# First Measurement of the Form Factors in the Decays D0 to rho- e+ nu_e and D+ to rho0 e+ nu_e

**ABSTRACT** Using the entire CLEO-c psi(3770) to DDbar event sample, corresponding to an

integrated luminosity of 818 pb^-1 and approximately 5.4 x 10^6 DDbar events,

we measure the form factors for the decays D0 to rho- e+ nu_e and D+ to rho0 e+

nu_e 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)/A_1(0) = 1.48 +- 0.15 +- 0.05 and A_2(0)/A_1(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 A_1(0),

A_2(0), and V(0). We also present a measurement of the branching fraction for

D^+ to omega e^+ nu_e with improved precision.

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arXiv:1112.2884v1 [hep-ex] 13 Dec 2011

CLNS 11/2075

CLEO 11-3

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

(CLEO Collaboration)

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)

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Abstract

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.

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The transition rate of charm semileptonic decays depends on the weak quark mixing

Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |Vcs| and |Vcd| [1], 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 [4]. Exploiting one of the

proposed double-ratio techniques [5], 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)) [6]:

dΓ

dq2dcosθπdcosθedχdmππ

3

8(4π)4G2

=

F|Vcd|2pρq2

+(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)

−2sin2θesin2θπcos2χH+(q2,mππ)H−(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. [7], we use the relativistic form

√m0Γ0(p/p0)

m2

M2

D

B(ρ → ππ)|BW(mππ)|2?

(1 + cosθe)2sin2θπ|H+(q2,mππ)|2

?

,(1)

BW(mππ) =

0− m2

ππ− im0Γ(mππ)

B(p)

B(p0), (2)

where m0 and Γ0 are the mass and width of the ρ meson [8], 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

H0(q2) =

2mππ

−4M2

M

M

V (q2), (3)

1

√q2

Dp2

?

(M2

D− m2

?

,

ππ− q2)MA1(q2)

ρ

A2(q2)(4)

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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)

A1(0)and r2=A2(0)

A1(0). (5)

We assume a simple pole form [9] 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[8] for the vector and axial form factors,

respectively. We have also explored a double-pole parametrization [3].

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 [10]. 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 [11].

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

BSL=Ntag,SL

Ntag

ǫtag,SL

ǫtag

=Ntag,SL/ǫ

Ntag

, (6)

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

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. [13]. 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 [13]. 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. [13]. 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−, K+π−π−,

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

beam/c2− c2m2

D)1/2, and ? pD+ = −? pD− [10], respectively.

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Page 5

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 [10], and two model

predictions: ISGW2 [2] and FK [3]. 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 [8].

Decay Mode

D0→ ρ−e+νe

D+→ ρ0e+νe

D+→ ωe+νe

ǫ (%)Ntag, SL

BSL

BSL(prev)

BSL(ISGW2) BSL(FK)

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

1.6+0.7

−0.6± 0.1

1.0

1.3

1.3

2.0

2.5

2.5

−0.22

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 [14] 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 [15] 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 [12]. 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−[16].

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 data.

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-

The signal

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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. [17], 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

than |pmiss|.

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. [18] 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

+5.7

−5.9%,

+5.5

−10.0%, 4.1%, for D0→ ρ−e+νe,

Without ambiguity, the four kinematic variables (q2, cosθπ, cosθe χ) are

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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

distributions.

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

interference.

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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 [8] 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. [17].

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 [3].

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 [8] and the lifetimes τD0 = (410.1 ± 1.5) × 10−15s and τD+ =

(1040 ± 7) × 10−15s [8], 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

−0.06.

Our branching fraction results are compared to previous measurements [10], with which

they are consistent, and theoretical predictions in Table I. The results are consistent with

isospin invariance:

be exact due to ρ0− ω interference [16]. Theoretical predictions from the ISGW2 model [2]

and a model (FK) which combines heavy-quark symmetry and properties of the chiral La-

grangian [3], are also listed in Table I. The branching fractions for ISGW2 are obtained

by combining the partial rates in Ref. [2] with |Vcd| and τDfrom PDG [8]. Our branching

fraction results are more consistent with the FK predictions than ISGW2.

Γ(D0→ρ−e+νe)

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 [8].

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

Council.

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Page 9

∗Now at: University of Science and Technology of China, Hefei 230026, People’s Republic of

China

†Now at: University of California, Santa Barbara, CA 93106, USA

‡Now at: Pacific Northwest National Laboratory, Richland, WA 99352

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