arXiv:1112.2884v1 [hep-ex] 13 Dec 2011
First Measurement of the Form Factors in the Decays
D0→ ρ−e+νeand D+→ ρ0e+νe
S. Dobbs,1Z. Metreveli,1K. K. Seth,1A. Tomaradze,1T. Xiao,1L. Martin,2
A. Powell,2G. Wilkinson,2H. Mendez,3J. Y. Ge,4G. S. Huang,4, ∗D. H. Miller,4
V. Pavlunin,4, †I. P. J. Shipsey,4B. Xin,4G. S. Adams,5D. Hu,5B. Moziak,5
J. Napolitano,5K. M. Ecklund,6J. Insler,7H. Muramatsu,7C. S. Park,7L. J. Pearson,7
E. H. Thorndike,7S. Ricciardi,8C. Thomas,2, 8M. Artuso,9S. Blusk,9R. Mountain,9
T. Skwarnicki,9S. Stone,9L. M. Zhang,9G. Bonvicini,10D. Cinabro,10A. Lincoln,10
M. J. Smith,10P. Zhou,10J. Zhu,10P. Naik,11J. Rademacker,11D. M. Asner,12, ‡
K. W. Edwards,12K. Randrianarivony,12G. Tatishvili,12, ‡R. A. Briere,13H. Vogel,13
P. U. E. Onyisi,14J. L. Rosner,14J. P. Alexander,15D. G. Cassel,15S. Das,15
R. Ehrlich,15L. Gibbons,15S. W. Gray,15D. L. Hartill,15B. K. Heltsley,15
D. L. Kreinick,15V. E. Kuznetsov,15J. R. Patterson,15D. Peterson,15D. Riley,15
A. Ryd,15A. J. Sadoff,15X. Shi,15W. M. Sun,15J. Yelton,16P. Rubin,17N. Lowrey,18
S. Mehrabyan,18M. Selen,18J. Wiss,18J. Libby,19M. Kornicer,20R. E. Mitchell,20
C. M. Tarbert,20D. Besson,21T. K. Pedlar,22D. Cronin-Hennessy,23and J. Hietala23
1Northwestern University, Evanston, Illinois 60208, USA
2University of Oxford, Oxford OX1 3RH, UK
3University of Puerto Rico, Mayaguez, Puerto Rico 00681
4Purdue University, West Lafayette, Indiana 47907, USA
5Rensselaer Polytechnic Institute, Troy, New York 12180, USA
6Rice University, Houston, Texas 77005, USA
7University of Rochester, Rochester, New York 14627, USA
8STFC Rutherford Appleton Laboratory,
Chilton, Didcot, Oxfordshire, OX11 0QX, UK
9Syracuse University, Syracuse, New York 13244, USA
10Wayne State University, Detroit, Michigan 48202, USA
11University of Bristol, Bristol BS8 1TL, UK
12Carleton University, Ottawa, Ontario, Canada K1S 5B6
13Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
14University of Chicago, Chicago, Illinois 60637, USA
15Cornell University, Ithaca, New York 14853, USA
16University of Florida, Gainesville, Florida 32611, USA
17George Mason University, Fairfax, Virginia 22030, USA
18University of Illinois, Urbana-Champaign, Illinois 61801, USA
19Indian Institute of Technology Madras,
Chennai, Tamil Nadu 600036, India
20Indiana University, Bloomington, Indiana 47405, USA
21University of Kansas, Lawrence, Kansas 66045, USA
22Luther College, Decorah, Iowa 52101, USA
23University of Minnesota, Minneapolis, Minnesota 55455, USA
(Dated: December 13, 2011)
Using the entire CLEO-c ψ(3770) → D¯D event sample, corresponding to an integrated luminosity
of 818 pb−1and approximately 5.4 × 106D¯D events, we measure the form factors for the decays
D0→ ρ−e+νe and D+→ ρ0e+νe for the first time and the branching fractions with improved
precision. A four-dimensional unbinned maximum likelihood fit determines the form factor ratios
to be: V (0)/A1(0) = 1.48 ± 0.15 ± 0.05 and A2(0)/A1(0) = 0.83 ± 0.11 ± 0.04. Assuming CKM
unitarity, the known D meson lifetimes and our measured branching fractions we obtain the form
factor normalizations A1(0), A2(0), and V (0). We also present a measurement of the branching
fraction for D+→ ωe+νewith improved precision.
The transition rate of charm semileptonic decays depends on the weak quark mixing
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |Vcs| and |Vcd| , and strong inter-
action effects binding quarks into hadrons parameterized by form factors.
In the decays D → ρe+νe, in the limit of negligible lepton mass, the hadronic current
is described by three dominant form factors: two axial and one vector, A1, A2, and V ,
respectively, which are functions of q2, the invariant mass of the lepton-neutrino system.
They are not amenable to unquenched LQCD calculations due to the large total decay
width of the ρ meson, but model predictions exist [2, 3]. No experimental information on
these form factors exists.
The helicity amplitudes for the rare decays B → V ℓ+ℓ−are related at leading order
in ΛQCD/mb to pseudoscalar-to-vector semileptonic transitions . Exploiting one of the
proposed double-ratio techniques , D → ρe+νeform factors, when combined with those
of D → K∗e+νeand B → V ℓ+ℓ−, can be used to extract |Vub| from B → ρe+νe.
The differential decay rate of D → ρe+νe can be expressed in terms of three helicity
amplitudes (H+(q2), H−(q2), and H0(q2)) :
+(1 − cosθe)2sin2θπ|H−(q2,mππ)|2+ 4sin2θecos2θπ|H0(q2,mππ)|2
+4sinθe(1 + cosθe)sinθπcosθπcosχH+(q2,mππ)H0(q2,mππ)
−4sinθe(1 − cosθe)sinθπcosθπcosχH−(q2,mππ)H0(q2,mππ)
where GFis the Fermi constant, pρis the momentum of the ρ in the D rest frame, B(ρ → ππ)
is a branching fraction, θπ is the angle between the π and the D direction in the ρ rest
frame, θeis the angle between the e+and the D direction in the e+νerest frame, χ is the
acoplanarity angle between the π+π−and e+νedecay planes, mππis the invariant mass of
the two pions, and BW(mππ) is the Breit-Wigner function that describes the ρ line shape.
Following Ref. , we use the relativistic form
B(ρ → ππ)|BW(mππ)|2?
(1 + cosθe)2sin2θπ|H+(q2,mππ)|2
where m0 and Γ0 are the mass and width of the ρ meson , p is the momentum of
the pion in the ππ rest frame, p0 is equal to p when mππ = m0, and B(p) is a Blatt-
Weisskopf form factor given by B(p) = 1/(1 + R2p2)1/2, with R = 3 GeV−1, and Γ(mππ) =
(p/p0)3(m0/mππ)Γ0[B(p)/B(p0)]2. The interference term between a possible s-wave ππ com-
ponent and the ρ amplitude has not been included in Eq. (1). Its absence is treated as a
source of systematic uncertainty on the measurement.
The helicity amplitudes are related to the form factors
H±(q2) = MA1(q2) ∓ 2MDpρ
where MDis the mass of the D meson and M = MD+ mππ. Since A1(q2) is common to all
three helicity amplitudes, it is natural to define two form factor ratios as
rV =V (0)
We assume a simple pole form  for A1(q2), A2(q2), and V (q2), where the pole mass is
MD∗(1−)= 2.01 GeV/c2and MD∗(1+)= 2.42 GeV/c2 for the vector and axial form factors,
respectively. We have also explored a double-pole parametrization .
We report herein the first measurement of the form factor ratios and absolute form factor
normalization in D → ρe+νe, and improved branching fraction measurements for these
decays and D+→ ωe+νe. (Throughout this Letter charge-conjugate modes are implied.)
These decays were studied previously using a smaller CLEO-c data sample . The data
sample used here consists of an integrated luminosity of 818 pb−1at the ψ(3770) resonance,
and includes about 3.0 × 106D0¯D0and 2.4 × 106D+D−events. The CLEO-c detector is
described in detail elsewhere .
The analysis technique was employed in previous CLEO-c studies [10, 12]. The presence of
two D mesons in a D¯D event allows a tag sample to be defined in which a¯D is reconstructed
in a hadronic decay mode. A sub-sample is then formed in which a positron and a set of
hadrons, as a signature of a semileptonic decay, are required in addition to the tag. The
semileptonic decay branching fraction BSLis given by
where Ntagand ǫtagare the yield and reconstruction efficiency, respectively, for the hadronic
tag, Ntag,SLand ǫtag,SLare those for the combined semileptonic decay and hadronic tag, and
ǫ = ǫtag,SL/ǫtagis the effective signal efficiency.
Candidate events are selected by reconstructing a¯D0or D−tag in the following hadronic
final states: K+π−, K+π−π0, and K+π−π−π+for neutral tags, and K0
lected based on two variables: ∆E ≡ ED− Ebeam, the difference between the energy of
the D−tag candidate ED and the beam energy Ebeam, and the beam-constrained mass
Mbc ≡ (E2
date. Selection criteria for tracks, π0, and K0
are described in Ref. . If multiple candidates are present in the same tag mode, one
candidate per tag charge with the smallest |∆E| is chosen. The yield of each tag mode is
obtained from fits to the Mbcdistributions . The data sample comprises 661232±879
and 481927±810 reconstructed neutral and charged tags, respectively.
After a tag is identified, we search for an e+and a ρ−(π−π0mode), ρ0(π+π−mode), or
ω (π+π−π0mode) recoiling against the tag following Ref. . A ρ → ππ candidate satisfies
|mππ− m0| < 150 MeV/c2. The combined tag and semileptonic candidates must account
for all tracks in the event. Semileptonic decays are identified with U ≡ Emiss− c|pmiss|,
where Emissand pmissare the missing energy and momentum of the D+meson. If the decay
products have been correctly identified, U is expected to be zero, since only a neutrino is
undetected. The resolution in U is improved by constraining the magnitude and direction of
the D+momentum to be pD+ = (E2
Due to the finite resolution of the detector, the distribution in U is approximately Gaussian,
with resolution ∼17 MeV for D0→ ρ−e+νeand D+→ ωe+νeand ∼8 MeV for D+→ ρ0e+νe.
Sπ−π0, K+π−π−π0, K0
Sπ−π−π+, and K−K+π−for charged tags. Tagged events are se-
beam/c4− |pD|2/c2)1/2, where pD is the measured momentum of the D−candi-
Scandidates used in the reconstruction of tags
D)1/2, and ? pD+ = −? pD− , respectively.
TABLE I. Signal efficiencies, yields, and branching fractions (BSL) for D0→ ρ−e+νe, D+→
ρ0e+νe, and D+→ ωe+νe, from this work, our previous (prev) measurements , and two model
predictions: ISGW2  and FK . All BSLare in units of 10−3. The uncertainties for ǫ and Ntag, SL
are statistical, while the uncertainties for branching fractions are statistical and systematic in that
order. The efficiencies include the ρ and ω decay branching fractions from the PDG .
ǫ (%)Ntag, SL
26.03 ± 0.02 304.6 ± 20.9
42.84 ± 0.03 447.4 ± 24.5
14.67 ± 0.03 128.5 ± 12.6
1.77 ± 0.12 ± 0.10
2.17 ± 0.12+0.12
1.82 ± 0.18 ± 0.07
1.94 ± 0.39 ± 0.13
2.1 ± 0.4 ± 0.1
To remove multiple candidates in each semileptonic mode one combination is chosen per tag
mode per tag charge, based on the proximity of the invariant masses of the ρ0, ρ+, or ω
candidates to their expected masses.
The U and invariant mass distributions for D0→ ρ−e+νe, D+→ ρ0e+νe, and D+→
ωe+νewith all tag modes combined are shown in Fig. 1. The yield for each of the three
modes is determined from a binned likelihood fit to the U distribution where the signal is
described by a modified Crystal Ball function with two power-law tails  which account
for initial- and final-state radiation (FSR) and mismeasured tracks. The signal parameters
are fixed with a GEANT-based Monte Carlo (MC) simulation  in fits to the data. The
background functions are determined by MC simulation that incorporates all available data
on D meson decays, which we refer to as “generic MC”. For D0→ ρ−e+νe, the backgrounds
arise mostly from D0→ K∗−e+νe, peaking at positive U and modeled with a Gaussian,
and events with misidentified tags, which are accounted for in the fit by a fourth order
polynomial. The backgrounds to D+→ ρ0e+νe has its largest contribution from D+→
¯K∗0e+ν,¯K∗0→ K−π+, with the peak at higher U due to charged kaons misidentified as
charged pions, and the peak at lower U from either decay-in-flight kaons or interactions with
detector material. We categorize the background components according to their shape in
U and parameterize the overall background shape using combinations of polynomials and
Gaussian functions. The background shape parameters are fixed in fits to the data, while the
background normalizations are allowed to float. The signal shapes for the invariant mass
distributions of the hadronic system are modeled with a Breit-Wigner function, and the
background shapes are modeled with generic MC. The peaking background for D+→ ρ0e+νe
arises from D+→ ωe+νe, ω → π+π−. Due to the tag, backgrounds from the non-D¯D
processes e+e−→ q¯ q, where q is a u, d, or s quark, e+e−→ τ+τ−, and e+e−→ ψ(2S)γ, are
negligible . The signal yields Ntag,SLare given in Table I.
