Catalog · Kaon · K002

K002 $\Delta m_K$

$K_L-K_S$ mass difference

Status REVIEWED VERIFIED Low Code: PARTIAL Priority Medium

PDG / equivalent values

Observable	Value	Year	Experiment / source	Provenance
$m(K_L^0$) - $m(K_S^0$)	$\Delta m_K = (0.5293 +/- 0.0009) x 10^10 hbar s^-1$	2026	Particle Data Group pdgLive DataBlock S013D	source ↑
$m(K_L^0$) - $m(K_S^0$)	OUR FIT not assuming CPT: 0.5289 +/- 0.0010	2026	Particle Data Group pdgLive DataBlock S013D	source ↑
Exploratory lattice $\Delta M_K$	$\Delta M_K = 3.19(41)(96) x 10^-12 MeV$	2014	Bai et al., arXiv:1406.0916	source ↑
Physical-mass lattice $\Delta m_K$ statistical uncertainty	statistical error approaches 9%	2023	Bigeng Wang, arXiv:2301.01387	source ↑

Why this constrains the RS scan

In anarchic warped flavor models, tree-level KK-gluon exchange generates $\Delta S=2$ four-quark operators after rotating to the quark-mass basis. $\Delta m_K$ constrains the real part of the new $K^0-\bar K^0$ mixing amplitude, while $\varepsilon_K$ constrains its imaginary part. The same VLL, VRR, SLL/SLR, and LR operator structures used for the repo's neutral-meson mixing lane apply here, and the LR matrix elements remain chirally enhanced. The important physics caveat is that the measured mass splitting contains a large long-distance Standard Model component, so the catalog should treat $\Delta m_K$ as a conservative magnitude bound unless a controlled SM subtraction is supplied.

What's changed since the original paper

Relative to the CFW 2008 baseline (cfw\_2008), the experimental side was updated by KTeV's final neutral-kaon analysis (ktev\_2011), which the PDG fit still uses. The more important post-2008 shift is theoretical: lattice calculations began attacking the long-distance contribution directly. Christ et al. (christ\_2013\_ld\_delta\_mk) developed the long-distance method, and Bai et al. (bai\_2014\_delta\_mk) reported a first complete lattice calculation, $\Delta M_K=3.19(41)(96)\times10^{-12}\,{\rm MeV}$, at heavier-than-physical masses. Later physical-quark-mass studies (wang\_2023\_delta\_mk) reached statistical uncertainty near $9\%$ while retaining important systematics. For BSM interpretation, FLAG 2024 (flag2024) and the physical-mass BSM kaon-mixing calculation (boyle\_2024\_bsm\_kaon\_mixing) update the bag inputs used to translate Wilson coefficients into matrix elements.

Validity and model dependence

The experimental number is precise, but $\Delta m_K$ is long-distance limited as a new-physics constraint. A conservative RS scan can require the absolute new-physics contribution not to exceed the measured splitting, or use only a fraction of it by policy. A sharper bound would require specifying the short-distance charm-loop treatment, the long-distance lattice input, and the CPT-fit convention. This entry is therefore robust as a conservative $\Delta F=2$ magnitude check, but not as a clean SM-subtracted exclusion.

Code coverage in this repo

PARTIAL. The methodology note lists $\Delta m_K$ among the five $\Delta F=2$ scan observables at docs/quark\_scan\_methodology\_note.tex:118. The legacy kaon helper sets the PDG mass splitting at quarkConstraints/deltaf2.py:615 and implements evaluate\_delta\_mk at quarkConstraints/deltaf2.py:755. The modern lane also has a K input displayed as $\Delta m_K$ at quarkConstraints/modern/inputs.py:958 and a bridge evaluator at quarkConstraints/modern/phenomenology.py:493. I keep the status partial because the legacy default kaon input is still the $\varepsilon_K$ entry at quarkConstraints/deltaf2.py:209, and the catalog has not audited a standalone long-distance policy.

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

Implementation difficulty

LOW. No new operator basis is needed for a conservative catalog or scan-side magnitude bound: the existing $\Delta F=2$ VLL/VRR/LR machinery covers the required Wilson coefficients. A future precision SM-subtracted $\Delta m_K$ likelihood would be a separate high-theory-effort project.

Reason: Existing $\Delta F = 2$ Wilson basis and kaon matrix-element machinery cover a conservative $\Delta m_{K}$ magnitude constraint; no new operator is required. Precision SM-subtracted treatment remains long-distance-theory-limited.

Key references

Process-local source keys: pdg\_live\_S013D\_20260516, ktev\_2011, christ\_2013\_ld\_delta\_mk, bai\_2014\_delta\_mk, wang\_2020\_physical\_delta\_mk, wang\_2023\_delta\_mk, boyle\_2024\_bsm\_kaon\_mixing, flag2024, and cfw\_2008.

display_value Delta m_K = (0.5293 +/- 0.0009) x 10^10 hbar s^-1

RESOLVED

Match snapshot line 13

L10:   S013D
L11:   VALUE (10^10 hbar s^-1) DOCUMENT ID  TECN  COMMENT
L12:   0.5289 +/- 0.0010  OUR FIT  Not assuming CPT
L13:   0.5293 +/- 0.0009  OUR FIT  Error includes scale factor of 1.3. Assuming CPT
L14:   0.52797 +/- 0.00195  ABOUZAID 2011  KTEV  Not assuming CPT
L15:   0.52699 +/- 0.00123  ABOUZAID 2011  KTEV  Assuming CPT
L16:

display_value OUR FIT not assuming CPT: 0.5289 +/- 0.0010

RESOLVED

Match snapshot line 12

L9: 
L10:   S013D
L11:   VALUE (10^10 hbar s^-1) DOCUMENT ID  TECN  COMMENT
L12:   0.5289 +/- 0.0010  OUR FIT  Not assuming CPT
L13:   0.5293 +/- 0.0009  OUR FIT  Error includes scale factor of 1.3. Assuming CPT
L14:   0.52797 +/- 0.00195  ABOUZAID 2011  KTEV  Not assuming CPT
L15:   0.52699 +/- 0.00123  ABOUZAID 2011  KTEV  Assuming CPT

display_value Delta M_K = 3.19(41)(96) x 10^-12 MeV

RESOLVED

Match snapshot line 18

L15:   domain-wall fermions, a 330 MeV pion mass, a 575 MeV kaon mass, and a
L16:   quenched charm quark of 949 MeV to implement GIM cancellation.  For those
L17:   heavier-than-physical masses, the result is Delta M_K =
L18:   3.19(41)(96) x 10^-12 MeV, where the first error is statistical and the
L19:   second estimates systematic discretization error.  The abstract emphasizes
L20:   that disconnected diagrams are important.
L21:

display_value statistical error approaches 9%

RESOLVED

Match snapshot line 17

L14:   The work reports completion of a lattice-QCD calculation of Delta m_K on a
L15:   64^3 x 128 lattice using 152 configurations with physical quark masses and
L16:   inverse lattice spacing 1/a = 2.36 GeV.  The abstract states that the
L17:   statistical error approaches 9%, while several sources of systematic error
L18:   may have more significant effects.
L19: