of 26
Measurement of Collins asymmetries in inclusive production of charged
pion pairs in
e
þ
e
annihilation at
BABAR
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3a,3b
G. Eigen,
4
B. Stugu,
4
D. N. Brown,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
M. J. Lee,
5
G. Lynch,
5
H. Koch,
6
T. Schroeder,
6
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
R. Y. So,
7
A. Khan,
8
V. E. Blinov,
9a,9c
A. R. Buzykaev,
9a
V. P. Druzhinin,
9a,9b
V. B. Golubev,
9a,9b
E. A. Kravchenko,
9a,9b
A. P. Onuchin,
9a,9c
S. I. Serednyakov,
9a,9b
Yu. I. Skovpen,
9a,9b
E. P. Solodov,
9a,9b
K. Yu. Todyshev,
9a,9b
A. N. Yushkov,
9a
D. Kirkby,
10
A. J. Lankford,
10
M. Mandelkern,
10
B. Dey,
11
J. W. Gary,
11
O. Long,
11
G. M. Vitug,
11
C. Campagnari,
12
M. Franco Sevilla,
12
T. M. Hong,
12
D. Kovalskyi,
12
J. D. Richman,
12
C. A. West,
12
A. M. Eisner,
13
W. S. Lockman,
13
B. A. Schumm,
13
A. Seiden,
13
D. S. Chao,
14
C. H. Cheng,
14
B. Echenard,
14
K. T. Flood,
14
D. G. Hitlin,
14
P. Ongmongkolkul,
14
F. C. Porter,
14
R. Andreassen,
15
Z. Huard,
15
B. T. Meadows,
15
B. G. Pushpawela,
15
M. D. Sokoloff,
15
L. Sun,
15
P. C. Bloom,
16
W. T. Ford,
16
A. Gaz,
16
U. Nauenberg,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17
,
W. H. Toki,
17
B. Spaan,
18
R. Schwierz,
19
D. Bernard,
20
M. Verderi,
20
S. Playfer,
21
D. Bettoni,
22a
C. Bozzi,
22a
R. Calabrese,
22a,22b
G. Cibinetto,
22a,22b
E. Fioravanti,
22a,22b
I. Garzia,
22a,22b
E. Luppi,
22a,22b
L. Piemontese,
22a
V. Santoro,
22a
R. Baldini-Ferroli,
23
A. Calcaterra,
23
R. de Sangro,
23
G. Finocchiaro,
23
S. Martellotti,
23
P. Patteri,
23
I. M. Peruzzi,
23
,
M. Piccolo,
23
M. Rama,
23
A. Zallo,
23
R. Contri,
24a,24b
E. Guido,
24a,24b
M. Lo Vetere,
24a,24b
M. R. Monge,
24a,24b
S. Passaggio,
24a
C. Patrignani,
24a,24b
E. Robutti,
24a
B. Bhuyan,
25
V. Prasad,
25
M. Morii,
26
A. Adametz,
27
U. Uwer,
27
H. M. Lacker,
28
P. D. Dauncey,
29
U. Mallik,
30
C. Chen,
31
J. Cochran,
31
W. T. Meyer,
31
S. Prell,
31
A. V. Gritsan,
32
N. Arnaud,
33
M. Davier,
33
D. Derkach,
33
G. Grosdidier,
33
F. Le Diberder,
33
A. M. Lutz,
33
B. Malaescu,
33
P. Roudeau,
33
A. Stocchi,
33
G. Wormser,
33
D. J. Lange,
34
D. M. Wright,
34
J. P. Coleman,
35
J. R. Fry,
35
E. Gabathuler,
35
D. E. Hutchcroft,
35
D. J. Payne,
35
C. Touramanis,
35
A. J. Bevan,
36
F. Di Lodovico,
36
R. Sacco,
36
G. Cowan,
37
J. Bougher,
38
D. N. Brown,
38
C. L. Davis,
38
A. G. Denig,
39
M. Fritsch,
39
W. Gradl,
39
K. Griessinger,
39
A. Hafner,
39
E. Prencipe,
39
K. R. Schubert,
39
R. J. Barlow,
40
,
G. D. Lafferty,
40
E. Behn,
41
R. Cenci,
41
B. Hamilton,
41
A. Jawahery,
41
D. A. Roberts,
41
R. Cowan,
42
D. Dujmic,
42
G. Sciolla,
42
R. Cheaib,
43
P. M. Patel,
43
,*
S. H. Robertson,
43
P. Biassoni,
44a,44b
N. Neri,
44a
F. Palombo,
44a,44b
L. Cremaldi,
45
R. Godang,
45
P. Sonnek,
45
D. J. Summers,
45
M. Simard,
46
P. Taras,
46
G. De Nardo,
47a,47b
D. Monorchio,
47a,47b
G. Onorato,
47a,47b
C. Sciacca,
47a,47b
M. Martinelli,
48
G. Raven,
48
C. P. Jessop,
49
J. M. LoSecco,
49
K. Honscheid,
50
R. Kass,
50
J. Brau,
51
R. Frey,
51
N. B. Sinev,
51
D. Strom,
51
E. Torrence,
51
E. Feltresi,
52a,52b
M. Margoni,
52a,52b
M. Morandin,
52a
M. Posocco,
52a
M. Rotondo,
52a
G. Simi,
52a
F. Simonetto,
52a,52b
R. Stroili,
52a,52b
S. Akar,
53
E. Ben-Haim,
53
M. Bomben,
53
G. R. Bonneaud,
53
H. Briand,
53
G. Calderini,
53
J. Chauveau,
53
Ph. Leruste,
53
G. Marchiori,
53
J. Ocariz,
53
S. Sitt,
53
M. Biasini,
54a,54b
E. Manoni,
54a
S. Pacetti,
54a,54b
A. Rossi,
54a
C. Angelini,
55a,55b
G. Batignani,
55a,55b
S. Bettarini,
55a,55b
M. Carpinelli,
55a,55b
,**
G. Casarosa,
55a,55b
A. Cervelli,
55a,55b
F. Forti,
55a,55b
M. A. Giorgi,
55a,55b
A. Lusiani,
55a,55c
B. Oberhof,
55a,55b
E. Paoloni,
55a,55b
A. Perez,
55a
G. Rizzo,
55a,55b
J. J. Walsh,
55a
D. Lopes Pegna,
56
J. Olsen,
56
A. J. S. Smith,
56
R. Faccini,
57a,57b
F. Ferrarotto,
57a
F. Ferroni,
57a,57b
M. Gaspero,
57a,57b
L. Li Gioi,
57a
G. Piredda,
57a
C. Bünger,
58
O. Grünberg,
58
T. Hartmann,
58
T. Leddig,
58
C. Voß,
58
R. Waldi,
58
T. Adye,
59
E. O. Olaiya,
59
F. F. Wilson,
59
S. Emery,
60
G. Hamel de Monchenault,
60
G. Vasseur,
60
Ch. Yèche,
60
F. Anulli,
61
,
††
D. Aston,
61
D. J. Bard,
61
J. F. Benitez,
61
C. Cartaro,
61
M. R. Convery,
61
J. Dorfan,
61
G. P. Dubois-Felsmann,
61
W. Dunwoodie,
61
M. Ebert,
61
R. C. Field,
61
B. G. Fulsom,
61
A. M. Gabareen,
61
M. T. Graham,
61
C. Hast,
61
W. R. Innes,
61
P. Kim,
61
M. L. Kocian,
61
D. W. G. S. Leith,
61
P. Lewis,
61
D. Lindemann,
61
B. Lindquist,
61
S. Luitz,
61
V. Luth,
61
H. L. Lynch,
61
D. B. MacFarlane,
61
D. R. Muller,
61
H. Neal,
61
S. Nelson,
61
M. Perl,
61
T. Pulliam,
61
B. N. Ratcliff,
61
A. Roodman,
61
A. A. Salnikov,
61
R. H. Schindler,
61
A. Snyder,
61
D. Su,
61
M. K. Sullivan,
61
J. Va
vra,
61
A. P. Wagner,
61
W. F. Wang,
61
W. J. Wisniewski,
61
M. Wittgen,
61
D. H. Wright,
61
H. W. Wulsin,
61
V. Ziegler,
61
W. Park,
62
M. V. Purohit,
62
R. M. White,
62
,
‡‡
J. R. Wilson,
62
A. Randle-Conde,
63
S. J. Sekula,
63
M. Bellis,
64
P. R. Burchat,
64
T. S. Miyashita,
64
E. M. T. Puccio,
64
