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EW002 $\Delta_{\rm CKM}^{(1)} = |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 - 1$

First-row CKM unitarity and the Cabibbo angle anomaly

Status SUBTLETY-ADDED VERIFIED High Code: PARTIAL Priority Medium

Why this constrains the RS scan

Warped and partially composite models can perturb the apparent CKM matrix through mixing with KK or vector-like quarks, non-universal $W$-quark couplings, and correlated shifts in $Z$-quark couplings. Thus EW002 probes a charged-current direction complementary to the existing $\Delta F=2$ neutral meson constraints. In lepton-extended variants, a new contribution to muon decay can also change the inferred $G_F$, feeding directly into beta- and kaon-decay CKM extractions.

What's changed since the original paper

Relative to arXiv:0804.1954, which emphasized RS-GIM and neutral-current flavor bounds with quoted KK-gluon scales near $21$ and $33\,\mathrm{TeV}$, the post-2008 story is a new charged-current precision problem. Hardy--Towner 2020 shifted the superallowed result to $|V_{ud}|=0.97373(31)$ after updated radiative-correction theory. PDG now quotes a closely related $|V_{ud}|=0.97367(32)$ and explicitly frames the first-row test as a possible $2$--$3\sigma$ inconsistency. FLAG 2024 supplies sub-percent lattice inputs for $f_+(0)$ and $f_K/f_\pi$, with QED and strong-isospin corrections now central to the comparison. The modern SMEFT literature treats the anomaly as a modified $W$-coupling problem correlated with electroweak precision and FCNC observables.

Validity and model dependence

The unitarity sum is experimentally clean but theory-limited. The limiting issues are the inner electroweak radiative correction and nuclear-structure terms in $V_{ud}$, lattice $f_+(0)$ and $f_K/f_\pi$, kaon radiative and isospin-breaking corrections, and correlations between $K_{\ell3}$ and $K_{\mu2}$. Its new-physics interpretation is global-fit-dependent: a shift can live in semileptonic charged currents, $G_F$, right-handed $W$ couplings, or correlated $Z$ and $\Delta F=2$ effects. In custodial RS, reduced $Z b_L b_L$ pressure can make this observable relatively more or less discriminating inside a combined EW fit; quote RS bounds only after specifying custodial protection, fermion embeddings, and brane kinetic terms.

Code coverage in this repo

PARTIAL. The repo fits and gates a unitary CKM target: PDG-derived per-element tolerances are documented at quarkConstraints/scan.py:42, the reduced observable set $(|V_{us}|,|V_{cb}|,|V_{ub}|,J)$ is implemented at quarkConstraints/fit.py:47, and the modern default CKM target is stored at quarkConstraints/modern/inputs.py:631. However, greps found no $V_{ud}$, superallowed-beta, $K_{\ell3}$, $f_+(0)$, or $\Delta_{\rm CKM}$ likelihood implementing EW002 as a constraint.

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

Implementation difficulty

HIGH. A production constraint needs new charged-current SMEFT or direct RS $W$-coupling matching, optional $G_F$ shifts for lepton extensions, and a correlated low-energy likelihood using beta, kaon, radiative-correction, and lattice inputs. This is outside the existing $\Delta F=2$ SLL/SLR/VLL/VRR/LR1/LR2 machinery.

Reason: A live EW002 constraint requires new charged-current SMEFT or direct RS W-quark matching, optional G_F-shift handling for lepton-sector variants, and a correlated beta/kaon/lattice/radiative-correction likelihood. The existing $\Delta F = 2$ basis and CKM fit gate do not cover this observable.

Key references

pdg\_2025\_vud\_vus\_ckm\_unitarity; flag\_2024\_vus\_ckm\_unitarity; hardy\_towner\_2020\_superallowed\_vud; bryman\_cirigliano\_crivellin\_inguglia\_2022\_lfu\_ckm; crivellin\_kirk\_kitahara\_mescia\_2022\_vlq\_caa; cfw\_2008\_rs\_flavor.

value 0.97367

RESOLVED

Match snapshot line 11

L8: Values used in EW002:
L9: - First-row test: |Vud|^2 + |Vus|^2 + |Vub|^2 = 1.
L10: - PDG says |Vub|^2 is about 1.7e-5 and is negligible for the Vud/Vus discussion.
L11: - Superallowed nuclear beta decays: Vud = 0.97367 with component errors
L12:   0.00011_exp, 0.00013_DeltaR^V, 0.00027_NS, total 0.00032.
L13: - Kaon semileptonic product: f_+(0)|Vus| = 0.21656(35).
L14: - FLAG-style lattice inputs quoted by PDG: f_+(0) = 0.9677(27) for

value_uncertainty 0.21656(35)

RESOLVED

Match snapshot line 13

L10: - PDG says |Vub|^2 is about 1.7e-5 and is negligible for the Vud/Vus discussion.
L11: - Superallowed nuclear beta decays: Vud = 0.97367 with component errors
L12:   0.00011_exp, 0.00013_DeltaR^V, 0.00027_NS, total 0.00032.
L13: - Kaon semileptonic product: f_+(0)|Vus| = 0.21656(35).
L14: - FLAG-style lattice inputs quoted by PDG: f_+(0) = 0.9677(27) for
L15:   Nf = 2+1 and f_+(0) = 0.9698(17) for Nf = 2+1+1.
L16: - K_l3 extraction: |Vus| = 0.22330 with errors 0.00004_exp+RC and

value 0.2233

RESOLVED

Match snapshot line 16

L13: - Kaon semileptonic product: f_+(0)|Vus| = 0.21656(35).
L14: - FLAG-style lattice inputs quoted by PDG: f_+(0) = 0.9677(27) for
L15:   Nf = 2+1 and f_+(0) = 0.9698(17) for Nf = 2+1+1.
L16: - K_l3 extraction: |Vus| = 0.22330 with errors 0.00004_exp+RC and
L17:   0.00004_lattice in the compact PDG line; the preceding sentence gives the
L18:   total 0.00053 breakdown as 0.00035_exp, 0.00039_LAT, 0.00008_RC+IB.
L19: - Kaon-sector average used by PDG for the first-row test: Vus = 0.22431(85)

value_uncertainty 0.22431(85)

