World Average Value for the V_cs from leptonic D_s⁺ 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 (cs in the case of D_s⁺ 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_s⁺→ l⁺ν) = G_F²/(8π)τ_Dsf_Ds²|V_cs|²M_DsM_l²(1-M_l²/M_Ds²)².

Here, M_Ds is the D_s meson mass, τ_Ds is its lifetime, M_l is the charged lepton mass, |V_cs| is the magnitude of the CKM matrix element, and G_F is the Fermi coupling constant. The parameter f_Ds is the D_s 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_Ds^LQCD=(249.9 ± 0.5) MeV that is used within this section to extract the magnitudes of the |V_cs| CKM matrix element from experimentally measured branching fractions of leptonic D_s⁺ meson decays.

Experimental results and world averages for branching fractions of purely leptonic Ds^+ decays and fDs|Vcs|. The first uncertainty is statistical and the second is experimental systematic. The third uncertainty in the case fDs|Vcs| is due to external inputs (dominated by the uncertainty of Ds meson's lifetime, τDs). We have recalculated Br(Ds^+→ τν) quoted by CLEO-c and BaBar using the latest values for branching fractions of τ decays to electron, muon, or pion and neutrinos given in the 2020 PDG edition . CLEO-c and BaBar include statistical uncertainty of number of Ds tags (denominator in the calculation of branching fraction) in the statistical uncertainty of measured branching fraction. We subtract this uncertainty from the statistical one and add it to the systematic uncertainty.
Mode	Branching Fraction	fDs|Vcs| (MeV)	Experiment	Source
μ^+ ν	(5.65 ± 0.44 ± 0.20) × 10^-3	249.8 ± 9.7 ± 4.4 ± 1.0	CLEO	J. P. Alexander et al. (CLEO Collaboration), Phys. Rev. D 79, 052001 (2009)
	(6.02 ± 0.37 ± 0.32) × 10^-3	257.8 ± 7.9 ± 6.9 ± 1.0	BaBar	P. del Amo Sanchez et al. (BaBar Collaboration), Phys. Rev. D 82, 091103 (2010)
	(5.31 ± 0.28 ± 0.20) × 10^-3	242.2 ± 6.4 ± 4.6 ± 1.0	Belle	A. Zupanc et al. (Belle Collaboration), JHEP 1309, 139 (2013)
	(5.17 ± 0.75 ± 0.21) × 10^-3	238.9 ± 17.3 ± 4.9 ± 0.9	BESIII	M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 94, 072004(2016)
	(5.35 ± 0.13 ± 0.16) × 10^-3	243.1 ± 3.0 ± 3.6 ± 1.0	BESIII	M. Ablikim et al. (BESIII Collaboration), arXiv:2102.11734
	(5.43 ± 0.11 ± 0.11) × 10^-3	244.9 ± 2.4 ± 2.5 ± 1.0	HFAG Average
τ^+(e^+) ν	(5.32 ± 0.47 ± 0.22) × 10^-2	245.4 ± 10.9 ± 5.1 ± 1.0	CLEO	P. U. E. Onyisi et al. (CLEO Collaboration), Phys. Rev. D 79, 052002 (2009)
τ^+(π^+) ν	(6.47 ± 0.80 ± 0.22) × 10^-2	270.1 ± 16.8 ± 4.6 ± 1.1		J. P. Alexander et al. (CLEO Collaboration), Phys. Rev. D 79, 052001 (2009)
τ^+(ρ^+) ν	(5.50 ± 0.54 ± 0.24) × 10^-2	249.8 ± 12.3 ± 5.5 ± 1.0		P. Naik et al. (CLEO Collaboration), Phys. Rev. D 80, 112004 (2009)
τ^+ ν	(5.59 ± 0.32 ± 0.14) × 10^-2	251.7 ± 7.2 ± 3.2 ± 1.0	CLEO
τ^+(e^+) ν	(5.09 ± 0.52 ± 0.68) × 10^-2	240.1 ± 12.3 ± 16.1 ± 1.0	BaBar	P. del Amo Sanchez et al. (BaBar Collaboration), Phys. Rev. D 82, 091103 (2010), Erratum ibid D. 91, 019901 (2015),
τ^+(μ^+) ν	(4.90 ± 0.46 ± 0.54) × 10^-2	235.7 ± 11.1 ± 13.0 ± 1.0
τ^+ ν	(4.96 ± 0.37 ± 0.57) × 10^-2	237.1 ± 8.8 ± 13.6 ± 1.0	BaBar
τ^+(e^+) ν	(5.38 ± 0.33 ^+0.35-0.31) × 10^-2	246.8 ± 7.6 ^+8.1-7.1 ± 1.0	Belle	A. Zupanc et al. (Belle Collaboration), JHEP 1309, 139 (2013)
τ^+(μ^+) ν	(5.86 ± 0.37 ^+0.34-0.59) × 10^-2	257.8 ± 8.1 ^+7.5-13.0 ± 1.0
τ^+(π^+) ν	(6.05 ± 0.43 ^+0.46-0.40) × 10^-2	261.7 ± 9.3 ^+10.0-8.7 ± 1.0
τ^+ ν	(5.70 ± 0.21 ± 0.31) × 10^-2	254.1 ± 4.7 ± 6.9 ± 1.8	Belle
τ^+(π^+) ν	(3.28 ± 1.83 ± 0.37) × 10^-2	193 ± 54 ± 11 ± 1.0	BESIII	M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 94, 072004(2016)
τ^+(π^+) ν	(5.21 ± 0.25 ± 0.17) × 10^-2	243.0 ± 5.8 ± 4.0 ± 1.0	BESIII	M. Ablikim et al. (BESIII Collaboration), arXiv:2102.11734
τ^+ ν	(5.40 ± 0.13 ± 0.19) × 10^-2	247.4 ± 3.1 ± 3.0 ± 1.0	HFAG average
μ^+ ν τ^+ ν		245.9 ± 1.9 ± 1.9 ± 1.0	HFAG average
e^+ ν	< 8.3 × 10^-5 at 90% C.L.		Belle	A. Zupanc et al. (Belle Collaboration), JHEP 1309, 139 (2013)

