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CKM angle γ measurements at LHCb
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2014 J. Phys.: Conf. Ser. 556 012034
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XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014)
IOP Publishing
Journal of Physics: Conference Series 556 (2014) 012034
doi:10.1088/1742-6596/556/1/012034
CKM angle γ measurements at LHCb
Alexis Vallier
Laboratoire de l'Accélérateur Linéaire, Université Paris-Sud 11, CNRS/IN2P3, Orsay, France.
E-mail: [email protected]
The CKM angle γ remains the least known parameter of the CKM mixing matrix.
The precise measurement of this angle, as a Standard Model benchmark, is a key goal of the
LHCb experiment. We present four recent CP violation studies related to the measurement of
γ , including amplitude analysis of B ± → DK ± decays, the ADS/GLW analysis of B 0 → DK ∗0
decays and the time-dependent analysis of Bs0 → Ds∓ K ± decays.
Abstract.
1. Introduction
The CKM angle γ, dened as γ ≡ arg −
, is the least known CKM parameter. B-factories
and the LHCb experiment measured γ with an uncertainty larger than 10 [1, 2, 3]. To compare,
global ts like CKMtter [4] and UTt [5] obtain an estimation of γ with an error about 2 . This
angle is directly measurable through tree processes, without signicant loop contribution. Hence
the extraction of its value is very clean and has a theoretical relative uncertainty lower than
10 [6]. Therefore a precise measurement of γ provides an excellent standard candle to check
the consistency of the CKM paradigm in the Standard Model and to probe some new physics.
The present paper summarises four measurements of γ performed by the LHCb collaboration
and presented at the BEACH 2014 conference.
∗
Vud Vub
∗
Vcd Vcb
◦
◦
−7
2. Time-integrated measurements of
Given its denition, γ is approximately the phase dierence between the quark transitions
b → c¯
us and b → u¯
cs and the interference between these two transitions is sensitive to this
angle. The interference is obtained by reconstructing the D and D¯ mesons produced in these
decays in an identical nal state (Fig. 1). In the case of a three body D meson decay a Dalitz
plot analysis can be carried out. This method is called GGSZ [7, 8] and two recent LHCb results
are reported in section 2.1. In the case of a two body D meson decay a counting analysis is
developed following the so called GLW [9, 10] or ADS [11, 12] methods. A recent ADS/GLW
result from LHCb is presented in section 2.2. All of these methods can be applied to the channels
B → DK and B → DK . Since these B mesons decays are self-tagged, no time-dependent
analysis is required.
γ
0
0
1
±
1
±
0
∗0
¯ 0 meson.
in the following D stands for either a D0 or a D
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
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Published under licence by IOP Publishing Ltd
1
XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014)
IOP Publishing
Journal of Physics: Conference Series 556 (2014) 012034
doi:10.1088/1742-6596/556/1/012034
Vcb
b
c
0
u¯ D
W−
Vub
b
u
¯0
c¯ D
W−
Vus
Vcs
s
Figure 1.
s
Feynman diagrams of b → c¯us (left) and b → u¯cs (right) decays.
The
decay amplitude h stands for either a charged pion or a charged
kaon can be written as
2.1. Measurements with 3-body D meson decays
B ± → D(KS0 h+ h− )K ±
AB ± (D) = AD (D) + ei(δB ±γ) AD¯ (D),
where A (A ) is the D (D¯ ) decay amplitude, D represents the D meson phase¯ space, δ is the
strong phase dierence and γ is the weak phase dierence between the D and D channels. The
D meson phase space is parameterised by two squared invariant masses, for instance m (K π )
and m (K π ) on a D meson Dalitz plot. The sensitivity to γ arises from large asymmetries
in some region of the Dalitz plot. In order to evaluate γ, the strong phase variation over the
Dalitz plot must be known. This can be done in two dierent ways: with a model-dependent
(MD) method using BaBar's amplitude model [13], or with a model-independent (MI) method
using CLEO-c measurements as inputs [14]. Both methods involve tting the D meson Dalitz
plot to extract the polar coordinates (r , δ , γ). The parameter r is the ratio of the
magnitudes of the suppressed and favoured B decay amplitudes. The polar coordinates are
not estimated directly from the Dalitz t, but the cartesian coordinates x = r cos(δ ± γ)
and y = r sin(δ ± γ).
