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Calculation of $K \to \pi\pi$ decay amplitudes with improved Wilson fermion in 2+1 flavor lattice QCD
High Energy Physics - Lattice · Submitted 30 Oct 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We obtain, for the first time with a Wilson-type fermion action, that ${\rm Re}A_0 = 60(36) \times10^{ -8}\,{\rm GeV}$ and ${\rm Im}A_0 =-67(56) \times10^{-12}\,{\rm GeV}$ for a matching scale $q^* =1/a$.
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The abstract, sentence by sentence
We present results for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ channels. This calculation is carried out on 480 gauge configurations in $N_f=2+1$ QCD generated over 12,000 trajectories with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$, $m_\pi=280\,{\rm MeV}$ and $m_K=580\,{\rm MeV}$ on a $32^3\times 64$ ($La=2.9\,{\rm fm}$) lattice. For the quark loops in the Penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver techniques work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${\rm Re}A_0 = 60(36) \times10^{ -8}\,{\rm GeV}$ and ${\rm Im}A_0 =-67(56) \times10^{-12}\,{\rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale is weak.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present results for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ channels. | contribution 0.86 | 0.94 | 0.18 | asserted 0.99 |
| s1 | This calculation is carried out on 480 gauge configurations in $N_f=2+1$ QCD generated over 12,000 trajectories with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$, $m_\pi=280\,{\rm MeV}$ and $m_K=580\,{\rm MeV}$ on a $32^3\times 64$ ($La=2.9\,{\rm fm}$) lattice. | method 1.00 | 0.92 | 0.03 | asserted 1.00 |
| s2 | For the quark loops in the Penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver techniques work very well for variance reduction. | method 0.99 | 0.85 | 0.03 | asserted 1.00 |
| s3 | We obtain, for the first time with a Wilson-type fermion action, that ${\rm Re}A_0 = 60(36) \times10^{ -8}\,{\rm GeV}$ and ${\rm Im}A_0 =-67(56) \times10^{-12}\,{\rm GeV}$ for a matching scale $q^* =1/a$. | result 1.00 | 0.97 | 0.80 | asserted 1.00 |
| s4 | The dependence on the matching scale is weak. | result 0.60 | 0.83 | 0.05 | asserted 1.00 |
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The paper
Original abstract
We present results for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ channels. This calculation is carried out on 480 gauge configurations in $N_f=2+1$ QCD generated over 12,000 trajectories with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$, $m_\pi=280\,{\rm MeV}$ and $m_K=580\,{\rm MeV}$ on a $32^3\times 64$ ($La=2.9\,{\rm fm}$) lattice. For the quark loops in the Penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver techniques work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${\rm Re}A_0 = 60(36) \times10^{ -8}\,{\rm GeV}$ and ${\rm Im}A_0 =-67(56) \times10^{-12}\,{\rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale is weak.
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What this paper does with the arXiv papers it cites
What it cites · 8 sentences without a confident label
Uses the method or tool of 4 papers
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In the present work we subtract it by imposing the relation [ref] ,
New lattice approaches to the $\Delta I=1/2$ rule · Calculation · 0.99
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where $δ^I(p)$ is the two-pion scattering phase shift for the iso-spin $I$ channel at the scattering momentum $p=√( E^2/4 - m_π^2)$, and $ϕ(q)$ is the analytic function defined in Ref. [ref] at $q=p ( 2π/L)$.
Weak transition matrix elements from finite-volume correlation functions · Results · 0.84
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For the calculation of the quark loop $Q(x,x)$, that is the quark propagator starting from the weak operator and ending at the same position in the type3 and type4 contractions, we use the stochastic method with the hopping parameter expansion technique (HPE) and the truncated solver method (TSM) proposed in Ref. [ref] .
Effective noise reduction techniques for disconnected loops in Lattice QCD · Calculation · 1.00
Uses data or values from 1 paper
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Our calculations are carried out on a subset of configurations previously generated by PACS-CS Collaboration with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $β=1.9$ on a $32^3×64$ lattice [ref] .
2+1 Flavor Lattice QCD toward the Physical Point · Introduction · 0.98
Compares with 2 papers
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We observe that the results for $A_0$ by RBC-UKQCD Collaboration at a similar quark mass $m_π=330 MeV$ [ref] have smaller errors than ours.
Practical methods for a direct calculation of $\Delta I=1/2$ $K$ to $\pi\pi$ Decay · Results · 0.99
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This appears different from the previous work by RBC-UKQCD collaboration with the domain wall fermion action in Refs. [ref] .
$K$ to $\pi\pi$ Decay amplitudes from Lattice QCD · Results · 0.94
Cites as support 2 papers
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This is also valid for the Wilson fermion action due to the $CPS$ symmetry as shown in Ref. [ref] and elaborated in Ref. [ref] .
Calculation of $K \to \pi\pi$ decay amplitudes with improved Wilson fermion · Calculation · 0.97
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This is also valid for the Wilson fermion action due to the $CPS$ symmetry as shown in Ref. [ref] and elaborated in Ref. [ref] .
Non-Perturbative Renormalization of Lattice Four-Fermion Operators without Power Subtractions · Calculation · 0.97
Cites as background 1 paper
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The results of a first direct calculation, on a $16^3×32$ lattice, was reported by RBC-UKQCD collaboration in Ref. [ref] at $m_π=422 MeV$ with the domain wall fermion action.
$K$ to $\pi\pi$ Decay amplitudes from Lattice QCD · Introduction · 0.85
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