M. S. Alam,
65
J. A. Ernst,
65
R. Gorodeisky,
66
N. Guttman,
66
D. R. Peimer,
66
A. Soffer,
66
S. M. Spanier,
67
J. L. Ritchie,
68
A. M. Ruland,
68
R. F. Schwitters,
68
B. C. Wray,
68
J. M. Izen,
69
X. C. Lou,
69
F. Bianchi,
70a,70b
F. De Mori,
70a,70b
A. Filippi,
70a
D. Gamba,
70a,70b
S. Zambito,
70a,70b
L. Lanceri,
71a,71b
L. Vitale,
71a,71b
F. Martinez-Vidal,
72
A. Oyanguren,
72
P. Villanueva-Perez,
72
H. Ahmed,
73
J. Albert,
73
Sw. Banerjee,
73
F. U. Bernlochner,
73
H. H. F. Choi,
73
G. J. King,
73
R. Kowalewski,
73
M. J. Lewczuk,
73
T. Lueck,
73
I. M. Nugent,
73
J. M. Roney,
73
R. J. Sobie,
73
N. Tasneem,
73
T. J. Gershon,
74
P. F. Harrison,
74
T. E. Latham,
74
H. R. Band,
75
S. Dasu,
75
Y. Pan,
75
R. Prepost,
75
and S. L. Wu
75
(
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
Facultat de Fisica, Departament ECM, Universitat de Barcelona, E-08028 Barcelona, Spain
3a
INFN Sezione di Bari, I-70126 Bari, Italy
3b
Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy
PHYSICAL REVIEW D
90,
052003 (2014)
1550-7998
=
2014
=
90(5)
=
052003(26)
052003-1
© 2014 American Physical Society
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
8
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
9a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
9b
Novosibirsk State University, Novosibirsk 630090, Russia
9c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
10
University of California at Irvine, Irvine, California 92697, USA
11
University of California at Riverside, Riverside, California 92521, USA
12
University of California at Santa Barbara, Santa Barbara, California 93106, USA
13
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
14
California Institute of Technology, Pasadena, California 91125, USA
15
University of Cincinnati, Cincinnati, Ohio 45221, USA
16
University of Colorado, Boulder, Colorado 80309, USA
17
Colorado State University, Fort Collins, Colorado 80523, USA
18
Fakultät Physik, Technische Universität Dortmund, D-44221 Dortmund, Germany
19
Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany
20
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
21
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
22a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
22b
Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy
23
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
24a
INFN Sezione di Genova, I-16146 Genova, Italy
24b
Dipartimento di Fisica, Università di Genova, I-16146 Genova, Italy
25
Indian Institute of Technology Guwahati, Guwahati, Assam 781 039, India
26
Harvard University, Cambridge, Massachusetts 02138, USA
27
Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
28
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
29
Imperial College London, London SW7 2AZ, United Kingdom
30
University of Iowa, Iowa City, Iowa 52242, USA
31
Iowa State University, Ames, Iowa 50011-3160, USA
32
Johns Hopkins University, Baltimore, Maryland 21218, USA
33
Laboratoire de l
Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,
Centre Scientifique d
Orsay, F-91898 Orsay Cedex, France
34
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
35
University of Liverpool, Liverpool L69 7ZE, United Kingdom
36
Queen Mary, University of London, London E1 4NS, United Kingdom
37
University of London, Royal Holloway and Bedford New College, Egham,
Surrey TW20 0EX, United Kingdom
38
University of Louisville, Louisville, Kentucky 40292, USA
39
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
40
University of Manchester, Manchester M13 9PL, United Kingdom
41
University of Maryland, College Park, Maryland 20742, USA
42
Laboratory for Nuclear Science, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
43
McGill University, Montréal, Québec H3A 2T8, Canada
44a
INFN Sezione di Milano, I-20133 Milano, Italy
44b
Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy
45
University of Mississippi, Oxford, Mississippi 38677, USA
46
Physique des Particules, Université de Montréal, Montréal, Québec H3C 3J7, Canada
47a
INFN Sezione di Napoli, I-80126 Napoli, Italy
47b
Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy
48
NIKHEF, National Institute for Nuclear Physics and High Energy Physics,
NL-1009 DB Amsterdam, The Netherlands
49
University of Notre Dame, Notre Dame, Indiana 46556, USA
50
Ohio State University, Columbus, Ohio 43210, USA
51
University of Oregon, Eugene, Oregon 97403, USA
52a
INFN Sezione di Padova, I-35131 Padova, Italy
52b
Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy
J. P. LEES
et al.
PHYSICAL REVIEW D
90,
052003 (2014)
052003-2
53
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS,
Université Pierre et Marie Curie-Paris 6, Université Denis Diderot-Paris 7, F-75252 Paris, France
54a
INFN Sezione di Perugia, I-06123 Perugia, Italy
54b
Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy
55a
INFN Sezione di Pisa, I-56127 Pisa, Italy
55b
Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy
55c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
56
Princeton University, Princeton, New Jersey 08544, USA
57a
