$\Bz\to\pip\pim\piz$ Time Dependent Dalitz analysis at BaBar
Untitled
0 → π+π−π0 Time Dependent Dalitz analysis at BaBar.
Gianluca Cavoto∗
INFN Sezione di Roma, Piazzale Aldo Moro 2, 00185 Rome, Italy
I present here results of a time-dependent analysis of the Dalitz structure of neutral B meson
decays to π+π−π0 from a dataset of 346 million BB̄ pairs collected at the Υ (4S) center of mass
energy by the BaBar detector at the SLAC PEP-II e+e− accelerator. No significant CP violation
effects are observed and 68% confidence interval is derived on the weak angle α to be [75,152]
I. INTRODUCTION
The time-dependent analysis of the B0 → π+π−π0
Dalitz plot (DP), dominated by the ρ(770) intermedi-
ate resonances, extracts simultaneously the strong tran-
sition amplitudes and the weak interaction phase α ≡
arg [−VtdV ∗tb/VudV ∗ub] of the Unitarity Triangle [1]. In
the Standard Model, a non-zero value for α is respon-
sible for the occurrence of mixing-induced CP violation
in this decay. ρ±π∓ is not a CP eigenstate, and four
flavor-charge configurations (B0(B0) → ρ±π∓) must be
considered. The corresponding isospin analysis [2] is un-
fruitful with the present statistics since two pentagonal
amplitude relations with 12 unknowns have to be solved
(compared to 6 unknowns for the π+π− and ρ+ρ− sys-
tems).
The differential B0 decay width with respect to the
Mandelstam variables s+, s− (i.e., the Dalitz plot [3])
reads dΓ(B0 → π+π−π0) = 1
(2π)3
|A3π |
ds+ds−, where
A3π (A3π) is the Lorentz-invariant amplitude of the
three-body decay B0 → π+π−π0 (B0 → π+π−π0). We
assume in the following that the amplitudes are dom-
inated by the three resonances ρ+, ρ− and ρ0 and we
write A3π = f+A
+ + f−A
− + f0A
0 and A3π = f+A+ +
−+f0A
0, where the fκ (with κ = {+,−, 0} denoting
the charge of the ρ from the decay of the B0 meson) are
functions of s+ and s− that incorporate the kinematic
and dynamical properties of the B0 decay into a (vec-
tor) ρ resonance and a (pseudoscalar) pion, and where
the Aκ are complex amplitudes that include weak and
strong transition phases and that are independent of the
Dalitz variables.
With ∆t ≡ t3π−ttag defined as the proper time interval
between the decay of the fully reconstructedB03π and that
of the other meson B0tag, the time-dependent decay rate
∗Electronic address: gianluca.cavoto@roma1.infn.it
when the tagging meson is a B0 (B0) is given by
|A±3π(∆t)|
e−|∆t|/τB0
|A3π |2 + |A3π|2 ∓
|A3π |2 − |A3π|2
cos(∆md∆t)
± 2Im
A3πA∗3π
sin(∆md∆t)
, (1)
where τB0 is the mean B
0 lifetime and ∆md is the B
oscillation frequency. Here, we have assumed that CP
violation in b mixing is absent (|q/p| = 1), ∆ΓBd = 0
and CPT is conserved. Inserting the amplitudes A3π and
A3π one obtains for the terms in Eq. (1)
|A3π |2 ± |A3π|2 =
κ∈{+,−,0}
|fκ|2U±κ +
κ<σ∈{+,−,0}
Re [fκf
κσ − Im [fκf∗σ ]U±,Imκσ
A3πA∗3π
κ∈{+,−,0}
|fκ|2Iκ +
κ<σ∈{+,−,0}
Re [fκf
σ ] I
κσ + Im [fκf
σ ] I
, (2)
The 27 real-valued coefficients defined in Tab.IV that
multiply the fκf
σ bilinears are determined by the fit.
Each of the coefficients is related in a unique way to phys-
ically more intuitive quantities, such as tree-level and
penguin-type amplitudes, the angle α, or the quasi-two-
body CP and dilution parameters [4] (cf. Section IVB).
We determine the quantities of interest in a subsequent
least-squares fit to the measured U and I coefficients.
