CKMfitter

Preliminary results as of Summer 2025
(CKM workshop 2025)

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The global CKM fit: Inputs and Numerical results

The global CKM fit in the large (ρ-bar,η-bar) plane

The global CKM fit in the small (ρ-bar,η-bar) plane (zoom)

The global CKM fit: Pulls

The global CKM fit in the large (ρ̅_M, η̅_M) plane with M = sb, ds, ct, ut, uc

The global CKM fit in the (|V_ud|,|V_us|) plane

The global CKM fit in the (|V_cd|,|V_cs|) plane

The global CKM fit in the (|V_ub|,|V_cb|) plane

The global CKM fit for α

The |V_td/V_ts| ratio from Radiative Penguins

The very rare Kaon decays

Numerical results:

The results of the global CKM analysis include:

Wolfenstein parameters,
UT angles and sides,
UT_sangle and apex,
CKM elements,
theory parameters,
rare branching fractions (B->lν, B->ll).

Numerical Results
(pdf)

The global CKM fit in the large (ρ-bar,η-bar) plane:

Constraints in the (ρ-bar,η-bar) plane. The red hashed region of the global combination corresponds to 68% CL.

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Constraints in the (ρ-bar,η-bar) plane. The |V_ub| constraint has been split in three contributions: |V_ub| from inclusive and exclusive semileptonic B decays (plain dark green), |V_ub| from B⁺→τ⁺ ν (hashed darker green), and |V_ub|/|V_cb| from Λ_b decays (hashed ligher green). The red hashed region of the global combination corresponds to 68% CL.

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The global CKM fit in the small (ρ-bar,η-bar) plane (zoom):

Zoomed constraints in the (ρ-bar,η-bar) plane.The red hashed region of the global combination corresponds to 68% CL.

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Zoomed constraints in the (ρ-bar,η-bar) plane. The |V_ub| constraint has been split in three contributions: |V_ub| from inclusive and exclusive semileptonic B decays (plain dark green), |V_ub| from B⁺→τ⁺ ν (hashed darker green), and |V_ub|/|V_cb| from Λ_b decays (hashed ligher green). The red hashed region of the global combination corresponds to 68% CL.

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Zoomed constraints in the (ρ-bar,η-bar) plane not including the angle measurements in the global fit.

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Constraints in the (ρ-bar,η-bar) plane including only the angle measurements.

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Constraints from CP conserving quantities (|V_ub / V_cb|, Δm_d, (Δm_d and Δm_s) and B⁺ →τ⁺ ν) in the (ρ-bar,η-bar) plane.

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Constraints from CP violating quantities (sin(2β), α, γ and ε_k) in the (ρ-bar,η-bar) plane.

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Constraints from "Tree" quantities in the (ρ-bar,η-bar) plane (involving γ(DK) and α from the isospin analysis with the help of sin2β (charmonium), which gives another tree only γ measurement (the only assumption is that the ΔI=3/2 b-->d EW penguin amplitude is negligible)).

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Constraints from "Tree" quantities in the (ρ-bar,η-bar) plane, with only input on |V_ub| from semileptonic decays (involving γ(DK) and α from the isospin analysis with the help of sin2β (charmonium), which gives another tree only γ measurement (the only assumption is that the ΔI=3/2 b-->d EW penguin amplitude is negligible)).

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Constraints from "Tree" quantities in the (ρ-bar,η-bar) plane (only γ(DK) is used).

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Constraints from "Tree" quantities in the (ρ-bar,η-bar) plane with only input on |V_ub| from semileptonic decays (ony γ(DK) is used).

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Constraints from "Tree" quantities in the (ρ-bar,η-bar) plane with only input on |V_ub| from exclusive semileptonic B decays (only γ(DK) is used).

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Constraints from "Tree" quantities in the (ρ-bar,η-bar) plane with only input on |V_ub| from inclusive semileptonic B decays (only γ(DK) is used).

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Constraints from "Loop" quantities in the (ρ-bar,η-bar) plane.

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Constraints in the (ρ-bar,η-bar) plane, not including the braching ratio of B⁺ → τ⁺ν in the global fit.

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Constraints in the (ρ-bar,η-bar) plane not including the measurement of sin2β in the global fit.