INFN Sezione di Roma, I-00185 Roma, Italy
57b
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy
58
Universität Rostock, D-18051 Rostock, Germany
59
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
60
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
61
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
62
University of South Carolina, Columbia, South Carolina 29208, USA
63
Southern Methodist University, Dallas, Texas 75275, USA
64
Stanford University, Stanford, California 94305-4060, USA
65
State University of New York, Albany, New York 12222, USA
66
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
67
University of Tennessee, Knoxville, Tennessee 37996, USA
68
University of Texas at Austin, Austin, Texas 78712, USA
69
University of Texas at Dallas, Richardson, Texas 75083, USA
70a
INFN Sezione di Torino, I-10125 Torino, Italy
70b
Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy
71a
INFN Sezione di Trieste, I-34127 Trieste, Italy
71b
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy
72
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
73
University of Victoria, Victoria, British Columbia V8W 3P6, Canada
74
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
75
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 24 September 2013; published 9 September 2014)
We present measurements of Collins asymmetries in the inclusive process
e
þ
e
ππ
X
, where
π
stands
for charged pions, at a center-of-mass energy of 10.6 GeV. We use a data sample of
468
fb
1
collected by
the
BABAR
experiment at the PEP-II
B
factory at SLAC, and consider pairs of charged pions produced in
opposite hemispheres of hadronic events. We observe clear asymmetries in the distributions of the
azimuthal angles in two distinct reference frames. We study the dependence of the asymmetry on several
kinematic variables, finding that it increases with increasing pion momentum and momentum transverse to
the analysis axis, and with increasing angle between the thrust and beam axis.
DOI:
10.1103/PhysRevD.90.052003
PACS numbers: 13.66.Bc, 13.87.Fh, 13.88.+e, 14.65.-q
I. INTRODUCTION
Parton fragmentation functions describe the probability
for a parton to fragment into a hadron carrying a certain
fraction
z
of the parton momentum. These functions
are denoted
D
i
h
ð
z
Þ
, where
i
represents the fragmenting
parton (
g; u;
u; d;
d;
...
), and
h
is the produced hadron.
Since the
D
i
h
ð
z
Þ
incorporate the long distance, nonpertur-
bative physics of the hadronization processes, they cannot
be calculated in perturbative QCD, but can be evolved
from a starting distribution at a defined energy scale.
Fragmentation processes have been studied in lepton-
hadron and hadron-hadron scattering, as well as in
e
þ
e
annihilation, which provides the cleanest environment
since no hadrons are present in the initial state. Due to
the large amount of experimental data collected at several
e
þ
e
facilities, mainly LEP
[1
3]
and SLC
[4
6]
at
high energies, and, recently, PEP-II
[7]
and KEKB
[8]
at
the center-of-mass energy
ffiffiffi
s
p
10
GeV, the unpolarized
functions are presently well known.
*
Deceased.
Presentaddress:UniversityofTabuk,Tabuk71491,SaudiArabia.
Also at Università di Perugia, Dipartimento di Fisica,
Perugia, Italy.
§
Present address: Laboratoire de Physique Nucláire et de
Hautes Energies, IN2P3/CNRS, Paris, France.
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, Sassari, Italy.
††
Also at INFN Sezione di Roma, Roma, Italy.
‡‡
Now at Universidad Técnica Federico Santa Maria,
Valparaiso 2390123, Chile.
MEASUREMENT OF COLLINS ASYMMETRIES IN
...
PHYSICAL REVIEW D
90,
052003 (2014)
052003-3
Transverse spin-dependent effects in fragmentation proc-
esses were first proposed by Collins
[9,10]
, who introduced
the chiral-odd polarized Collins fragmentation function
H
1
. It describes the relation between the transverse spin of
the fragmenting quark and the azimuthal distribution of
final-state hadrons around the quark momentum (spin-orbit
correlation). In the transverse-momentum approach,
H
1
depends on the hadron fractional energy
z
¼
2
E
h
=
ffiffiffi
s
p
,
where
E
h
and
ffiffiffi
s
p
=
2
are, respectively, the hadron and beam
energy in the center-of-mass system, and on the magnitude
of the hadron transverse momentum
P
h
with respect to the
three-momentum of the fragmenting quark.
The number density for finding a spinless hadron
h
, with
mass
M
h
, produced from a transversely polarized quark
(
q
) is defined in terms of the leading-twist unpolarized
D
q
1
,
and Collins
H
q
1
fragmentation functions, as
[11]
D
q
h
ð
z;
P
h
Þ¼
D
q
1
ð
z; P
h
2
Þþ
H
q
1
ð
z; P
h
2
Þ
ð
ˆ
k
×
P
h
Þ
·
S
q
zM
h
:
ð
1
Þ
The term containing the Collins function depends on the
spin vector of the quark
S
q
, and introduces an azimuthal
asymmetry in the distribution of hadrons around the quark
three-momentum direction
ˆ
k
. The triple product of Eq.
(1)
,
in fact, produces a cos
φ
modulation, where
φ
is the
azimuthal angle between the plane perpendicular to the
quark spin, and the plane determined by
P
h