II. DALITZ MODEL
The ρ resonances are assumed to be the sum of the
ground state ρ(770) and the radial excitations ρ(1450)
and ρ(1700), with resonance parameters determined by
a combined fit to τ+ → ντπ+π0 and e+e− → π+π−
data [5]. Since the hadronic environment is different in
B decays, we cannot rely on this result and therefore de-
termine the relative ρ(1450) and ρ(1700) amplitudes si-
multaneously with the CP parameters from the fit. Vari-
ations of the other parameters and possible contributions
http://arxiv.org/abs/0704.0571v1
0 0.2 0.4 0.6 0.8 1
Interference
FIG. 1: Square Dalitz plots for Monte-Carlo generated B0 →
π+π−π0 decays.The decays have been simulated without any
detector effect and the amplitudes A+, A− and A0 have all
been chosen equal to 1 in order to have destructive inter-
ferences at equal ρ masses. The main overlap regions be-
tween the charged and neutral ρ bands are indicated by the
hatched areas. Dashed lines in both plots correspond to√
s+,−,0 = 1.5 GeV/c
2: the central region of the Dalitz plot
contains almost no signal event.
to the B0 → π+π−π0 decay other than the ρ’s are studied
as part of the systematic uncertainties (Section IVA).
Following Ref. [5], the ρ resonances are parameterized
in fκ by a modified relativistic Breit-Wigner function in-
troduced by Gounaris and Sakurai (GS) [6].
Large variations occurring in small areas of the Dalitz
plot are very difficult to describe in detail. These re-
gions are particularly important since this is where the
interference, and hence our ability to determine the
strong phases, occurs. We therefore apply the trans-
formation ds+ ds− −→ | detJ | dm′ dθ′, which defines
the Square Dalitz plot (SDP). The new coordinates are
m′ ≡ 1
arccos
−mmin
, θ′ ≡ 1
θ0, where
m0 is the invariant mass between the charged tracks,
mmax0 = mB0 − mπ0 and mmin0 = 2mπ+ are the kine-
matic limits of m0 and θ0 is the ρ
0 helicity angle; θ0 is
defined by the angle between the π+ in the ρ0 rest frame
and the ρ0 flight direction in the B0 rest frame. J is
the Jacobian of the transformation that zooms into the
kinematic boundaries of the Dalitz plot, shown in Fig.1 .
III. ANALYSIS DESCRIPTION
The U and I coefficients and the B0 → π+π−π0 event
yield are determined by a maximum-likelihood fit of the
signal model to the selected candidate events. Kinematic
and event shape variables exploiting the characteristic
properties of the events are used in the fit to discriminate
signal from background.
A. Signal and background parametrization
We reconstruct B0 → π+π−π0 candidates from pairs
of oppositely-charged tracks, which are required to form
a good quality vertex, and a π0 candidate. In order to
ensure that all events are within the Dalitz plot bound-
aries, we constrain the three-pion invariant mass to the
B mass.
A B-meson candidate is characterized kinemat-
ically by the energy-substituted mass mES =
s+ p0 · pB)2/E20 − p2B]
2 and energy difference
∆E = E∗B − 12
s, where (EB,pB) and (E0,p0) are
the four-vectors of the B-candidate and the initial
electron-positron system, respectively. The asterisk
denotes the Υ (4S) frame, and s is the square of the
invariant mass of the electron-positron system. We
require 5.272 < mES < 5.288GeV/c
2. The ∆E res-
olution exhibits a dependence on the π0 energy and
therefore varies across the Dalitz plot. We account
for this effect by introducing the transformed quantity
∆E′ = (2∆E − ∆E+ − ∆E−)/(∆E+ − ∆E−), with
∆E±(m0) = c± − (c± ∓ c̄) (m0/mmax0 )2, where m0 is
strongly correlated with the energy of π0. We use the val-
ues c̄ = 0.045GeV, c− = −0.140GeV, c+ = 0.080GeV,
mmax0 = 5.0GeV, and require −1 < ∆E′ < 1.
Backgrounds arise primarily from random combina-
tions in continuum qq̄ events. To enhance discrimination
between signal and continuum, we use a neural network
(NN) [7] to combine discriminating topological variables.
The time difference ∆t is obtained from the measured
distance between the z positions (along the beam direc-
tion) of the B03π and B
tag decay vertices, and the boost
βγ = 0.56 of the e+e− system: ∆t = ∆z/βγc. To deter-
mine the flavor of the B0tag we use the B flavor tagging
algorithm of Ref. [8]. This produces six mutually exclu-
sive tagging categories.
Events with multiple B candidates passing the full se-
lection occur in 16% (ρ±π∓) and 9% (ρ0π0) of the time,
according to signal MC. If the multiple candidates have
different π0 candidates, we choose the B candidate with
the reconstructed π0 mass closest to the nominal π0 mass;
in the case that both candidates have the same π0, we
pick the first one.
The signal efficiency determined from MC simulation
is 24% for B0 → ρ±π∓ and B0 → ρ0π0 events, and 11%
for non-resonant B0 → π+π−π0 events.
Of the selected signal events, 22% of B0 → ρ±π∓,
13% of B0 → ρ0π0, and 6% of non-resonant events are
misreconstructed. Misreconstructed events occur when a
track or neutral cluster from the tagging B is assigned
to the reconstructed signal candidate. This occurs most
often for low-momentum tracks and photons and hence
the misreconstructed events are concentrated in the cor-
ners of the Dalitz plot. Since these are also the areas
where the ρ resonances overlap strongly, it is important
to model the misreconstruced events correctly.
We use MC simulated events to study the background
from other B decays. More than a hundred channels
were considered in preliminary studies, of which twenty-
nine are included in the final likelihood model. For each
mode, the expected number of selected events is com-
puted by multiplying the selection efficiency (estimated
using MC simulated decays) by the world average branch-
ing fraction (or upper limit), scaled to the dataset lumi-
nosity (310 fb−1). The selected on-resonance data sample
is assumed to consist of signal, continuum-background
and B-background components, separated by the flavor
and tagging category of the tag side B decay. The sig-
nal likelihood consists of the sum of a correctly recon-
structed (“truth-matched”, TM) component and a mis-
reconstructed (“self-cross-feed”, SCF) component.
B. Dalitz and ∆t distribution
The Dalitz plot PDFs require as input the Dalitz plot-
dependent relative selection efficiency, ǫ = ǫ(m′, θ′), and
SCF fraction, fSCF = fSCF(m
′, θ′). Both quantities are
taken from MC simulation.
Away from the Dalitz plot corners the efficiency is uni-
form, while it decreases when approaching the corners,
where one of the three particles in the final state is close
to rest so that the acceptance requirements on the par-
ticle reconstruction become restrictive. Combinatorial
backgrounds and hence SCF fractions are large in the
corners of the Dalitz plot due to the presence of soft neu-
tral clusters and tracks.
The width of the dominant ρ(770) resonance is large
compared to the mass resolution for TM events (about
8MeV/c2 core Gaussian resolution). We therefore neglect
resolution effects in the TM model. Misreconstructed
events have a poor mass resolution that strongly varies
across the Dalitz plot. It is described in the fit by a
2 × 2-dimensional resolution function, convoluted with
signal Dalitz PDF.
The ∆t resolution function for signal and B-
background events is a sum of three Gaussian distribu-
tions, with parameters determined by a fit to fully recon-
structed B0 decays [8].
The Dalitz plot- and ∆t-dependent PDFs factorize for
the charged-B-backgroundmodes, but not necessarily for
the neutral-B background due to B0B0 mixing.
The charged B-background contribution to the likeli-
hood parametrizes tag-“charge” correlation (represented
by an effective flavor-tag-versus-Dalitz-coordinate cor-
relation), and therefore possible direct CP violation in
these events.
The Dalitz plot PDFs are obtained from MC simula-
tion and are described with the use of non-parametric
functions. The ∆t resolution parameters are determined
by a fit to fully reconstructed B+ decays.
The neutral-B background is parameterized with
PDFs that depend on the flavor tag of the event and,
depending on the final states they can show correla-
tions between the flavor tag and the Dalitz coordinate.
The Dalitz plot PDFs are obtained from MC simulation
and are described with the use of non-parametric func-
tions. For neutral-B background, the signal ∆t resolution
model is assumed.
The Dalitz plot of the continuum events is
parametrized with an empirical shape. extracted from
on-resonance events selected in the mES sidebands and
corrected for feed-through from B decays. The contin-
uum ∆t distribution is parameterized as the sum of three
Gaussian distributions with common mean and three dis-
tinct widths that scale the ∆t per-event error, all deter-
mined by the fit.
IV. RESULTS
The maximum-likelihood fit results in a B0 → π+π−π0
event yield of 1847 ± 69, where the error is statistical
only. For the U and I coefficients, the results are given
together with their statistical and systematic errors in
Table IV. The signal is dominated by B0 → ρ±π∓ de-
cays. We observe an excess of ρ0π0 events, which is in
agreement with our previous upper limit [9], and the lat-
est measurement from the Belle collaboration [10]. The
result for the ρ(1450) amplitude is in agreement with the
findings in τ and e+e− decays [5]. For the relative strong
phase between the ρ(770) and the ρ(1450) amplitudes we
find (171± 23)◦ (statistical error only), which is compat-
ible with the result from τ and e+e− data.
A. Systematics studies
The most important contribution to the systematic un-
certainty stems from the modeling of the Dalitz plot dy-
namics for signal. We evaluated this by observing the
difference between the true values and Monte Carlo fit re-
sults, in which events are generated based on an alterna-
tive model. The alternative fit model has, in addition, a
uniform Dalitz distribution for the non-resonance events
and possible resonances including f0(980), f2(1270), and
a low mass S-wave σ. The fit does not find significant
number of any of those decays. However, the inclusion
of a low mass π+π− S-wave component significantly de-
grades our ability to identify ρ0π0 events. .
We vary the mass and width of the ρ(770), ρ(1450),
and ρ(1700) within ranges that exceed twice the errors
found for these parameters in the fits to τ and e+e−
data [5], and assign the observed differences in the mea-
sured U and I coefficients as systematic uncertainties.
To validate the fitting tool, we perform fits on large MC
samples with the measured proportions of signal, contin-
uum and B-background events. No significant biases are
observed in these fits, and the statistical uncertainties on
the fit parameters are taken as systematic uncertainties
”Quasi twobody” U±κ = |Aκ|2 ± |Aκ|2
U+0 ρ
0π0 fit fraction 0.237 ± 0.053 ± 0.043
U+− ρ
−π+ fit fraction 1.33± 0.11 ± 0.04
U−0 Direct CPV (ρ
0π0) −0.055± 0.098 ± 0.13
U−− Direct CPV (ρ
−π+) −0.30± 0.15 ± 0.03
U−+ Direct CPV (ρ
+π−) 0.53± 0.15 ± 0.04
”Quasi twobody” Iκ = Im
AκAκ∗
I0 Int. Mixing CPV ρ
0π0 −0.028± 0.058 ± 0.02
I− Int. Mixing CPV ρ
−π+ −0.03± 0.10 ± 0.03
I+ Int. Mixing CPV ρ
+π− −0.039± 0.097 ± 0.02
”Interference” U
±,Re(Im)
κσ = Re(Im)
AκAσ∗ ± AκAσ∗
+− 0.62± 0.54 ± 0.72
+− 0.13± 0.94 ± 0.17
+− 0.38± 0.55 ± 0.28
+− 2.14± 0.91 ± 0.33
+0 0.03± 0.42 ± 0.12
+0 −0.75± 0.40 ± 0.15
+0 −0.93± 0.68 ± 0.08
+0 −0.47± 0.80 ± 0.3
−0 −0.03± 0.40 ± 0.23
−0 −0.52± 0.32 ± 0.08
−0 0.24± 0.61 ± 0.2
−0 −0.42± 0.73 ± 0.28
”Interference” IReκσ = Re
AκAσ∗ −AσAκ∗
IRe+− −0.1 ± 1.9 ± 0.3
IRe+0 0.2 ± 1.1 ± 0.4
IRe−0 0.92± 0.91 ± 0.4
”Interference” IImκσ = Im
AκAσ∗ + AσAκ∗
IIm+− −1.9 ± 1.1 ± 0.1
IIm+0 −0.1 ± 1.1 ± 0.3
IIm−0 0.7 ± 1.0 ± 0.3
TABLE I: Definitions and results for the 26 U and I observ-
ables extracted from the fit. We determine the relative values
of U and I coefficients to U++ .
Another major source of systematic uncertainty is the
B-background model. The expected event yields from
the background modes are varied according to the uncer-
tainties in the measured or estimated branching fractions
Since B-backgroundmodes may exhibit CP violation, the
corresponding parameters are varied within appropriate
uncertainty ranges.
Continuum Dalitz plot PDF is extrapolated form mES
sideband, and large samples of off-resonance data with
loosened requirements on ∆E and the NN are used to
compare the distributions of m′ and θ′ between the mES
sideband and the signal region. No significant differences
are found. We assign as systematic error the effect seen
when weighting the continuum Dalitz plot PDF by the
ratio of both data sets. This effect is mostly statistical
in origin.
