CLNS 07/1990
CLEO 07-02
Measurement of B(D+s → ℓ
+ν) and the Decay Constant fD+s
T. K. Pedlar
Luther College, Decorah, Iowa 52101
D. Cronin-Hennessy, K. Y. Gao, J. Hietala, Y. Kubota, T. Klein,
B. W. Lang, R. Poling, A. W. Scott, A. Smith, and P. Zweber
University of Minnesota, Minneapolis, Minnesota 55455
S. Dobbs, Z. Metreveli, K. K. Seth, and A. Tomaradze
Northwestern University, Evanston, Illinois 60208
J. Ernst
State University of New York at Albany, Albany, New York 12222
K. M. Ecklund
State University of New York at Buffalo, Buffalo, New York 14260
H. Severini
University of Oklahoma, Norman, Oklahoma 73019
W. Love and V. Savinov
University of Pittsburgh, Pittsburgh, Pennsylvania 15260
O. Aquines, A. Lopez, S. Mehrabyan, H. Mendez, and J. Ramirez
University of Puerto Rico, Mayaguez, Puerto Rico 00681
G. S. Huang, D. H. Miller, V. Pavlunin, B. Sanghi, I. P. J. Shipsey, and B. Xin
Purdue University, West Lafayette, Indiana 47907
G. S. Adams, M. Anderson, J. P. Cummings, I. Danko, D. Hu, B. Moziak, and J. Napolitano
Rensselaer Polytechnic Institute, Troy, New York 12180
Q. He, J. Insler, H. Muramatsu, C. S. Park, E. H. Thorndike, and F. Yang
University of Rochester, Rochester, New York 14627
M. Artuso, S. Blusk, J. Butt, S. Khalil, J. Li, N. Menaa, R. Mountain,
http://arxiv.org/abs/0704.0437v3
S. Nisar, K. Randrianarivony, R. Sia, T. Skwarnicki, S. Stone, and J. C. Wang
Syracuse University, Syracuse, New York 13244
G. Bonvicini, D. Cinabro, M. Dubrovin, and A. Lincoln
Wayne State University, Detroit, Michigan 48202
D. M. Asner, K. W. Edwards, and P. Naik
Carleton University, Ottawa, Ontario, Canada K1S 5B6
R. A. Briere, T. Ferguson, G. Tatishvili, H. Vogel, and M. E. Watkins
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
J. L. Rosner
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637
N. E. Adam, J. P. Alexander, D. G. Cassel, J. E. Duboscq, R. Ehrlich, L. Fields,
L. Gibbons, R. Gray, S. W. Gray, D. L. Hartill, B. K. Heltsley, D. Hertz, C. D. Jones,
J. Kandaswamy, D. L. Kreinick, V. E. Kuznetsov, H. Mahlke-Krüger, D. Mohapatra,
P. U. E. Onyisi, J. R. Patterson, D. Peterson, J. Pivarski, D. Riley, A. Ryd,
A. J. Sadoff, H. Schwarthoff, X. Shi, S. Stroiney, W. M. Sun, and T. Wilksen
Cornell University, Ithaca, New York 14853
S. B. Athar, R. Patel, and J. Yelton
University of Florida, Gainesville, Florida 32611
P. Rubin
George Mason University, Fairfax, Virginia 22030
C. Cawlfield, B. I. Eisenstein, I. Karliner, D. Kim,
N. Lowrey, M. Selen, E. J. White, and J. Wiss
University of Illinois, Urbana-Champaign, Illinois 61801
R. E. Mitchell and M. R. Shepherd
Indiana University, Bloomington, Indiana 47405
D. Besson
University of Kansas, Lawrence, Kansas 66045
(CLEO Collaboration)
(Dated: April 2, 2003)
Abstract
We examine e+e− → D−s D
s andD
s interactions at 4170 MeV using the CLEO-c detector in
order to measure the decay constant f
. We use the D+s → ℓ
+ν channel, where the ℓ+ designates
either a µ+ or a τ+, when the τ+ → π+ν. Analyzing both modes independently, we determine
B(D+s → µ
+ν) = (0.594±0.066±0.031)%, and B(D+s → τ
+ν) = (8.0±1.3±0.4)%. We also analyze
them simultaneously to find an effective value of Beff(D+s → µ
+ν) = (0.638± 0.059± 0.033)% and
extract f
= 274 ± 13 ± 7 MeV. Combining with our previous determination of B(D+ → µ+ν),
we also find the ratio f
/fD+ = 1.23± 0.11± 0.04. We compare to current theoretical estimates.
Finally, we find B(D+s → e
+ν) < 1.3 × 10−4 at 90% confidence level.
PACS numbers: 13.20.Fc, 13.66.Bc
I. INTRODUCTION
To extract precise information on the size of Cabibbo-Kobayashi-Maskawa matrix ele-
ments from B − B mixing measurements, the “decay constants” for Bd and Bs mesons or
their ratio, fBd/fBs , must be well known [1]. These factors are related to the overlap of the
heavy and light quark wave-functions at zero spatial separation. Indeed, the recent measure-
ment of B0s −B
s mixing by CDF [2] that can now be compared to the very well measured B
mixing [3] has pointed out the urgent need for precise values [4]. Decay constants have been
calculated theoretically. The most promising of these calculations are based on lattice-gauge
theory that includes light quark loops [5], often called “unquenched.” In order to ensure that
these theories can adequately predict fBs/fBd it is useful to check the analogous ratio in
charm decays f
/fD+ . We have previously measured fD+ [6, 7]. Here we present the most
precise measurement to date of f
and the ratio f
/fD+ .