and
ˆ
k
,as
shown in Fig.
1
. In the literature, the amplitude of this
modulation is called the Collins asymmetry or the Collins
effect.
Experimental evidence for a nonzero Collins function
was found by the HERMES
[12,13]
and COMPASS
[14,15]
Collaborations, from the analysis of semi-inclusive
deep inelastic scattering (SIDIS) of leptons off transversely
polarized hydrogen and/or deuteron targets. The cross
sections for these processes are proportional to the con-
volution of the Collins function and the chiral-odd trans-
versity function
[16]
. The latter is the least well known
among the twist-two parton distribution functions of the
nucleon, and plays a crucial role in understanding the spin
properties. In order to extract the transversity and the
Collins functions from SIDIS data, independent measure-
ment of at least one of them is needed.
In
e
þ
e
annihilation to a quark-antiquark pair, each
quark spin is unknown: the Collins asymmetry in a single
jet (
q
hX
) will be zero. However, the Collins effect
can be observed when the products of the quark and
antiquark fragmentation are simultaneously considered.
Experimentally, this is achieved by studying the process
e
þ
e
q
q
h
1
h
2
X
, where
h
1
(
h
2
) is a spinless hadron
(
π
or
K
) coming from the fragmenting quark
q
ð
q
Þ
.
Events with a two-jet topology are selected, and the
correlation between the azimuthal angles of pairs of
hadrons detected in opposite hemispheres with respect to
the plane perpendicular to the jet axis is analyzed. The
resulting azimuthal asymmetry is proportional to the
product of two Collins functions.
The first efforts to measure Collins asymmetries in
e
þ
e
annihilation, by studying
Z
0
2
jets events, were per-
formed by the DELPHI Collaboration
[17]
, while the
first observation was obtained by the Belle Collaboration
[18
20]
, from a study of inclusive production of charged
pion pairs at a center-of-mass (c.m.) energy of approx-
imately 10.6 GeV. Assuming the universality of the Collins
function
[21,22]
, a global analysis of SIDIS and
e
þ
e
annihilation data has been carried out by the authors of
Refs.
[23,24]
, allowing the simultaneous extraction of the
transversity and Collins functions for the pion system.
We report the measurements of the azimuthal modulation
due to the Collins effect (Collins asymmetries) in the
process
e
þ
e
q
q
ππ
X
, where
π
stands for charged
pion and
q
for a light quark:
u; d; s
. We reproduce the Belle
analysis
[19,20]
of the azimuthal asymmetries as a function
of the pions fractional energy in two reference frames. We
also perform a new measurement of the asymmetries as a
function of the transverse momentum
p
t
of pions with
respect to the analysis axis.
II. ANALYSIS OVERVIEW
Charged pions are selected in opposite jets of hadronic
events according to the thrust axis of the event
[25,26]
,
which permits the identification of two hemispheres (called
1 and 2, respectively, along and opposite to the thrust axis
direction) and to label the two pions as
π
1
and
π
2
. The
analysis is performed in two convenient reference frames:
the thrust reference frame, defined in Sec.
II A
, and the
second hadron momentum frame, defined in Sec.
II B
.
This choice follows the scheme outlined by authors of
Refs.
[27,28]
, and allows a direct comparison of our results
with the Belle measurements. Section
III
summarizes the
data sets used, while Sec.
IV
describes in detail the event
and track selection. The analysis method is discussed in
Secs.
V
and
VI
. Dilutions of the asymmetries induced by
background sources and by detector effects not related to
FIG. 1 (color online). Definition of the azimuthal angle
φ
for a
quark with transverse spin
S
q
which fragments into a spinless
hadron of momentum
P
h
with a component
P
h
transverse to the
quark momentum
k
.
J. P. LEES
et al.
PHYSICAL REVIEW D
90,
052003 (2014)
052003-4
the Collins function are discussed in Secs.
VII
and
VIII
,
respectively. Studies of possible systematic effects are
summarized in Sec.
IX
, while the final results on Collins
asymmetry for light quark fragmentation are reported
in Sec.
X
.
A. Thrust reference frame: RF12
As mentioned in Sec.
I
, the Collins asymmetry manifests
itself as an azimuthal modulation of two final state pions
around the fragmenting quark-antiquark momentum. The
q
q
direction is not accessible to a direct measurement and
is approximated by the thrust axis of the event
[26]
. The
kinematics in the
e
þ
e
c.m. system corresponding to
e
þ
e
π
1
π
2
X
, together with the definition of the two
azimuthal angles, are schematically represented in Fig.
2
.
We refer to this frame as the thrust reference frame or
RF12, since the thrust axis serves as reference axis for the
azimuthal angles. The correlation of the quark and anti-
quark Collins functions in opposite hemispheres gives a
product of two modulations for the azimuthal angles
φ
1
and
φ
2
, resulting in a cos
ð
φ
1
þ
φ
2
Þ
modulation. The azimuthal
angles are defined as
φ
1
;
2
¼
sign
½
ˆ
n
·
ˆ
u
×
ˆ
n
Þ
×
ð
ˆ
n
×
ˆ
P
1
;
2
Þg
× arccos

