of 7
Tests of
CPT
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,
8a,8b,8c
A. R. Buzykaev,
8a
V. P. Druzhinin,
8a,8b
V. B. Golubev,
8a,8b
E. A. Kravchenko,
8a,8b
A. P. Onuchin,
8a,8b,8c
S. I. Serednyakov,
8a,8b
Yu. I. Skovpen,
8a,8b
E. P. Solodov,
8a,8b
K. Yu. Todyshev,
8a,8b
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. Röhrken,
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,
16a
C. Bozzi,
16a
R. Calabrese,
16a,16b
G. Cibinetto,
16a,16b
E. Fioravanti,
16a,16b
I. Garzia,
16a,16b
E. Luppi,
16a,16b
V. Santoro,
16a
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,
35a
N. Neri,
35a
F. Palombo,
35a,35b
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,
42a
M. Margoni,
42a,42b
M. Posocco,
42a
M. Rotondo,
42a
G. Simi,
42a,42b
F. Simonetto,
42a,42b
R. Stroili,
42a,42b
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,
44a,44b
E. Manoni,
44a
A. Rossi,
44a
G. Batignani,
45a,45b
S. Bettarini,
45a,45b
M. Carpinelli,
45a,45b
G. Casarosa,
45a,45b
M. Chrzaszcz,
45a
F. Forti,
45a,45b
M. A. Giorgi,
45a,45b
A. Lusiani,
45a,45c
B. Oberhof,
45a,45b
E. Paoloni,
45a,45b
M. Rama,
45a
G. Rizzo,
45a,45b
J. J. Walsh,
45a
A. J. S. Smith,
46
F. Anulli,
47a
R. Faccini,
47a,47b
F. Ferrarotto,
47a
F. Ferroni,
47a,47b
A. Pilloni,
47a,47b
G. Piredda,
47a
C. Bünger,
48
S. Dittrich,
48
O. Grünberg,
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,
60a,60b
F. De Mori,
60a,60b
A. Filippi,
60a
D. Gamba,
60a,60b
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
(
B
A
B
AR
Collaboration)
1
Laboratoire d
Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie,
CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
3
INFN Sezione di Bari and Dipartimento di Fisica, Università 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 California, Berkeley, California 94720, USA
6
Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany
7
University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
8a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
8b
Novosibirsk State University, Novosibirsk 630090, Russia
8c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
9
University of California at Irvine, Irvine, California 92697, USA
10
University of California at Riverside, Riverside, California 92521, USA
11
University of California at Santa Cruz, Institute for Particle 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
16a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
16b
Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy
17
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
18
INFN Sezione di Genova, I-16146 Genova, Italy
PHYSICAL REVIEW D
94,
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=
2016
=
94(1)
=
011101(7)
011101-1
© 2016 American Physical Society
RAPID COMMUNICATIONS
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, Saudi Arabia
23
Johns Hopkins University, Baltimore, Maryland 21218, USA
24
Laboratoire de l
Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,
Centre Scientifique d
Orsay, F-91898 Orsay Cedex, France
25
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
26
University of Liverpool, Liverpool L69 7ZE, United Kingdom
27
Queen Mary, University of London, London E1 4NS, United Kingdom
28
University of London, Royal Holloway and Bedford New College,
Egham, Surrey TW20 0EX, United Kingdom
29
University of Louisville, Louisville, Kentucky 40292, USA
30
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
31
University of Manchester, Manchester M13 9PL, United Kingdom
32
University of Maryland, College Park, Maryland 20742, USA
33
Massachusetts Institute of Technology, Laboratory for Nuclear Science,
Cambridge, Massachusetts 02139, USA
34
McGill University, Montréal, Québec H3A 2T8, Canada
35a
INFN Sezione di Milano, I-20133 Milano, Italy
35b
Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy
36
University of Mississippi, University, Mississippi 38677, USA
37
Université de Montréal, Physique des Particules, Montréal, Québec H3C 3J7, Canada
38
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Università di Napoli Federico II,
I-80126 Napoli, Italy
39
NIKHEF, National Institute for Nuclear Physics and High Energy Physics,
NL-1009 DB Amsterdam, Netherlands
40
University of Notre Dame, Notre Dame, Indiana 46556, USA
41
Ohio State University, Columbus, Ohio 43210, USA
42a
INFN Sezione di Padova, I-35131 Padova, Italy
42b
Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy
43
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS,
Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7,
F-75252 Paris, France
44a
INFN Sezione di Perugia, I-06123 Perugia, Italy
44b
Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy
45a
INFN Sezione di Pisa, I-56127 Pisa, Italy
45b
Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy
45c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
46
Princeton University, Princeton, New Jersey 08544, USA
47a
INFN Sezione di Roma, I-00185 Roma, Italy
47b
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy
48
Universität Rostock, D-18051 Rostock, Germany
49
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
50
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
51
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
52
University of South Carolina, Columbia, South Carolina 29208, 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 Aviv 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
60a
INFN Sezione di Torino, I-10125 Torino, Italy
60b
Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy
61
INFN Sezione di Trieste and Dipartimento di Fisica, Università di Trieste,
I-34127 Trieste, Italy
62
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
63
University of Victoria, Victoria, British Columbia V8W 3P6, Canada
J. P. LEES
et al.
PHYSICAL REVIEW D
94,
011101(R) (2016)
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64
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
65
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 16 May 2016; published 18 July 2016)
Using the eight time dependences e
Γ
t
ð
1
þ
C
i
cos
Δ
mt
þ
S
i
sin
Δ
mt
Þ
for the decays
Υ
ð
4
S
Þ
B
0
̄
B
0
f
j
f
k
,
with the decay into a flavor-specific state
f
j
¼
l

X
before or after the decay into a
CP
eigenstate
f
k
¼
c
̄
cK
S;L
, as measured by the
BABAR
experiment, we determine the three
CPT
-sensitive parameters
Re
ð
z
Þ
and Im
ð
z
Þ
in
B
0
-
̄
B
0
mixing and
j
̄
A=A
j
in
B
0
c
̄
cK
0
decays. We find Im
ð
z
Þ¼
0
.
010

0
.
030

0
.
013
,
Re
ð
z
Þ¼
0
.
065

0
.
028

0
.
014
, and
j
̄
A=A
0
.
999

0
.
023

0
.
017
, in agreement with
CPT
symmetry.
DOI:
10.1103/PhysRevD.94.011101
I. INTRODUCTION
The discovery of
CP
violation in 1964
[1]
motivated
searches for
T
and
CPT
violation. Since
CPT
¼
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
¼
ψ
1
j
B
0
ψ
2
j
̄
B
0
i
;
ð
1
Þ
using the Schrödinger equation
_
Ψ
¼
i
H
Ψ
;
ð
2
Þ
where the Hamiltonian
H
is given by two constant
Hermitian matrices,
H
ij
¼
m
ij
þ
i
Γ
ij
=
2
. In this evolution,
CP
violation is described by three parameters,
j
q=p
j
,
Re
ð
z
Þ
, and Im
ð
z
Þ
, defined by
j
q=p
1
2
Im
ð
m

12
Γ
12
Þ
4
j
m
12
j
2
þj
Γ
12
j
2
;
z
¼
ð
m
11
m
22
Þ
i
ð
Γ
11
Γ
22
Þ
=
2
Δ
m
i
ΔΓ
=
2
;
ð
3
Þ
where
Δ
m
¼
m
ð
B
H
Þ
m
ð
B
L
Þ
2
j
m
12
j
and
ΔΓ
¼
Γ
ð
B
H
Þ
Γ
ð
B
L
Þ
þ
2
j
Γ
12
j
or
2
j
Γ
12
j
are the mass and
the width differences of the two mass eigenstates
(
H
¼
heavy,
L
¼
light) of the Hamiltonian,
B
H
¼ð
p
ffiffiffiffiffiffiffiffiffiffiffi
1
þ
z
p
B
0
q
ffiffiffiffiffiffiffiffiffiffiffi
1
z
p
̄
B
0
Þ
=
ffiffiffi
2
p
;
B
L
¼ð
p
ffiffiffiffiffiffiffiffiffiffiffi
1
z
p
B
0
þ
q
ffiffiffiffiffiffiffiffiffiffiffi
1
þ
z
p
̄
B
0
Þ
=
ffiffiffi
2
p
:
ð
4
Þ
Note that we use the convention with
þ
q
for the light and
q
for the heavy eigenstate. If
j
q=p
j
1
, the evolution
violates the discrete symmetries
CP
and
T
.If
z
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
¼j
q=p
j
1
. Throughout the
following, we neglect contributions of orders
r
2
,
z
2
,
r
z
, and
higher.
The
T
-sensitive mixing parameter
j
q=p
j
has been
determined in several experiments, the present world
average
[7]
being
j
q=p
1
þð
0
.
8

0
.
8
Þ
×
10
3
.The
CPT
-sensitive parameter Im
ð
z
Þ
has been determined by
analyzing the time dependence of dilepton events in the
decay
Υ
ð
4
S
Þ
B
0
̄
B
0
ð
l
þ
ν
X
Þð
l
̄
ν
X
Þ
;the
BABAR
result
[8]
is Im
ð
z
Þ¼ð
13
.
9

7
.
3

3
.
2
Þ
×
10
3
. Since
ΔΓ
is very small, dilepton events are only sensitive to
the product Re
ð
z
Þ
ΔΓ
. Therefore, Re
ð
z
Þ
has so far only
been determined by analyzing the time dependence of
the decays
Υ
ð
4
S
Þ
B
0
̄
B
0
with one
B
meson decaying
into
l
ν
X
and the other one into
c
̄
cK
. With
88
×
10
6
B
̄
B
events,
BABAR
measured Re
ð
z
Þ¼ð
19

48

47
Þ
×
10
3
in 2004
[9]
, while Belle used
535
×
10
6
B
̄
B
events to measure Re
ð
z
Þ¼ð
19

37

33
Þ
×
10
3
in
2012
[10]
.
In our present analysis, we use the final data set of the
BABAR
experiment
[11,12]
with
470
×
10
6
B
̄
B
events for a
new determination of Re
ð
z
Þ
and Im
ð
z
Þ
. 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
Δ
S
¼
Δ
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
;
*
Present address: Wuhan University, Wuhan 43072, China.
Present address: Università di Bologna and INFN Sezione di
Bologna, I-47921 Rimini, Italy.
Present address: University of Huddersfield, Huddersfield
HD1 3DH, United Kingdom.
§
Present address: University of South Alabama, Mobile,
Alabama 36688, USA.
Also at Università di Sassari, I-07100 Sassari, Italy.
TESTS OF
CPT
SYMMETRY IN
B
0
-
̄
B
0
...
PHYSICAL REVIEW D
94,
011101(R) (2016)
011101-3
RAPID COMMUNICATIONS
(2)
CP
violation in
K
0
-
̄
K
0
mixing is negligible, i.e.,
K
0
S
¼ð
K
0
þ
̄
K
0
Þ
=
ffiffiffi
2
p
,
K
0
L
¼ð
K
0
̄
K
0
Þ
=
ffiffiffi
2
p
.
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
ð
4
S
Þ
decays,
B
0
and
̄
B
0
mesons
are produced in the entangled state
ð
B
0
̄
B
0
̄
B
0
B
0
Þ
=
ffiffiffi
2
p
.
When the first meson decays into
f
¼
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
¼
t
2
t
1
0
:
ð
5
Þ
Note that
t
is the only relevant time here; it is the
evolution time of the single-meson state
B
2
in its rest
frame.
The present analysis does not start from raw data but uses
intermediate results from Ref.
[6]
where, as mentioned
above, we used our final data set for the demonstration of
large
T
violation. This was shown in four time-dependent
transition-rate differences
R
ð
B
j
B
i
Þ
R
ð
B
i
B
j
Þ
;
ð
6
Þ
where
B
i
¼
B
0
or
̄
B
0
, and
B
j
¼
B
þ
or
B
. The two states
B
i
were defined by flavor-specific decays
[13]
denoted as
B
0
l
þ
X
,
̄
B
0
l
X
. The state
B
þ
was defined 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
þ
j
B
0
, which requires the additional assumption
(3)
j
̄
A=A
1
.
In the same 2012 analysis, we demonstrated that
CPT
symmetry is unbroken within uncertainties by measuring
the four rate differences
R
ð
B
j
B
i
Þ
R
ð
̄
B
i
B
j
Þ
:
ð
7
Þ
For both measurements in Eqs.
(6)
and
(7)
, expressions
R
i
ð
t
Þ¼
N
i
e
Γ
t
ð
1
þ
C
i
cos
Δ
mt
þ
S
i
sin
Δ
mt
Þ
;
ð
8
Þ
i
¼
1
...
8
, were fitted to the four time-dependent rates
where the
l
X
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
ΔΓ
¼
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
Δ
τ
, equal to
t
c
̄
cK
t
l
X
if the
l
X
decay occurred first, and equal to
t
l
X
t
c
̄
cK
with
c
̄
cK
as the first decay. After the fits, the
T
-violating and
CPT
-testing rate differences were evalu-
ated from the obtained
S
i
and
C
i
results. The
CPT
test
showed no
CPT
violation, i.e., it was compatible with
z
¼
0
, but no results for Re
ð
z
Þ
and Im
ð
z
Þ
were given
in 2012.
Our present analysis uses the eight measured time
dependences in the 2012 analysis, i.e., the 16 results
C
i
and
S
i
, for determining
z
. This is possible without
assumption (3) since we do not need to use the concept
of states
B
þ
and
B
. We are therefore able to determine the
decay parameter
j
̄
A=A
j
in addition to the mixing parameters
Re
ð
z
Þ
and Im
ð
z
Þ
. As in 2012, we use
ΔΓ
¼
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) that the amplitudes
A
and
̄
A
have a single
weak phase,
only two more parameters
j
̄
A=A
j
and Im
ð
q
̄
A=pA
Þ
are
required in addition to
j
q=p
j
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
ð
q
̄
A=pA
Þ¼
0
[14]
, and
CPT
symmetry requires
j
̄
A=A
1
[15]
.
II. 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
¼
A
ð
B
0
f
Þ
and
̄
A
f
¼
A
ð
̄
B
0
f
Þ
, using
λ
f
¼
q
̄
A
f
=
ð
pA
f
Þ
and approximating
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1
z
2
p
¼
1
, the
rates are given by
R
ð
B
0
f
Þ¼
j
A
f
j
2
e
Γ
t
4
1
z
þ
λ
f
Þ
e
i
Δ
mt
e
ΔΓ
t=
4
þð
1
þ
z
λ
f
Þ
e
ΔΓ
t=
4
j
2
;
R
ð
̄
B
0
f
Þ¼
j
̄
A
f
j
2
e
Γ
t
4
1
þ
z
þ
1
=
λ
f
Þ
e
i
Δ
mt
e
ΔΓ
t=
4
þð
1
z
1
=
λ
f
Þ
e
ΔΓ
t=
4
j
2
:
ð
9
Þ
For the
CP
eigenstates
c
̄
cK
0
L
ð
CP
¼þ
1
Þ
and
c
̄
cK
0
S
ð
CP
¼
1
Þ
with
A
S
ð
L
Þ
¼
A
½
B
0
c
̄
cK
0
S
ð
L
Þ