In the Standard Model (SM), the only way for a Ds meson to decay purely leptonically is
via annihilation through a virtual W+, as depicted in Fig. 1. The decay rate is given by [8]
Γ(D+s → ℓ
+ν) =
m2ℓMD+s
|Vcs|
, (1)
where M
is the D+s mass, mℓ is the mass of the charged final state lepton, GF is the Fermi
coupling constant, and |Vcs| is a Cabibbo-Kobayashi-Maskawa matrix element with a value
we take equal to 0.9738 [3].
FIG. 1: The decay diagram for D+s → ℓ
In this paper we analyze both D+s → µ
+ν and D+s → τ
+ν, τ+ → π+ν. In both Ds
decays the charged lepton must be produced with the wrong helicity because the Ds is a
spin-0 particle, and the final state consists of a naturally left-handed spin-1/2 neutrino and
a naturally right-handed spin-1/2 anti-lepton. Because the τ+ has a mass close to that of
the D+s , the helicity suppression is broken with respect to the µ
+ decay, but there is an
additional large phase space suppression.
New physics can affect the expected widths; any undiscovered charged bosons would
interfere with the SM W+. These effects may be difficult to ascertain, since they would
simply change the value of the decay constants. The ratio f
/fD+ is much better predicted
in the SM than the values individually, so deviations from the the SM expectation are more
easily seen. Any such discrepancies would point to beyond the SM charged bosons. For
example, Akeroyd predicts that the presence of a charged Higgs boson would suppress this
ratio significantly [9].
We can also measure the ratio of decay rates to different leptons, and the SM predictions
then are fixed only by well-known masses. For example, for τ+ν to µ+ν:
Γ(D+s → τ
Γ(D+s → µ
. (2)
Using measured masses [3], this expression yields a value of 9.72 with a negligibly small
error. Any deviation in R from the value predicted by Eq. 2 would be a manifestation of
physics beyond the SM. This could occur if any other charged intermediate boson existed
that affected the decay rate differently than mass-squared. Then the couplings would be
different for muons and τ ’s. This would be a clear violation of lepton universality [10].
Previous measurements of fD+s have been hampered by a lack of statistical precision, and
relatively large systematic errors [11, 12, 13, 14, 15, 16]. One large systematic error source has
been the lack of knowledge of the absolute branching fraction of the normalization channel,
usually D+s → φπ
+ [17]. The results we report here will not have this limitation.
II. EXPERIMENTAL METHOD
A. Selection of Ds Candidates
The CLEO-c detector [18] is equipped to measure the momenta and directions of charged
particles, identify them using specific ionization (dE/dx) and Cherenkov light (RICH) [19],
detect photons and determine their directions and energies.
In this study we use 314 pb−1 of data produced in e+e− collisions using the Cornell Elec-
tron Storage Ring (CESR) and recorded near a center-of-mass energy (ECM) of 4.170 GeV. At
this energy the e+e− annihilation cross-section into D∗+s D
s is approximately 1 nb,
while the cross-section for D+s D
s is about a factor of 20 smaller. In addition, D mesons
are produced mostly as D∗D∗, with a cross-section of ∼5 nb, and also in D∗D +DD∗ final
states with a cross-section of ∼2 nb. The DD cross-section is a relatively small ∼0.2 nb
[20]. There also appears to be DD
π production. The underlying light quark “continuum”
background is about 12 nb. The relatively large cross-sections, relatively large branching
fractions and sufficient luminosities allow us to fully reconstruct one Ds as a “tag,” and ex-
amine the properties of the other. In this paper we designate the tag as a D−s and examine
the leptonic decays of the D+s , though, in reality, we use both charges for tags and signals.
Track requirements, particle identification, π0, η, and K0S selection criteria are the same as
those described in Ref. [6], except that we now require a minimum momentum of 700 MeV/c
for a track to be identified using the RICH.
We also use several resonances that decay via the strong interaction. Here we select
intervals in invariant mass within ±10 MeV of the known mass for η′ → π+π−η, ±10 MeV
for φ → K+K−, ±100 MeV for K∗0 → K−π+, and ±150 MeV for ρ− → π−π0.
We reconstruct tags from either directly produced Ds mesons or those that result from
the decay of a D∗s . The beam constrained mass, mBC, is formed by using the beam energy
to construct the Ds candidate mass via the formula
mBC =
E2beam − (
−→p i)
2, (3)
where i runs over all the final state particles. If we ignore the photon from the D∗s → γDs
decay, and reconstruct the mBC distribution, we obtain the distribution from Monte Carlo
simulation shown in Fig. 2. The narrow peak occurs when the reconstructed Ds does not
come from the D∗s decay, but is directly produced.
FIG. 2: The beam constrained mass mBC from Monte Carlo simulation of e
+e− → D+s D
s , D
φπ± at an ECM of 4170 MeV. The narrow peak is from the D
s and the wider one fromD
s → γD
(The distributions are not centered at the D+s or D
s masses, because the reconstructed particles
are assumed to have the energy of the beam.)
Rather than selecting events based on only mBC, we first select an interval that accepts
most of the events, 2.015 < mBC < 2.067 GeV, and examine the invariant mass. Distributions
from data for the 8 tag decay modes we use in this analysis are shown in Fig. 3. Note that
the resolution in invariant mass is excellent, and the backgrounds not abysmally large, at
least in these modes. To determine the number of D−s events we fit the invariant mass
distributions to the sum of two Gaussians centered at the D−s mass, a function we refer to
as “two-Gaussian.” The r.m.s. resolution (σ) is defined as
σ ≡ f1σ1 + (1− f1)σ2, (4)
where σ1 and σ2 are the individual widths of each of the two Gaussians and f1 is the fractional
area of the first Gaussian. The number of tags in each mode is listed in Table I. We will
later use sidebands of the signal peaks shown in Fig. 3 for part of the background estimate.
