CLNS 07/1989
CLEO 07-01
Measurement of the Decay Constant f
using D+
→ ℓ+ν
M. Artuso,1 S. Blusk,1 J. Butt,1 S. Khalil,1 J. Li,1 N. Menaa,1 R. Mountain,1 S. Nisar,1
K. Randrianarivony,1 R. Sia,1 T. Skwarnicki,1 S. Stone,1 J. C. Wang,1 G. Bonvicini,2 D. Cinabro,2
M. Dubrovin,2 A. Lincoln,2 D. M. Asner,3 K. W. Edwards,3 P. Naik,3 R. A. Briere,4 T. Ferguson,4
G. Tatishvili,4 H. Vogel,4 M. E. Watkins,4 J. L. Rosner,5 N. E. Adam,6 J. P. Alexander,6
D. G. Cassel,6 J. E. Duboscq,6 R. Ehrlich,6 L. Fields,6 L. Gibbons,6 R. Gray,6 S. W. Gray,6
D. L. Hartill,6 B. K. Heltsley,6 D. Hertz,6 C. D. Jones,6 J. Kandaswamy,6 D. L. Kreinick,6
V. E. Kuznetsov,6 H. Mahlke-Krüger,6 D. Mohapatra,6 P. U. E. Onyisi,6 J. R. Patterson,6 D. Peterson,6
J. Pivarski,6 D. Riley,6 A. Ryd,6 A. J. Sadoff,6 H. Schwarthoff,6 X. Shi,6 S. Stroiney,6 W. M. Sun,6
T. Wilksen,6 S. B. Athar,7 R. Patel,7 J. Yelton,7 P. Rubin,8 C. Cawlfield,9 B. I. Eisenstein,9
I. Karliner,9 D. Kim,9 N. Lowrey,9 M. Selen,9 E. J. White,9 J. Wiss,9 R. E. Mitchell,10
M. R. Shepherd,10 D. Besson,11 T. K. Pedlar,12 D. Cronin-Hennessy,13 K. Y. Gao,13 J. Hietala,13
Y. Kubota,13 T. Klein,13 B. W. Lang,13 R. Poling,13 A. W. Scott,13 A. Smith,13 P. Zweber,13
S. Dobbs,14 Z. Metreveli,14 K. K. Seth,14 A. Tomaradze,14 J. Ernst,15 K. M. Ecklund,16
H. Severini,17 W. Love,18 V. Savinov,18 O. Aquines,19 A. Lopez,19 S. Mehrabyan,19 H. Mendez,19
J. Ramirez,19 G. S. Huang,20 D. H. Miller,20 V. Pavlunin,20 B. Sanghi,20 I. P. J. Shipsey,20
B. Xin,20 G. S. Adams,21 M. Anderson,21 J. P. Cummings,21 I. Danko,21 D. Hu,21 B. Moziak,21
J. Napolitano,21 Q. He,22 J. Insler,22 H. Muramatsu,22 C. S. Park,22 E. H. Thorndike,22 and F. Yang22
(CLEO Collaboration)
Syracuse University, Syracuse, New York 13244
Wayne State University, Detroit, Michigan 48202
Carleton University, Ottawa, Ontario, Canada K1S 5B6
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637
Cornell University, Ithaca, New York 14853
University of Florida, Gainesville, Florida 32611
George Mason University, Fairfax, Virginia 22030
University of Illinois, Urbana-Champaign, Illinois 61801
Indiana University, Bloomington, Indiana 47405
University of Kansas, Lawrence, Kansas 66045
Luther College, Decorah, Iowa 52101
University of Minnesota, Minneapolis, Minnesota 55455
Northwestern University, Evanston, Illinois 60208
State University of New York at Albany, Albany, New York 12222
State University of New York at Buffalo, Buffalo, New York 14260
University of Oklahoma, Norman, Oklahoma 73019
University of Pittsburgh, Pittsburgh, Pennsylvania 15260
University of Puerto Rico, Mayaguez, Puerto Rico 00681
Purdue University, West Lafayette, Indiana 47907
Rensselaer Polytechnic Institute, Troy, New York 12180
University of Rochester, Rochester, New York 14627
(Dated: November 1, 2018)
We measure the decay constant f
using the D+s → ℓ
+ν channel, where the ℓ+ designates either
a µ+ or a τ+, when the τ+ → π+ν. Using both measurements we find f
= 274 ± 13 ± 7 MeV.
Combining with our previous determination of fD+ , we compute the ratio fD+
/fD+ = 1.23±0.11±
0.04. We compare with theoretical estimates.
PACS numbers: 13.20.Fc, 13.66.Bc
To extract precise information on the size of CKM
matrix elements from Bd and Bs mixing measure-
ments the ratio of “decay constants,” that are re-
lated to the heavy and light quark wave-function
overlap at zero separation, must be well known [1].
Recent measurement of B0s mixing by CDF [2] has
shown the urgent need for precise numbers. De-
cay constants have been calculated for both B and
http://arxiv.org/abs/0704.0629v3
D mesons using several methods, including lattice
QCD [3]. Here we present the most precise measure-
ment to date of f
, and combined with our previ-
ous determination of fD+ [4, 5], we find fD+
/fD+ .