ˆ
u
×
ˆ
n
j
ˆ
u
×
ˆ
n
j
·
ˆ
n
×
P
1
;
2
j
ˆ
n
×
P
1
;
2
j

;
ð
2
Þ
where
ˆ
u
is a unit vector defined along the direction of
the electron beam,
ˆ
n
is the thrust axis, and
P
1
;
2
is the
three-momentum vector of the pion detected in the first (
π
1
)
or in the second (
π
2
) hemisphere.
The differential cross section depends on the fractional
energies
z
1
and
z
2
of the two pions, and on the sum of the
azimuthal angles
φ
1
and
φ
2
. It can be written as
[28]
d
σ
ð
e
þ
e
π
1
π
2
X
Þ
d
z
1
d
z
2
d
φ
1
d
φ
2
d cos
θ
th
¼
X
q;
q
3
α
2
s
e
2
q
4
z
2
1
z
2
2
×
1
þ
cos
2
θ
th
Þ
D
q;
½
0

1
ð
z
1
Þ
D
q;
½
0

1
ð
z
2
Þþ
sin
2
θ
th
cos
ð
φ
1
þ
φ
2
Þ
H
q;
½
1

1
ð
z
1
Þ
H
q;
½
1

1
ð
z
2
Þg
;
ð
3
Þ
where the summation runs over all quark flavors accessible
at the c.m. energy
ffiffiffi
s
p
,
e
q
is the charge of the quark
q
in
units of
e
, and the antiquark fragmentation function is
denoted by a bar. The so-called transverse moments of the
fragmentation functions are defined as
[28]
F
½
n

ð
z
Þ¼
Z
d
j
k
2
T
j

j
k
T
j
M
π

n
F
ð
z;
k
2
T
Þ
;
ð
4
Þ
with
F
D
q
1
,
D
q
1
,
H
q
1
, and
H
q
1
. In this equation, the pion
transverse momentum has been rewritten in terms of the
quark intrinsic transverse momentum
k
T
1
:
P
h
¼
z
k
T
, and
M
π
is the pion mass. The Collins asymmetry can be
extracted by measuring the cosine modulation of the
distribution of the quantity
ð
φ
1
þ
φ
2
Þ
on top of the uniform
distribution due to the unpolarized part of the fragmentation
function. Dividing the full
ð
φ
1
þ
φ
2
Þ
range into one
hundred intervals, we define the normalized azimuthal
distribution as
R
12
ð
φ
1
þ
φ
2
Þ¼
N
ð
φ
1
þ
φ
2
Þ
h
N
12
i
;
ð
5
Þ
with
N
ð
φ
1
þ
φ
2
Þ
the di-pion yield in each
ð
φ
1
þ
φ
2
Þ
subdivision, and
h
N
12
i
the average bin contents.
B. Second-pion reference frame: RF0
The azimuthal asymmetries can also be measured in a
different reference frame: Following Ref.
[27]
, we use the
direction of the observed pion
π
2
as the reference axis, and
we define the scattering plane by the beam axis and the
momentum
P
2
of that pion, as illustrated in Fig.
3
. Also
in this frame, the kinematic variables are calculated in the
e
þ
e
c.m. system, but only one azimuthal angle,
φ
0
,is
defined:
FIG. 2 (color online). Thrust reference frame (RF12). The
azimuthal angles
φ
1
and
φ
2
are the angles between the scattering
plane and the transverse pion momenta
p
ti
around the thrust axis.
The polar angle
θ
th
is defined as the angle between the beam axis
and the thrust axis
ˆ
n
. The pion transverse momenta
p
ti
used in the
analysis differ from the corresponding
P
h
, which refer to the true
q
q
direction.
1
Throughout the paper we use the subscript
T
to denote the
transverse component of a vector to the di-pion axis in the frame
where they are collinear, while the superscript
indicates the
component transverse to the
q
q
axis.
MEASUREMENT OF COLLINS ASYMMETRIES IN
...
PHYSICAL REVIEW D
90,
052003 (2014)
052003-5
φ
0
¼
sign
½
P
2
·
ˆ
u
×
P
2
Þ
×
ð
P
2
×
P
1
Þg
× arccos

ˆ
u
×
P
2
j
ˆ
u
×
P
2
j
·
P
2
×
P
1
j
P
2
×
P
1
j

:
ð
6
Þ
We refer to this frame as the second-pion reference frame,
or RF0. At leading order in the strong coupling
α
s
, the
differential cross section is given by
[27]
d
σ
ð
e
þ
e
π
1
π
2
X
Þ
d
z
1
d
z
2
d
2
q
T
d cos
ð
θ
2
Þ
d
φ
0
¼
3
α
2
s
z
2
1
z
2
2
4
×