To select our sample of tag events, we require the invariant masses, shown in Fig. 3,
to be within ± 2.5σ (±2σ for the ηρ− mode) of the known D−s mass. Then we look for
an additional photon candidate in the event that satisfies our shower shape requirement.
FIG. 3: Invariant mass of D−s candidates in the decay modes (a) K
+K−π−, (b) KSK
−, (c) ηπ−,
(d) η′π−, (e) φρ−, (f) π+π−π−, (g) K∗−K∗0, and (h) ηρ−, after requiring the total energy of the
D−s candidate to be consistent with the beam energy. The curves are fits to two-Gaussian signal
functions plus a polynomial background.
TABLE I: Tagging modes and numbers of signal and background events, within ±2.5σ of the
D−s mass for all modes, except ηρ
− (±2σ), determined from two-Gaussian fits to the invariant
mass plots, and the number of photon tags in each mode, within ±2.5σ of the Ds mass-squared
determined from fits of the MM∗2 distributions (see text) to a signal Crystal Ball function (see
text) and a 5th order Chebychev background polynomial, and the associated backgrounds.
Mode Invariant Mass MM∗2
Signal Background Signal Background
K+K−π− 13871±262 10850 8053± 211 13538
− 3122±79 1609 1933±88 2224
ηπ−; η → γγ 1609 ± 112 4666 1024±97 3967
η′π−; η′ → π+π−η, η → γγ 1196±46 409 792±69 1052
φρ−; φ → K+K−, ρ− → π−π0 1678±74 1898 1050±113 3991
π+π−π− 3654±199 25208 2300±187 15723
K∗−K∗0; K∗− → K0Sπ
−, K∗0 → K+π− 2030±98 4878 1298±130 5672
ηρ−; η → γγ, ρ− → π−π0 4142±281 20784 2195±225 17353
Sum 31302 ± 472 70302 18645±426 63520
Regardless of whether or not the photon forms a D∗s with the tag, for real D
sDs events, the
missing mass squared, MM∗2, recoiling against the photon and the D−s tag should peak at
the D+s mass-squared. We calculate
MM∗2 = (ECM − EDs −Eγ)
−→pCM −
−→pDs −
−→p γ
, (5)
where ECM (
−→pCM) is the center-of-mass energy (momentum), EDs (
−→pDs) is the energy (mo-
mentum) of the fully reconstructed D−s tag, and Eγ (
−→p γ) is the energy (momentum) of the
additional photon. In performing this calculation we use a kinematic fit that constrains the
decay products of the D−s to the known Ds mass and conserves overall momentum and en-
ergy. All photon candidates in the event are used, except for those that are decay products
of the D−s tag candidate.
The MM∗2 distributions from the selected D−s event sample are shown in Fig. 4. We fit
these distributions to determine the number of tag events. This procedure is enhanced by
having information on the shape of the signal function. One possibility is to use the Monte
Carlo simulation for this purpose, but that would introduce a relatively large systematic
error. Instead, we use our relatively large sample of fully reconstructed DsD
s events, where
we use the same decay modes listed in Table I; we find these events and then examine the
signal shape in data when one Ds is ignored. The MM
∗2 distribution from this sample is
shown in Fig. 5. The signal is fit to a Crystal Ball function [21, 22]. The σ parameter, that
represents the width of the distribution, is found to be 0.032±0.002 GeV2. We do expect
this to vary somewhat depending on the final state, but we do not expect the parameters
that fix the shape of the tail to change, since they depend mostly on beam radiation and the
properties of photon detection.
We fit the MM∗2 distributions for each mode using the Crystal Ball function with fixed
tail parameters, but allowing σ to float, and a 5th order Chebyshev polynomial background.
We find a total of 18645±426 events within a ±2.5σ interval defined by the fit to each
FIG. 4: The MM∗2 distribution from events with a photon in addition to the D−s tag for the modes:
(a) K+K−π−, (b) K0SK
−, (c) ηπ−, (d) η′π−, (e) φρ−, (f) π+π−π−, (g) K∗−K∗0, and (h) ηρ−. The
curves are fits to the Crystal Ball function and a 5th order Chebychev background function.
mode. There is also a small enhancement of 4.8% on our ability to find tags in µ+ν (or τ+ν,
τ+ → π+ν) events (tag bias) as compared with generic events, determined by Monte Carlo
simulation, to which we assign a systematic error of 21% giving a correction of (4.8± 1.0)%.
An overall systematic error of 5% on the number of tags is assigned by changing the fitting
range, using 4th order and 6th order Chebychev background polynomials, and allowing the
FIG. 5: The MM∗2 distribution from a sample of fully reconstructed D−s D
s events where one Ds
is ignored. The curve is a fit to the Crystal Ball function.
parameters of the tail of the fitting function to float.
B. Signal Reconstruction
We next describe the search for D+s → µ
+ν. Candidate events are selected that contain
only a single extra track with opposite sign of charge to the tag. The track must make an
angle >35.9◦ with respect to the beam line, and in addition we require that there not be any
neutral cluster detected in the calorimeter with energy greater than 300 MeV. These cuts
are highly effective in reducing backgrounds. The photon energy cut is especially useful to
reject D+s → π
+π0, should this mode be significant, and D+s → ηπ
Since we are searching for events where there is a single missing neutrino, the missing
mass squared, MM2, evaluated by taking into account the observed µ+, D−s , and γ should
peak at zero; the MM2 is computed as
MM2 = (ECM − EDs −Eγ − Eµ)
−→pCM −
−→pDs −
−→p γ −
−→p µ
, (6)
where Eµ (
−→p µ) are the energy (momentum) of the candidate muon track and all variables
are the same as defined in Eq. 5.
