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Eur. Phys. J. C (2014) 74:2914

DOI 10.1140/epjc/s10052-014-2914-3

Regular Article - Experimental Physics

Constraints on exclusive branching fractions B

→ X

ν)

from moment measurements in inclusive B → X

lν decays

Florian U. Bernlochner

, Dustin Biedermann

2,a

, Heiko Lacker

, Thomas Lück

University of Victoria, Victoria, British Columbia, V8W 3P, Canada

Institut für Physik, Humboldt Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany

Received: 5 March 2014 / Accepted: 19 May 2014 / Published online: 24 June 2014

Abstract As an alternative to direct measurements, we

extract the branching fractions B

→ X

ν)

with

= D, D

∗

, D



, D



, D

∗

and non-resonant

ﬁnal states (D

(∗)

π)

, from a ﬁt to electron energy, hadronic

mass and combined hadronic mass–energy moments mea-

sured in inclusive B → X

lν decays. The ﬁt is per-

formed by constraining the sum of exclusive branching

fractions to the measured B(B

→ X

ν) value, and

with different sets of additional constraints for the directly

measured branching fractions. There is no ﬁt scenario in

which a single branching fraction can close the gap between

B(B

→ X

ν) and the sum of known branching frac-

tions B

→ X

ν). The ﬁtted B(B

→ D

∗0

ν)

is found to be signiﬁcantly larger than its direct measure-

ment. B(B

→ D

ν) is in good agreement with the

direct measurement; when B(B

→ D

∗0

ν) is constrained

the ﬁtted B(B

→ D

ν) increases. Within large uncer-

tainties, B(B

→ D

0

ν) agrees with direct measure-

ments. Depending on the ﬁt scenario, B(B

→ D

ν) is

consistent with or larger than its direct measurement. The

ﬁt is not able to easily disentangle B

→ D

ν and

→ D

ν, and tends to increase the sum of these two

branching fractions. B(B

→ (D

(∗)

π)

ν) with non-

resonant (D

(∗)

π)

ﬁnal states is found to be of the order

0.3 %. No indication is found for signiﬁcant contributions

from so far unmeasured B

→ D

(∗)0

ν decays.

1 Introduction

The Cabibbo–Kobayashi–Maskawa (CKM) quark mixing

matrix [1] governs the weak coupling strength between up-

Charge conjugation is always implied.

e-mail: biedermd@physik.hu-berlin.de

and down-type quarks. The CKM-matrix element |V

| can

be extracted from semileptonic B-meson decays B → X

lν

with hadronic ﬁnal states X

containing mesons with charm

whereby inclusive or exclusive ﬁnal states can be used. For

analyses of exclusive ﬁnal states, such as B → D

(∗)

lν

decays, good knowledge about the overall composition of the

ﬁnal states is crucial. Precise understanding of semilep-

tonic B → X

lν decays is also of utmost importance for the

precision determination of |V

| from B → X

lν decays,

since B → X

lν decays represent the main source of back-

ground events in this kind of analyses.

Precise knowledge of B → X

lν decays has also re-

levance for new physics searches. A B

ABAR measurement of

B(B→D

(∗)

−

ν)

B(B→D

(∗)

−

ν)

, where l ∈{e,μ}, exceeds the Standard Model

expectations by 3.4 σ [2]. In this analysis, an important sys-

tematic uncertainty originates from the detailed knowledge of

the composition of B → D

∗∗

lν decays, where D

∗∗

denotes

the four 1P states of non-strange charmed mesons. In particu-

lar, D

∗∗

decays to D

(∗)

ππ states are seen to have a large

impact on the measured ratio.

1.1 Branching fraction measurements

Great efforts have been made in measuring the inclu-

sive and exclusive branching fractions of B → X

lν

transitions. Exclusive branching fractions B(B → X

lν)

have been determined for the hadronic ﬁnal states X

∈

{D, D

∗

, D



, D

(∗)

π}. Averages for these mea-

surements and for the inclusive branching fraction B(B →

lν) are provided by the Heavy Flavor Averaging Group

(HFAG) [3], which we use in this paper. In Table 1, we quote

all branching fractions for semileptonic decays of charged B-

mesons, for which measurements exist and which are used

in our analysis. Thereby, we assume that the corresponding

branching fractions for semileptonic decays of neutral B-

mesons can be obtained by applying isospin invariance of

123

2914 Page 2 of 36 Eur. Phys. J. C (2014) 74:2914

Table 1 Semileptonic branching fractions

B(B

→ X

(i)

ν) taken or

calculated from HFAG averages [3] as described in the text

Decay Branching fraction (%)

→ D

ν 2.30 ± 0.10

→ D

∗0

ν 5.34 ± 0.12

→ D

ν 0.652 ± 0.071

→ D

ν 0.284 ± 0.032

→ D

0

ν 0.195 ± 0.060

→ D

ν 0.435 ± 0.075

→ (D

(∗)

π)

ν 0.17 ± 0.14

→ X

ν 10.90 ± 0.14

strong interactions. That is, the decay rates for semileptonic

decays of charged and neutral B-mesons are set to be equal.

The B(B

→ D

ν)and B(B

→ D

∗0

ν)values quoted

in Table 1 are calculated from the HFAG (isospin) averages

provided for B(B

→ D

(∗)−

ν) [3]as

B(B

→ D

(∗)0

ν) = τ

B(B

→ D

(∗)−

ν), (1.1)

with

= 1.079 ± 0.007 (1.2)

being the ratio of lifetimes τ

and τ

of charged and neutral

B-mesons, respectively [3]. Since for B(B → X

lν) quoted

in Ref. [3] charged and neutral B-meson decays were used,

we calculate B(B

→ X

ν) according to

B(B

→ X

ν) = τ

+ 1

1 + f

B(B → X

lν), (1.3)

with

B(ϒ(4S) → B

−

)

B(ϒ(4S) → B

)

= 1.065 ± 0.026 (1.4)

being the measured ratio of ϒ(4S) branching fractions into

charged and neutral B-meson pairs as quoted in Ref. [3].

In case of X

being one of the D

∗∗

mesons D

, D



,orD

, only product branching-fractions B(B

→

∗∗

(∗)−

ν) = B(B

→ D

∗∗

ν) × B(D

∗∗

→

(∗)−

) are available [3]. In these cases, we have to cor-

rect for the branching fraction B(D

∗∗

→ D

(∗)−

) to obtain

B(B

→ D

∗∗

ν). To do so we assume strong-isospin sym-

metry. As a result, in case of a D

∗∗

two-body decay, we

account for the decay mode of the D

∗∗

in which the created

u quark is interchanged by a d quark by introducing a mul-

tiplicative factor of

. Furthermore, we make the following

assumptions: the D

can only decay into Dπ ,theD



only

into D

∗

π,theD

-meson only into D

∗

π and Dππ, and the

-meson into Dπ and D

∗

π.

In cases of D

decays we use the average of measurements

for the ratio (see Appendix B)

B(B

→ D

(Dππ)π

)

B(B

→ D

∗

π)π

)

= 0.53 ± 0.14, (1.5)

relying on Refs. [4–7], and obtain B(B

→ D

ν) accord-

ing to

B(B

→ D

ν) =



1 +

B(B

→ D

(Dππ)π

)

B(B

→ D

∗

π)π

)



×B(B

→ D

∗

π)l

ν). (1.6)

For the D

-meson we use the measured ratio [8]

B(D

→ D

−

)

B(D

→ D

∗+

−

)

= 1.56 ± 0.16, (1.7)

and calculate B(B

→ D

ν) in an analogous way as

B(B

→ D

ν).

Measurements for B

→ D

(∗)

πl

ν have been performed

as well. Using the HFAG averages of these branching frac-

tions [3] together with the B(B

→ D

∗∗

(∗)

π)l

ν) aver-

ages [3] we determine the branching fraction for semilep-

tonic decays into non-resonant (nr) ﬁnal states (D

(∗)

π)

For these cases, we calculate the isospin average according

to Eq. 9.1 as described in Appendix C.

The values for B(B

→ X

ν), with X

being D

, D



, D

, and (D

(∗)

π)

, obtained in this way, are quoted in

Table 1.

1.2 Puzzles and possible solutions

Some serious problems arise from the quoted branching frac-

tions:

– The most obvious puzzle and in the following denoted as

“gap problem” results from the fact that the sum of the

directly measured exclusive branching fractions does not

saturate the measured inclusive branching fraction, i.e.

B(B

→ X

ν) = (10.90 ± 0.14) %

=



i=D,D

∗

∗∗

→ X

ν) = (9.2 ± 0.2) %.

(1.8)

Even if the branching fraction for decays into non-

resonant (D

(∗)

π)

, B(B

→ (D

(∗)

π)

ν), is taken

into consideration as well, the gap can not be closed.

