include the full error budget (statistical, chiral, continuum, isospin-breaking, QED) so that the unitarity test can be reproduced from the published numbers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. We appreciate the recognition of the methodological advantage provided by combining the Wilson unitary action with the mixed-action setup to improve control over the continuum extrapolation.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation consists of a standard N_f=2+1 lattice QCD computation of f_K/f_π in the iso-symmetric limit using a Wilson unitary action combined with an established mixed-action regularization from independent prior works. Continuum extrapolation control is achieved by the combination of regularizations, which is a conventional methodological step without any reduction of the reported ratio to a fitted parameter defined by the target observable. The subsequent extraction of |V_us|/|V_ud| and the CKM unitarity test incorporate external isospin-breaking and QED corrections following established practice. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the chain; the central results remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard lattice QCD assumptions about the existence of a continuum limit and the validity of the cited mixed-action regularization for controlling discretization effects. No new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The continuum limit of lattice QCD with the chosen actions exists and can be reached by controlled extrapolation
    Implicit in every lattice calculation that performs a continuum extrapolation.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. $F_K/F_\pi$ as a precision test of a new four flavor Domain Wall Fermion action

    hep-lat 2026-05 unverdicted novelty 5.0

    New four-flavor smeared Möbius Domain Wall Fermion ensembles yield F_K/F_pi = 1.1962(34) as a precision test for inexpensive chiral fermion calculations in lattice QCD.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · cited by 1 Pith paper · 27 internal anchors

  1. [1]

    Bussone, A

    (ALPHA), A. Bussone, A. Conigli, J. Frison, G. Herdo´ ıza, C. Pena, D. Preti et al.,Hadronic physics from a Wilson fermion mixed-action approach: Setup and scale setting,2510.20450

    work page arXiv

  2. [2]

    Bussone, A

    (Alpha), A. Bussone, A. Conigli, J. Frison, G. Herdo´ ıza, C. Pena, D. Preti et al.,Hadronic physics from a Wilson fermion mixed-action approach: charm quark mass andD (s) meson decay constants,Eur. Phys. J. C84(2024) 506, [2309.14154]

    work page arXiv 2024

  3. [3]

    W. J. Marciano and A. Sirlin,Radiative corrections to pi(lepton 2) decays,Phys. Rev. Lett. 71(1993) 3629–3632

    work page 1993

  4. [4]

    A note on isospin violation in Pl2(gamma) decays

    V. Cirigliano and H. Neufeld,A note on isospin violation in Pl2(gamma) decays,Phys. Lett. B700(2011) 7–10, [1102.0563]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  5. [5]

    Navas et al.,Review of particle physics,Phys

    (Particle Data Group), S. Navas et al.,Review of particle physics,Phys. Rev. D110 (2024) 030001

    work page 2024

  6. [6]

    Di Carlo, D

    M. Di Carlo, D. Giusti, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo et al., Light-meson leptonic decay rates in lattice QCD+QED,Phys. Rev. D100(2019) 034514, [1904.08731]

    work page arXiv 2019

  7. [7]

    Boyle et al

    P. Boyle et al.,Isospin-breaking corrections to light-meson leptonic decays from lattice simulations at physical quark masses,JHEP02(2023) 242, [2211.12865]

    work page arXiv 2023

  8. [8]

    FLAG Review 2024

    (Flavour Lattice A veraging Group (FLAG)), Y. Aoki et al.,FLAG Review 2024, 2411.04268

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  9. [9]

    Simulation of QCD with N_f=2+1 flavors of non-perturbatively improved Wilson fermions

    M. Bruno et al.,Simulation of QCD with N f =2+1 flavors of non-perturbatively improved Wilson fermions,JHEP02(2015) 043, [1411.3982]. 18

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  10. [10]

    CLS 2+1 flavor simulations at physical light- and strange-quark masses

    D. Mohler, S. Schaefer and J. Simeth,CLS 2+1 flavor simulations at physical light- and strange-quark masses,EPJ Web Conf.175(2018) 02010, [1712.04884]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  11. [11]