We also make use of a set of kinematical constraints and fit each event to two hypotheses
one of which is that the D−s tag is the daughter of a D
s and the other that the D
s decays
into γD+s , with the D
s subsequently decaying into µ
+ν. The kinematical constraints, in the
center-of-mass frame, are
−→pDs +
−→pD∗s = 0 (7)
ECM = EDs + ED∗s
ED∗s =
M2D∗s −M
or EDs =
M2D∗s −M
MD∗s −MDs = 143.6 MeV.
In addition, we constrain the invariant mass of the D−s tag to the known Ds mass. This
gives us a total of 7 constraints. The missing neutrino four-vector needs to be determined,
so we are left with a three-constraint fit. We perform a standard iterative fit minimizing χ2.
As we do not want to be subject to systematic uncertainties that depend on understanding
the absolute scale of the errors, we do not make a χ2 cut but simply choose the photon and
the decay sequence in each event with the minimum χ2.
In this analysis, we consider three separate cases: (i) the track deposits < 300 MeV in the
calorimeter, characteristic of a non-interacting pion or a muon; (ii) the track deposits > 300
MeV in the calorimeter, characteristic of an interacting pion, and is not consistent with
being an electron; (iii) the track satisfies our electron selection criteria defined below. Then
we separately study the MM2 distributions for these three cases. The separation between
muons and pions is not complete. Case (i) contains 99% of the muons but also 60% of the
pions, while case (ii) includes 1% of the muons and 40% of the pions [7]. Case (iii) does not
include any signal but is used later for background estimation. For cases (i) and (ii) we insist
that the track not be identified as a kaon. For electron identification we require a match
between the momentum measurement in the tracking system and the energy deposited in
the CsI calorimeter and we also require that dE/dx and RICH information be consistent
with expectations for an electron.
C. The Expected MM2 Spectrum
For the µ+ν final state the MM2 distribution can be modeled as the sum of two Gaussians
centered at zero (see Eq. 4). A Monte Carlo simulation of the MM2 for the φπ− subset of
K+K−π− tags is shown in Fig. 6 both before and after the fit. The fit changes the resolution
from σ=0.032 GeV2 to σ=0.025 GeV2, a 22% improvement, without any loss of events.
We check the resolution using data. The mode D+s → K
K+ provides an excellent testing
ground.1 We search for events with at least one additional track identified as a kaon using
the RICH detector, in addition to a D−s tag. The MM
2 distribution is shown in Fig. 7.
Fitting this distribution to a two-Gaussian shape gives a MM2 resolution of 0.025 GeV2 in
agreement with Monte Carlo simulation.
For the τ+ν, τ+ → π+ν final state a Monte Carlo simulation of the MM2 spectrum is
shown in Fig. 8. The extra missing neutrino results in a smeared distribution.
1 In this paper the notation K
K+ refers to the sum of K0K+ and K0K+ final states.
FIG. 6: The MM2 resolution from Monte Carlo simulation for D+s → µ
+ν utilizing a φπ− tag and
a γ from either D∗s decay, both before the kinematic fit (a) and after (b).
D. MM2 Spectra in Data
The MM2 distributions from data are shown in Fig. 9. The overall signal region we
consider is −0.05 < MM2 < 0.20 GeV2. The higher limit is imposed to exclude background
from ηπ+ and K0π+ final states. There is a clear peak in Fig. 9(i) due to D+s → µ
Furthermore, the region between the µ+ν peak and 0.20 GeV2 has events that we will show
are dominantly due to the D+s → τ
+ν decay. The events in Fig. 9(ii) below 0.20 GeV2 are
also dominantly due to τ+ν decay.
The specific signal regions are defined as follows: for µ+ν, −0.05 <MM2 < 0.05 GeV2,
corresponding to ±2σ; for τν, τ+ → π+ν, in case (i) 0.05 <MM2 < 0.20 GeV2 and in case
(ii) −0.05 <MM2 < 0.20 GeV2. In these regions we find 92, 31, and 25 events, respectively.
E. Background Evaluations
We consider the background arising from two sources: one from real D+s decays and the
other from the background under the single-tag signal peaks. For the latter, we obtain
the background from data. We define side-bands of the invariant mass signals shown in
Fig. 2 starting at 4σ on the low and high sides of the invariant mass peaks for all modes.
The intervals extend away from the signal peaks by approximately the same width used
in selecting the signal, 5σ, so as to ensure that the number of background events in the
sidebands accurately reflects the numbers under the signal peaks. Thus the amount of data
corresponds to twice the number of background events under the signal peaks, except for the
ηπ− and ηρ− modes, where the signal widths are so wide that we chose narrower side-bands
only equaling the data. We analyze these events in exactly the same manner as those in the
signal peak.2
2 The D
mass used in the fit is chosen to be the middle of the relevant sideband interval.
FIG. 7: The MM2 distribution for events with an identified K+ track. The kinematic fit has been
applied. The curve is a fit to the sum of two Gaussians centered at the square of the K0 mass and
a linear background.
The backgrounds are given here as the sum of two numbers, the first being the number
from all modes, except ηπ− and ηρ−, and the second being the number from these modes.
