Catalog · Kaon · K001

K001 $\epsilon_K$

Indirect CP violation in K0-K0bar mixing

Status REVIEWED VERIFIED Low Code: YES Priority High

PDG / equivalent values

Observable	Value	Year	Experiment / source	Provenance
$\|\epsilon\|$	$\|\epsilon\| = (2.228 +/- 0.011) x 10^-3$	2026	Particle Data Group, 2026 Review of Particle Physics, CP Violation in K0_L Decays	source ↑
$\|\epsilon_K\|_{SM}$	2.16(18) x 10^-3	2020	Brod, Gorbahn, and Stamou, Phys. Rev. Lett. 125, 171803 (2020)	source ↑

Why this constrains the RS scan

In anarchic RS models, KK-gluon and electroweak gauge exchange generate misaligned four-quark operators for $K^0$-$\bar K^0$ mixing. The left-right matrix elements are chirally enhanced, and generic complex phases make $\varepsilon_K$ one of the strongest constraints on the KK scale. The CFW baseline (CsakiFalkowskiWeiler:RSFlavor2008) quotes KK-gluon masses of about $21$ TeV in standard RS and about $33$ TeV in the composite pseudo-Goldstone variant.

What's changed since the original paper

Since the 2008 CFW baseline, the experimental central value has remained precise. The main updates are theory and lattice inputs. BGS (BrodGorbahnStamou:EpsilonK2020) reformulated the SM prediction with manifest CKM unitarity and reduced the charm short-distance uncertainty, giving the $2.161\times10^{-3}$ central value used in the repository's legacy $\Delta F=2$ path. FLAG 2024 (FLAG2024:BK) updates the kaon bag averages. The catalog should therefore keep the experimental number, the SM theory budget, and the lattice inputs separate.

Validity and model dependence

As an RS constraint this is robust for generic tree-level $\Delta F=2$ new physics with arbitrary CP phases. Model dependence enters through the matching convention, QCD running, hadronic matrix elements, and the choice of how much of $|\varepsilon_K^{\rm exp}-\varepsilon_K^{\rm SM}|$ is assigned as the allowed NP budget. It is not a standalone measurement of one Wilson coefficient; it constrains the imaginary part of the full $K$-mixing amplitude.

Code coverage in this repo

Coverage status: YES. The legacy implementation registers epsilon\_k in quarkConstraints/deltaf2.py:209--212, stores the PDG and BGS constants at quarkConstraints/deltaf2.py:622--623, and evaluates the observable in quarkConstraints/deltaf2.py:729. The QCD-running wrapper is at quarkConstraints/deltaf2.py:775. The modern policy surface also enumerates epsilon\_K at quarkConstraints/modern/phenomenology.py:23 and routes it through the bridge evaluator at quarkConstraints/modern/phenomenology.py:458 and quarkConstraints/modern/phenomenology.py:1320. Tests exercise the evaluator at tests/test\_epsilon\_k\_physics.py:166 and tests/test\_epsilon\_k\_physics.py:421.

Linked evidence (opens GitHub blob at flavor-catalog-website/2026q2):

Implementation difficulty

Implementation difficulty: LOW. The needed $\Delta F=2$ operator basis, hadronic contraction, QCD-running wrapper, and pass/fail budget already exist. A future production cleanup should reconcile the legacy and modern frozen input constants, but no new operator, lattice machinery, or exclusive-mode calculation is needed for catalog-level coverage.

Reason: Existing $\Delta F = 2$ SLL/SLR/VLL/VRR/LR1/LR2-style kaon-mixing machinery already covers epsilon_K; catalog promotion mainly needs input/provenance reconciliation.

Key references

Process-local source keys before bibliography consolidation: PDG2026:EpsilonK, BrodGorbahnStamou:EpsilonK2020, FLAG2024:BK, and CsakiFalkowskiWeiler:RSFlavor2008.

display_value |epsilon| = (2.228 +/- 0.011) x 10^-3

AMBIGUOUS

Match 1 of 2 snapshot line 9

L6: 
L7: Relevant extracted values:
L8: - The review states that CP violation was observed in K0 decays at a level given by epsilon approximately 2.3e-3.
L9: - Equation 68.8a gives |epsilon| = (2.228 +/- 0.011) x 10^-3.
L10: - Equation 68.8b gives phi_epsilon = (43.5 +/- 0.5) degrees.
L11: - Equation 68.8f gives A_L = (3.32 +/- 0.06) x 10^-3.
L12:

Match 2 of 2 snapshot line 14

L11: - Equation 68.8f gives A_L = (3.32 +/- 0.06) x 10^-3.
L12: 
L13: Notes for K001:
L14: Use |epsilon| = (2.228 +/- 0.011) x 10^-3 as the canonical experimental
L15: epsilon_K magnitude unless the checker selects a PDG data-block value with
L16: newer metadata. The phase and semileptonic asymmetry are context values, not
L17: the headline K001 constraint.