In the Standard Model (SM) purely leptonic Ds
decay proceeds via annihilation through a virtual
W+. The decay rate is given by [6]
s → ℓ
|Vcs|
where M
is the D+s mass, mℓ is the lepton mass,
GF is the Fermi constant, and |Vcs| is a CKM matrix
element with a value of 0.9738 [7].
In this Letter we report measurements of both
B(D+s → µ
+ν) and B(D+s → τ
+ν), when τ+ → π+ν
(D+s → π
+νν). More details are given in a compan-
ion paper [8]. The ratio Γ(D+s → τ
+ν)/Γ(D+s →
µ+ν) predicted in the SM via Eq. 1 depends only
on well-known masses, and equals 9.72; any devia-
tion would be a manifestation of new physics as it
would violate lepton universality [9]. New physics
can also affect the expected widths; any undiscov-
ered charged bosons would interfere with the SM
W+ [10].
The CLEO-c detector [11] is equipped to mea-
sure the momenta of charged particles, identify
them using dE/dx and Cherenkov imaging (RICH)
[12], detect photons and determine their directions
and energies. We use 314 pb−1 of data produced
in e+e− collisions using CESR near 4.170 GeV.
Here the cross-section for our analyzed sample,
D∗+s D
s , is ∼1 nb. Other charm produc-
tion totals ∼7 nb [13], and the underlying light-
quark “continuum” is ∼12 nb. We fully reconstruct
oneD−s as a “tag,” and examine the properties of the
D+s . (Charge conjugate decays are used.) Track se-
lection, particle identification, π0, η, and K0S criteria
are the same as those described in Ref. [4], except
that RICH identification now requires a minimum
momentum of 700 MeV/c.
Tag modes are listed in Table I. For resonance de-
cays 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 require tags to have
momentum consistent with coming from DsD
s pro-
duction. The distribution for the K+K−π− mode
(44% of all the tags) is shown in Fig. 1.
To select tags, we first fit the invariant mass dis-
tributions to the sum of two Gaussians centered
at MDs . The r.m.s. resolution (σ) is defined as
σ ≡ f1σ1 + (1 − f1)σ2, where σ1 and σ2 are the in-
dividual widths and f1 is the fractional area of the
FIG. 1: Invariant mass of K+K−π− candidates after
requiring the total energy to be consistent with the beam
energy. The curve shows a fit to a two-Gaussian signal
function plus a polynomial background.
TABLE I: Tagging modes and numbers of signal and
background events, within cuts, from two-Gaussian fits
to the invariant mass plots, and the number of γ tags in
each mode, within ±2.5σ from a fit to the signal Crys-
tal Ball function (see text) and a 5th order Chebychev
background polynomial and the associated background.
Mode Invariant Mass MM∗2
Signal Bkgrnd Signal Bkgrnd
K+K−π− 13871±262 10850 8053± 211 13538
− 3122±79 1609 1933±88 2224
ηπ− 1609± 112 4666 1024±97 3967
η′π− 1196±46 409 792±69 1052
φρ− 1678±74 1898 1050±113 3991
π+π−π− 3654±199 25208 2300±187 15723
K∗−K∗0 2030±98 4878 1298±130 5672
ηρ− 4142±281 20784 2195±225 17353
Sum 31302 ± 472 70302 18645±426 63520
first Gaussian. We require the invariant masses to be
within ± 2.5σ (±2σ for the ηρ− mode) of MDs . We
have a total of 31302±472 tag candidates. Then we
add a γ candidate that satisfies our shower shape re-
quirement. Regardless of whether or not the γ forms
a D∗s with the tag, for real D
sDs events, the missing
mass squared, MM∗2, recoiling against the γ and the
D−s tag should peak at M
. We calculate
MM∗2 = (ECM − EDs − Eγ)
−→pCM −
−→pDs −
where ECM (
−→pCM) is the center-of-mass energy (mo-
mentum), EDs (
−→pDs) is the energy (momentum) of
the fully reconstructed D−s tag, Eγ (
−→pγ) is the en-
ergy (momentum) of the additional γ. We use a
kinematic fit that constrains the decay products of
the D−s to MDs and conserves overall momentum
and energy. All γ’s in the event are used, except for
those that are decay products of the D−s tag.
The MM∗2 distribution from K+K−π− tags is
shown in Fig. 2. We fit all the modes individually
to determine the number of tag events. This proce-
dure is enhanced by having information on the shape
of the signal function. We use fully reconstructed
D−s D
s events, and examine the signal shape when
one Ds is ignored. The signal is fit to a Crystal Ball
function [14], which determines σ and the shape of
the tail. Though σ varies somewhat between modes,
the tail parameters don’t change, since they depend
on beam radiation and γ energy resolution.
FIG. 2: The MM∗2 distribution from events with a γ
in addition to the K+K−π− tag. The curve is a fit to
the Crystal Ball function and a 5th order Chebychev
background function.
Fits of MM∗2 in each mode when summed show
18645±426 events within a ±2.5σ interval (see Ta-
ble I). There is a small enhancement of (4.8± 1.0)%
in our ability to find tags in µ+ν (or π+νν) events
(tag bias) as compared with generic events. Addi-
tional systematic errors are evaluated by changing
the fitting range, using 4th and 6th order Chebychev
background polynomials, and allowing the parame-
ters of the tail of the fitting function to float, leading
to an overall systematic uncertainty of 5%.