For case (i) we find 2.5+1 background in the µ+ν signal region and 2.5+0 background in the
τ+ν region. For case (ii) we find 2+1 events. Our total sideband background summing over
all of these cases is 9.0±2.3. The numbers of signal and background events due to false D−s
tags as evaluated from sidebands are given in Table II.
TABLE II: Numbers of events in the signal region, and background events evaluated from sideband
regions.
Case Region (GeV2) Signal Background
i -0.05<MM2 < 0.05 92 3.5±1.4
i 0.05<MM2 < 0.20 31 2.5±1.1
ii -0.05<MM2 < 0.20 25 3.0±1.3
Sum -0.05<MM2 < 0.20 148 9.0±2.3
This entire procedure was checked by performing the same study on a sample of Monte
Carlo generated at an ECM of 4170 MeV that includes known charm and continuum produc-
tion cross-sections. The Monte Carlo sample corresponds to an integrated luminosity that is
four times larger than the data. We find the number of background events predicted directly
by examining the decay generator of the simulation is 28 and the sideband method yields
FIG. 8: The MM2 distribution from Monte-Carlo simulation for D+s → τ
+ν, τ+ → π+ν at an ECM
of 4170 MeV.
22. These are slightly smaller than found in the data, but consistent within errors. We note
that the Monte Carlo is far from perfect as many branching fractions are unknown and so
are estimated.
The background from real D+s decays is studied by identifying each possible source mode
by mode. For the µ+ν final state, the only possible background within the signal region is
D+s → π
+π0. This mode has not been studied previously. We show in Fig. 10 the π+π0
invariant mass spectrum from a 195 pb−1 subsample of our data. We do not see a signal
and set an upper limit < 1.1 × 10−3 at 90% confidence level. Recall that any such events
are also heavily suppressed by the extra photon energy cut of 300 MeV. There are also some
D+s → τ
+ν, τ+ → π+ν events that occur in the signal region. Using the SM expected ratio
of decay rates from Eq. 2 we calculate a contribution of 7.4 π+νν events that we will treat
as part of the signal.
For the τ+ν, τ+ → π+ν final state the real D+s backgrounds include, in addition to
the π+π0 background discussed above, semileptonic decays, possible π+π0π0 decays, and
other τ+ decays. Semileptonic decays involving muons are equal to those involving electrons
shown in Fig. 9(c). Since no electron events appear in the signal region, the background from
muons is taken to be zero. The π+π0π0 background is estimated by considering the π+π+π−
final state whose measured branching fraction is (1.02±0.12)% [17]. This mode has large
contributions from f0(980)π
+ and other π+π− resonant structures at higher mass [23]. The
π+π0π0 mode will also have these contributions, but the MM2 opposite to the π+ will be at
large mass. The only component that can potentially cause background is the non-resonant
component measured by FOCUS as (17±4)% [23]. This background has been evaluated by
Monte Carlo simulation as have backgrounds from other τ+ decays, and each is listed in
Table III.
FIG. 9: The MM2 distributions from data for events with a D−s reconstructed in a tag mode, an
additional positively charged track and no neutral energy clusters above 300 MeV. For case (i)
when the single track deposits < 300 MeV of energy in the calorimeter. The peak near zero is
from D+s → µ
+ν events. Case (ii): track deposits > 300 MeV in the crystal calorimeter but is not
consistent with being an electron. Case (iii): the track is identified as an electron.
III. LEPTONIC BRANCHING FRACTIONS
The sum of MM2 distributions for case (i) and case (ii), corresponding to the sum of
D+s → µ
+ν and D+s → τ
+ν, τ+ → π+ν candidates, is compared in Fig. 11 with the expected
shape, assuming the SM value of R as given in Eq. 2 for the ratio of τ+ν to µ+ν rates.
The curve is normalized to the total number of events below MM2 <0.2 GeV2. Besides
the prominent µ+ν peak and τ+ν; τ+ → π+ν shoulder, there is an enhancement between
0.25-0.35 GeV2, due to K0π+ and ηπ+ final states, where the decay products other than
FIG. 10: The invariant π+π0 mass. The upper curve shows a fit using a background polynomial
plus Gaussian signal functions, where the width of the Gaussian is fixed to a value determined by
Monte Carlo simulation. The lower curve shows just the background polynomial.
TABLE III: Backgrounds in the D+s → τ
+ν, τ+ → π+ν sample for correctly reconstructed tags,
case (i) for 0.05<MM2 < 0.20 GeV2 and case (ii) for -0.05<MM2 < 0.20 GeV2.
Source B(%) # of events case (i) # of events case(ii) Sum
D+s → Xµ
+ν 8.2 0+1.8
−0 0 0
D+s → π
+π0π0 1.0 0.03±0.04 0.08±0.03 0.11±0.04
D+s → τ
+ν 6.4
τ+ → π+π0ν 1.5 0.55±0.22 0.64±0.24 1.20±0.33
τ+ → µ+νν 1.0 0.37±0.15 0 0.37±0.15
Sum 1.0+1.8
−0 0.7±0.2 1.7
the π+ escape detection. The data are consistent with our expectation that the region
−0.05 <MM2 <0.2 GeV2 contains mostly signal. Recall there are 148 total events only 10.7
of which we estimate are background, 9.0 from fake D−s tags and 1.7 from real tags and D
decays. Above 0.2 GeV2 other, larger backgrounds enter.