123

Eur. Phys. J . C (2014) 74:2914 Page 3 of 36 2914

– A more subtle problem and commonly referred to as the

“

vs.

puzzle” [9] concerns the sector of B → D

∗∗

lν

decays. Theoretical deliberations [9–11] suggest that the

branching fraction of B → D

lν decays should

be about one order of magnitude larger than B(B →



lν). The measured values are in clear contradic-

tion to this expectation.

It should be noted though that a quark-model based cal-

culation essentially agrees with the measured values of

all B → D

∗∗

lν transitions [12]. If correct this would be

in contrast to the stated “

vs.

puzzle” [9–11].

– Furthermore, the branching fraction of B

→ D

0

decays which is given in Table 1 is the result of a weighted

average of three measurements from DELPHI [13], Belle

[14] and B

ABAR [15]:

– B (B

→ D

0

∗−

ν)

= (0.74 ± 0.17 ± 0.18) % (DELPHI),

– B (B

→ D

0

∗−

ν)

= (−0.03 ± 0.06 ± 0.07) % (Belle),

– B (B

→ D

0

∗−

ν)

= (0.27 ± 0.04 ± 0.04) %(B

ABAR).

When averaging these three measurements one obtains

a χ

over degrees of freedom (dof )ofχ

/dof =

corresponding to a conﬁdence level of 0.1 %. Possibly,

at least one measurement underestimates the uncertainty,

thus the average might be biased and the uncertainty on

the weighted average might be underestimated.

Possible experimental issues that might be the source for

these puzzles are:

– Exclusive decay channels B → X

lν into ﬁnal states

not measured yet could contribute signiﬁcantly to the

inclusive semileptonic decay rate. Such transitions could

be for example B → D

(∗)

lν, where D

(∗)

might be the

recently discovered resonances D(2550) and D

∗

(2600)

[16]. In Ref. [17] a rough estimation suggested that a

combined branching fraction of about 1 % could be real-

ized in nature for B → D

(∗)

lν whereas in Ref. [18]it

was argued that such a large branching fraction would be

difﬁcult to understand theoretically.

– It is possible that not all D

∗∗

decay channels were incor-

porated when determining B(B → D

lν) from

the measured product branching fractions. For example,

there might be D

→ D

∗

ππ, D

→ D

∗

ππ (upper

limits are given in Ref. [4]), D

→ D



π and

→ Dη decays with sizeable branching fractions [19].

If true this would relax both the “gap problem” and the

“

vs.

puzzle” at the same time.

– Another possibility would be that the branching fraction

of B → D

(∗)

lν decays is experimentally underestimated,

which would ease the “gap problem” but not the “

vs.

puzzle”. However, B(B → D

(∗)

lν) is measured with

high precision. As a consequence, one would need to

enlarge these branching fractions signiﬁcantly more than

it is allowed by the quoted uncertainty in order to relax

the “gap problem”.

One possible effect that could lead to a biased estimate

of the B → D

∗

lν branching fraction is an overesti-

mate of the reconstruction efﬁciency of t he low-energy

pion appearing in the D

∗

decay to a D and a π.It

should be noted though that the experimental measure-

ment that has a very strong weight in the average extracts

B(B → D

∗

lν) from a global ﬁt to kinematical distribu-

tions without relying on the reconstruction of the low-

energy pion from the D

∗

decay [20].

If there is no experimental problem with the reconstruc-

tion of the low-energy pions or other issues relevant

to the analyses, another explanation of underestimated

B → D

(∗)

lν branching fractions could be overestimated

D and/or D

∗

branching fractions. However, D-meson

branching fractions are very well determined by experi-

ments running on the ψ(3770) resonance, such as CLEO-

c or BES-III. Since the ψ(3770) decays into D

D, abso-

lute branching-fraction measurements are possible by

tagging one D-meson and measuring the decay of the

other one into a speciﬁc ﬁnal state.

For the D

∗

-meson, possible electromagnetic decays not

measured yet are D

∗

→ De

−

and D

∗

→ Dγγ. These

decays would have to compete at least with D

∗

→ Dγ

in order to have a sizeable effect on B(B → D

∗

lν).This

would come as a real surprise since one would expect

a rate suppression of these decays of the order the ﬁne-

structure constant α ≈ 1/137 with respect to D

∗

→ Dγ .