Candidate µ+ν events are required to have only a
single additional track oppositely charged to the tag
with an angle >35.9◦ with respect to the beam line.
We also require that there not be any neutral en-
ergy cluster detected of more than 300 MeV, which
is especially useful to reject D+s → π
+π0 and ηπ+
decays. Since here we are searching for events in
which there is a single missing ν, the missing mass
squared, MM2, should peak at zero:
MM2 = (ECM − EDs − Eγ − Eµ)
−→pCM −
−→pDs −
−→pγ −
where Eµ (
−→pµ) are the energy (momentum) of the
candidate µ+ track.
We also make use of a set of kinematical con-
straints and fit each event to two hypotheses:
(1) the D−s tag is the daughter of a D
s and
(2) the D∗+s decays into γD
s . The kinemati-
cal constraints, in the center-of-mass frame, are
−→pDs +
−→pD∗
= 0, ECM = EDs + ED∗
, ED∗
ECM/2+
−M2Ds
/2ECM or EDs = ECM/2−
−M2Ds
/2ECM, MD∗
− MDs = 143.6 MeV.
In addition, we constrain the invariant mass of the
D−s tag to MDs . This gives a total of 7 constraints.
The missing ν four-vector needs to be determined,
so we are left with a three-constraint fit. We perform
an iterative fit minimizing χ2. To eliminate system-
atic uncertainties that depend on understanding the
absolute scale of the errors, we do not make a χ2 cut
but simply choose the γ and the decay sequence in
each event with the minimum χ2.
We consider three separate cases: (i) the track de-
posits < 300 MeV in the calorimeter, characteristic
of a non-interacting pion or a µ+; (ii) the track de-
posits > 300 MeV in the calorimeter, characteristic
of an interacting pion; or (iii) the track satisfies our
electron selection criteria. 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 [5]. Case (iii) does not include any signal but
is used for background estimation. For cases (i) and
(ii) we insist that the track not be identified as an
electron or a kaon. Electron candidates have a match
between the momentum measured in the tracking
system and the energy deposited in the CsI calorime-
ter, and dE/dx and RICH measurements consistent
with this hypothesis.
For the µ+ν final state the MM2 distribution is
modeled as the sum of two Gaussians centered at
zero. A Monte Carlo (MC) simulation of the MM2
shows σ=0.025 GeV2 after the fit. We check the
resolution using the D+s → K
K+ mode. We search
for events with at least one additional track identi-
fied as a kaon using the RICH detector, in addition
to a D−s tag. The MM
2 resolution is 0.025 GeV2 in
agreement with the simulation.
In the π+νν final state, the extra missing ν re-
sults in a smeared MM2 distribution that is almost
triangular in shape starting near -0.05 GeV2, peak-
ing near 0.10 GeV2, and ending at 0.75 GeV2.
FIG. 3: The MM2 distributions from data usingD−s tags,
and one additional opposite-sign charged track and no
extra energetic showers, for cases (i), (ii), and (iii).
The MM2 distributions from data are shown in
Fig. 3. The overall signal region is -0.05 < MM2 <
0.20 GeV2. The upper limit is chosen to prevent
background from ηπ+ and K0π+ final states. The
peak in Fig. 3(i) is due to D+s → µ
+ν. Below 0.20
GeV2 in both (i) and (ii) we have π+νν events.
The specific signal regions are: for µ+ν, −0.05 <
MM2 < 0.05 GeV2, corresponding to ±2σ; for π+νν,
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.
We consider backgrounds from two sources: one
from real D+s decays and the other from the back-
ground under the single-tag signal peaks. For the
latter, we estimate the background from data using
side-bands of the invariant mass, shown in Fig. 1.
For case (i) we find 3.5 (properly normalized) back-
ground events in the µ+ν region and 2.5 back-
grounds in the τ+ν region; for case (ii) we find 3
events. Our total background estimate summing
over all of these cases is 9.0±2.3 events.
The background from real D+s decays is evaluated
by identifying specific sources. For µ+ν the only
possible background is D+s → π
+π0. Using a 195
pb−1 subsample of our data, we limit the branching
fraction as < 1.1 × 10−3 at 90% C.L. [8]. This low
rate coupled with the extra γ veto yields a negligible
contribution. The real D+s backgrounds for π
are listed in Table II. Using the SM expected ratio of
decay rates we calculate a contribution of 7.4 π+νν
events.
TABLE II: Event backgrounds in the π+νν sample from
real D+s decays.