The number of real µ+ν events Nµν is related to the number of events detected in the
signal region Ndet (92), the estimated background Nbkgrd (3.5), the number of tags, Ntag, and
the branching fractions as
Nµν ≡ Ndet −Nbkgrd = Ntag · ǫ
ǫ′B(D+s → µ
+ν) + ǫ′′B(D+s → τ
+ν; τ+ → π+ν)
, (8)
where ǫ (80.1%) includes the efficiency for reconstructing the single charged track including
final state radiation (77.8%), the (98.3±0.2)% efficiency of not having another unmatched
cluster in the event with energy greater than 300 MeV, and for the fact that it is easier
to find tags in µ+ν events than in generic decays by 4.8%, as determined by Monte Carlo
simulation. The efficiency labeled ǫ′ (91.4%) is the product of the 99.0% muon efficiency
FIG. 11: The sum of case (i) and case (ii) MM2 distributions (histogram) compared to the predicted
shape (curve) for the sum of D+s → µ
+ν and τ+ν, τ+ → π+ν. The curve is normalized to the total
number of events below MM2 <0.2 GeV2.
for depositing less than 300 MeV in the calorimeter and 92.3% acceptance of the MM2 cut
of |MM2| < 0.05 GeV2. The quantity ǫ′′ (7.9%) is the fraction of τ+ν; τ+ → π+ν events
contained in the µ+ν signal window (13.2%) times the 60% acceptance for a pion to deposit
less than 300 MeV in the calorimeter.
The two D+s branching fractions in Eq. 8 are related as
B(D+s → τ
+ν; τ+ → π+ν) = R · B(τ+ → π+ν)B(D+s → µ
+ν) = 1.059 · B(D+s → µ
+ν) , (9)
where we take the Standard Model ratio for R as given in Eq. 2 and B(τ+ →
π+ν)=(10.90±0.07)% [3]. This allows us to solve Eq. 8. Since Ndet= 92, Nbkgrd=3.5±1.4,
and Ntag = 18645± 426± 1081, we find
B(D+s → µ
+ν) = (0.594± 0.066± 0.031)%. (10)
We can also sum the µ+ν and τ+ν contributions, where we restrict ourselves to the MM2
region below 0.20 GeV2 and above -0.05 GeV2. Eq. 8 still applies. The number of events
in the signal region and the number of background events changes to 148 and 10.7+2.9
−2.3,
respectively. The efficiency ǫ′ becomes 96.2%, and ǫ′′ increases to 45.2%. Using this method,
we find an effective branching fraction of
Beff(D+s → µ
+ν) = (0.638± 0.059± 0.033)%. (11)
The systematic errors on these branching fractions are given in Table IV. The error on
track finding is determined from a detailed comparison of the simulation with double tag
events where one track is ignored. “Minimum ionization” indicates the error due to the
requirement that the charged track deposit no more than 300 MeV in the calorimeter; it
is determined using two-body D0 → K−π+ decays (see Ref. [7]). The error on the photon
veto efficiency, due to the 300 MeV/c extra shower energy cut, is determined using Monte
Carlo simulation. The Monte Carlo was cross-checked using a sample of fully reconstructed
D+D− events and comparing the inefficiency due to additional photons with energy above
300 MeV/c. These events have no real extra photons above 300 MeV/c; those that are
present are due to interactions of the D± decay products in the detector material. The error
on the number of tags of ±5% has been discussed earlier. In addition there is a small error
of ±0.6% on the τ+ν branching fraction due to the uncertainty on the τ+ decay fraction
to π+ν. Additional systematic errors arising from the background estimates are negligible.
Note that the minimum ionization error does not apply to the summed branching fraction
given in Eq. 11; in this case the total systematic error is 5.1%.
TABLE IV: Systematic errors on determination of the D+s → µ
+ν branching fraction.
Error Source Size (%)
Track finding 0.7
Photon veto 1
Minimum ionization∗ 1
Number of tags 5
Total 5.2
*-Not applicable for summed rate
We also analyze the τ+ν final state independently. We use different MM2 regions for
cases (i) and (ii) defined above. For case (i) we define the signal region to be the interval
0.05<MM2 <0.20 GeV2, while for case (ii) we define the signal region to be the interval
-0.05<MM2 <0.20 GeV2. Case (i) includes the µ+ν signal, so we must exclude the region
close to zero MM2, while for case (ii) we are specifically selecting pions so the signal region
can be larger. The upper limit on MM2 is chosen to avoid background from the tail of the
K0π+ peak. The fractions of the MM2 range accepted are 32% and 45% for case (i) and (ii),
respectively.
We find 31 signal and 3.5+1.7
−1.1 background events for case (i) and 25 signal and 5.1±1.6
background events for case (ii). The branching fraction, averaging the two cases is
B(D+s → τ
+ν) = (8.0± 1.3± 0.4)%. (12)
Lepton universality in the Standard Model requires that the ratio R from Eq. 2 be equal to
a value of 9.72. We measure
Γ(D+s → τ
Γ(D+s → µ
= 13.4± 2.6± 0.2 . (13)
Here the systematic error is dominated by the uncertainty on the minimum ionization cut
that we use to separate the µ+ν and τ+ν regions at 300 MeV. We take this error as 2%, since
a change here affects both the numerator and denominator. The ratio is consistent with the
Standard Model prediction. Current results on D+ leptonic decays also show no deviations
[26]. The absence of any detected electrons opposite to our tags allows us to set an upper
limit of
B(D+s → e
+ν) < 1.3× 10−4 (14)
at 90% confidence level; this is also consistent with Standard Model predictions and lepton
universality.