ð
1
þ
cos
2
θ
2
Þ
F
ð
D
1
ð
z
1
Þ
D
1
ð
z
2
ÞÞ þ
sin
2
θ
2
cos
ð
2
φ
0
Þ
×
F

ð
2
ˆ
h
·
k
T
ˆ
h
·
p
T
k
T
·
p
T
Þ
H
1
ð
z
1
Þ
H
1
ð
z
2
Þ
M
2
π

;
ð
7
Þ
where
j
q
T
Q
t
is the transverse momentum of the virtual
photon from
e
þ
e
annihilation in the frame where
P
1
and
P
2
are collinear
[28]
.
F
is a convolution integral over the
transverse momenta
P
1
¼
z
1
k
T
and
P
2
¼
z
2
p
T
, with
k
T
and
p
T
the transverse momenta of the two fragmenting
quarks,
F
ð
X
X
Þ¼
X
a;
a
e
2
a
Z
d
2
k
T
d
2
p
T
δ
2
ð
p
T
þ
k
T
q
T
Þ
×
X
ð
z
1
;z
2
1
k
2
T
Þ
X
ð
z
2
;z
2
2
p
2
T
Þ
;
ð
8
Þ
and
ˆ
h
is the unit vector in the direction of the transverse
momentum of the first hadron relative to the axis defined by
the second hadron.
In this frame, the modulation due to the Collins effect is
in the cosine of twice the azimuthal angle
φ
0
, and the
normalized distribution is defined as
R
0
ð
2
φ
0
Þ¼
N
ð
2
φ
0
Þ
h
N
0
i
:
ð
9
Þ
The differential cross sections in Eq.
(3)
and Eq.
(7)
for
the two reference frames are related to each other.
Integrating the first equation over
φ
1
and
φ
2
, and the
second over
φ
0
and
q
T
, the same unpolarized cross section
is obtained. A similar relation exists for the Collins
contributions. However, due to the additional convolution
over the transverse momentum
q
T
, the intrinsic transverse
momentum dependence of the Collins function should
be known to exploit it. The majority of authors assume
that the Collins function is a Gaussian in
k
T
,asisthe
unpolarized fragmentation function, with a different width
from the unpolarized case. More details can be found, for
example, in Refs.
[29
31]
.
C. Unlike, like, and charged pion pairs
The possibility to select pion pairs with equal or opposite
charges allows sensitivity to
favored
and
disfavored
fragmentation processes. A favored fragmentation function
describes the fragmentation of a quark of flavor
q
into a
hadron containing a valence quark of the same flavor: for
example,
u
π
þ
and
d
π
. Vice versa, we refer to
u
π
and
d
π
þ
as disfavored fragmentation proc-
esses. Therefore, the production of pion pairs with opposite
charge from the fragmentation of a
u
u
pair (
e
þ
e
u
u
π

π
X
) can proceed through two favored fragmen-
tation processes (
u
π
þ
and
u
π
) or through two
disfavored fragmentation processes (
u
π
and
u
π
þ
).
Neglecting heavy quark fragmentation and introducing the
favored functions
D
fav
ð
z
Þ¼
D
π
þ
u
ð
z
Þ¼
D
π
d
ð
z
Þ
, and
D
fav
ð
z
Þ¼
D
π
u
ð
z
Þ¼
D
π
þ
d
ð
z
Þ
, as well as the disfavored
functions
D
dis
ð
z
Þ¼
D
π
u
ð
z
Þ¼
D
π
þ
d
ð
z
Þ¼
D
π

s
ð
z
Þ
and
D
dis
ð
z
Þ¼
D
π
þ
u
ð
z
Þ¼
D
π
d
ð
z
Þ¼
D
π

s
ð
z
Þ
, the cross section
for charged pion pair production can be written as
[31]
N
U
ð
φ
Þ¼
d
σ
ð
e
þ
e
π

π
X
Þ
d
Ω
d
z
1
d
z
2
5
9
D
fav
ð
z
1
Þ
D
fav
ð
z
2
Þþ
7
9
D
dis
ð
z
1
Þ
D
dis
ð
z
2
Þ
N
L
ð
φ
Þ¼
d
σ
ð
e
þ
e
π