Source B(%) case (i) case (ii) Sum
D+s → Xµ
+ν 8.2 0+1.8
0 0+1.8
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
0.7±0.2 1.7+1.8
The event yield in the signal region, Ndet (92), is
related to the number of tags, Ntag, the branching
fractions, and the background Nbkgrd (3.5) as
Ndet −Nbkgrd = Ntag · ǫ[ǫ
′B(D+s → µ
+ν) (3)
+ǫ′′B(D+s → π
+νν)],
where ǫ (80.1%) includes the efficiencies (77.8%) for
reconstructing the single charged track including fi-
nal state radiation, (98.3)% for not having another
unmatched cluster in the event with energy greater
than 300 MeV, and the correction for the tag bias
(4.8%); ǫ′ (91.4%) is the product of the 99.0% µ+
calorimeter efficiency and the 92.3% acceptance of
the MM2 cut of |MM2| < 0.05 GeV2; ǫ′′ (7.6%) is
the fraction of π+νν events contained in the µ+ν sig-
nal window (13.2%) times the 60% acceptance for a
pion to deposit less than 300 MeV in the calorime-
ter. Using B(τ+ → π+ν) of (10.90±0.07)% [7], the
ratio of the π+νν to µ+ν widths is 1.059; we find:
B(D+s → µ
+ν) = (0.594± 0.066± 0.031)%. (4)
We can also sum the µ+ν and τ+ν contributions
for −0.05 < MM2 < 0.02 GeV2. Equation 3 still ap-
plies. The number of signal and background events
changes to 148 and 10.7, respectively. ǫ′ becomes
96.2%, and ǫ′′ increases to 45.2%. The effective
branching fraction, assuming lepton universality, is
Beff(D+s → µ
+ν) = (0.638± 0.059± 0.033)%. (5)
The systematic errors on these branching fractions
are dominated by the error on the number of tags
(5%). Other errors include: (a) track finding (0.7%),
determined from a detailed comparison of the sim-
ulation with double tag events where one track is
ignored; (b) the error due to the requirement that
the charged track deposit no more than 300 MeV
in the calorimeter (1%), determined using two-body
D0 → K−π+ decays [5]; (c) the γ veto efficiency
(1%), determined by extrapolating measurements on
fully reconstructed events. Systematic errors arising
from the background estimates are negligible. The
total systematic error for Eq. 4 is 5.2%, and is 5.1%
for Eq. 5 as (b) doesn’t apply here.
We also analyze the τ+ν final state independently.
For case (i) we define the signal region to be the in-
terval 0.05<MM2 <0.20 GeV2, while for case (ii)
-0.05<MM2 <0.20 GeV2. The upper limit on MM2
is chosen to avoid background from the tail of the
K0π+ peak. The fractions of the MM2 range ac-
cepted are 32% and 45% for case (i) and (ii), respec-
tively.
We find 31 [25] events in the signal region with a
background of 3.5 [5.1] events for case (i) [(ii)]. The
branching fraction, averaging the two cases is
B(D+s → τ
+ν) = (8.0± 1.3± 0.4)%, (6)
where the systematic error includes a contribution
of 0.06% from the uncertainty on B(τ+ → π+ν).
We measure 13.4 ± 2.6 ± 0.2 for the ratio of τ+ν
to µ+ν rates using Eq. 4. Here the systematic er-
ror is dominated by the uncertainty on the mini-
mum ionization cut. We also set an upper limit of
B(D+s → e
+ν) < 1.3 × 10−4 at 90% C.L. Both of
these results are consistent with SM predictions and
lepton universality.
We perform an overall check of our procedures by
measuring B(D+s → K
K+). We compute the MM2
(Eq. 2) using events with an additional charged track
identified as a kaon. These track candidates have
momenta of approximately 1 GeV/c; here the RICH
has a pion to kaon fake rate of 1.1% with a kaon
detection efficiency of 88.5% [12]. For this study,
we do not veto events with extra charged tracks, or
γ’s, because of the presence of the K0. We deter-
mine B(D+s → K
K+) = (2.90 ± 0.19 ± 0.18)%.
This method gives a result in good agreement with
preliminary CLEO-c results using double tags of
(3.00 ± 0.19 ± 0.10)% [15]; these results are not in-
dependent.
We also performed the entire analysis on a MC
sample that is 4 times larger than the data sam-
ple. The input branching fraction is 0.5% for µ+ν
and 6.57% for τ+ν, while our analysis measured
(0.514±0.027)% for the case (i) µ+ν signal and
(0.521±0.024)% for µ+ν and τ+ν combined.
Using B(D+s → µ
+ν) from Eq. 5, and Eq. 1 with
a Ds lifetime of (500±7)×10
−15 s [7], we extract
= 274± 13± 7 MeV. (7)
We combine with our previous result fD+ =
222.6± 16.7+2.8
−3.4 MeV [4], and find
/fD+ = 1.23± 0.11± 0.04. (8)
Lattice QCD predictions for f
and the ratio
/fD+ have been summarized by Onogi [16]. Our
measurements are consistent with most calculations;
examples are unquenched Lattice that predicts 249±
3± 16 MeV and 1.24± 0.01± 0.07 for the ratio [17],
while a recent quenched prediction gives 266± 10±
18 MeV and 1.13 ± 0.03 ± 0.05 [18]. There is no
evidence yet for any suppression in the ratio due to
the presence of a virtual charged Higgs [10].
The CLEO-c determination of f
is the most
accurate to date and consistent with other measure-
ments [7, 8]. It also does not rely on the indepen-
dent determination of any normalization mode (e.g.
φπ+). (We note that a preliminary CLEO-c result
using D+s → τ
+ν, τ+ → e+νν [19] is consistent with
these results.)
We gratefully acknowledge the effort of the CESR
staff in providing us with excellent luminosity and
running conditions. This work was supported by the
A.P. Sloan Foundation, the National Science Foun-
dation, the U.S. Department of Energy, and the Nat-
ural Sciences and Engineering Research Council of
Canada.
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  We measure the decay constant fDs using the Ds -> l+ nu channel, where the l+
designates either a mu+ or a tau+, when the tau+ -> pi+ nu. Using both
measurements we find fDs = 274 +-13 +- 7 MeV. Combining with our previous
determination of fD+, we compute the ratio fDs/fD+ = 1.23 +- 0.11 +- 0.04. We
compare with theoretical estimates.