RESOLVED

Match snapshot line 19

L16: - K_l3 extraction: |Vus| = 0.22330 with errors 0.00004_exp+RC and
L17:   0.00004_lattice in the compact PDG line; the preceding sentence gives the
L18:   total 0.00053 breakdown as 0.00035_exp, 0.00039_LAT, 0.00008_RC+IB.
L19: - Kaon-sector average used by PDG for the first-row test: Vus = 0.22431(85)
L20:   with scale factor S = 2.5.
L21: - PDG unitarity consistency check: |Vud|^2 + |Vus|^2 + |Vub|^2 =
L22:   0.9983(6)(4), about an overall 2 sigma deviation; PDG notes it increases

value 0.9983

RESOLVED

Match snapshot line 22

L19: - Kaon-sector average used by PDG for the first-row test: Vus = 0.22431(85)
L20:   with scale factor S = 2.5.
L21: - PDG unitarity consistency check: |Vud|^2 + |Vus|^2 + |Vub|^2 =
L22:   0.9983(6)(4), about an overall 2 sigma deviation; PDG notes it increases
L23:   to about 3 sigma if nuclear-structure uncertainties are ignored.
L24: 
L25: Relevance notes:

value_uncertainty 0.21654(41)

RESOLVED

Match snapshot line 10

L7: 
L8: Values used in EW002:
L9: - Experimental kaon inputs in Sec. 5.1:
L10:   |Vus| f_+(0) = 0.21654(41).
L11:   |Vus/Vud| f_K+-/f_pi+- = 0.27599(41).
L12: - Direct lattice average for semileptonic form factor:
L13:   f_+(0) = 0.9698(17) for Nf = 2+1+1.

value_uncertainty 0.9698(17)

RESOLVED

Match snapshot line 13

L10:   |Vus| f_+(0) = 0.21654(41).
L11:   |Vus/Vud| f_K+-/f_pi+- = 0.27599(41).
L12: - Direct lattice average for semileptonic form factor:
L13:   f_+(0) = 0.9698(17) for Nf = 2+1+1.
L14:   f_+(0) = 0.9677(27) for Nf = 2+1.
L15: - Direct lattice average for charged decay-constant ratio:
L16:   f_K+-/f_pi+- = 1.1934(19) for Nf = 2+1+1.

value_uncertainty 0.22328(58)

RESOLVED

Match snapshot line 19

L16:   f_K+-/f_pi+- = 1.1934(19) for Nf = 2+1+1.
L17:   f_K+-/f_pi+- = 1.1916(34) for Nf = 2+1.
L18: - Extraction from f_+(0), Nf = 2+1+1:
L19:   |Vus| = 0.22328(58).
L20: - Conversion from f_K+-/f_pi+-, Nf = 2+1+1:
L21:   |Vus|/|Vud| = 0.23126(50).
L22: - |Vub| input for the first-row test:

value_uncertainty 0.99802(66)

RESOLVED

Match snapshot line 25

L22: - |Vub| input for the first-row test:
L23:   |Vub| = 3.82(24)e-3.
L24: - First-row sum using Nf = 2+1+1 f_+(0) and the PDG beta-decay Vud:
L25:   |Vu|^2 = 0.99802(66), about a 3.0 sigma deviation from unity.
L26: - First-row sum using Nf = 2+1+1 f_K+-/f_pi+- and the same Vud:
L27:   |Vu|^2 = 0.99888(67), about a 1.7 sigma deviation from unity.
L28: - Nf = 2+1 comparison:

value_uncertainty 0.99888(67)

RESOLVED

Match snapshot line 27

L24: - First-row sum using Nf = 2+1+1 f_+(0) and the PDG beta-decay Vud:
L25:   |Vu|^2 = 0.99802(66), about a 3.0 sigma deviation from unity.
L26: - First-row sum using Nf = 2+1+1 f_K+-/f_pi+- and the same Vud:
L27:   |Vu|^2 = 0.99888(67), about a 1.7 sigma deviation from unity.
L28: - Nf = 2+1 comparison:
L29:   f_+(0) plus beta-decay Vud gives |Vu|^2 = 0.99824(69), about 2.5 sigma;
L30:   f_K+-/f_pi+- gives |Vu|^2 = 0.99903(72), about 1.4 sigma.

value_uncertainty 0.97373 +/- 0.00031

RESOLVED

Match snapshot line 15

L12:   measurements from 174 references.
L13: - Fifteen corrected Ft values reach precision of 0.3% or better.
L14: - The averaged Ft value combined with the muon lifetime gives
L15:   Vud = 0.97373 +/- 0.00031.
L16: - The result is lower than the 2015 survey by about one standard deviation and
L17:   has a 50% larger uncertainty because of updated radiative-correction theory.
L18: - The lower Vud increases the tension in the top-row CKM unitarity test.

value about 3 sigma

RESOLVED

Match snapshot line 13

L10: 
L11: Values used in EW002:
L12: - The abstract states two Cabibbo-angle tensions: Vus determinations from
L13:   K_mu2, K_l3, and tau decays disagree at about 3 sigma, and beta-decay plus
L14:   averaged Vus inputs imply a first-row unitarity deficit also at about
L15:   3 sigma.
L16: - Their input table includes: