arXiv:1605.04545v2 [hep-ex] 13 Jul 2016
B
A
B
AR
-PUB-16/002
SLAC-PUB-16525
Tests of
CP T
symmetry in
B
0
-
B
0
mixing and in
B
0
→
c
cK
0
decays
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3
G. Eigen,
4
D. N. Brown,
5
Yu. G. Kolomensky,
5
H. Koch,
6
T. Schroeder,
6
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
R. Y. So,
7
V. E. Blinov
abc
,
8
A. R. Buzykaev
a
,
8
V. P. Druzhinin
ab
,
8
V. B. Golubev
ab
,
8
E. A. Kravchenko
ab
,
8
A. P. Onuchin
abc
,
8
S. I. Serednyakov
ab
,
8
Yu. I. Skovpen
ab
,
8
E. P. Solodov
ab
,
8
K. Yu. Todyshev
ab
,
8
A. J. Lankford,
9
J. W. Gary,
10
O. Long,
10
A. M. Eisner,
11
W. S. Lockman,
11
W. Panduro Vazquez,
11
D. S. Chao,
12
C. H. Cheng,
12
B. Echenard,
12
K. T. Flood,
12
D. G. Hitlin,
12
J. Kim,
12
T. S. Miyashita,
12
P. Ongmongkolkul,
12
F. C. Porter,
12
M. R ̈ohrken,
12
Z. Huard,
13
B. T. Meadows,
13
B. G. Pushpawela,
13
M. D. Sokoloff,
13
L. Sun,
13,
∗
J. G. Smith,
14
S. R. Wagner,
14
D. Bernard,
15
M. Verderi,
15
D. Bettoni
a
,
16
C. Bozzi
a
,
16
R. Calabrese
ab
,
16
G. Cibinetto
ab
,
16
E. Fioravanti
ab
,
16
I. Garzia
ab
,
16
E. Luppi
ab
,
16
V. Santoro
a
,
16
A. Calcaterra,
17
R. de Sangro,
17
G. Finocchiaro,
17
S. Martellotti,
17
P. Patteri,
17
I. M. Peruzzi,
17
M. Piccolo,
17
A. Zallo,
17
S. Passaggio,
18
C. Patrignani,
18,
†
B. Bhuyan,
19
U. Mallik,
20
C. Chen,
21
J. Cochran,
21
S. Prell,
21
H. Ahmed,
22
A. V. Gritsan,
23
N. Arnaud,
24
M. Davier,
24
F. Le Diberder,
24
A. M. Lutz,
24
G. Wormser,
24
D. J. Lange,
25
D. M. Wright,
25
J. P. Coleman,
26
E. Gabathuler,
26
D. E. Hutchcroft,
26
D. J. Payne,
26
C. Touramanis,
26
A. J. Bevan,
27
F. Di Lodovico,
27
R. Sacco,
27
G. Cowan,
28
Sw. Banerjee,
29
D. N. Brown,
29
C. L. Davis,
29
A. G. Denig,
30
M. Fritsch,
30
W. Gradl,
30
K. Griessinger,
30
A. Hafner,
30
K. R. Schubert,
30
R. J. Barlow,
31,
‡
G. D. Lafferty,
31
R. Cenci,
32
A. Jawahery,
32
D. A. Roberts,
32
R. Cowan,
33
R. Cheaib,
34
S. H. Robertson,
34
B. Dey
a
,
35
N. Neri
a
,
35
F. Palombo
ab
,
35
L. Cremaldi,
36
R. Godang,
36,
§
D. J. Summers,
36
P. Taras,
37
G. De Nardo,
38
C. Sciacca,
38
G. Raven,
39
C. P. Jessop,
40
J. M. LoSecco,
40
K. Honscheid,
41
R. Kass,
41
A. Gaz
a
,
42
M. Margoni
ab
,
42
M. Posocco
a
,
42
M. Rotondo
a
,
42
G. Simi
ab
,
42
F. Simonetto
ab
,
42
R. Stroili
ab
,
42
S. Akar,
43
E. Ben-Haim,
43
M. Bomben,
43
G. R. Bonneaud,
43
G. Calderini,
43
J. Chauveau,
43
G. Marchiori,
43
J. Ocariz,
43
M. Biasini
ab
,
44
E. Manoni
a
,
44
A. Rossi
a
,
44
G. Batignani
ab
,
45
S. Bettarini
ab
,
45
M. Carpinelli
ab
,
45,
¶
G. Casarosa
ab
,
45
M. Chrzaszcz
a
,
45
F. Forti
ab
,
45
M. A. Giorgi
ab
,
45
A. Lusiani
ac
,
45
B. Oberhof
ab
,
45
E. Paoloni
ab
,
45
M. Rama
a
,
45
G. Rizzo
ab
,
45
J. J. Walsh
a
,
45
A. J. S. Smith,
46
F. Anulli
a
,
47
R. Faccini
ab
,
47
F. Ferrarotto
a
,
47
F. Ferroni
ab
,
47
A. Pilloni
ab
,
47
G. Piredda
a
,
47
C. B ̈unger,
48
S. Dittrich,
48
O. Gr ̈unberg,
48
M. Heß,
48
T. Leddig,
48
C. Voß,
48
R. Waldi,
48
T. Adye,
49
F. F. Wilson,
49
S. Emery,
50
G. Vasseur,
50
D. Aston,
51
C. Cartaro,
51
M. R. Convery,
51
J. Dorfan,
51
W. Dunwoodie,
51
M. Ebert,
51
R. C. Field,
51
B. G. Fulsom,
51
M. T. Graham,
51
C. Hast,
51
W. R. Innes,
51
P. Kim,
51
D. W. G. S. Leith,
51
S. Luitz,
51
V. Luth,
51
D. B. MacFarlane,
51
D. R. Muller,
51
H. Neal,
51
B. N. Ratcliff,
51
A. Roodman,
51
M. K. Sullivan,
51
J. Va’vra,
51
W. J. Wisniewski,
51
M. V. Purohit,
52
J. R. Wilson,
52
A. Randle-Conde,
53
S. J. Sekula,
53
M. Bellis,
54
P. R. Burchat,
54
E. M. T. Puccio,
54
M. S. Alam,
55
J. A. Ernst,
55
R. Gorodeisky,
56
N. Guttman,
56
D. R. Peimer,
56
A. Soffer,
56
S. M. Spanier,
57
J. L. Ritchie,
58
R. F. Schwitters,
58
J. M. Izen,
59
X. C. Lou,
59
F. Bianchi
ab
,
60
F. De Mori
ab
,
60
A. Filippi
a
,
60
D. Gamba
ab
,
60
L. Lanceri,
61
L. Vitale,
61
F. Martinez-Vidal,
62
A. Oyanguren,
62
J. Albert,
63
A. Beaulieu,
63
F. U. Bernlochner,
63
G. J. King,
63
R. Kowalewski,
63
T. Lueck,
63
I. M. Nugent,
63
J. M. Roney,
63
N. Tasneem,
63
T. J. Gershon,
64
P. F. Harrison,
64
T. E. Latham,
64
R. Prepost,
65
and S. L. Wu
65
(The
B
A
B
AR
Collaboration)
1
Laboratoire d’Annecy-le-Vieux de Physique des Particules
(LAPP),
Universit ́e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vie
ux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament
ECM, E-08028 Barcelona, Spain
3
INFN Sezione di Bari and Dipartimento di Fisica, Universit`
a di Bari, I-70126 Bari, Italy
4
University of Bergen, Institute of Physics, N-5007 Bergen,
Norway
5
Lawrence Berkeley National Laboratory and University of Ca
lifornia, Berkeley, California 94720, USA
6
Ruhr Universit ̈at Bochum, Institut f ̈ur Experimentalphys
ik 1, D-44780 Bochum, Germany
7
University of British Columbia, Vancouver, British Columb
ia, Canada V6T 1Z1
8
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630
090
a
,
Novosibirsk State University, Novosibirsk 630090
b
,
2
Novosibirsk State Technical University, Novosibirsk 6300
92
c
, Russia
9
University of California at Irvine, Irvine, California 926
97, USA
10
University of California at Riverside, Riverside, Califor
nia 92521, USA
11
University of California at Santa Cruz, Institute for Parti
cle Physics, Santa Cruz, California 95064, USA
12
California Institute of Technology, Pasadena, California
91125, USA
13
University of Cincinnati, Cincinnati, Ohio 45221, USA
14
University of Colorado, Boulder, Colorado 80309, USA
15
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS
/IN2P3, F-91128 Palaiseau, France
16
INFN Sezione di Ferrara
a
; Dipartimento di Fisica e Scienze della Terra, Universit`a
di Ferrara
b
, I-44122 Ferrara, Italy
17
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, I
taly
18
INFN Sezione di Genova, I-16146 Genova, Italy
19
Indian Institute of Technology Guwahati, Guwahati, Assam,
781 039, India
20
University of Iowa, Iowa City, Iowa 52242, USA
21
Iowa State University, Ames, Iowa 50011, USA
22
Physics Department, Jazan University, Jazan 22822, Kingdo
m of Saudi Arabia
23
Johns Hopkins University, Baltimore, Maryland 21218, USA
24
Laboratoire de l’Acc ́el ́erateur Lin ́eaire, IN2P3/CNRS et
Universit ́e Paris-Sud 11,
Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France
25
Lawrence Livermore National Laboratory, Livermore, Calif
ornia 94550, USA
26
University of Liverpool, Liverpool L69 7ZE, United Kingdom
27
Queen Mary, University of London, London, E1 4NS, United Kin
gdom
28
University of London, Royal Holloway and Bedford New Colleg
e, Egham, Surrey TW20 0EX, United Kingdom
29
University of Louisville, Louisville, Kentucky 40292, USA
30
Johannes Gutenberg-Universit ̈at Mainz, Institut f ̈ur Ker
nphysik, D-55099 Mainz, Germany
31
University of Manchester, Manchester M13 9PL, United Kingd
om
32
University of Maryland, College Park, Maryland 20742, USA
33
Massachusetts Institute of Technology, Laboratory for Nuc
lear Science, Cambridge, Massachusetts 02139, USA
34
McGill University, Montr ́eal, Qu ́ebec, Canada H3A 2T8
35
INFN Sezione di Milano
a
; Dipartimento di Fisica, Universit`a di Milano
b
, I-20133 Milano, Italy
36
University of Mississippi, University, Mississippi 38677
, USA
37
Universit ́e de Montr ́eal, Physique des Particules, Montr ́
eal, Qu ́ebec, Canada H3C 3J7
38
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche,
Universit`a di Napoli Federico II, I-80126 Napoli, Italy
39
NIKHEF, National Institute for Nuclear Physics and High Ene
rgy Physics, NL-1009 DB Amsterdam, The Netherlands
40
University of Notre Dame, Notre Dame, Indiana 46556, USA
41
Ohio State University, Columbus, Ohio 43210, USA
42
INFN Sezione di Padova
a
; Dipartimento di Fisica, Universit`a di Padova
b
, I-35131 Padova, Italy
43
Laboratoire de Physique Nucl ́eaire et de Hautes Energies,
IN2P3/CNRS, Universit ́e Pierre et Marie Curie-Paris6,
Universit ́e Denis Diderot-Paris7, F-75252 Paris, France
44
INFN Sezione di Perugia
a
; Dipartimento di Fisica, Universit`a di Perugia
b
, I-06123 Perugia, Italy
45
INFN Sezione di Pisa
a
; Dipartimento di Fisica,
Universit`a di Pisa
b
; Scuola Normale Superiore di Pisa
c
, I-56127 Pisa, Italy
46
Princeton University, Princeton, New Jersey 08544, USA
47
INFN Sezione di Roma
a
; Dipartimento di Fisica,
Universit`a di Roma La Sapienza
b
, I-00185 Roma, Italy
48
Universit ̈at Rostock, D-18051 Rostock, Germany
49
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX
11 0QX, United Kingdom
50
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, F
rance
51
SLAC National Accelerator Laboratory, Stanford, Californ
ia 94309 USA
52
University of South Carolina, Columbia, South Carolina 292
08, USA
53
Southern Methodist University, Dallas, Texas 75275, USA
54
Stanford University, Stanford, California 94305, USA
55
State University of New York, Albany, New York 12222, USA
56
Tel Aviv University, School of Physics and Astronomy, Tel Av
iv, 69978, Israel
57
University of Tennessee, Knoxville, Tennessee 37996, USA
58
University of Texas at Austin, Austin, Texas 78712, USA
59
University of Texas at Dallas, Richardson, Texas 75083, USA
60
INFN Sezione di Torino
a
; Dipartimento di Fisica, Universit`a di Torino
b
, I-10125 Torino, Italy
61
INFN Sezione di Trieste and Dipartimento di Fisica, Univers
it`a di Trieste, I-34127 Trieste, Italy
62
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spa
in
63
University of Victoria, Victoria, British Columbia, Canad
a V8W 3P6
64
Department of Physics, University of Warwick, Coventry CV4
7AL, United Kingdom
65
University of Wisconsin, Madison, Wisconsin 53706, USA
3
Using the eight time dependences e
−
Γ
t
(1+
C
i
cos ∆
mt
+
S
i
sin ∆
mt
) for the decays
Υ
(4
S
)
→
B
0
B
0
→
f
j
f
k
, with the decay into a flavor-specific state
f
j
=
ℓ
±
X
before or after the decay into a
CP
eigenstate
f
k
=
c
cK
S,L
, as measured by the
B
A
B
AR
experiment, we determine the three
CP T
-
sensitive parameters Re (
z
) and Im (
z
) in
B
0
-
B
0
mixing and
|
A/A
|
in
B
0
→
c
cK
0
decays. We find
Im (
z
) = 0
.
010
±
0
.
030
±
0
.
013, Re (
z
) =
−
0
.
065
±
0
.
028
±
0
.
014, and
|
A/A
|
= 0
.
999
±
0
.
023
±
0
.
017,
in agreement with
CP T
symmetry.
PACS numbers: 11.30.Er, 13.20.He, 13.25.Hw, 14.40.Nd
INTRODUCTION
The discovery of
CP
violation in 1964 [1] motivated
searches for
T
and
CPT
violation. Since
CPT
=
CP
×
T
,
violation of
CP
means that
T
or
CPT
or both are also
violated. For the
K
0
system, the two contributions were
first determined [2] in 1970, by using the Bell-Steinberger
unitarity relation [3] for
CP
violation in
K
0
-
K
0
mixing:
T
was violated with about 5
σ
significance and no
CPT
violation was observed. Large
CP
violation in the
B
0
system was discovered in 2001 [4, 5] in the interplay of
B
0
-
B
0
mixing and
B
0
→
c
cK
0
decays, but an explicit
demonstration of
T
violation was given only recently
[6]. In the present analysis, we test
CPT
symmetry
quantitatively in
B
0
-
B
0
mixing and in
B
0
→
c
cK
0
decays.
Transitions in the
B
0
-
B
0
system are well described by
the quantum-mechanical evolution of a two-state wave
function
Ψ =
ψ
1
|
B
0
i
+
ψ
2
|
B
0
i
,
(1)
using the Schr ̈odinger equation
̇
Ψ =
−
i
H
Ψ
,
(2)
where the Hamiltonian
H
is given by two constant Hermi-
tian matrices,
H
ij
=
m
ij
+ iΓ
ij
/
2. In this evolution,
CP
violation is described by three parameters,
|
q/p
|
, Re (
z
),
and Im (
z
), defined by
|
q/p
|
= 1
−
2 Im (
m
∗
12
Γ
12
)
4
|
m
12
|
2
+
|
Γ
12
|
2
,
z
=
(
m
11
−
m
22
)
−
i (Γ
11
−
Γ
22
)
/
2
∆
m
−
i ∆Γ
/
2
,
(3)
where ∆
m
=
m
(
B
H
)
−
m
(
B
L
)
≈
2
|
m
12
|
and ∆Γ =
Γ(
B
H
)
−
Γ(
B
L
)
≈
+2
|
Γ
12
|
or
−
2
|
Γ
12
|
are the mass
and the width differences of the two mass eigenstates
(
H
=heavy,
L
=light) of the Hamiltonian,
B
H
= (
p
√
1 +
z
B
0
−
q
√
1
−
z
B
0
)
/
√
2
,
B
L
= (
p
√
1
−
z
B
0
+
q
√
1 +
z
B
0
)
/
√
2
.
(4)
Note that we use the convention with +
q
for the light
and
−
q
for the heavy eigenstate. If
|
q/p
| 6
= 1, the
evolution violates the discrete symmetries
CP
and
T
.
If
z
6
= 0, it violates
CP
and
CPT
. The normalizations
of the two eigenstates, as given in Eq. (4), are precise
in the lowest order of
r
and
z
, where
r
=
|
q/p
| −
1.
Throughout the following, we neglect contributions of
orders
r
2
,
z
2
,
r
z
, and higher.
The
T
-sensitive mixing parameter
|
q/p
|
has been
determined in several experiments, the present world
average [7] being
|
q/p
|
= 1 + (0
.
8
±
0
.
8)
×
10
−
3
. The
CPT
-sensitive parameter Im (
z
) has been determined
by analyzing the time dependence of dilepton events in
the decay
Υ
(4
S
)
→
B
0
B
0
→
(
ℓ
+
νX
) (
ℓ
−
νX
); the
B
A
B
AR
result [8] is Im (
z
) = (
−
13
.
9
±
7
.
3
±
3
.
2)
×
10
−
3
. Since
∆Γ is very small, dilepton events are only sensitive to
the product Re (
z
)∆Γ. Therefore, Re (
z
) has so far only
been determined by analyzing the time dependence of
the decays
Υ
(4
S
)
→
B
0
B
0
with one
B
meson decaying
into
ℓνX
and the other one into
c
cK
. With 88
×
10
6
B
B
events,
B
A
B
AR
measured Re (
z
) = (19
±
48
±
47)
×
10
−
3
in 2004 [9], while Belle used 535
×
10
6
B
B
events to
measure Re (
z
) = (19
±
37
±
33)
×
10
−
3
in 2012 [10].
In our present analysis, we use the final data set of the
B
A
B
AR
experiment [11, 12] with 470
×
10
6
B
B
events for
a new determination of Re (
z
) and Im (
z
). As in Refs. [9,
10], this is based on
c
cK
decays with amplitudes
A
for
B
0
→
c
cK
0
and
A
for
B
0
→
c
c
K
0
, using the following two
assumptions:
(1)
c
cK
decays obey the ∆
S
= ∆
B
rule, i. e.,
B
0
states
do not decay into
c
c
K
0
, and
B
0
states do not decay
into
c
cK
0
;
(2)
CP
violation in
K
0
-
K
0
mixing is negligible,
i. e.
K
0
S
= (
K
0
+
K
0
)
/
√
2,
K
0
L
= (
K
0
−
K
0
)
/
√
2.
The
CPT
-sensitive parameters are determined from
the measured time dependences of the four decay rates
B
0
,
B
0
→
c
cK
0
S
,K
0
L
. In
Υ
(4
S
) decays,
B
0
and
B
0
mesons
are produced in the entangled state (
B
0
B
0
−
B
0
B
0
)
/
√
2.
When the first meson decays into
f
=
f
1
at time
t
1
, the
state collapses into the two states
f
1
and
B
2
. The later
decay
B
2
→
f
2
at time
t
2
depends on the state
B
2
and,
because of
B
0
-
B
0
mixing, on the decay-time difference
t
=
t
2
−
t
1
≥
0
.
(5)
Note that
t
is the only relevant time here, it is the evolu-
tion time of the single-meson state
B
2
in its rest frame.
4
The present analysis does not start from raw data but
uses intermediate results from Ref. [6] where, as men-
tioned above, we used our final data set for the demon-
stration of large
T
violation. This was shown in four
time-dependent transition-rate differences
R
(
B
j
→
B
i
)
−
R
(
B
i
→
B
j
)
(6)
where
B
i
=
B
0
or
B
0
, and
B
j
=
B
+
or
B
−
. The two
states
B
i
were defined by flavor-specific decays [13] de-
noted as
B
0
→
ℓ
+
X
,
B
0
→
ℓ
−
X
. The state
B
+
was de-
fined as the remaining state
B
2
after a
c
cK
0
S
decay, and
B
−
as
B
2
after a
c
cK
0
L
decay. In order to use the two
states for testing
T
symmetry in Eq. (6), they must be
orthogonal;
h
B
+
|
B
−
i
= 0, which requires the additional
assumption
(3)
|
A/A
|
= 1 .
In the same 2012 analysis, we demonstrated that
CPT
symmetry is unbroken within uncertainties by measuring
the four rate differences
R
(
B
j
→
B
i
)
−
R
(
B
i
→
B
j
)
.
(7)
For both measurements in Eqs. (6) and (7), expressions
R
i
(
t
) =
N
i
e
−
Γ
t
(1 +
C
i
cos ∆
mt
+
S
i
sin ∆
mt
)
,
(8)
i
= 1
...
8, were fitted to the four time-dependent rates
where the
ℓX
decay precedes the
c
cK
decay, and to
the four rates where the order of the decays is inverted.
The rate ansatz in Eq. (8) requires ∆Γ = 0. The time
t
≥
0 in these expressions is the time between the first
and the second decay of the entangled
B
0
B
0
pair as
defined in Eq. (5). In our 2012 analysis, we named
it ∆
τ
, equal to
t
c
c
K
−
t
ℓX
if the
ℓX
decay occurred
first, and equal to
t
ℓX
−
t
c
cK
with
c
cK
as first decay.
After the fits, the
T
-violating and
CPT
-testing rate
differences were evaluated from the obtained
S
i
and
C
i
results. The
CPT
test showed no
CPT
violation, i. e., it
was compatible with
z
= 0, but no results for Re (
z
) and
Im (
z
) were given in 2012.
Our present analysis uses the eight measured time de-
pendences in the 2012 analysis, i. e. the 16 results
C
i
and
S
i
, for determining
z
. This is possible without assump-
tion (3) since we do not need to use the concept of states
B
+
and
B
−
. We are therefore able to determine the de-
cay parameter
|
A/A
|
in addition to the mixing parame-
ters Re (
z
) and Im (
z
). As in 2012, we use ∆Γ = 0, but we
show at the end of this analysis that the final results are
independent of this constraint. Accepting assumptions
(1) and (2), and in addition
(4) the amplitudes
A
and
A
have a single weak phase,
only two more parameters
|
A/A
|
and Im (
q
A/pA
) are
required in addition to
|
q/p
|
and
z
for a full description
of
CP
violation in time-dependent
B
0
→
c
cK
0
decays.
In this framework,
T
symmetry requires Im (
q
A/pA
) = 0
[14], and
CPT
symmetry requires
|
A/A
|
= 1 [15].
B-MESON DECAY RATES
The time-dependent rates of the decays
B
0
,
B
0
→
c
cK
are sensitive to both symmetries
CPT
and
T
in
B
0
-
B
0
mixing and in
B
0
decays. For decays into final states
f
with amplitudes
A
f
=
A
(
B
0
→
f
) and
A
f
=
A
(
B
0
→
f
),
using
λ
f
=
q
A
f
/
(
pA
f
) and approximating
√
1
−
z
2
= 1,
the rates are given by
R
(
B
0
→
f
) =
|
A
f
|
2
e
−
Γ
t
4
∣
∣
∣
(1
−
z
+
λ
f
) e
i∆
mt
e
∆Γ
t/
4
+ (1 +
z
−
λ
f
) e
−
∆Γ
t/
4
∣
∣
∣
2
,
R
(
B
0
→
f
) =
|
A
f
|
2
e
−
Γ
t
4
∣
∣
∣
(1 +
z
+ 1
/λ
f
) e
i∆
mt
e
∆Γ
t/
4
+ (1
−
z
−
1
/λ
f
) e
−
∆Γ
t/
4
∣
∣
∣
2
.
(9)
For the
CP
eigenstates
c
cK
0
L
(
CP
= +1) and
c
cK
0
S
(
CP
=
−
1) with
A
S
(
L
)
=
A
[
B
0
→
c
cK
0
S
(
L
)
] and
A
S
(
L
)
=
A
[
B
0
→
c
cK
0
S
(
L
)
], assumptions (1) and (2) give
A
S
=
A
L
=
A/
√
2 and
A
S
=
−
A
L
=
A/
√
2. In the
following, we only need to use
λ
S
=
−
λ
L
=
λ
. Setting
∆Γ = 0 and keeping only first-order terms in the small
quantities
|
λ
| −
1,
z
, and
r
=
|
q/p
| −
1, this leads to rate
expressions as given in Eq. (8) with coefficients
5
S
1
=
S
(
ℓ
−
X,c
cK
L
) =
2 Im (
λ
)
1 +
|
λ
|
2
−
Re (
z
)Re (
λ
)Im (
λ
) + Im (
z
)[Re (
λ
)]
2
,
C
1
= +
1
− |
λ
|
2
2
−
Re (
λ
) Re (
z
)
−
Im (
λ
) Im (
z
)
,
S
2
=
S
(
ℓ
+
X,c
cK
L
) =
−
2 Im(
λ
)
1 +
|
λ
|
2
−
Re (
z
)Re (
λ
)Im (
λ
)
−
Im (
z
)[Re (
λ
)]
2
,
C
2
=
−
1
− |
λ
|
2
2
+ Re (
λ
) Re (
z
)
−
Im (
λ
) Im (
z
)
,
S
3
=
S
(
ℓ
−
X,c
cK
S
) =
−
2 Im(
λ
)
1 +
|
λ
|
2
−
Re (
z
)Re (
λ
)Im (
λ
) + Im (
z
)[Re (
λ
)]
2
,
C
3
= +
1
− |
λ
|
2
2
+ Re (
λ
) Re (
z
) + Im (
λ
) Im (
z
)
,
S
4
=
S
(
ℓ
+
X,c
cK
S
) =
2 Im (
λ
)
1 +
|
λ
|
2
−
Re (
z
)Re (
λ
)Im (
λ
)
−
Im (
z
)[Re (
λ
)]
2
,
C
4
=
−
1
− |
λ
|
2
2
−
Re (
λ
) Re (
z
) + Im (
λ
) Im (
z
)
.
(10)
The four other rates
R
5
(
t
)
· · ·
R
8
(
t
) with
c
cK
as the first
decay and
t
ℓX
−
t
c
c
K
=
t
follow from the same two-
decay-time expression [16, 17] as the rates
R
1
...R
4
with
t
c
c
K
−
t
ℓX
=
t
. Therefore, the rates
R
5
(
c
cK
L
,ℓ
−
X
),
R
6
(
c
cK
L
,ℓ
+
X
),
R
7
(
c
cK
S
,ℓ
−
X
), and
R
8
(
c
cK
S
,ℓ
+
X
)
are given by Eq. (8) with the coefficients
S
i
=
−
S
i
−
4
, C
i
= +
C
i
−
4
for
i
= 5
,
6
,
7
,
and 8
.
(11)
The
S
i
and
C
i
results from our 2012 analysis, including
uncertainties and correlation matrices, have been pub-
lished as Supplemental Material [18] in Tables II, III, and
IV. For completeness, we include in Table I the results
and the uncertainties.
TABLE I: Input values from the Supplemental Material [18]
of Ref. [6]. The second column gives the two decays with their
sequence in decay time.
i
decay pairs
S
i
σ
stat
σ
sys
C
i
σ
stat
σ
sys
1
ℓ
−
X, c
cK
L
0.51 0.17 0.11
−
0
.
01 0.13 0.08
2
ℓ
+
X, c
cK
L
−
0
.
69 0.11 0.04
−
0
.
02 0.11 0.08
3
ℓ
−
X, c
cK
S
−
0
.
76 0.06 0.04 0.08 0.06 0.06
4
ℓ
+
X, c
cK
S
0.55 0.09 0.06 0.01 0.07 0.05
5
c
cK
L
, ℓ
−
X
−
0
.
83 0.11 0.06 0.11 0.12 0.08
6
c
cK
L
, ℓ
+
X
0.70 0.19 0.12 0.16 0.13 0.06
7
c
cK
S
, ℓ
−
X
0.67 0.10 0.08 0.03 0.07 0.04
8
c
cK
S
, ℓ
+
X
−
0
.
66 0.06 0.04
−
0
.
05 0.06 0.03
FIT RESULTS
The relations between the 16 observables
y
i
=
S
1
· · ·
C
8
in Eqs. (10) and (11) and the four parameters
p
1
=
(1
− |
λ
|
2
)
/
2,
p
2
= 2 Im (
λ
)
/
(1 +
|
λ
|
2
),
p
3
= Im (
z
), and
p
4
= Re (
z
) are approximately linear. Therefore, the
four parameters can be determined in a two-step lin-
ear
χ
2
fit using matrix algebra. The first-step fit de-
termines
p
1
and
p
2
by fixing Re (
λ
) and Im (
λ
) in the
products Re (
z
)Re (
λ
), Im (
z
)Im (
λ
), Im (
z
)[Re (
λ
)]
2
, and
Re (
z
)Re (
λ
)Im (
λ
). After fixing these terms, the relation
between the vectors
y
and
p
is strictly linear,
y
=
M
1
p,
(12)
where
M
1
uses Im (
λ
) = 0
.
67 and Re (
λ
) =
−
0
.
74, moti-
vated by the results of analyses assuming
CPT
symmetry
[7]. With this ansatz,
χ
2
is given by
χ
2
= (
M
1
p
−
ˆ
y
)
T
G
(
M
1
p
−
ˆ
y
)
,
(13)
where ˆ
y
is the measured vector of observables, and the
weight matrix
G
is taken to be
G
= [
C
stat
(
y
) +
C
sys
(
y
)]
−
1
,
(14)
where
C
stat
(
y
) and
C
sys
(
y
) are the statistical and system-
atic covariance matrices, respectively. The minimum of
χ
2
is reached for
ˆ
p
=
M
1
ˆ
y
with
M
1
= (
M
T
1
G M
1
)
−
1
M
T
1
G ,
(15)
and the uncertainties of ˆ
p
are given by the covariance
matrices
C
stat
(
p
) =
M
1
C
stat
(
y
)
M
T
1
,
C
sys
(
p
) =
M
1
C
sys
(
y
)
M
T
1
,
(16)
6
with the property
C
stat
(
p
) +
C
sys
(
p
) = (
M
T
1
G M
1
)
−
1
.
(17)
This first-step fit yields
p
1
= 0
.
001
±
0
.
023
±
0
.
017
,
p
2
= 0
.
689
±
0
.
030
±
0
.
015
.
(18)
This leads to
|
λ
|
= 1
−
p
1
= 0
.
999
±
0
.
023
±
0
.
017
,
Im (
λ
) = (1
−
p
1
)
p
2
= 0
.
689
±
0
.
034
±
0
.
019
,
Re (
λ
) =
−
(1
−
p
1
)
√
1
−
p
2
2
=
−
0
.
723
±
0
.
043
±
0
.
028
,
(19)
where the negative sign of Re (
λ
) is motivated by four
measurements [19–22]. The results of all four favor
cos 2
β >
0, and in Ref. [22] cos 2
β <
0 is excluded with
4.5
σ
significance.
In the second step, we fix the two
λ
values according
to the
p
1
and
p
2
results of the first step, i.e. to the
central values in Eqs. (19). Equations (12) to (17) are
then applied again, replacing
M
1
with the new relations
matrix
M
2
. This gives the same results for
p
1
and
p
2
as
in Eq. (18), and
p
3
= Im (
z
) = 0
.
010
±
0
.
030
±
0
.
013
,
p
4
= Re (
z
) =
−
0
.
065
±
0
.
028
±
0
.
014
,
(20)
with a
χ
2
value of 6.9 for 12 degrees of freedom.
The Re (
z
) result deviates from 0 by 2
.
1
σ
. The result
for
|
λ
|
can be easily converted into
|
A/A
|
by using the
world average of measurements for
|
q/p
|
. With
|
q/p
|
=
1
.
0008
±
0
.
0008 [7], we obtain
|
A/A
|
= 0
.
999
±
0
.
023
±
0
.
017
,
(21)
in agreement with
CPT
symmetry. Using the matrix
algebra in Eqs. (12) to (17) allows us to determine
the separate statistical and systematic covariance
matrices of the final results, in agreement with
the condition
C
stat
(
p
) +
C
sys
(
p
) = (
M
T
G M
)
−
1
,
where
M
relates
y
and
p
after convergence of
the fit. The statistical correlation coefficients are
ρ
[
|
A/A
|
,
Im (
z
)] = 0
.
03,
ρ
[
|
A/A
|
,
Re (
z
)] = 0
.
44,
and
ρ
[Re (
z
)
,
Im (
z
)] = 0
.
03. The systematic cor-
relation coefficients are
ρ
[
|
A/A
|
,
Im (
z
)] = 0
.
03,
ρ
[
|
A/A
|
,
Re (
z
)] = 0
.
48, and
ρ
[Re (
z
)
,
Im (
z
)] =
−
0
.
15.
ESTIMATING THE INFLUENCE OF ∆Γ
Using an accept/reject algorithm, we have performed
two “toy simulations”, each with
∼
2
×
10
6
events, i.e.
t
values sampled from the distributions
e
−
Γ
t
[1 + Re (
λ
) sinh(∆Γ
t/
2) + Im (
λ
) sin(∆
mt
)]
,
(22)
with ∆Γ = 0 for one simulation and ∆Γ = 0
.
01Γ for
the other one, corresponding to one standard deviation
from the present world average [7]. For both simulations
we use Im (
λ
) = 0
.
67 and Re (
λ
) =
−
0
.
74 and sample
t
values between 0 and +5
/
Γ. We then fit the two samples,
binned in intervals of ∆
t
= 0
.
25
/
Γ, to the expressions
N
e
−
Γ
t
[1 +
C
cos(∆
mt
) +
S
sin(∆
mt
)]
,
(23)
with three free parameters
N
,
C
and
S
. The fit results
agree between the two simulations within 0
.
002 for
C
and 0
.
008 for
S
. We, therefore, conclude that omission
of the sinh term in Ref. [6] has a negligible influence on
the three final results of this analysis.
CONCLUSION
Using 470
×
10
6
B
B
events from
B
A
B
AR
, we determine
Im (
z
) = 0
.
010
±
0
.
030
±
0
.
013
,
Re (
z
) =
−
0
.
065
±
0
.
028
±
0
.
014
,
|
A/A
|
= 0
.
999
±
0
.
023
±
0
.
017
,
where the first uncertainties are statistical and the
second uncertainties are systematic. All three results are
compatible with
CPT
symmetry in
B
0
-
B
0
mixing and
in
B
→
c
cK
decays. The uncertainties on Re (
z
) are com-
parable with those obtained by Belle in 2012 [10] with
535
×
10
6
B
B
events, Re (
z
) =
−
0
.
019
±
0
.
037
±
0
.
033.
The uncertainties on Im (
z
) are considerably larger,
as expected, than those obtained by
B
A
B
AR
in 2006
[8] with dilepton decays from 232
×
10
6
B
B
events,
Im (
z
) =
−
0
.
014
±
0
.
007
±
0
.
003. The result of the
present analysis for Re (
z
)
,
−
0
.
065
±
0
.
028
±
0
.
014,
supersedes the
B
A
B
AR
result of 2004 [9].
ACKNOWLEDGEMENTS
We thank H.-J. Gerber (ETH Zurich) and T. Ruf
(CERN) for very useful discussions on
T
and
CPT
sym-
metry. We are grateful for the excellent luminosity and
machine conditions provided by our PEP-II colleagues,
and for the substantial dedicated effort from the com-
puting organizations that support
B
A
B
AR
. The collab-
orating institutions wish to thank SLAC for its support
and kind hospitality. This work is supported by DOE and
NSF (USA), NSERC (Canada), CEA and CNRS-IN2P3
(France), BMBF and DFG (Germany), INFN (Italy),
FOM (The Netherlands), NFR (Norway), MES (Russia),
7
MINECO (Spain), STFC (United Kingdom), BSF (USA-
Israel). Individuals have received support from the Marie
Curie EIF (European Union) and the A. P. Sloan Foun-
dation (USA).
∗
Now at: Wuhan University, Wuhan 43072, China
†
Now at: Universit`a di Bologna and INFN Sezione di
Bologna, I-47921 Rimini, Italy
‡
Now at: University of Huddersfield, Huddersfield HD1
3DH, UK
§
Now at: University of South Alabama, Mobile,
Alabama 36688, USA
¶
Also at: Universit`a di Sassari, I-07100 Sassari, Italy
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