CLNS 07/1989
CLEO 07-01
Measurement of the Decay Constant f
using D+
→ ℓ+ν
M. Artuso,1 S. Blusk,1 J. Butt,1 S. Khalil,1 J. Li,1 N. Menaa,1 R. Mountain,1 S. Nisar,1
K. Randrianarivony,1 R. Sia,1 T. Skwarnicki,1 S. Stone,1 J. C. Wang,1 G. Bonvicini,2 D. Cinabro,2
M. Dubrovin,2 A. Lincoln,2 D. M. Asner,3 K. W. Edwards,3 P. Naik,3 R. A. Briere,4 T. Ferguson,4
G. Tatishvili,4 H. Vogel,4 M. E. Watkins,4 J. L. Rosner,5 N. E. Adam,6 J. P. Alexander,6
D. G. Cassel,6 J. E. Duboscq,6 R. Ehrlich,6 L. Fields,6 L. Gibbons,6 R. Gray,6 S. W. Gray,6
D. L. Hartill,6 B. K. Heltsley,6 D. Hertz,6 C. D. Jones,6 J. Kandaswamy,6 D. L. Kreinick,6
V. E. Kuznetsov,6 H. Mahlke-Krüger,6 D. Mohapatra,6 P. U. E. Onyisi,6 J. R. Patterson,6 D. Peterson,6
J. Pivarski,6 D. Riley,6 A. Ryd,6 A. J. Sadoff,6 H. Schwarthoff,6 X. Shi,6 S. Stroiney,6 W. M. Sun,6
T. Wilksen,6 S. B. Athar,7 R. Patel,7 J. Yelton,7 P. Rubin,8 C. Cawlfield,9 B. I. Eisenstein,9
I. Karliner,9 D. Kim,9 N. Lowrey,9 M. Selen,9 E. J. White,9 J. Wiss,9 R. E. Mitchell,10
M. R. Shepherd,10 D. Besson,11 T. K. Pedlar,12 D. Cronin-Hennessy,13 K. Y. Gao,13 J. Hietala,13
Y. Kubota,13 T. Klein,13 B. W. Lang,13 R. Poling,13 A. W. Scott,13 A. Smith,13 P. Zweber,13
S. Dobbs,14 Z. Metreveli,14 K. K. Seth,14 A. Tomaradze,14 J. Ernst,15 K. M. Ecklund,16
H. Severini,17 W. Love,18 V. Savinov,18 O. Aquines,19 A. Lopez,19 S. Mehrabyan,19 H. Mendez,19
J. Ramirez,19 G. S. Huang,20 D. H. Miller,20 V. Pavlunin,20 B. Sanghi,20 I. P. J. Shipsey,20
B. Xin,20 G. S. Adams,21 M. Anderson,21 J. P. Cummings,21 I. Danko,21 D. Hu,21 B. Moziak,21
J. Napolitano,21 Q. He,22 J. Insler,22 H. Muramatsu,22 C. S. Park,22 E. H. Thorndike,22 and F. Yang22
(CLEO Collaboration)
Syracuse University, Syracuse, New York 13244
Wayne State University, Detroit, Michigan 48202
Carleton University, Ottawa, Ontario, Canada K1S 5B6
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637
Cornell University, Ithaca, New York 14853
University of Florida, Gainesville, Florida 32611
George Mason University, Fairfax, Virginia 22030
University of Illinois, Urbana-Champaign, Illinois 61801
Indiana University, Bloomington, Indiana 47405
University of Kansas, Lawrence, Kansas 66045
Luther College, Decorah, Iowa 52101
University of Minnesota, Minneapolis, Minnesota 55455
Northwestern University, Evanston, Illinois 60208
State University of New York at Albany, Albany, New York 12222
State University of New York at Buffalo, Buffalo, New York 14260
University of Oklahoma, Norman, Oklahoma 73019
University of Pittsburgh, Pittsburgh, Pennsylvania 15260
University of Puerto Rico, Mayaguez, Puerto Rico 00681
Purdue University, West Lafayette, Indiana 47907
Rensselaer Polytechnic Institute, Troy, New York 12180
University of Rochester, Rochester, New York 14627
(Dated: November 1, 2018)
We measure the decay constant f
using the D+s → ℓ
+ν channel, where the ℓ+ designates either
a µ+ or a τ+, when the τ+ → π+ν. Using both measurements we find f
= 274 ± 13 ± 7 MeV.
Combining with our previous determination of fD+ , we compute the ratio fD+
/fD+ = 1.23±0.11±
0.04. We compare with theoretical estimates.
PACS numbers: 13.20.Fc, 13.66.Bc
To extract precise information on the size of CKM
matrix elements from Bd and Bs mixing measure-
ments the ratio of “decay constants,” that are re-
lated to the heavy and light quark wave-function
overlap at zero separation, must be well known [1].
Recent measurement of B0s mixing by CDF [2] has
shown the urgent need for precise numbers. De-
cay constants have been calculated for both B and
http://arxiv.org/abs/0704.0629v3
D mesons using several methods, including lattice
QCD [3]. Here we present the most precise measure-
ment to date of f
, and combined with our previ-
ous determination of fD+ [4, 5], we find fD+
/fD+ .