and
̄
A
S
ð
L
Þ
¼
A
½
̄
B
0
c
̄
cK
0
S
ð
L
Þ

, assumptions (1) and (2) give
A
S
¼
A
L
¼
A=
ffiffiffi
2
p
and
̄
A
S
¼
̄
A
L
¼
̄
A=
ffiffiffi
2
p
. In the
following, we only need to use
λ
S
¼
λ
L
¼
λ
. Setting
ΔΓ
¼
0
and keeping only first-order terms in the
small quantities
j
λ
j
1
,
z
, and
r
¼j
q=p
j
1
, this
leads to rate expressions as given in Eq.
(8)
with
coefficients
J. P. LEES
et al.
PHYSICAL REVIEW D
94,
011101(R) (2016)
011101-4
RAPID COMMUNICATIONS
S
1
¼
S
ð
l
X; c
̄
cK
L
Þ
¼
2
Im
ð
λ
Þ
1
þj
λ
j
2
Re
ð
z
Þ
Re
ð
λ
Þ
Im
ð
λ
Þþ
Im
ð
z
Þ½
Re
ð
λ
Þ
2
;
C
1
¼þ
1
j
λ
j
2
2
Re
ð
λ
Þ
Re
ð
z
Þ
Im
ð
λ
Þ
Im
ð
z
Þ
;
S
2
¼
S
ð
l
þ
X; c
̄
cK
L
Þ
¼
2
Im
ð
λ
Þ
1
þj
λ
j
2
Re
ð
z
Þ
Re
ð
λ
Þ
Im
ð
λ
Þ
Im
ð
z
Þ½
Re
ð
λ
Þ
2
;
C
2
¼
1
j
λ
j
2
2
þ
Re
ð
λ
Þ
Re
ð
z
Þ
Im
ð
λ
Þ
Im
ð
z
Þ
;
S
3
¼
S
ð
l
X; c
̄
cK
S
Þ
¼
2
Im
ð
λ
Þ
1
þj
λ
j
2
Re
ð
z
Þ
Re
ð
λ
Þ
Im
ð
λ
Þþ
Im
ð
z
Þ½
Re
ð
λ
Þ
2
;
C
3
¼þ
1
j
λ
j
2
2
þ
Re
ð
λ
Þ
Re
ð
z
Þþ
Im
ð
λ
Þ
Im
ð
z
Þ
;
S
4
¼
S
ð
l
þ
X; c
̄
cK
S
Þ
¼
2
Im
ð
λ
Þ
1
þj
λ
j
2
Re
ð
z
Þ
Re
ð
λ
Þ
Im
ð
λ
Þ
Im
ð
z
Þ½
Re
ð
λ
Þ
2
;
C
4
¼
1
j
λ
j
2
2
Re
ð
λ
Þ
Re
ð
z
Þþ
Im
ð
λ
Þ
Im
ð
z
Þ
:
ð
10
Þ
The four other rates
R
5
ð
t
Þ
R
8
ð
t
Þ
with
c
̄
cK
as the
first decay and
t
l
X
t
c
̄
cK
¼
t
follow from the same two-
decay-time expression
[16,17]
as the rates
R
1
...
R
4
with
t
c
̄
cK
t
l
X
¼
t
. Therefore, the rates
R
5
ð
c
̄
cK
L
;
l
X
Þ
,
R
6
ð
c
̄
cK
L
;
l
þ
X
Þ
,
R
7
ð
c
̄
cK
S
;
l
X
Þ
, and
R
8
ð
c
̄
cK
S
;
l
þ
X
Þ
are
given by Eq.
(8)
with the coefficients
S
i
¼
S
i
4
;C
i
¼þ
C
i
4
for
i
¼
5
;
6
;
7
;
and
8
:
ð
11
Þ
The
S
i
and
C
i
results from our 2012 analysis, including
uncertainties and correlation matrices, have been published
as Supplemental Material
[18]
of Ref.
[6]
in Tables II
IV.
For completeness, we include in Table
I
the results and the
uncertainties.
III. FIT RESULTS
The relations between the 16 observables
y
i
¼
S
1

C
8
in Eqs.
(10)
and
(11)
and the four parameters
p
1
¼ð
1
j
λ
j
2
Þ
=
2
,
p
2
¼
2
Im
ð
λ
Þ
=
ð
1
þj
λ
j
2
Þ
,
p
3
¼
Im
ð
z
Þ
,
and
p
4
¼
Re
ð
z
Þ
are approximately linear. Therefore, the
four parameters can be determined in a two-step linear
χ
2
fit
using matrix algebra. The first-step fit determines
p
1
and
p
2
by fixing Re
ð
λ
Þ
and Im
ð
λ
Þ
in the products Re
ð
z
Þ
Re
ð
λ
Þ
,
Im
ð
z
Þ
Im
ð
λ
Þ
,Im
ð
z
Þ½
Re
ð
λ
Þ
2
, and Re
ð
z
Þ
Re
ð
λ
Þ
Im
ð
λ
Þ
. After
fixing these terms, the relation between the vectors
y
and
p
is strictly linear,
y
¼
M
1
p;
ð
12
Þ
where
M
1
uses Im
ð
λ
Þ¼
0
.
67
and Re
ð
λ
Þ¼
0
.
74
, moti-
vated by the results of analyses assuming
CPT
symmetry
[7]
. With this ansatz,
χ
2
is given by
χ
2
¼ð
M
1
p
ˆ
y
Þ
T
G
ð
M
1
p
ˆ
y
Þ
;
ð
13
Þ
where
ˆ
y
is the measured vector of observables, and the
weight matrix
G
is taken to be
G
¼½
C
stat
ð
y
Þþ
C
sys
ð
y
Þ
1
;
ð
14
Þ
where
C
stat
ð
y
Þ
and
C
sys
ð
y
Þ
are the statistical and systematic
covariance matrices, respectively. The minimum of
χ
2
is
reached for
ˆ
p
¼
M
1
ˆ
y
with
M
1
¼ð
M
T
1
GM
1
Þ
1
M
T
1
G;
ð
15
Þ
and the uncertainties of
ˆ
p
are given by the covariance
matrices
C
stat
ð
p
Þ¼
M
1
C
stat
ð
y
Þ
M
T
1
;
C
sys
ð
p
Þ¼
M
1
C
sys
ð
y
Þ
M
T
1
;
ð
16
Þ
with the property
C
stat
ð
p
Þþ
C
sys
ð
p
Þ¼ð
M
T
1
GM
1
Þ
1
:
ð
17
Þ
This first-step fit yields
p
1
¼
0
.
001

0
.
023

0
.
017
;
p
2
¼
0
.
689

0
.
030

0
.
015
:
ð
18
Þ
This leads to
j
λ
1
p
1
¼
0
.
999

0
.
023

0
.
017
;
Im
ð
λ
Þ¼ð
1
p
1
Þ
p
2
¼
0
.
689

0
.
034

0
.
019
;
Re
ð
λ
Þ¼
ð
1
p
1
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1
p
2
2
q
¼
0
.
723

0
.
043

0
.
028
;
ð
19
Þ
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
þ
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
þ
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
þ
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
þ
X
0
.
66
0.06 0.04
0
.
05
0.06 0.03
TESTS OF
CPT
SYMMETRY IN
B
0
-
̄
B
0
...
PHYSICAL REVIEW D
94,
011101(R) (2016)
011101-5
RAPID COMMUNICATIONS
where the negative sign of Re
ð
λ
Þ
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
¼
Im
ð
z
Þ¼
0
.
010

0
.
030

0
.
013
;
p
4
¼
Re
ð
z
Þ¼
0
.
065

0
.
028

0
.
014
;
ð
20
Þ
with a
χ
2
value of 6.9 for 12 degrees of freedom.
The Re
ð
z
Þ
result deviates from 0 by
2
.
1
σ
. The result
for
j
λ
j
can be easily converted into
j
̄
A=A
j
by using the
world average of measurements for
j
q=p
j
. With
j
q=p
1
.
0008