related_phase phi_epsilon = (43.5 +/- 0.5) degrees

RESOLVED

Match snapshot line 10

L7: Relevant extracted values:
L8: - The review states that CP violation was observed in K0 decays at a level given by epsilon approximately 2.3e-3.
L9: - Equation 68.8a gives |epsilon| = (2.228 +/- 0.011) x 10^-3.
L10: - Equation 68.8b gives phi_epsilon = (43.5 +/- 0.5) degrees.
L11: - Equation 68.8f gives A_L = (3.32 +/- 0.06) x 10^-3.
L12: 
L13: Notes for K001:

related_semileptonic_asymmetry A_L = (3.32 +/- 0.06) x 10^-3

RESOLVED

Match snapshot line 11

L8: - The review states that CP violation was observed in K0 decays at a level given by epsilon approximately 2.3e-3.
L9: - Equation 68.8a gives |epsilon| = (2.228 +/- 0.011) x 10^-3.
L10: - Equation 68.8b gives phi_epsilon = (43.5 +/- 0.5) degrees.
L11: - Equation 68.8f gives A_L = (3.32 +/- 0.06) x 10^-3.
L12: 
L13: Notes for K001:
L14: Use |epsilon| = (2.228 +/- 0.011) x 10^-3 as the canonical experimental

total_display_value 2.16(18) x 10^-3

RESOLVED

Match snapshot line 12

L9: Relevant extracted values:
L10: - Abstract: epsilon_K = 2.16(6)(8)(15) x 10^-3. The three uncertainties are QCD short-distance, long-distance, and parametric.
L11: - TeX-source equation gives |epsilon_K| = (2.161 +/- 0.153_param. +/- 0.064_eta_tt +/- 0.008_eta_ut +/- 0.027_Bhat_K +/- 0.052_xi_s +/- 0.046_kappa_epsilon) x 10^-3.
L12: - The same equation groups this as (2.161 +/- 0.153_param. +/- 0.076_non-pert. +/- 0.065_pert.) x 10^-3 = 2.16(18) x 10^-3.
L13: - The phenomenological formula uses kappa_epsilon = 0.94(2) and, in the original 2020 input set, Bhat_K = 0.7625(97).
L14: 
L15: Notes for K001:

hat_B_K_Nf21 0.7533(91)

RESOLVED

Match snapshot line 10

L7: 
L8: Relevant extracted values from BK/BK.tex in arXiv source v3:
L9: - The review states that the expression of epsilon_K in terms of B_K is derived in the three-flavour theory.
L10: - For N_f = 2+1, the FLAG average is Bhat_K = 0.7533(91).
L11: - For N_f = 2+1, the converted values are B_K^MSbar(2 GeV) = 0.5503(66) and B_K^MSbar(3 GeV) = 0.5324(64).
L12: - For N_f = 2+1+1, the FLAG entry quotes Bhat_K = 0.717(18)(16), B_K^MSbar(2 GeV) = 0.524(13)(12), and B_K^MSbar(3 GeV) = 0.507(13)(11).
L13:

B_K_MSbar_2GeV_Nf21 0.5503(66)

RESOLVED

Match snapshot line 11

L8: Relevant extracted values from BK/BK.tex in arXiv source v3:
L9: - The review states that the expression of epsilon_K in terms of B_K is derived in the three-flavour theory.
L10: - For N_f = 2+1, the FLAG average is Bhat_K = 0.7533(91).
L11: - For N_f = 2+1, the converted values are B_K^MSbar(2 GeV) = 0.5503(66) and B_K^MSbar(3 GeV) = 0.5324(64).
L12: - For N_f = 2+1+1, the FLAG entry quotes Bhat_K = 0.717(18)(16), B_K^MSbar(2 GeV) = 0.524(13)(12), and B_K^MSbar(3 GeV) = 0.507(13)(11).
L13: 
L14: Notes for K001:

B_K_MSbar_3GeV_Nf21 0.5324(64)

RESOLVED

Match snapshot line 11

L8: Relevant extracted values from BK/BK.tex in arXiv source v3:
L9: - The review states that the expression of epsilon_K in terms of B_K is derived in the three-flavour theory.
L10: - For N_f = 2+1, the FLAG average is Bhat_K = 0.7533(91).
L11: - For N_f = 2+1, the converted values are B_K^MSbar(2 GeV) = 0.5503(66) and B_K^MSbar(3 GeV) = 0.5324(64).
L12: - For N_f = 2+1+1, the FLAG entry quotes Bhat_K = 0.717(18)(16), B_K^MSbar(2 GeV) = 0.524(13)(12), and B_K^MSbar(3 GeV) = 0.507(13)(11).
L13: 
L14: Notes for K001:

hat_B_K_Nf211 0.717(18)(16)

RESOLVED

Match snapshot line 12

L9: - The review states that the expression of epsilon_K in terms of B_K is derived in the three-flavour theory.
L10: - For N_f = 2+1, the FLAG average is Bhat_K = 0.7533(91).
L11: - For N_f = 2+1, the converted values are B_K^MSbar(2 GeV) = 0.5503(66) and B_K^MSbar(3 GeV) = 0.5324(64).
L12: - For N_f = 2+1+1, the FLAG entry quotes Bhat_K = 0.717(18)(16), B_K^MSbar(2 GeV) = 0.524(13)(12), and B_K^MSbar(3 GeV) = 0.507(13)(11).
L13: 
L14: Notes for K001:
L15: The live legacy code uses B_1_K = 0.5503, matching the N_f = 2+1