IV. CHECKS OF THE METHOD
We perform an overall check of our procedures by measuring B(D+s → K
K+). For this
measurement we compute the MM2 (Eq. 6) using events with an additional charged track
but here identified as a kaon. These track candidates have momenta of approximately 1
GeV/c; here our RICH detector has a pion to kaon fake rate of 1.1% with a kaon detection
efficiency of 88.5% [19]. For this study, we do not veto events with extra charged tracks, or
neutral energy deposits >300 MeV, because of the presence of the K0.
The MM2 distribution is shown in Fig. 7. The peak near 0.25 GeV2 is due to the decay
mode of interest. We fit this to a linear background from 0.02-0.50 GeV2 plus a two-Gaussian
signal function. The fit yields 375±23±18 events. Events from the ηπ+ mode where the
π+ fakes a K+ are very rare and would not peak at the proper MM2. Since ηK+ could in
principle contribute a background in this region, we searched for this final state in a 195 pb−1
subsample of the data. Not finding any signal, we set an upper limit of B(D+s → ηK
2.8 × 10−3 at 90% confidence level, approximately a factor of ten below our measurement.
This final state would peak at a MM2 of 0.30 GeV2 and would cause an asymmetric tail on
the high side of the peak. Since we see no evidence for an asymmetry in the K
K+ peak
we ignore the ηK+ final state from here on. In order to compute the branching fraction we
must include the efficiency of detecting the kaon track 76.2%, including radiation [24], the
particle identification efficiency of 88.5%, and take into account that it is easier to detect
tags in events containing a K
K+ decay than in the average DsD
s event due to the track
and photon multiplicities, which gives a 3% correction.3 These rates are estimated by using
Monte Carlo simulation. We determine
B(D+s → K
K+) = (2.90± 0.19± 0.18)%, (15)
where the systematic errors are listed in Table V. We estimate the error from the signal
shape by taking the change in the number of events when varying the signal width of the
two-Gaussian function by ±1σ. The error on the background shapes is given by varying the
shape of the background fit. The error on the particle identification efficiency is measured
using two-body D0 decays [19]. The other errors are the same as described in Table IV.
Again, the largest component of the systematic error arises from the number of tag events
(5%). In fact, to use this result as a check on our procedures, we need only consider the
3 The tag bias is less here than in the µ+ν case because of the K0 decays and interactions in the detector.
systematic errors that are different here than in the µ+ν case. Those are due only to the
signal and background shapes and the particle identification cut. Those systematic errors
amount to 3.7% or ±0.11 in the branching fraction.
To determine absolute branching fractions of charm mesons, CLEO-c uses a method
where both particles are fully reconstructed (so called “double tags”) and the rates are
normalized using events where only one particle is fully reconstructed. Our preliminary
result using this method for B(D+s → KSK
+)=(1.50±0.09 ± 0.05)%, which when doubled
becomes (3.00±0.19±0.10)% [25]. This is in excellent agreement with the number in Eq. 15.
These results are not independent.
TABLE V: Systematic errors on determination of the D+s → K
K+ branching fraction.
Error Source Size (%)
Signal shape 3
Background shape 2
Track finding 0.7
PID cut 1.0
Number of tags 5
Total 6.3
We also performed the entire analysis on a Monte Carlo sample that corresponds to
an integrated luminosity four 4 times larger than the data sample. The input branching
fraction in the Monte Carlo is 0.5% for µ+ν and 6.57% for τ+ν, while our analysis measured
B(D+s → µ
+ν) =(0.514±0.027)% for the case (i) µ+ν signal and (0.521±0.024)% for µ+ν
and τ+ν combined. We also find (6.6±0.6)% for the τ+ν rate.
V. THE DECAY CONSTANT
Using our most precise value for B(D+s → µ
+ν) from Eq. 11, that is derived using both
our µ+ν and τ+ν samples, and Eq. 1 with a Ds lifetime of (500±7)×10
−15 s [3], we extract
fD+s = 274± 13± 7 MeV. (16)
We combine with our previous result [6]
fD+ = 222.6± 16.7
−3.4 MeV (17)
and find a value for
= 1.23± 0.11± 0.04, (18)
where only a small part of the systematic error cancels in the ratio of our two measurements.
VI. CONCLUSIONS
Theoretical models that predict f
and the ratio
are listed in Table VI. Our result
for fDs is slightly higher than most theoretical expectations. We are consistent with lattice
gauge theory, and most other models, for the ratio of decay constants. There is no evidence
at this level of precision for any suppression in the ratio due to the presence of a virtual
charged Higgs [9].
TABLE VI: Theoretical predictions of f
, fD+ , and fD+s
/fD+ . QL indicates quenched lattice
calculations.
Model f
(MeV) fD+ (MeV) fD+s
Lattice (HPQCD+UKQCD) [27] 241± 3 208 ± 4 1.162 ± 0.009
Lattice (FNAL+MILC+HPQCD) [28] 249± 3± 16 201 ± 3± 17 1.24 ± 0.01± 0.07
QL (QCDSF) [29] 220 ± 6± 5± 11 206 ± 6± 3± 22 1.07 ± 0.02± 0.02
QL (Taiwan) [30] 266 ± 10 ± 18 235 ± 8± 14 1.13 ± 0.03± 0.05
QL (UKQCD) [31] 236 ± 8+17
−14 210± 10
−16 1.13 ± 0.02
+0.04
−0.02
QL [32] 231± 12+6
−1 211± 14
−12 1.10 ± 0.02
QCD Sum Rules [33] 205 ± 22 177± 21 1.16 ± 0.01± 0.03
QCD Sum Rules [34] 235 ± 24 203± 20 1.15 ± 0.04
Field Correlators [35] 210 ± 10 260± 10 1.24 ± 0.03
Quark Model [36] 268 234 1.15
Quark Model [37] 248±27 230±25 1.08±0.01
LFQM (Linear) [38] 211 248 1.18
LFQM (HO) [38] 194 233 1.20
LF-QCD [39] 253 241 1.05
Potential Model [40] 241 238 1.01
Isospin Splittings [41] 262± 29
By using a theoretical prediction for f
/fD+ we can derive a value for the ratio of CKM
elements |Vcd/Vcs|. Taking the value from Ref. [27] of 1.162± 0.009, we find
|Vcd/Vcs| = 0.2171± 0.021± 0.0017 , (19)
where the first error is due the statistical and systematic errors of the experiment and the
second is due to the stated error on the theoretical prediction. This value is expected to be
almost equal to the ratio of the CKM elements |Vus|/|Vud|
We now compare with previous measurements. The branching fractions, modes, and
derived values of f
are listed in Table VII. Our values are shown first. We are generally
consistent with previous measurements, although ours are more precise.