    Lattice QCD with a chirally twisted mass term

    (Alpha), R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz,Lattice QCD with a chirally twisted mass term,JHEP08(2001) 058, [hep-lat/0101001]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  12. [12]

    C. Pena, S. Sint and A. Vladikas,Twisted mass QCD and lattice approaches to the Delta I = 1/2 rule,JHEP09(2004) 069, [hep-lat/0405028]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  13. [13]

    Properties and uses of the Wilson flow in lattice QCD

    M. L¨ uscher,Properties and uses of the Wilson flow in lattice QCD,JHEP08(2010) 071, [1006.4518]. [Erratum: JHEP 03, 092 (2014)]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  14. [14]

    High-precision scale setting in lattice QCD

    (BMW), S. Bors´ anyi, S. D¨ urr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg et al., High-precision scale setting in lattice QCD,JHEP09(2012) 010, [1203.4469]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  15. [15]

    Monte Carlo errors with less errors

    (ALPHA), U. Wolff,Monte Carlo errors with less errors,Comput. Phys. Commun.156 (2004) 143–153, [hep-lat/0306017]. [Erratum: Comput.Phys.Commun. 176, 383 (2007)]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  16. [16]

    Automatic differentiation for error analysis of Monte Carlo data

    A. Ramos,Automatic differentiation for error analysis of Monte Carlo data,Comput. Phys. Commun.238(2019) 19–35, [1809.01289]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  17. [17]

    Setting the scale for the CLS $2 + 1$ flavor ensembles

    M. Bruno, T. Korzec and S. Schaefer,Setting the scale for the CLS2 + 1flavor ensembles, Phys. Rev. D95(2017) 074504, [1608.08900]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  18. [18]

    Kuberski,Low-mode deflation for twisted-mass and RHMC reweighting in lattice QCD, Comput

    S. Kuberski,Low-mode deflation for twisted-mass and RHMC reweighting in lattice QCD, Comput. Phys. Commun.300(2024) 109173, [2306.02385]

    work page arXiv 2024

  19. [19]

    E. T. Neil and J. W. Sitison,Model averaging approaches to data subset selection,Phys. Rev. E108(2023) 045308, [2305.19417]

    work page arXiv 2023

  20. [20]

    Lattice QCD with open boundary conditions and twisted-mass reweighting

    M. Luscher and S. Schaefer,Lattice QCD with open boundary conditions and twisted-mass reweighting,Comput. Phys. Commun.184(2013) 519–528, [1206.2809]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  21. [21]

    A. D. Kennedy, I. Horvath and S. Sint,A New exact method for dynamical fermion computations with nonlocal actions,Nucl. Phys. B Proc. Suppl.73(1999) 834–836, [hep-lat/9809092]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  22. [22]

    M. A. Clark and A. D. Kennedy,The RHMC algorithm for two flavors of dynamical staggered fermions,Nucl. Phys. B Proc. Suppl.129(2004) 850–852, [hep-lat/0309084]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  23. [23]

    M. A. Clark and A. D. Kennedy,Accelerating dynamical fermion computations using the rational hybrid Monte Carlo (RHMC) algorithm with multiple pseudofermion fields,Phys. Rev. Lett.98(2007) 051601, [hep-lat/0608015]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  24. [24]

    Charm and strange quark in openqcd simulations

    M. L¨ uscher, “Charm and strange quark in openqcd simulations. ” http://luscher.web.cern.ch/luscher/openQCD/, 2019

    work page 2019

  25. [25]

    The pion and proton mass in finite volume

    G. Colangelo, A. Fuhrer and C. Haefeli,The pion and proton mass in finite volume,Nucl. Phys. B Proc. Suppl.153(2006) 41–48, [hep-lat/0512002]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  26. [26]

    Finite volume effects for meson masses and decay constants

    G. Colangelo, S. Durr and C. Haefeli,Finite volume effects for meson masses and decay constants,Nucl. Phys. B721(2005) 136–174, [hep-lat/0503014]. 19