This section presents the model-dependent analysis of the
B → D(K π π )K signal, using a sample of proton-proton collision data at a centre-ofmass energy of 7 TeV corresponding to an integrated luminosity of 1 fb . Full details can
be found in Ref. [15]. The analysis is carried out in two distinct stages. First a t to the B
meson reconstructed invariant mass is performed on the selected B → D(K π π )K and
B → D(K π π )π candidates. This t determines the signal and background fractions in
the data sample. A total yield of 637 B → D(K π π )K signal events and 8866 events of
the B → D(K π π )π control channel are found. Then a t to the Daliz plot determines the
CP violation observables (x , y ). The signal and background yields and the parameters of the
B invariant mass probability distribution function are xed to the values obtained in the rst
stage. The model used to described the amplitude of the D → K π π decay over the phase
space is the one determined by the BaBar collaboration in Ref. [13]. Fitting simultaneously
the distributions in the D → K π π phase space for the B → D(K π π )K and the
B → D(K π π )π candidates enables to take into account the variation of eciency over
the phase space. The B → D(K π π )π decay is a good proxy to get the eciency variation,
since it has a kinematic topology similar to the signal one and CP violation can be neglected in
this channel. The resulting values of the cartesian coordinates are:
D
0
¯
D
0
B
0
0
2
2
B±
B±
B
±
±
2.1.1.
±
B±
±
±
0 + −
S
0 + −
S
±
±
0 + −
S
±
±
0 + −
S
0 + −
S
±
0.048+0.009
−0.007
0 + −
S
±
±
x− = +0.027 ± 0.044+0.010
−0.008 ± 0.001,
y− = +0.013 ±
±
±
0
0 + −
S
0 + −
S
±
0
±
B
−1
±
0 + −
S
±
B±
B
Model-dependent analysis
0 + −
±
S
±
0 +
S
0 −
S
x+ = −0.084 ± 0.045 ± 0.009 ± 0.005,
y+ = −0.032 ± 0.048+0.010
−0.009 ± 0.008,
± 0.003,
2
XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014)
IOP Publishing
Journal of Physics: Conference Series 556 (2014) 012034
doi:10.1088/1742-6596/556/1/012034
where the rst uncertainty is statistical, the second systematic and the third due to the amplitude
model used to describe the D → K π π decay. The leading experimental systematic errors
are due to the eciency and background description uncertainties. The constraints obtained on
the polar coordinates are r = 0.06 ± 0.04, δ = (115 ) and γ = (84 ) . These results are
consistent with those of the LHCb model-independent analysis based on the same data set [16].
This section presents the model-independent analysis of the
B → D(K h h )K decays, using a sample of proton-proton collision data at a centre-ofmass energy of 7 and 8 TeV corresponding to a total integrated luminosity of 3 fb . Full details
can be found in Ref. [17]. There is a signicant improvement compared to the former results in
Ref. [16], thanks to the increased statistics and a better analysis technique. To know the strong
phase variation over the D → K h h phase space the measurements made by CLEO-c, in
a particular binning scheme, is used [14]. In this way the analysis is a counting experiment in
bins of the Dalitz plot. The expected number of D from B events falling in a particular bin
labelled ±i (the ± sign comes from the phase symmetry with respect to the Dalitz diagonal) can
be expressed as
i
h
p
(1)
N = h F + (x + y )F + 2 F F (x c ∓ y s ) ,
where c and s are the averaged cosine and sine of the strong phase dierence in bin i (CLEO-c
inputs), F is the expected fraction of pure D events in bin i taking into account the eciency
prole over the phase space, and h is a normalisation factor. The F parameters are determined from the B → D µ ν control mode. This is an excellent proxy because the sample
has a high purity, a high statistics and the D meson is tagged thanks to the slow pion in the
D → D π decay. Some corrections are applied from simulated data to account for reconstruction and selection discrepancies between the B → D µ ν and the B → DK decays.
The t is performed in two steps. First the phase space integrated B mass t determines the
total signal yields (around 2600) and xes the model used in the second step. This last t is
made in each Dalitz bin with all the parameters in Eq. (1) xed but the normalisation factor
and the (x , y ) observables. The resulting values of the cartesian coordinates (Fig. 2) are the
most precise to date:
0 + −
S
0
B±
+41 ◦
−51
B
+49 ◦
−42
2.1.2. Model-independent analysis
±
0 + −
±
S
−1
0 + −
S
0
+
+
±i
i
+
B
∓i
2
+
2
+
i −i
±i
+ ±i
+ ±i
i
0
i
0
∗± ∓
+
B
i
µ
0
∗+
0 +
∗± ∓
0
±
±
µ
±
±
±
x− = (2.5 ± 2.5 ± 1.0 ± 0.5) × 10−2 ,
y− = (7.5 ± 2.9 ± 0.5 ± 1.4) × 10−2 ,
x+ = (−7.7 ± 2.4 ± 1.0 ± 0.4) × 10−2 ,
y+ = (−2.2 ± 2.5 ± 0.4 ± 1.0) × 10−2 ,
where the rst uncertainty is statistical, the second systematic and the third due to the experimental knowledge of the (c , s ) parameters. Compared to the 1 fb measurement [16], the
statistical uncertainty is reduced thanks to the larger data sample, the experimental systematic
is reduced by using the new control mode B → D µ ν , and the (c , s ) systematic is also
improved by the increased LHCb sample size. The results are: r = 0.080 , γ = (62 )
and δ = (62 ) .
i
−1
i
∗± ∓
0
µ
i
B±
B
+14 ◦
−15
i
+0.019
−0.021
+15 ◦
−14
This section presents the ADS/GLW analysis of the B → DK decays, using a sample of
proton-proton collision data at a centre-of-mass energy of 7 and 8 TeV corresponding to a
total integrated luminosity of 3 fb . Full details can be found in Ref. [18]. Compared to
the B → DK decays, both Cabibbo favoured and suppressed diagrams are color suppressed,
which brings about a higher interference amplitude (r is larger than r ). Hence a better
sensitivity to γ is expected. However this neutral channel is experimentally more challenging.
For the GLW modes the D mesons are reconstructed in two CP eigenstate: K K and π π .
2.2. Measurements with 2-body D meson decays
0
∗0
−1
±
±
B0
B±
+
3
−
+ −
y
XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014)
IOP Publishing
Journal of Physics: Conference Series 556 (2014) 012034
doi:10.1088/1742-6596/556/1/012034
0.3
LHCb
0.2
B
−
0.1
Condence levels at 39.3%,
86.5% and 98.9% probability for (x , y )
and (x , y ) as measured in B →
DK decays (statistical uncertainties
only). The parameters (x , y ) relate
to B decays and (x , y ) refer to B
decays. The stars represent the best t
central values.
Figure 2.
0
-0.1
B+
±
-0.2
−
+
±
−
+
+
-0.3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
x
−
+
+
−
−
For the ADS modes the D mesons are reconstructed in the K π and K π nal states. From
these decays several observables sensitive to γ can be built. For instance in the GLW modes the
CP asymmetries
+ −
Ahh
d ≡
− +
¯ 0 → D(h+ h− )K
¯ ∗0 ) − Γ(B 0 → D(h+ h− )K ∗0 )
Γ(B
2rB 0 κ sin δB sin γ
=
2
0
+
−
∗0
0
+
−
∗0
¯
¯
1 + rB 0 + 2rB 0 κ cos δB cos γ
Γ(B → D(h h )K ) + Γ(B → D(h h )K )
are measured. The parameter κ is the coherence factor introduced to account for the eect of
the non resonant B → DK π contribution in the K signal region. And in the ADS mode
the ratio of suppressed B → D(π¯ K )K to favoured B → D(K π )K partial widths are
measured separately for B and B :
+ −
0
0
0
Γ(B 0
Γ(B 0
¯0
Γ(B
¯0
Γ(B
Rd+ ≡
Rd− ≡
∗0
+
−
∗0
+ −
0
∗0
0
2 + r 2 + 2r r κ cos(δ + δ + γ)
rB
→ D(π + K − )K ∗0 )
0
B
D
B0 D
D
=
,
2
2
+
−
∗0
→ D(K π )K )
1 + rB 0 rD + 2rB 0 rD κ cos(δB − δD + γ)
2 + r 2 + 2r r κ cos(δ + δ − γ)
¯ ∗0 )
rB
→ D(π − K + )K
0
B
D
B0 D
D
.
=
2
2
¯ ∗0 )
1 + rB 0 rD + 2rB 0 rD κ cos(δB − δD − γ)
→ D(K − π + )K
The parameters r and δ are the magnitude ratio and the phase dierence, respectively, between
the amplitudes
of the D → K π and D → K π decays. The signicances of the combined
¯ signals for the B → D(K K )K , B → D(π π )K and B → D(π K )K
B and B
decay modes are 8.6 σ, 5.8 σ and 2.9 σ respectively. Once the production and eciency asymmetries are taken into account the results are:
D
0
D
0
0
+ −
0
− +
0
−
+
∗0
+ −
0
∗0
0
+
−
∗0
Rd+ = 0.06 ± 0.03 ± 0.01,
Rd− = 0.06 ± 0.03 ± 0.01,
AKK
= −0.20 ± 0.15 ± 0.02,
d
ππ
Ad
= −0.09 ± 0.22 ± 0.02,
where the rst uncertainties are statistical and the second systematic. A , R and R are rst
measurements and the A result supersedes the former LHCb one [19]. From these measurements the value of r (proper to the B → DK channel) is found to be r = 0.240 .
This is the most precise measurement to date.