Most measurements of D+s → ℓ
+ν are normalized with respect to B(D+s → φπ
+) ≡ Bφπ.
An exception is the OPAL measurement which is normalized to the Ds fraction in Z
0 events
that is derived from an overall fit to heavy flavor data at LEP [42]. It still, however, relies on
absolute branching fractions that are hidden by this procedure, and the estimated error on
the normalization is somewhat smaller than that indicated by the error on Bφπ available at
TABLE VII: Our results for B(D+s → µ
+ν), B(D+s → τ
+ν), and f
compared with previous
measurements. Results have been updated for the new value of the Ds lifetime [3]. ALEPH
combines both measurements to derive a value for the decay constant.
Exp. Mode B Bφπ (%) fD+s
(MeV)
CLEO-c µ+ν (5.94 ± 0.66 ± 0.31) · 10−3 264 ± 15± 7
CLEO-c τ+ν (8.0± 1.3 ± 0.4) · 10−2 310 ± 25± 8
CLEO-c combined - 274 ± 13± 7
CLEO [11] µ+ν (6.2 ± 0.8± 1.3± 1.6) · 10−3 3.6±0.9 273± 19± 27± 33
BEATRICE [12] µ+ν (8.3 ± 2.3± 0.6± 2.1) · 10−3 3.6±0.9 312± 43± 12± 39
ALEPH [13] µ+ν (6.8± 1.1 ± 1.8) · 10−3 3.6±0.9 282± 19± 40
ALEPH [13] τ+ν (5.8± 0.8 ± 1.8) · 10−2
L3 [14] τ+ν (7.4 ± 2.8± 1.6± 1.8) · 10−2 299± 57± 32± 37
OPAL [15] τ+ν (7.0± 2.1 ± 2.0) · 10−2 283± 44± 41
BaBar [16] µ+ν (6.74 ± 0.83 ± 0.26 ± 0.66) · 10−3 4.71±0.46 283 ± 17± 7± 14
the time of their publication. The L3 measurement is normalized taking the fraction of Ds
mesons produced in c quark fragmentation as 0.11±0.02, and the ratio of D∗s/Ds production
of 0.65±0.10. The ALEPH results use Bφπ for their µ
+ν results and a similar procedure as
OPAL for their τ+ν results. We note that the recent BaBar result uses a larger Bφπ than
the other results.
The CLEO-c determination of fD+s is the most accurate to date. It also does not rely
on the independent determination of any normalization mode. (We note that a preliminary
CLEO-c result using D+s → τ
+ν, τ+ → e+νν [43] is consistent with these results.)
VII. ACKNOWLEDGMENTS
We gratefully acknowledge the effort of the CESR staff in providing us with excellent
luminosity and running conditions. D. Cronin-Hennessy and A. Ryd thank the A.P. Sloan
Foundation. This work was supported by the National Science Foundation, the U.S. Depart-
ment of Energy, and the Natural Sciences and Engineering Research Council of Canada.
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	Introduction
	Experimental Method
	Selection of Ds Candidates
	Signal Reconstruction
	The Expected MM2 Spectrum
	MM2 Spectra in Data
	Background Evaluations
	Leptonic Branching Fractions
	Checks of the Method
	The Decay Constant
	Conclusions
	Acknowledgments
	References

  We examine e+e- --> Ds- Ds*+ and Ds*- Ds+ interactions at 4170 MeV using the
CLEO-c detector in order to measure the decay constant fDs+. We use the Ds+ -->
l+ nu channel, where the l+ designates either a mu+ or a tau+, when the tau+
--> pi+ nu. Analyzing both modes independently, we determine B(Ds+ --> mu+ nu)
= (0.594 +- 0.066 +- 0.031)%, and B(Ds+ --> tau+ nu) = (8.0 +- 1.3 +- 0.4)%. We
also analyze them simultaneously to find an effective value of B{eff}(Ds+ -->
mu+ nu) = (0.638 +- 0.059 +- 0.033)% and extract fDs = (274 +- 13 +- 7) MeV.
Combining with our previous determination of B(D+ -> mu+ nu), we also find the
ratio fDs/fD+ = 1.23 +- 0.11 +- 0.04. We compare to current theoretical
estimates. Finally, we find B(Ds+ --> e+ nu) < 1.3 x10^{-4} at 90% confidence
level.

Introduction
	Experimental Method
	Selection of Ds Candidates
	Signal Reconstruction
	The Expected MM2 Spectrum
	MM2 Spectra in Data
	Background Evaluations
	Leptonic Branching Fractions
	Checks of the Method
	The Decay Constant
	Conclusions
	Acknowledgments
	References