π

X
Þ
d
Ω
d
z
1
d
z
2
5
9
D
fav
ð
z
1
Þ
D
dis
ð
z
2
Þþ
5
9
D
dis
ð
z
1
Þ
D
fav
ð
z
2
Þþ
2
9
D
dis
ð
z
1
Þ
D
dis
ð
z
2
Þ
N
C
ð
φ
Þ¼
d
σ
ð
e
þ
e
ππ
X
Þ
d
Ω
d
z
1
d
z
2
¼
N
U
ð
φ
Þþ
N
L
ð
φ
Þ
5
9
½
D
fav
ð
z
1
Þþ
D
dis
ð
z
1
Þ½
D
fav
ð
z
2
Þþ
D
dis
ð
z
2
Þ þ
4
9
D
dis
ð
z
1
Þ
D
dis
ð
z
2
Þð
10
Þ
where
π
stands for a generic charged pion,
φ
is the
azimuthal angle
φ
1
þ
φ
2
in RF12 or
φ
0
in RF0, d
Ω
¼
d
φ
d cos
θ
with
θ
the polar angle of the analysis axis, and the
upper index indicates unlike (U), like (L), and charged (C)
sign pion pairs.
III.
BABAR
EXPERIMENT AND DATA SAMPLE
The results presented here are based on a sample of data
collected with the
BABAR
detector at the PEP-II
asymmetric-energy
e
þ
e
collider, at the SLAC National
Accelerator Laboratory. A total integrated luminosity of
J. P. LEES
et al.
PHYSICAL REVIEW D
90,
052003 (2014)
052003-6
about
468
fb
1
[32]
is used, consisting of
424
fb
1
collected at the peak of the
Υ
ð
4
S
Þ
resonance, and about
44
fb
1
collected 40 MeV below the peak.
The
BABAR
detector is described in detail in referen-
ces
[33,34]
. Charged particle momenta are measured by a
combination of a 5-layer, double-sided silicon vertex
tracker (SVT), and a 40-layer drift chamber (DCH) that
covers 92% of the solid angle in the c.m. frame, both
located inside a 1.5 T superconducting solenoidal magnet.
Discrimination between charged pions, kaons, and protons
is obtained from measurements of the specific ionization
(d
E=
d
x
) in the tracking system, and from the Cherenkov
light collected by an internally reflecting ring-imaging
Cherenkov detector (DIRC). The DIRC covers 84% of
the c.m. solid angle in the central region of the
BABAR
detector and has a 95% (95%) identification efficiency for
pions (kaons) with momenta above 1.5 GeV.
2
Photons and
electrons are identified and their energies measured with a
high resolution CsI(Tl) crystal electromagnetic calorimeter.
Muons are identified in the instrumented flux return, which
consists of 18 layers of steel interleaved with single-gap
resistive plate chambers or limited-streamer tubes.
Detailed Monte Carlo (MC) simulation is used to test
and optimize the selection criteria, to study the detector
acceptance, and to estimate the contribution of various
background sources. The simulation package J
ETSET
[35]
is
used to generate hadronic events in nonresonant
e
þ
e
annihilation. Separate MC samples are generated for light
quarks,
e
þ
e
q
q
ð
q
¼
u; d; s
Þ
, called generic
uds
MC,
and charm quarks,
e
þ
e
c
c
. Samples of
B
B
events with
generic
B
decays are generated with the E
VT
G
EN
[36]
package. Finally,
τ
þ
τ
and
μ
þ
μ
event samples are
produced with the KK2F
[37]
generator, and
μ
þ
μ
γ
events with A
FK
Q
ED
[38]
. The generated events undergo
a full detector simulation based on G
EANT
4
[39]
and are
analyzed in the same way as the experimental data. No
transverse spin effects are implemented in the MC gen-
eration, so it can be used to evaluate detector biases. In
addition, the
uds
MC samples are reweighted to simulate
Collins asymmetries and to study the analyzing power of
the method.
IV. EVENT AND TRACK SELECTION
We focus on the measurement of the Collins effect in
light quark fragmentation, as the helicity is conserved only
in the approximation of massless quarks, and the correla-
tion between the fragmenting quark and antiquark may be
lost for heavy quarks. In this section, we summarize the
event and track selection requirements.
Multihadronic events are selected by requiring at least
three reconstructed charged particles and the value of the
second divided by the zeroth Fox-Wolfram moment
[40]
,
calculated from charged tracks only,
R
0
2
<
0
.
98
. To sup-
press backgrounds from
e
þ
e
τ
þ
τ
,
γγ
processes, and
events characterized by emission of a very energetic photon
via initial state radiation, we require the visible energy of
the event in the laboratory frame (
E
vis
), defined as the sum
of the energies of all reconstructed charged tracks and
neutral candidates, to be higher than 7 GeV.
Only good-quality reconstructed tracks with momenta
transverse to the beam direction of at least 0.1 GeV are
considered for the asymmetry measurements. Every track is
required to originate from the vicinity of the interaction
point (IP) by requiring the distance of closest approach to
the IP in the transverse plane
d
XY
<
0
.
2
cm and along the
electron beam
j
d
Z
j
<
1
.
5
cm, and to fall within the detector
acceptance region:
0
.
41
<
θ
lab
<
2
.
54
rad, where
θ
lab
is the
polar angle of the track with respect to the beamline axis.
The thrust of the event is calculated using tracks with
relaxed cuts
d
XY
<
1
.
5
cm and
j
d
Z
j
<
10
cm, as well as
neutral candidates lying within the calorimeter fiducial
region with an energy greater than 0.030 GeV. To avoid
possible biases originating from the different forward/
backward detector configuration, the sign of the thrust
axis is chosen at random.
Since the correlation between the
q
and the
q
spin is lost
in the case of emission of energetic gluons, we select the
two-jet topology and suppress
e
þ
e
q
qg
events by
requiring a value of the event thrust
T>
0
.
8
. As shown
in Fig.
4
the distribution of the thrust for
uds
events peaks
at values higher than 0.85, but has a long tail at lower
values, which is mainly due to hard gluon radiation. The
requirement
T>
0
.
8
also removes the majority of the more
spherical
B
B
events produced in
Υ
ð
4
S
Þ
decays. Events with
charm quarks have a shape similar to the light quarks; their
contribution to the asymmetry is evaluated and subtracted
as described in Sec.
VII
.
Events from the
e
þ
e
τ
þ
τ
reaction populate the
region at higher thrust values
T
and lower
E
vis
, as is evident
from Fig.
5
, which shows a scatter plot of
E
vis
vs
T
for
FIG. 3 (color online). Second-pion reference frame (RF0). The
azimuthal angle
φ
0
is defined as the angle between the plane
spanned by the beam axis and the second-pion momentum
P
2
,
and the transverse momentum
p
t
0
of the first pion around the
second-pion direction. The polar angle
θ
2
is defined as the angle
between the beam axis and the momentum
P
2
of the second pion.
2
Natural units are used throughout this article.
MEASUREMENT OF COLLINS ASYMMETRIES IN
...
PHYSICAL REVIEW D
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052003 (2014)
052003-7
events having at least one good hadron pair. The small
accumulation visible at lower energies and
T>
0
.
94
is due
to
τ
þ
τ
events, and it is removed by applying a cut around
this region, as indicated by the line in Fig.
5
.
Radiative
e
þ
e
e
þ
e
γ
and
e
þ
e
μ
þ
μ
γ
events are
the sources of the background peaking at
E
vis
¼
12
GeV
and concentrated in particular at high
T
. This kind of
background is suppressed by requiring at least three
charged hadrons in the event. However, some
μ
þ
μ
γ