KK
d
B0
ππ
d
∗0
0
3. Time-dependent measurement of
+
d
B0
−
d
+0.055
−0.048
This section presents the time-dependent analysis of the B → D K decays, using a sample of
proton-proton collision data at a centre-of-mass energy of 7 TeV corresponding to an integrated
γ
0
s
4
∓
s
±
10
1
1-CL
data
total
±
B0s → Ds K
−
B0s → Ds π+
B0s →D(s*)− (π+, ρ+ )
0
(B0d, Λb ) → X
Combinatorial
LHCb
2
10
±
Candidates / ( 0.1 ps)
XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014)
IOP Publishing
Journal of Physics: Conference Series 556 (2014) 012034
doi:10.1088/1742-6596/556/1/012034
LHCb
+28
0.8
115−43
0.6
0.4
68.3%
1
0.2
4
2
4
6
8
6
8
10
12
14
τ (Bs→ DsK) (ps)
95.5%
0
0
20
40
60
80 100 120 140 160 180
γ [°]
10
12
14
τ (B0s → Ds K± ) [ps]
±
2
0
-2
2
Result of the decay-time
candidates.
Figure 3.
Bs0 → Ds∓ K ±
cFit
Frequentist interpretation of the γ
measurement in term of condence interval.
to the
Figure 4.
luminosity of 1 fb . Full details can be found in Ref. [20]. In addition to the same tree level
processes as in the time-integrated analysis (Fig. 1), the eect of the B mixing occurs. Hence
the interference between mixing and decay amplitudes in B → D K is sensitive to the CP
violating phase (γ − 2β ) where β ≡ arg(−V V /V V ). The time-dependent decay rates
depend on the CP observables:
C =
,
A =
, A =
,
S =
, S=
,
where r is the magnitude of the amplitude ratio |A(B¯ → D K )/A(B → D K )|, δ
the strong phase dierence and (γ − 2β ) the weak phase dierence. This analysis uses an
independent measurement of φ [21] and assumes φ = −2β to interpret the results in terms
of γ. To discriminate the signal and background components a 3D t is performed on the B
and D masses along with the log-likelihood dierence between the kaon and pion hypothesis for
the companion particle (K for B → D K signal and π for B → D π control mode).
Then the output of this multivariate t is used for the decay-time t. Two ts are performed: a
background subtracted t, called , using the
[22, 23] determined by the multivariate
t; and a classical t, called (Fig. 3) where all signal and background time distributions are
described. The results of these two ts are in excellent agreement:
Parameter
tted value
tted value
−1
s
f
f
∗
ts tb
s
2
1−rD
sK
2
1+rD
sK
2rDs K sin(δ−(γ−2βs ))
2
1+rD
sK
∆Γ
f
f¯
∗
cs cb
−2rDs K cos(δ−(γ−2βs ))
2
1+rD
sK
−2rDs K sin(δ+(γ−2βs ))
2
1+rDs K
s
±
−2rDs K cos(δ+(γ−2βs ))
2
1+rD
sK
∆Γ
f¯
−
s
0
s
Ds K
∓
s
0
s
+
0
s
−
s
+
s
s
s
s
s
s
±
0
s
sFit
∓
s
±
+
− +
s
0
s
sWeights
cFit
Cf
A∆Γ
f
A∆Γ
f¯
Sf
Sf¯
sFit
cFit
0.52 ± 0.25 ± 0.04
0.29 ± 0.42 ± 0.17
0.14 ± 0.41 ± 0.18
−0.09 ± 0.31 ± 0.06
−0.36 ± 0.34 ± 0.06
0.53 ± 0.25 ± 0.04
0.37 ± 0.42 ± 0.20
0.20 ± 0.41 ± 0.20
−1.09 ± 0.33 ± 0.08
−0.36 ± 0.34 ± 0.08
The rst uncertainties are statistical and the second systematic. The main sources of systematic
arise from the trigger-induced time-dependent eciency, Γ and ∆Γ . These results can be
interpreted as a condence interval γ = (115 ) at 68% CL (Fig. 4). This is the rst
measurement of γ with B → D K decays.
0
s
∓
s
+28 ◦
−43
±
5
s
s
XI International Conference on Hyperons, Charm and Beauty Hadrons (BEACH 2014)
IOP Publishing
Journal of Physics: Conference Series 556 (2014) 012034
doi:10.1088/1742-6596/556/1/012034
4. Conclusion
The latest LHCb results on the CKM angle γ are reported: the model dependent GGSZ analysis
of B → DK decays, the update to the full available data set of the model indepent GGSZ
analysis of B → DK decays, the ADS/GLW analysis of B → DK decays and the
rst γ measurement with B → D K decays. Using these results and the corresponding
improvements, the next combination of the LHCb γ measurements should yield a signicant
reduction of the uncertainty.
±
±
±
±
0
0
s
∓
s
∗0
±
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6

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