In the Standard Model (SM) purely leptonic Ds
decay proceeds via annihilation through a virtual
W+. The decay rate is given by [6]
s → ℓ
|Vcs|
where M
is the D+s mass, mℓ is the lepton mass,
GF is the Fermi constant, and |Vcs| is a CKM matrix
element with a value of 0.9738 [7].
In this Letter we report measurements of both
B(D+s → µ
+ν) and B(D+s → τ
+ν), when τ+ → π+ν
(D+s → π
+νν). More details are given in a compan-
ion paper [8]. The ratio Γ(D+s → τ
+ν)/Γ(D+s →
µ+ν) predicted in the SM via Eq. 1 depends only
on well-known masses, and equals 9.72; any devia-
tion would be a manifestation of new physics as it
would violate lepton universality [9]. New physics
can also affect the expected widths; any undiscov-
ered charged bosons would interfere with the SM
W+ [10].
The CLEO-c detector [11] is equipped to mea-
sure the momenta of charged particles, identify
them using dE/dx and Cherenkov imaging (RICH)
[12], detect photons and determine their directions
and energies. We use 314 pb−1 of data produced
in e+e− collisions using CESR near 4.170 GeV.
Here the cross-section for our analyzed sample,
D∗+s D
s , is ∼1 nb. Other charm produc-
tion totals ∼7 nb [13], and the underlying light-
quark “continuum” is ∼12 nb. We fully reconstruct
oneD−s as a “tag,” and examine the properties of the
D+s . (Charge conjugate decays are used.) Track se-
lection, particle identification, π0, η, and K0S criteria
are the same as those described in Ref. [4], except
that RICH identification now requires a minimum
momentum of 700 MeV/c.
Tag modes are listed in Table I. For resonance de-
cays 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 require tags to have
momentum consistent with coming from DsD
s pro-
duction. The distribution for the K+K−π− mode
(44% of all the tags) is shown in Fig. 1.
To select tags, we first fit the invariant mass dis-
tributions to the sum of two Gaussians centered
at MDs . The r.m.s. resolution (σ) is defined as
σ ≡ f1σ1 + (1 − f1)σ2, where σ1 and σ2 are the in-
dividual widths and f1 is the fractional area of the
FIG. 1: Invariant mass of K+K−π− candidates after
requiring the total energy to be consistent with the beam
energy. The curve shows a fit to a two-Gaussian signal
function plus a polynomial background.
TABLE I: Tagging modes and numbers of signal and
background events, within cuts, from two-Gaussian fits
to the invariant mass plots, and the number of γ tags in
each mode, within ±2.5σ from a fit to the signal Crys-
tal Ball function (see text) and a 5th order Chebychev
background polynomial and the associated background.
Mode Invariant Mass MM∗2
Signal Bkgrnd Signal Bkgrnd
K+K−π− 13871±262 10850 8053± 211 13538
− 3122±79 1609 1933±88 2224
ηπ− 1609± 112 4666 1024±97 3967
η′π− 1196±46 409 792±69 1052
φρ− 1678±74 1898 1050±113 3991
π+π−π− 3654±199 25208 2300±187 15723
K∗−K∗0 2030±98 4878 1298±130 5672
ηρ− 4142±281 20784 2195±225 17353
Sum 31302 ± 472 70302 18645±426 63520
first Gaussian. We require the invariant masses to be
within ± 2.5σ (±2σ for the ηρ− mode) of MDs . We
have a total of 31302±472 tag candidates. Then we
add a γ candidate that satisfies our shower shape re-
quirement. Regardless of whether or not the γ forms
a D∗s with the tag, for real D
sDs events, the missing
mass squared, MM∗2, recoiling against the γ and the
D−s tag should peak at M
. We calculate
MM∗2 = (ECM − EDs − Eγ)
−→pCM −
−→pDs −
where ECM (
−→pCM) is the center-of-mass energy (mo-
mentum), EDs (
−→pDs) is the energy (momentum) of
the fully reconstructed D−s tag, Eγ (
−→pγ) is the en-
ergy (momentum) of the additional γ. We use a
kinematic fit that constrains the decay products of
the D−s to MDs and conserves overall momentum
and energy. All γ’s in the event are used, except for
those that are decay products of the D−s tag.
The MM∗2 distribution from K+K−π− tags is
shown in Fig. 2. We fit all the modes individually
to determine the number of tag events. This proce-
dure is enhanced by having information on the shape
of the signal function. We use fully reconstructed
D−s D
s events, and examine the signal shape when
one Ds is ignored. The signal is fit to a Crystal Ball
function [14], which determines σ and the shape of
the tail. Though σ varies somewhat between modes,
the tail parameters don’t change, since they depend
on beam radiation and γ energy resolution.
FIG. 2: The MM∗2 distribution from events with a γ
in addition to the K+K−π− tag. The curve is a fit to
the Crystal Ball function and a 5th order Chebychev
background function.
Fits of MM∗2 in each mode when summed show
18645±426 events within a ±2.5σ interval (see Ta-
ble I). There is a small enhancement of (4.8± 1.0)%
in our ability to find tags in µ+ν (or π+νν) events
(tag bias) as compared with generic events. Addi-
tional systematic errors are evaluated by changing
the fitting range, using 4th and 6th order Chebychev
background polynomials, and allowing the parame-
ters of the tail of the fitting function to float, leading
to an overall systematic uncertainty of 5%.