– Reconstructing B → D



lν and B → D

lν with



→ D

∗

π is not an easy experimental task as the

and the D



are very broad resonances and therefore

hard to distinguish from non-resonant (D

(∗)

π)

ﬁnal

states. Therefore, the correct values for B(B → D



lν)

and B(B → D

lν) could be indeed smaller than the

HFAG averages, which would relax the “

vs.

puzzle”,

but not the “gap problem”.

– Non-resonant decays B → (D

(∗)

π)

lν could ﬁll the

gap. If this is true, this would suggest a serious prob-

lem in t he B → D

(∗)

πlν and/or B → D

∗∗

(∗)

π)lν

analysis since the B → D

(∗)

πlν together with the

B → D

∗∗

(∗)

π)lν results leave only a small space

for B → (D

(∗)

π)

lν decays. In addition, theoretical

expectations do not support a large branching fraction

for non-resonant B → (D

(∗)

π)

lν decays [17].

– There might be contributions from yet to be discovered

B → (D

(∗)

ππ)

lν or B → (D

(∗)

η)

lν decays, which

would ease the “gap problem”. Such decays have not

123

2914 Page 4 of 36 Eur. Phys. J. C (2014) 74:2914

been observed yet and we did not investigate them in our

analysis since our general ﬁndings do not prefer large

contributions from high-mass states like B → D

(∗)

lν

or from non-resonant decays B → (D

(∗)

π)

lν so that

we don’t expect signiﬁcant contributions from B →

(∗)

ππ)

lν or B → (D

(∗)

η)

lν decays either. More-

over, by adding too many free parameters to the problem

our analysis would loose in sensitivity.

Kinematical distributions of the lepton energy E

and the

hadronic invariant mass m

measured in inclusive B →

lν decays are sensitive to the composition of exclusive

ﬁnal states containing mesons with charm. Usually, moments

of these kinematical distributions are used to extract non-

perturbative parameters of a Heavy Quark Expansion (HQE)

[21–26] with the aim to measure the CKM matrix element

(e.g. Ref. [27]) with highest precision. In this paper,

we make use of such moment measurements to ﬁt exclusive

branching fractions B(B

→ X

ν) with the aim to shed

additional light on a solution to the puzzles described above.

We investigate the contributions to the inclusive branching

fraction from exclusive ﬁnal states X

= D, D

∗

, D



, D

, (D

(∗)

π)

, and D

(∗)

. Hereby, we assume that D

0

and

∗0

can be identiﬁed with the observed D(2550), respec-

tively, D

∗

(2600) state. One should stress that a moment of

a kinematical distribution for any speciﬁc exclusive decay

B → X

lν, with X

being a resonant state such as D, D

∗

, D



, D

,orD

(∗)

, does not depend on the branching frac-

tions of such a resonance decaying into speciﬁc ﬁnal states.

Therefore, branching-fraction values found by the ﬁt being

larger than the directly measured values may indicate that the

decay branching-fractions assumed are overestimated.

In Sect. 2, we describe the moments entering our analysis

as ﬁt inputs. Section 3 provides information concerning the

Monte-Carlo events used for the calculation of the moments

for an exclusive decay. In Sect. 4, we outline the ﬁt procedure

and its validation, and we present the ﬁt results in Sect. 5.In

the last section we give a summary.

2 Moments in semileptonic decays

For our analysis we use three different kinds of moments:

moments of the electron–energy spectrum, of the hadronic

mass spectrum and of the combined hadronic energy-

mass spectrum, which were measured at the experiments

ABAR [27], Belle [28,29], CLEO [30], and DELPHI [31].

In Table 2, we quote the moment measurements to which we

ﬁt the branching fractions.

In the following, moments which correspond to a single

decay mode we refer to as “exclusive moments”, while when

summing over exclusive decay modes we refer to the term

“inclusive moments”.

Table 2 Experimentally measured moments used to constrain exclusive

semileptonic branching fractions in B → X

lν decays

Exp. E

cut

(GeV) or p

cut

(GeV/c) Ref.