Other systematic effects due to the signal PDFs com-
prise uncertainties in the PDF parameterization, the
treatment of misreconstructed events, the tagging per-
0 50 100 150
α (deg)
B A B A R
P R E L I M I N A R Y
FIG. 2: Confidence level functions for α. Indicated by the
dashed horizontal lines are the confidence level (C.L.) values
corresponding to 1σ and 2σ, respectively.
formance, and the modeling of the signal contributions
and are estimated using arious data control samples.
B. Intepretation of the results
The U and I coefficients are related to the quasi-two-
body parameters (Tab.IVB) defined in Ref. [4], explic-
itly accounting for the presence of interference effects,
and are thus exact even for a ρ with finite width. The
systematic errors are dominated by the uncertainty on
the CP content of the B-related backgrounds. One can
transform the experimentally convenient, namely uncor-
related, direct CP -violation parameters C and Aρπ into
the physically more intuitive quantities A+−ρπ and A−+ρπ .
The significance, including systematic uncertainties and
calculated by using a mininum χ2 method, for the ob-
servation of non-zero direct CP violation is at the 3.0σ
level.
C = (C+ + C−)/2 0.154 ± 0.090 ± 0.037
S = (S+ + S−)/2 0.01± 0.12 ± 0.028
∆C = (C+ − C−)/2 0.377 ± 0.091 ± 0.021
∆S = (S+ − S−)/2 0.06 ± 0.13 ± 0.029
Aρπ =
−0.142± 0.041 ± 0.015
A+−ρπ = |κ
+−|2−1
|κ+−|2+1
0.03 ± 0.07± 0.03
A−+ρπ = |κ
−+|2−1
|κ−+|2+1
−0.38+0.15−0.16 ± 0.07
TABLE II: Quasi twobody parameters definition and results,
where C± =
and S± =
; κ+− = (q/p)(A−/A+)
and κ−+ = (q/p)(A+/A−), so that A+−ρπ (A−+ρπ ) involves only
diagrams where the ρ (π) meson is emitted by the W bo-
son. A+−ρπ and A−+ρπ are evaluated as −
Aρπ+C+Aρπ∆C
1+∆C+AρπC
Aρπ−C−Aρπ∆C
1−∆C−AρπC
. Their correlation coefficient is 0.62.
The measurement of the resonance interference terms
allows us to constrain the relative phase δ+− =
arg (A+∗A−) between the amplitudes of the decays B0 →
ρ−π+ and B0 → ρ+π−. This constraint can be improved
with the use of strong isospin symmetry. The amplitudes
Aκ represent the sum of tree-level (T κ) and penguin-
type (P κ) amplitudes, which have different CKM fac-
tors. Here we denote by κ the charge conjugate of κ,
where 0 = 0. We define [11] Aκ = T κe−iα + P κ and
Aκ = T κe+iα + P κ, where the magnitudes of the CKM
factors have been absorbed in the T κ, P κ, T κ and P κ.
Using strong isospin symmetry and neglecting isospin-
breaking effects, one can identify P 0 = −(P+ + P−)/2
and 9 unknowns have to be determined by the fit.
We find for the solution that is favored by the fit
δ+− = (34 ± 29)◦, where the errors include both sta-
tistical and systematic effects, but only a marginal con-
straint on δ+− is obtained for C.L. < 0.05.
Finally, following the same procedure, we can also de-
rive a constraint on α. The resulting C.L. function versus
α is given in Fig. 2 and includes systematic uncertain-
ties. Ignoring the mirror solution at α + 180◦, we find
α ∈ (75◦, 152◦) at 68% C.L. No constraint on α is
achieved at two sigma and beyond.
V. CONCLUSIONS
We have presented the preliminary measurement of
CP -violating asymmetries in B0 → π+π−π0 decays dom-
inated by the ρ resonance. The results are obtained from
a data sample of 346 million Υ (4S) → BB decays. We
perform a time-dependent Dalitz plot analysis. From the
measurement of the coefficients of 26 form factor bilin-
ears we determine the three CP -violating and two CP -
conserving quasi-two-body parameters, where we find a
3.0σ evidence of direct CP violation. Taking advantage
of the interference between the ρ resonances in the Dalitz
plot, we derive constraints on the relative strong phase
between B0 decays to ρ+π− and ρ−π+, and on the an-
gle α of the Unitarity Triangle. These measurements are
consistent with the expectation from the CKM fit [12].
Acknowledgments
The author wishes to thank the conference organizers
for an enjoyable and well-organized workshop. This work
is supported by the Istituto Nazionale di Fisica Nucle-
are (INFN) and the United State Department of Energy
(DOE) under contract DE-AC02-76SF00515.
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