Candidate µ+ν events are required to have only a
single additional track oppositely charged to the tag
with an angle >35.9◦ with respect to the beam line.
We also require that there not be any neutral en-
ergy cluster detected of more than 300 MeV, which
is especially useful to reject D+s → π
+π0 and ηπ+
decays. Since here we are searching for events in
which there is a single missing ν, the missing mass
squared, MM2, should peak at zero:
MM2 = (ECM − EDs − Eγ − Eµ)
−→pCM −
−→pDs −
−→pγ −
where Eµ (
−→pµ) are the energy (momentum) of the
candidate µ+ track.
We also make use of a set of kinematical con-
straints and fit each event to two hypotheses:
(1) the D−s tag is the daughter of a D
s and
(2) the D∗+s decays into γD
s . The kinemati-
cal constraints, in the center-of-mass frame, are
−→pDs +
−→pD∗
= 0, ECM = EDs + ED∗
, ED∗
ECM/2+
−M2Ds
/2ECM or EDs = ECM/2−
−M2Ds
/2ECM, MD∗
− MDs = 143.6 MeV.
In addition, we constrain the invariant mass of the
D−s tag to MDs . This gives a total of 7 constraints.
The missing ν four-vector needs to be determined,
so we are left with a three-constraint fit. We perform
an iterative fit minimizing χ2. To eliminate system-
atic uncertainties that depend on understanding the
absolute scale of the errors, we do not make a χ2 cut
but simply choose the γ and the decay sequence in
each event with the minimum χ2.
We consider three separate cases: (i) the track de-
posits < 300 MeV in the calorimeter, characteristic
of a non-interacting pion or a µ+; (ii) the track de-
posits > 300 MeV in the calorimeter, characteristic
of an interacting pion; or (iii) the track satisfies our
electron selection criteria. 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 [5]. Case (iii) does not include any signal but
is used for background estimation. For cases (i) and
(ii) we insist that the track not be identified as an
electron or a kaon. Electron candidates have a match
between the momentum measured in the tracking
system and the energy deposited in the CsI calorime-
ter, and dE/dx and RICH measurements consistent
with this hypothesis.
For the µ+ν final state the MM2 distribution is
modeled as the sum of two Gaussians centered at
zero. A Monte Carlo (MC) simulation of the MM2
shows σ=0.025 GeV2 after the fit. We check the
resolution using the D+s → K
K+ mode. We search
for events with at least one additional track identi-
fied as a kaon using the RICH detector, in addition
to a D−s tag. The MM
2 resolution is 0.025 GeV2 in
agreement with the simulation.
In the π+νν final state, the extra missing ν re-
sults in a smeared MM2 distribution that is almost
triangular in shape starting near -0.05 GeV2, peak-
ing near 0.10 GeV2, and ending at 0.75 GeV2.
FIG. 3: The MM2 distributions from data usingD−s tags,
and one additional opposite-sign charged track and no
extra energetic showers, for cases (i), (ii), and (iii).
The MM2 distributions from data are shown in
Fig. 3. The overall signal region is -0.05 < MM2 <
0.20 GeV2. The upper limit is chosen to prevent
background from ηπ+ and K0π+ final states. The
peak in Fig. 3(i) is due to D+s → µ
+ν. Below 0.20
GeV2 in both (i) and (ii) we have π+νν events.
The specific signal regions are: for µ+ν, −0.05 <
MM2 < 0.05 GeV2, corresponding to ±2σ; for π+νν,
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.
We consider backgrounds from two sources: one
from real D+s decays and the other from the back-
ground under the single-tag signal peaks. For the
latter, we estimate the background from data using
side-bands of the invariant mass, shown in Fig. 1.
For case (i) we find 3.5 (properly normalized) back-
ground events in the µ+ν region and 2.5 back-
grounds in the τ+ν region; for case (ii) we find 3
events. Our total background estimate summing
over all of these cases is 9.0±2.3 events.
The background from real D+s decays is evaluated
by identifying specific sources. For µ+ν the only
possible background is D+s → π
+π0. Using a 195
pb−1 subsample of our data, we limit the branching
fraction as < 1.1 × 10−3 at 90% C.L. [8]. This low
rate coupled with the extra γ veto yields a negligible
contribution. The real D+s backgrounds for π
are listed in Table II. Using the SM expected ratio of
decay rates we calculate a contribution of 7.4 π+νν
events.
TABLE II: Event backgrounds in the π+νν sample from
real D+s decays.