The second row of Fig. 1 shows the mπ−π0, mπ+π−, and mπ+π−π0 distributions with |U| <
60 MeV for the three signal modes, respectively. The peaking background at mπ−π0 ∼
0.49 GeV/c2arises from D0→ K−e+ν with K−→ π−π0. The small background peak at
mπ+π− ∼ 0.78 GeV/c2is due to D+→ ωe+νewith ω → π+π−.
The absolute branching fractions in Table I are obtained using Eq. (6).
efficiencies ǫ are determined by MC simulation, and have been weighted by the tag yields in
The systematic uncertainties for the branching fractions of D0→ ρ−e+νe and D+→
ρ0e+νe are dominated by uncertainties in the line shape of the ρ (5.0%), and the non-
resonant background (−1.5% for D0→ ρ−e+νe and −8.4% for D+→ ρ0e+νe). The un-
FIG. 1. Fits to the U and hadron invariant mass distributions in data (filled circles with error bars)
for (a) and (d) D0→ ρ−e+νe, ρ−→ π−π0; (b) and (e) D+→ ρ0e+νe, ρ0→ π+π−; and (c) and
(f) D+→ ωe+νe, ω → π+π−π0. The solid line represents the fit of the sum of the signal function
and background function to the data. The dashed line indicates the background contribution. The
arrows indicate a ±48 MeV region around the K0
Smass, which has been removed for display.
certainty due to the line shape of the ρ is estimated by (1) requiring |U| < 60 MeV and
fitting the mππdistribution, (2) varying the selection criterion |mππ−m0| < 150 MeV. The
uncertainty due to the non-resonant background is obtained by performing a form factor
fit, with an additional interfering non-resonant D → ππe+νe(s-wave) component modeled
following Ref. , then integrating over the kinematic variables to recalculate the branching
fractions. The unknown form factors in D+→ ωe+νeare the dominant uncertainty in its
branching fraction (3.0%). The remaining systematic uncertainties include the track and π0
finding efficiencies, positron and charged hadron identification, the number of tags, the no-
additional-track requirement, the shape of the signal and background functions, and the MC
FSR and form factor modeling. These estimates are added in quadrature to obtain the total
systematic uncertainties on the branching fractions:
D+→ ρ0e+νe, and D+→ ωe+νe, respectively.
A form factor analysis is performed for D → ρe+νe. We calculate the energy and mo-
mentum of the neutrino using Eν= Emissand |pν| = Emiss, because Emissis better measured
measured with resolutions of (0.021 GeV2/c4, 0.020, 0.048, 0.024) for D0→ ρ−e+νe, and
(0.013 GeV2/c4, 0.013, 0.037, 0.019) for D+→ ρ0e+νe.
A four-dimensional maximum likelihood fit in a manner similar to Ref.  is performed
in the space of q2, cosθπ, cosθe, and χ. The technique makes possible a multidimensional
fit to variables modified by experimental acceptance and resolution taking into account cor-
relations among the variables. The signal probability density function for the likelihood
−10.0%, 4.1%, for D0→ ρ−e+νe,
Without ambiguity, the four kinematic variables (q2, cosθπ, cosθe χ) are
FIG. 2. Projections of the combined ρ−and ρ0data (points with statistical error bars) and the fit
(solid histogram) onto q2, cosθe, cosθπ, and χ. The dashed lines show the sum of the background
function is estimated at each data point using signal MC events by sampling the MC distri-
bution at the reconstructed level in a search volume around the data point, then weighting
by the ratio of the decay distribution for the trial values of rV and r2to that of the gener-
ated distribution. The search volumes are one tenth the full kinematic range of each of the
four dimensions. Large MC samples are generated to ensure that each search volume has
sufficient statistics. The background probability density function is modeled using events
from the generic MC. Due to the low statistics of the background in the generic MC, we
reduce the four dimensional space to lower dimensional subspaces. Due to the correlation
between q2and cosθe, the two subspaces are chosen to be (q2, cosθe) and (cosθπ, χ). The
background normalization is fixed in the fits to the values measured in the determination of
the branching fractions.