BABAR 0.6, 0.8, 1.0, 1.2, 1.5 [27]

m

 BABAR 1.1, 1.3, 1.5, 1.7, 1.9 [27]

m

 BABAR 0.8, 1.2, 1.4, 1.6, 1.8 [27]

m

 BABAR 0.9, 1.1, 1.5, 1.7, 1.9 [27]

m

 BABAR 0.8, 1.0, 1.2, 1.6, 1.8 [27]

m

 BABAR 0.9, 1.1, 1.3, 1.5, 1.9 [27]

m

 BABAR 0.8, 1.0, 1.2, 1.4, 1.6 [27]

n

 BABAR 0.8–1.9, in steps of 0.1 [27]

n

 BABAR 0.8–1.9, in steps of 0.1 [27]

n

 BABAR 0.8–1.9, in steps of 0.1 [27]

Belle 1.0, 1.4 [28]

Belle 0.6, 1.4 [28]

Belle 0.8, 1.2 [28]

Belle 0.6, 1.2 [28]

m

 Belle 0.7–1.9, in steps of 0.2 [29]

m

 Belle 0.7–1.9, in steps of 0.2 [29]

m

centr

 CLEO 1.0, 1.5 [30]

m

centr

 CLEO 1.0, 1.5 [30]

DELPHI 0.0 [31]

We calculate the theoretical prediction for these moments

from Monte-Carlo (MC) simulated events using the follow-

ing estimators, where the nomenclature is based on Ref. [27]:

– The estimator for t he ﬁrst electron–energy moment M

is given by

cut

) =E

cut



cut



cut

, (2.1)

where E

cut

is the lower electron–energy cut-off above

which the electron energies are included in the calcula-

tion of the moment and E

is the energy of the electron

of the i-th event in the B-meson rest frame. To switch

between different form-factor models in exclusive decays

we introduce the event weights g

For higher moments, the estimator is given by

cut

) =(E −E

cut

)





cut

−E

cut

)



cut

(2.2)

with k > 1.

123

Eur. Phys. J . C (2014) 74:2914 Page 5 of 36 2914

For later convenience, the exclusive and inclusive moments

are arranged in vectors:

M =

⎛

⎜

⎝

cut

)

cut

)

cut

)

cut

)

cut

)

cut

)

⎞

⎟

⎠

⎛

⎜

⎝

E

cut

E

cut

(E −E)



cut

(E −E)



cut

(E −E)



cut

(E −E)



cut

⎞

⎟

⎠

. (2.3)

Here, E

cut

denotes again the corresponding lower

electron–energy cut-off.

We deﬁne in addition the vector

E=

⎛

⎜

⎝

E

cut

E

cut

E



cut

E



cut

E



cut

E



cut

⎞

⎟

⎠

. (2.4)

– The non-central moments of the hadronic mass spectrum

in B → X

lν decays are deﬁned as the mean of powers

of the invariant hadronic mass. Again they are measured

as a function of a lower lepton (e or μ) momentum cut-off

cut

in the B-meson rest frame.

The estimators of the mass moments are given by:

m



cut



> p

cut



> p

cut

, (2.5)

with m

being the invariant hadronic mass of event i.

The estimator of the central mass moments are deﬁned

m

centr



cut



> p

cut

−

)



> p

cut

, (2.6)

as well as

m

centr



cut



> p

cut

−

)



> p

cut

. (2.7)

Here, i runs over all events for which p

> p

cut

, where

is the lepton momentum in the semileptonic decay,

cut

the cut-off momentum, g

the event weight and

(

+ 3m

∗

)

/4 = 1.973 GeV/c

, with m

and

∗

themassesoftheD meson and D

∗

meson, respec-

tively.

Again, these moments are written in form of a vector:

m=

⎛

⎜

⎝

m

cut

m

cut

m



cut

m



cut

m



cut

m



cut

⎞

⎟

⎠

, (2.8)

whereby the vector of central moments with respect to

is deﬁned analogously

m

centr

⎛

⎜

⎝

m

centr



cut

m

centr



cut

m

centr



cut

m

centr



cut

⎞

⎟

⎠

. (2.9)

–InRef.[32] a measurement of combined mass–energy

moments was proposed. These moments are for instance

better controlled theoretically and therefore they may

result in a more reliable extraction of higher-order non-

perturbative HQE parameters. Hence, a more accu-

rate determination of the Standard Model parameters

|, the charm quark mass m

and the bottom quark

mass m

should be possible. The ﬁrst three even com-

bined mass–energy moments were measured by the

ABAR collaboration [27]. Here, we use the following

estimators for the prediction of these moments:

n



cut



> p

cut

> p

cut



− 2

E



)

k/2

(2.10)

where i runs over all events for which p

> p

cut

, E

denotes the hadronic energy and m

the invariant mass

of the hadronic system X

, p

is the momentum of the

123

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