We use the measurements of the branching fraction Br(D_s⁺→ μ⁺ν) from CLEO, BaBar, Belle and BESIII and obtain its world average (WA) value of

Br(D_s⁺→ μ⁺ν)^WA = (5.43 ± 0.16) × 10^-3.

The WA value for Br(D_s⁺→ τ⁺ν) is also calculated from CLEO, BaBar, Belle and BESIII measurements. CLEO made separate measurements for leptonic τ decays and hadronic decay to ρ; BaBar made measurements using the leptonic τ decays only; and Belle made measurements using the leptonic τ decays and the hadronic decay to π. Combining all of them we obtain the WA value of

Br(D_s⁺→ τ⁺ν)^WA = (5.40 ± 0.23) × 10^-2.

The ratio of the branching fractions of leptonic D_s⁺ decays is thus found to be

R = Br(D_s⁺→ τ⁺ν)/Br(D_s⁺→ μ⁺ν) = 9.94 ± 0.52,

and is consistent with the value expected in the SM (from lepton universality), R^SM = 9.75 ± 0.01.

From the average values of branching fractions of muonic and tauonic decays we determine from the equation on top of this page the product of D_s meson decay constant and the |V_cs| CKM matrix element to be (figure)

f_Ds |V_cs| = (245.9 ± 2.9) MeV,

where the uncertainty includes the uncertainties on Br(D_s⁺→ μ⁺ν)^WA and Br(D_s⁺→ τ⁺ν)^WA and external inputs. To obtain the average we have taken into account the correlations within each experiment for the uncertainties related to: normalization, tracking, particle identification, signal and background parameterizations, and peaking background contributions. Using the LQCD value for the decay constant, f_Ds^LQCD=(249.9 ± 0.5) MeV, we finally obtain the CKM matrix element Vcs to be

|V_cs|_{Ds→ lν} = 0.9839 ± 0.0115(exp.) ± 0.0020(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→ Klν decays, |V_cs|_{D→ Klν} = 0.9447 ± 0.0043(exp.) ± 0.0137(LQCD). Averaging both, assuming 100% correlation between the LQCD uncertainties, and none between the experimental uncertainties, gives (figure)

|V_cs|_{D(s)→ (K)lν} = 0.9683 ± 0.0091.

Assuming unitarity of the CKM matrix, the value of the element relevant in the case of leptonic D_s⁺ decays is known from the global fit of the CKM matrix, |V_cs| = 0.973394^+0.000074_-0.000096. This value can be used to extract the D_s meson decay constant from the experimentally measured product f_Ds|V_cs|. This leads to the experimentally measured D_s meson decay constant to be:

f_Ds = 252.6 ± 3.0 MeV,

which is in agreement with the LQCD determination given above. Using the D meson decay constant determined from the leptonic decays, we find the ratio of the D_s and D meson decay constants to be

f_Ds/f_D = 1.232 ± 0.030.

The ratio is in agreement with the average of LQCD calculations, f_Ds/f_D = 1.1716 ± 0.0032, at the level of 1.8σ.

WA value for f_Ds|V_cs|. 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_cs| determined from the leptonic and semileptonic D meson decays and from W decays and indirect determination from the global fit assuming CKM unitarity. (Click on figure for higher resolution.)

External parameter values as taken from the 2020 Particle Data Book:
            G_F /(ℏc)³ = (1.1663787 ± 0.0000006) × 10^-5 GeV^-2
M_μ = (0.1056583745 ± 0.0000000024) GeV/c²
M_τ = (1.77686 ± 0.00012) GeV/c²
            M_Ds = (1.96835 ± 0.00007) GeV/c²
τ_Ds = (504 ± 4) × 10^-15 s

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