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  27. [27]

    Husung,Logarithmic corrections to O(a) and O(a 2) effects in lattice QCD with Wilson or Ginsparg–Wilson quarks,Eur

    N. Husung,Logarithmic corrections to O(a) and O(a 2) effects in lattice QCD with Wilson or Ginsparg–Wilson quarks,Eur. Phys. J. C83(2023) 142, [2206.03536]. [Erratum: Eur.Phys.J.C 83, 144 (2023)]

    work page arXiv 2023

  28. [28]

    Frison, (2023), arXiv:2302.06550 [hep-lat]

    J. Frison,Towards fully bayesian analyses in Lattice QCD,2302.06550

    work page arXiv

  29. [29]

    & Sommer, R

    M. Bruno and R. Sommer,On fits to correlated and auto-correlated data,Comput. Phys. Commun.285(2023) 108643, [2209.14188]

    work page arXiv 2023

  30. [30]

    pi/K -> e nu branching ratios to O(e^2 p^4) in Chiral Perturbation Theory

    V. Cirigliano and I. Rosell,pi/K —>e anti-nu(e) branching ratios to O(e**2 p**4) in Chiral Perturbation Theory,JHEP10(2007) 005, [0707.4464]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  31. [31]

    Huet al.(CLQCD), Phys

    (CLQCD), Z.-C. Hu et al.,Quark masses and low-energy constants in the continuum from the tadpole-improved clover ensembles,Phys. Rev. D109(2024) 054507, [2310.00814]

    work page arXiv 2024

  32. [32]

    (QCDSF–UKQCD), V. G. Bornyakov, R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow et al.,Flavour breaking effects in the pseudoscalar meson decay constants, Phys. Lett. B767(2017) 366–373, [1612.04798]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    Leptonic decay-constant ratio $f_K/f_\pi$ from lattice QCD using 2+1 clover-improved fermion flavors with 2-HEX smearing

    S. D¨ urr et al.,Leptonic decay-constant ratiofK/fπfrom lattice QCD using 2+1 clover-improved fermion flavors with 2-HEX smearing,Phys. Rev. D95(2017) 054513, [1601.05998]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  34. [34]

    Domain wall QCD with physical quark masses

    (RBC, UKQCD), T. Blum et al.,Domain wall QCD with physical quark masses,Phys. Rev. D93(2016) 074505, [1411.7017]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  35. [35]

    Results for light pseudoscalar mesons

    (MILC), A. Bazavov et al.,Results for light pseudoscalar mesons,PoSLA TTICE2010 (2010) 074, [1012.0868]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  36. [36]

    (BMW), S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth et al.,The ratio FK/Fpi in QCD,Phys. Rev. D81(2010) 054507, [1001.4692]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  37. [37]

    High Precision determination of the pi, K, D and D_s decay constants from lattice QCD

    (HPQCD, UKQCD), E. Follana, C. T. H. Davies, G. P. Lepage and J. Shigemitsu,High Precision determination of the pi, K, D and D(s) decay constants from lattice QCD,Phys. Rev. Lett.100(2008) 062002, [0706.1726]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  38. [38]

    Radiative corrections in weak semi-leptoni processes at low energy: a two-step matching determination

    S. Descotes-Genon and B. Moussallam,Radiative corrections in weak semi-leptonic processes at low energy: A Two-step matching determination,Eur. Phys. J. C42(2005) 403–417, [hep-ph/0505077]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  39. [39]

    Four-point correlator constraints on electromagnetic chiral parameters and resonance effective Lagrangians

    B. Ananthanarayan and B. Moussallam,Four-point correlator constraints on electromagnetic chiral parameters and resonance effective Lagrangians,JHEP06(2004) 047, [hep-ph/0405206]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  40. [40]

    Review of lattice results concerning low-energy particle physics

    S. Aoki et al.,Review of lattice results concerning low-energy particle physics,Eur. Phys. J. C 77(2017) 112, [1607.00299]. 20

    work page internal anchor Pith review Pith/arXiv arXiv 2017