World Average Value for the V_cd from leptonic D⁺ decays
(updated 15 June 2021)

People working on this: Hailong Ma

The branching fraction of leptonic decays of pseudo-scalar mesons that proceed via the annihilation of the initial quark-antiquark pair (cd in the case of D⁺ meson) into a virtual W⁺ that finally materializes as an antilepton-neutrino pair (lν, where l = e, μ, or τ) is given in the Standard Model by

Br(D⁺→ l⁺ν) = G_F²/(8π)τ_Df_D²|V_cd|²M_DM_l²(1-M_l²/M_D²)².

Here, M_D is the D meson mass, τ_D is its lifetime, M_l is the charged lepton mass, |V_cd| is the magnitude of the CKM matrix element, and G_F is the Fermi coupling constant. The parameter f_D is the D meson decay constant and is related to the wave-function overlap of the meson's constituent quark and anti-quark. Within the SM, the decay constants have been predicted using several methods, the most precise being the lattice gauge theory (LQCD) calculations. The Flavor Lattice Averaging Group combines all LQCD calculations and provides averaged value for f_D^LQCD=(212.0 ± 0.7) MeV that is used within this section to extract the magnitudes of the |V_cd| CKM matrix element from experimentally measured branching fractions of leptonic D⁺ meson decays.

Experimental results and world averages for branching fractions of purely leptonic D^+ decays and fD|Vcd|. The first uncertainty is statistical and the second is experimental systematic. The third uncertainty in the case fD|Vcd| is due to external inputs (dominated by the uncertainty of D meson's lifetime, τD).
Mode	Branching Fraction	fD|Vcd| (MeV)	Experiment	Source
μ^+ ν	(3.95 ± 0.35 ± 0.09) × 10^-4	47.1 ± 2.1 ± 0.5 ± 0.2	CLEO	B. I. Eisenstein et al. (CLEO Collaboration), Phys. Rev. D 78, 052003 (2008)
	(3.71 ± 0.19 ± 0.06) × 10^-4	45.7 ± 1.2 ± 0.4 ± 0.2	BESIII	M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 89, 051104 (2014)
	(3.77 ± 0.17 ± 0.05) × 10^-4	46.1 ± 1.0 ± 0.3 ± 0.2	HFAG Average
τ^+ ν	(1.20 ± 0.24 ± 0.12) × 10^-3	50.4 ± 5.0 ± 2.5 ± 0.2	BESIII	M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 123, 211802 (2019)
μ^+ ν+τ^+ ν		46.2 ± 1.0 ± 0.3 ± 0.2	HFAG Average
e^+ ν	< 8.8 × 10^-6 at 90% C.L.		CLEO	B. I. Eisenstein et al. (CLEO Collaboration), Phys. Rev. D 78, 052003 (2008)

We use the two measurements of the branching fraction Br(D⁺→ μ⁺ν) from CLEO and BESIII to calculate its world average (WA) value. Uncertainties of the two measurements are considered to be uncorrelated. We obtain

Br(D⁺→ μ⁺ν)^WA = (3.77 ± 0.17) × 10^-4.

Combining this with

Br(D⁺→ τ⁺ν)^WA = (1.20 ± 0.27) × 10^-3.

The ratio of the branching fractions of the leptonic D⁺ decays is determined to be

R = Br(D⁺→ τ⁺ν)/Br(D⁺→ μ⁺ν) = 3.18 ± 0.73,

and is consistent with the value expected in the SM (from lepton universality), R^SM = 2.67 ± 0.01.

Based on these two branching fractions, we determine the product of the decay constant and the CKM matrix element using the equation from the top of the page to be (figure)

f_D|V_cd| = (46.2 ± 1.0) MeV,

where the uncertainty includes the uncertainty on Br(D⁺→ μ⁺ν)^WA and external inputs. Using the LQCD value for the decay constant, f_D^LQCD=(212.0 ± 0.7) MeV, we finally obtain the CKM matrix element Vcd to be

|V_cd|_{D→ lν} = 0.2181 ± 0.0049(exp.) ± 0.0007(LQCD),

where the uncertainties are from the experiments and lattice calculations, respectively. The value is found to be consistent with the one determined from the semileptonc D→ πlν decays, |V_cd|_{D→ πlν} = 0.2249 ± 0.0028(exp.) ± 0.0055(LQCD). Averaging both, assuming 100% correlation between the LQCD uncertainties, and none between the experimental uncertainties, gives (figure)

|V_cd|_{D→ (π)lν} = 0.2208 ± 0.0040.

Assuming unitarity of the CKM matrix, the value of the element relevant in the case of leptonic D⁺ decays is known from the global fit of the CKM matrix, |V_cd| = 0.22529^+0.00041_-0.00032. This value can be used to extract the D meson decay constant from the experimentally measured product f_D|V_cd|. This leads to the experimentally measured D meson decay constant to be:

f_D = 205.1 ± 4.4 MeV,

which is in agreement with the LQCD determination given above.

WA value for f_D|V_cd|. For each point, the first error listed is the statistical
and the second error is the systematic error. (Click on figure for higher resolution.)

Comparison of magnitudes of the CKM matrix element |V_cd| determined from the leptonic and semileptonic D meson decays and from neutrino scattering data and indirect determination from the global fit assuming CKM unitarity. (Click on figure for higher resolution.)

External parameter values as taken from the 2018 Particle Data Book:
            G_F /(ℏc)³ = (1.1663787 ± 0.0000006) × 10^-5 GeV^-2
            M_μ = (0.1056583745 ± 0.0000000024) GeV/c²
            M_D = (1.86966 ± 0.00005) GeV/c²
            τ_D = (1040 ± 7) × 10^-15 s

---

This page is maintained by Hailong Ma and was last updated