Using the above method, a simultaneous fit is made to the isospin-conjugate modes
D0→ ρ−e+νeand D+→ ρ0e+νe. We find rV = 1.48 ± 0.15 and r2= 0.83 ± 0.11, with a
correlation coefficient ρV 2= −0.18. The confidence level of the fit is determined to be 5.0%
by comparing the negative log-likelihood from the data to the distribution from toy MC fits.
Fig. 2 shows the q2, cosθe, cosθπ, and χ projections for the combined ρ−and ρ0data and
the fit. We also make fits to the two modes separately. The results are consistent. We note
that the difference between the data and the fit projection for cosθπmight be due to s-wave
We have considered the following sources of systematic uncertainty in the form factor
measurement. Our estimate of their magnitude are given in parentheses for rV and r2,
respectively. The uncertainty associated with background modeling (0.01, 0.02) is estimated
by changing the normalization of the three largest background components by a factor of
two in each semileptonic mode. The uncertainty due to imperfect knowledge of the ρ line
shape (0.01, 0.02) is estimated by modifying the ρ line shape by increasing and decreasing
the population of signal MC events below and above the nominal ρ mass  by 20%. The
uncertainty due to non-resonant background (0.01, 0.02) is obtained by repeating the fit with
an additional interfering non-resonant D → ππe+νecomponent (s-wave) following Ref. .
The procedure for extracting the form factor parameters is tested using the generic MC
sample, from which events are drawn randomly to form mock data samples, each equivalent
in size to the data sample. When backgrounds are absent, the measured form factor ratios
are consistent with the input values. In the presence of background, a small statistically
significant shift is observed. Its magnitude is taken as the uncertainty due to possible bias
in the form factor fitter (0.03, 0.02). The uncertainty associated with the unknown q2
dependence of the form factors (0.03, 0.02) is estimated by introducing a second pole .
Adding all sources of systematic uncertainty in quadrature, the final result is rV = 1.48±
0.15 ± 0.05 and r2 = 0.83 ± 0.11 ± 0.04. Using |Vcd| = 0.2252 ± 0.0007 obtained using
CKM unitarity constraints  and the lifetimes τD0 = (410.1 ± 1.5) × 10−15s and τD+ =
(1040 ± 7) × 10−15s , we combine our form factor ratio and branching fraction results to
obtain A1(0) = 0.56 ± 0.01+0.02
−0.03, A2(0) = 0.47 ± 0.06 ± 0.04, and V (0) = 0.84 ± 0.09+0.05
Our branching fraction results are compared to previous measurements , with which
they are consistent, and theoretical predictions in Table I. The results are consistent with
be exact due to ρ0− ω interference . Theoretical predictions from the ISGW2 model 
and a model (FK) which combines heavy-quark symmetry and properties of the chiral La-
grangian , are also listed in Table I. The branching fractions for ISGW2 are obtained
by combining the partial rates in Ref.  with |Vcd| and τDfrom PDG . Our branching
fraction results are more consistent with the FK predictions than ISGW2.
2Γ(D+→ρ0e+νe)= 1.03 ± 0.09+0.08
−0.02. Isospin symmetry is not expected to
The FK model predicts A1(0) = 0.61, A2(0) = 0.31, and V (0) = 1.05. These values are
compatible with our form factor measurements. No other experimental form factor results
on these decays exist. Our values of rV and r2are very similar to the current PDG average
of D+→¯K∗0e+ν form factor ratios rV = 1.62 ± 0.08 and r2= 0.83 ± 0.05 .
In summary, we have made the first measurement of the form factor ratios and absolute
form factor normalization in D → ρe+νe, and improved branching fraction measurements
for these decays and D+→ ωe+νe. Our branching fractions are consistent with our previous
measurements but with improved precision. The form factor measurement in D → ρe+νeis
the first in a semileptonic Cabibbo-suppressed pseudoscalar-to-vector transition.
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 Foundation, the U.S. Department of Energy, the Natural Sciences and
Engineering Research Council of Canada, and the U.K. Science and Technology Facilities
∗Now at: University of Science and Technology of China, Hefei 230026, People’s Republic of Download full-text
†Now at: University of California, Santa Barbara, CA 93106, USA
‡Now at: Pacific Northwest National Laboratory, Richland, WA 99352
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