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Pulls

Pulls for various inputs or parameters involved in the Standard Model global fit. Each pull (in units of σ) is computed by taking the square root of the difference between χ²_min obtained including or not including direct information on the quantity. This corresponds to consider Δχ²_X;min=χ²_{with data on X;min}- χ²_{without data X;min} as a random variable distributed with 1 degree of freedom, and reinterpret the probability of reaching the observed value in units of σ. The presence of a plateau in the Rfit model for systematic uncertainties may lead to a vanishing pull for some quantities even in cases where the predicted and observed values are not identical. Some of the pulls presented in this plot are correlated [this is for instance the case for sin 2β and Br(B→τν)].

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The global CKM fit in the large (ρ̅_M, η̅_M) plane
with M = sb, ds, ct, ut, uc:

The constraints can be expressed in the unitarity triangles associated with the different mesons of interest, with the relative coordinates of the upper appex of each triangle defined as
ρ̅_d₁d₂+iη̅_d₁d₂ =-(V_ud₁V_ud₂^*)/(V_cd₁V_cd₂^*), ρ̅_u₁u₂+iη̅_u₁u₂ =-(V_u₁dV_u₂d^*)/(V_u₁sV_u₂s^*).

Constraints in the (ρ̅_sb, η̅_sb) plane. The red hashed region of the global combination corresponds to 68% CL.

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Constraints in the (ρ̅_ds, η̅_ds) plane. The red hashed region of the global combination corresponds to 68% CL.

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Constraints in the (ρ̅_tc, η̅_tc) plane. The red hashed region of the global combination corresponds to 68% CL.

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Constraints in the (ρ̅_tu, η̅_tu) plane. The red hashed region of the global combination corresponds to 68% CL.

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Constraints in the (ρ̅_cu, η̅_cu) plane. The red hashed region of the global combination corresponds to 68% CL.

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The global CKM fit in the (|V_ud|,|V_us|) plane:

Constraints in the (|V_ud|,|V_us|) plane. The indirect constraints (coming from b transitions) are related to |V_ud| and |V_us| through unitarity. The yellow region of the global combination corresponds to 68% CL.

Deviation from unitarity (based only on direct determinations) : |V_ud|²+|V_us|²+|V_ub|²-1 = -0.00233 [+0.00221 -0.00020] (1σ) |V_ud|²+|V_us|²+|V_ub|²-1 = -0.00233 [+0.00235 -0.00041] (2σ) |V_ud|²+|V_us|²+|V_ub|²-1 = -0.00233 [+0.00236 -0.00062] (3σ)

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The global CKM fit in the (|V_cd|,|V_cs|) plane:

Constraints in the (\|V_cd\|,\|V_cs\|) plane. The indirect constraints (combing from b and s transitions) are related to \|V_cd\| and \|V_cs\| through unitarity. The direct constraints combine leptonic and semileptonic D and D_s decays as well as information from neutrino-nucleaon scattering and W → cs decays. The yellow region of the global combination corresponds to 68% CL.	eps png
Constraints in the (\|V_cd\|,\|V_cs\|) plane where direct constraints involve only leptonic D and D_s decays with our inputs for lattice averages for f_D and f_Ds.	eps png
Constraints in the (\|V_cd\|,\|V_cs\|) plane where direct constraints involve only semileptonic D and D_s decays with our inputs for lattice averages F_{D → π}(0) and F_{D → K}(0).	eps png
Constraints in the (\|V_cd\|,\|V_cs\|) plane where direct constraints involve only information from neutrino-nucleaon scattering and W→ cs decays (no lattice input).	eps png

The global CKM fit in the (|V_ub|,|V_cb|) plane:

Constraints in the (\|V_ub\|,\|V_cb\|) plane. The horizontal and vertical coloured bands represent our average of the determinations from semileptonic B decays. The white bands with solid (dashed) borders correspond to the determination from exclusive (inclusive) semileptonic B decays. The diagonal coloured band corresponds to the determination of \|V_ub\|/\|V_cb\| from Λ_b decays. The rainbow oval region indicates the indirect determination of \|V_ub\| and \|V_cb\| from the global fit, without any information from semileptonic or leptonic decays of b-hadrons.	eps png
Constraints on \|V_ub\| from inclusive and exclusive B decays, as well as our average, compared to the indirect determination from the global fit.	eps png
Constraints on \|V_cb\| from inclusive and exclusive B decays, as well as our average, compared to the indirect determination from the global fit.	eps png

The global CKM fit for α:

Combined constraint on the CKM angle α (Φ₂) from the Isospin analyses of B→ππ, B→ρπ and B→ρρ compared to the prediction of the global CKM fit (not including the α-related measurements).

