People working on this: Hailong
Ma
The branching fraction of leptonic
decays of pseudoscalar mesons that proceed via the annihilation of the initial
quarkantiquark pair (cd in the case of D^{+} meson) into a virtual W^{+}
that finally materializes as an antileptonneutrino pair (lν, where l = e,
μ, or τ) is given in the Standard Model by
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 wavefunction overlap of
the meson's constituent quark and antiquark. 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 f_{D}V_{cd}. 

Mode 
Branching Fraction 
f_{D}V_{cd} (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/c^{2}
M_{D} = (1.86966 ±
0.00005) GeV/c^{2}
τ_{D} = (1040 ±
7) × 10^{15} s
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