0
.
0008
[7]
, we obtain
j
̄
A=A
0
.
999

0
.
023

0
.
017
;
ð
21
Þ
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
ð
p
Þþ
C
sys
ð
p
Þ¼ð
M
T
GM
Þ
1
, where
M
relates
y
and
p
after convergence of the fit. The statistical correlation
coefficients are
ρ
½j
̄
A=A
j
;
Im
ð
z
Þ ¼
0
.
03
,
ρ
½j
̄
A=A
j
;
Re
ð
z
Þ ¼
0
.
44
, and
ρ
½
Re
ð
z
Þ
;
Im
ð
z
Þ ¼
0
.
03
. The systematic correla-
tion coefficients are
ρ
½j
̄
A=A
j
;
Im
ð
z
Þ ¼
0
.
03
,
ρ
½j
̄
A=A
j
;
Re
ð
z
Þ ¼
0
.
48
, and
ρ
½
Re
ð
z
Þ
;
Im
ð
z
Þ ¼
0
.
15
.
IV. ESTIMATING THE INFLUENCE OF
Δ
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
½
1
þ
Re
ð
λ
Þ
sinh
ð
ΔΓ
t=
2
Þþ
Im
ð
λ
Þ
sin
ð
Δ
mt
Þ
;
ð
22
Þ
with
ΔΓ
¼
0
for one simulation and
ΔΓ
¼
0
.
01
Γ
for the
other one, corresponding to one standard deviation from the
present world average
[7]
. For both simulations we use
Im
ð
λ
Þ¼
0
.
67
and Re
ð
λ
Þ¼
0
.
74
and sample
t
values
between 0 and
þ
5
=
Γ
. We then fit the two samples, binned
in intervals of
Δ
t
¼
0
.
25
=
Γ
, to the expressions
N
e
Γ
t
½
1
þ
C
cos
ð
Δ
mt
Þþ
S
sin
ð
Δ
mt
Þ
;
ð
23
Þ
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.
V. CONCLUSION
Using
470
×
10
6
B
̄
B
events from
BABAR
, we determine
Im
ð
z
Þ¼
0
.
010

0
.
030

0
.
013
;
Re
ð
z
Þ¼
0
.
065

0
.
028

0
.
014
;
j
̄
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
ð
z
Þ
are compa-
rable with those obtained by Belle in 2012
[10]
with
535
×
10
6
B
̄
B
events, Re
ð
z
Þ¼
0
.
019

0
.
037

0
.
033
.
The uncertainties on Im
ð
z
Þ
are considerably larger, as
expected, than those obtained by
BABAR
in 2006
[8]
with
dilepton decays from
232
×
10
6
B
̄
B
events, Im
ð
z
Þ¼
0
.
014

0
.
007

0
.
003
. The result of the present analysis
for Re
ð
z
Þ
;
0
.
065

0
.
028

0
.
014
, supersedes the
BABAR
result of 2004
[9]
.
ACKNOWLEDGMENTS
We thank H.-J. Gerber (ETH Zurich) and T. Ruf
(CERN) for very useful discussions on
T
and
CPT
symmetry. We are grateful for the excellent luminosity
and machine conditions provided by our PEP-II col-
leagues, and for the substantial dedicated effort from the
computing organizations that support
BABAR
. The col-
laborating institutions 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 (Netherlands), NFR (Norway), MES (Russia),
MINECO (Spain), STFC (United Kingdom), and BSF
(USA-Israel). Individuals have received support from the
Marie Curie EIF (European Union) and the A. P. Sloan
Foundation (USA).
J. P. LEES
et al.
PHYSICAL REVIEW D
94,
011101(R) (2016)
011101-6
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TESTS OF
CPT
SYMMETRY IN
B
0
-
̄
B
0
...
PHYSICAL REVIEW D
94,
011101(R) (2016)
011101-7
RAPID COMMUNICATIONS