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Constraints on the CKM angle α (Φ₂) from B→ππ (BABAR, Belle incl. Belle II and WA) compared to the prediction of the global CKM fit (not including the α-related measurements).

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Constraint on the reduced isospin amplitude
a^+- = A^+-/A⁺⁰ (and CP conjugate) in the complex plane for the B/B →ππ systems. The individual constraint from the B⁰→π⁺π^- channel and from the B⁰→π⁺π⁰ chanel observables are indicated by the yellow and green circular areas, respectively. The corresponding isospin triangular relation is represented by the black triangle.

B→ππ system	B →ππ system
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Constraints on the CKM angle α (Φ₂) from B→ρρ (BABAR, Belle incl. Belle II and WA) compared to the prediction of the global CKM fit (not including the α-related measurements).

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Constraint on the reduced isospin amplitude
a^+- = A^+-/A⁺⁰ (and CP conjugate) in the complex plane for the B/B →ρρ systems. The individual constraint from the B⁰→ρ⁺ρ^- channel and from the B⁰→ρ⁺ρ⁰ chanel observables are indicated by the yellow and green circular areas, respectively. The corresponding isospin triangular relation is represented by the black triangle.

B→ρρ system	B →ρρ system
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Constraint on the CP-violating observables (C⁰⁰,S⁰⁰) in the B⁰ →π⁰π⁰ decay (not including the direct measurement). The world-average value for the time-integrated parameter, C⁰⁰, is indicated by the hatched gray band.

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One-dimensional projection on the yet-unmeasured time-dependent observable, S⁰⁰, predicted to be,
S⁰⁰=0.79 (+0.10,-0.15) (1σ)

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|V_td/V_ts| from Radiative Penguins :

Constraint on the ratio |V_td/V_ts| from the oscillation frequencies, Δm_d/s, compared to the branching fractions of exclusive b →(d/s)γ radiative decays. The latter includes the recent LHCb measurement of B⁰→ρ⁰γ branching fraction and the first evidence for B_s →K^*0γ decays . The indirect CKM fit value (not including the oscillation data) is indicated by the blue point.

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The current precision on |V_td/V_ts| derived from radiative data is limited by the theoretical knowledge of the hadronic form-factors for B⁰→ρ⁰, B_s→K^*0 and B⁰→K^*0 transitions. Their relative values, estimated from QCD factorisation and light-cone sum rules, are taken from P. Ball et all, 2007. Conversely, the B⁰→ρ⁰ (left) and B_s→K^*0 (right) form-factors relative to the B⁰→K^*0 one, can be constrained by the global CKM fit including the radiative data, and compared to their theoretical determination.

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Similarly, the theoretical decay-constant and bag-parameter can simultaneously be constrained in the plane (f_{B_d}√B̂_{B_d}, f_{B_d}) using the B⁰ oscillation frequency, Δm_d, and the branching ratio of the B^-→τ^-ν leptonic decay. The precise determinations of those theoretical quantities derived from Lattice QCD is represented by the black contour.

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Global CKM fit constraint on the SU(3)-breaking ratio
ξ = f_{B_d}√B̂_{B_d} / f_{B_s}√B̂_{B_s} (Lattice QCD excluded) compared to its value derived from Lattice QCD.

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Very rare Kaon decays:

Global CKM fit prediction for the branching fractions of the rare K→πνν decays:
B(K⁺→π⁺νν) = (8.6 ± 0.7) 10^-11
B(K_L→π⁰νν) = (3.0 ± 0.1) 10^-11
The experimental measurement reported by NA62 (arXiv) for the first observation of the charged mode is indicated.

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Constraints on the R_t side of the Unitarity Triangle in the (ρ-bar,η-bar) plane, from the B-mesons oscillations (orange), the Radiative data (green), and the rare charged kaon decay, K⁺→π⁺νν (gray).

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