Source B(%) case (i) case (ii) Sum
D+s → Xµ
+ν 8.2 0+1.8
0 0+1.8
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
0.7±0.2 1.7+1.8
The event yield in the signal region, Ndet (92), is
related to the number of tags, Ntag, the branching
fractions, and the background Nbkgrd (3.5) as
Ndet −Nbkgrd = Ntag · ǫ[ǫ
′B(D+s → µ
+ν) (3)
+ǫ′′B(D+s → π
+νν)],
where ǫ (80.1%) includes the efficiencies (77.8%) for
reconstructing the single charged track including fi-
nal state radiation, (98.3)% for not having another
unmatched cluster in the event with energy greater
than 300 MeV, and the correction for the tag bias
(4.8%); ǫ′ (91.4%) is the product of the 99.0% µ+
calorimeter efficiency and the 92.3% acceptance of
the MM2 cut of |MM2| < 0.05 GeV2; ǫ′′ (7.6%) is
the fraction of π+νν events contained in the µ+ν sig-
nal window (13.2%) times the 60% acceptance for a
pion to deposit less than 300 MeV in the calorime-
ter. Using B(τ+ → π+ν) of (10.90±0.07)% [7], the
ratio of the π+νν to µ+ν widths is 1.059; we find:
B(D+s → µ
+ν) = (0.594± 0.066± 0.031)%. (4)
We can also sum the µ+ν and τ+ν contributions
for −0.05 < MM2 < 0.02 GeV2. Equation 3 still ap-
plies. The number of signal and background events
changes to 148 and 10.7, respectively. ǫ′ becomes
96.2%, and ǫ′′ increases to 45.2%. The effective
branching fraction, assuming lepton universality, is
Beff(D+s → µ
+ν) = (0.638± 0.059± 0.033)%. (5)
The systematic errors on these branching fractions
are dominated by the error on the number of tags
(5%). Other errors include: (a) track finding (0.7%),
determined from a detailed comparison of the sim-
ulation with double tag events where one track is
ignored; (b) the error due to the requirement that
the charged track deposit no more than 300 MeV
in the calorimeter (1%), determined using two-body
D0 → K−π+ decays [5]; (c) the γ veto efficiency
(1%), determined by extrapolating measurements on
fully reconstructed events. Systematic errors arising
from the background estimates are negligible. The
total systematic error for Eq. 4 is 5.2%, and is 5.1%
for Eq. 5 as (b) doesn’t apply here.
We also analyze the τ+ν final state independently.
For case (i) we define the signal region to be the in-
terval 0.05<MM2 <0.20 GeV2, while for case (ii)
-0.05<MM2 <0.20 GeV2. The upper limit on MM2
is chosen to avoid background from the tail of the
K0π+ peak. The fractions of the MM2 range ac-
cepted are 32% and 45% for case (i) and (ii), respec-
tively.
We find 31 [25] events in the signal region with a
background of 3.5 [5.1] events for case (i) [(ii)]. The
branching fraction, averaging the two cases is
B(D+s → τ
+ν) = (8.0± 1.3± 0.4)%, (6)
where the systematic error includes a contribution
of 0.06% from the uncertainty on B(τ+ → π+ν).
We measure 13.4 ± 2.6 ± 0.2 for the ratio of τ+ν
to µ+ν rates using Eq. 4. Here the systematic er-
ror is dominated by the uncertainty on the mini-
mum ionization cut. We also set an upper limit of
B(D+s → e
+ν) < 1.3 × 10−4 at 90% C.L. Both of
these results are consistent with SM predictions and
lepton universality.
We perform an overall check of our procedures by
measuring B(D+s → K
K+). We compute the MM2
(Eq. 2) using events with an additional charged track
identified as a kaon. These track candidates have
momenta of approximately 1 GeV/c; here the RICH
has a pion to kaon fake rate of 1.1% with a kaon
detection efficiency of 88.5% [12]. For this study,
we do not veto events with extra charged tracks, or
γ’s, because of the presence of the K0. We deter-
mine B(D+s → K
K+) = (2.90 ± 0.19 ± 0.18)%.
This method gives a result in good agreement with
preliminary CLEO-c results using double tags of
(3.00 ± 0.19 ± 0.10)% [15]; these results are not in-
dependent.
We also performed the entire analysis on a MC
sample that is 4 times larger than the data sam-
ple. The input branching fraction is 0.5% for µ+ν
and 6.57% for τ+ν, while our analysis measured
(0.514±0.027)% for the case (i) µ+ν signal and
(0.521±0.024)% for µ+ν and τ+ν combined.
Using B(D+s → µ
+ν) from Eq. 5, and Eq. 1 with
a Ds lifetime of (500±7)×10
−15 s [7], we extract
= 274± 13± 7 MeV. (7)
We combine with our previous result fD+ =
222.6± 16.7+2.8
−3.4 MeV [4], and find
/fD+ = 1.23± 0.11± 0.04. (8)
Lattice QCD predictions for f
and the ratio
/fD+ have been summarized by Onogi [16]. Our
measurements are consistent with most calculations;
examples are unquenched Lattice that predicts 249±
3± 16 MeV and 1.24± 0.01± 0.07 for the ratio [17],
while a recent quenched prediction gives 266± 10±
18 MeV and 1.13 ± 0.03 ± 0.05 [18]. There is no
evidence yet for any suppression in the ratio due to
the presence of a virtual charged Higgs [10].
The CLEO-c determination of f
is the most
accurate to date and consistent with other measure-
ments [7, 8]. It also does not rely on the indepen-
dent determination of any normalization mode (e.g.
φπ+). (We note that a preliminary CLEO-c result
using D+s → τ
+ν, τ+ → e+νν [19] is consistent with
these results.)
We gratefully acknowledge the effort of the CESR
staff in providing us with excellent luminosity and
running conditions. This work was supported by the
A.P. Sloan Foundation, the National Science Foun-
dation, the U.S. Department of Energy, and the Nat-
ural Sciences and Engineering Research Council of
Canada.
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