events, with the initial state photon converting to a
e
þ
e
pair, can pass this selection. These events are characterized
by small multiplicity and by two very energetic muons. We
reduce this contamination to a negligible level by requiring
that for events with multiplicity lower than five, the two
most energetic tracks are not identified as muons, and no
electrons are present.
Reconstructed tracks in the selected events are used for
the study of the Collins asymmetries if they are identified as
pions and fail to pass specific muon and electron selectors.
The efficiencies estimated for the latter are about 70% and
98%, and the pion misidentification rate of about 2% and
4%, for muons and electrons, respectively.
Two-body decays of
b
b
bound states, mainly produced
via initial state radiation, generate a significant amount of
unlike-sign pairs, with both tracks of c.m. momentum
above 4.5 GeV. On the other hand, we expect to have very
small signal from fragmentation processes with two such
energetic tracks. In particular no like-sign pairs are
observed in the data sample with
z
1
and
z
2
above 0.9.
We therefore limit the study to tracks with
z<
0
.
9
.
The residual contributions of all other background
sources (
c
c
,
B
B
, and
τ
þ
τ
) are evaluated, and the measured
asymmetry corrected as described in Sec.
VII
.
The fragmentation functions depend on the light cone
momentum fraction
z
of the produced hadron with respect
to the fragmenting quark
[27]
, which is equivalent to the
fractional energy at large c.m. energy and not too small
values of
z
[23]
,
2
E
h
ffiffiffi
s
p
¼
z
þ
P
2
zs
z:
It may be of interest to extend the study also for very
low
z
values, in order to assess when this approxi-
mation fails. On the other hand, low momentum tracks
pose severe experimental difficulties due to the association
of the hadrons to the incorrect jet. For these reasons, the
measurement of Collins asymmetry is performed only for
candidate pions with
z>
0
.
15
.
The selected pions are separated in opposite hemispheres
according to the thrust axis (
ˆ
n
), and are combined if they
satisfy the following condition
W
hemi
¼ð
P
1
·
ˆ
n
Þð
P
2
·
ˆ
n
Þ
<
0
;
ð
11
Þ
where
P
1
;
2
are the pions momenta. For pairs with values of
W
hemi
near to zero there is a higher probability that one of
the two tracks has been assigned to the wrong hemisphere.
This effect is particularly evident for pions with low
fractional energies. The requirement that the pions are
emitted within a cone of
45
° around the thrust axis removes
the ambiguous tracks.
One of the most important contributions to azimuthal
asymmetries not connected to the Collins effect originates
from low energy gluon radiation
e
þ
e
q
qg
, which is not
completely removed by the event selection. As reported in
Refs.
[28,41]
, the angular distribution of the gluon radiation
process
e
þ
e
q
qg
h
1
h
2
X
is given by
dN
d
Ω
Q
2
t
s
þ
Q
2
t
sin
2
θ
cos
ð
2
φ
Þ
:
ð
12
Þ
In addition, all the formalism used so far is valid in the
region where the transverse momentum
Q
t
is small
T
0.8
0.85
0.9
0.95
1
(GeV)
vis
E
7
8
9
10
11
12
13
14
0
50
100
150
200
250
300
FIG. 5 (color online). Total visible energy of the event in the
laboratory frame vs the thrust value for the on-resonance data
sample. The events at high thrust value and low total energy are
due to the
e
þ
e
τ
þ
τ
process. The black line is the cut applied
in the analysis in order to remove this background. The peak at
E
vis
12
GeV and high thrust values, is due to radiative BhaBha
and
μ
þ
μ
ð
γ
Þ
events.
T
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Entries/0.005
1
2
3
4
5
6
3
10
×
1
2
3
4
5
6
Legend
uds
b
b
c
c
-
τ
+
τ
FIG. 4 (color online). Thrust distributions for simulated events
under the
Υ
ð
4
S
Þ
resonance for multihadronic events with at least
one pair of good quality tracks:
e
þ
e
B
B
(horizontal lines),
e
þ
e
c
c
(asterisks),
e
þ
e
q
q; q
¼
uds
(white histogram)
and
e
þ
e
ττ
(vertical lines). The samples are normalized to an
arbitrary luminosity.
J. P. LEES
et al.
PHYSICAL REVIEW D
90,
052003 (2014)
052003-8
compared to
ffiffiffi
s
p
ð
Q
2
t
s
Þ
[28]
, and a safe compromise is to
require
Q
t
<
3
.
5
GeV.
The same selection is applied to same-charge and
opposite-charge pion pairs. About
10
8
pion pairs are
selected and used in the analysis.
V. NORMALIZED AZIMUTHAL DISTRIBUTIONS
Following Eqs.
(3)
and
(7)
, the azimuthal distributions of
the normalized yields
R
α
, defined in Sec.
II
, can be
parametrized as
R
i
α
¼
b
α
þ
a
α
cos
ð
β
α
Þ
;
ð
13
Þ
where
α
¼
0
;
12
indicates the reference frame,
i
¼
U; L; C
the charge combination of the pion pair, and
β
is the
azimuthal angle combination
φ
1
þ
φ
2
or
2
φ
0
, according to
the frame used. The parameter
b
α
should be consistent with
unity, while
a
α
gives the amplitude of the asymmetries. The
normalized azimuthal distributions, presented in Fig.
6
for
MC and data samples, are strongly affected by detector
acceptances and show apparent modulations. This is clearly
visible in the simulated sample, for which a flat distribution
is expected since the polarized
D
ð
z
Þ
are not implemented in
the MC generator. However, the
R
L
and
R
U
distributions
are almost coincident in the MC sample [Fig.
6(a)
], while a
clear difference is observed in data [Fig.
6(b)
]. This
difference is the observable effect of the azimuthal asym-
metry produced by the Collins effect.
Detector effects depend on the jet direction. When the
q
q
pair is created at low polar angle with respect to the beam
axis, there is a higher probability that part of the jet falls
outside the detector coverage, and the thrust can be badly
reconstructed. The result is a distortion of the distribution,
as visible in Fig.
7
, which shows
R
U
and
R
L
in the RF0
frame for different intervals of cos
ð
θ
th
Þ
. The same effect is
also visible in the RF12 frame. The triangles in Fig.
7
also
show the residual effects of gluon radiation to be small.
We can parameterize the acceptance effects on the nor-
malized distribution as an additional contribution to the
cos
ð
β
α
Þ
modulation, whose amplitude varies with
θ
:
a
ε
α
ð
θ
Þ
.
Therefore, Eq.
(13)
becomes
R
i
α
¼ð
1
þ
a
ε
α
ð
θ
Þ
cos
ð
β
α
ÞÞ
·
ð
b
α
þ
a
α
cos
ð
β
α
ÞÞ
¼
b
α
þ½
a
α
þ
a
ε
α
ð
θ
Þ
b
α

cos
ð
β
α
Þþ
a
α
a
ε
α
ð
θ
Þ
cos
2
ð
β
α
Þ
;
ð
14
Þ
and shows a coupling between the Collins and detector
acceptance effects proportional to cos
2
ð
β
α
Þ
.
In principle, it would be possible to estimate detector
acceptance effects with simulated events, and correct the
asymmetries measured in the data sample, but this pro-
cedure would introduce large uncertainties. All these
considerations suggest the possibility to form a suitable
double ratio of azimuthal distributions, in order to reduce
the effect of detector acceptance and perform a measure-
ment almost independent from simulation.
VI. DOUBLE RATIOS
Given the difficulties in separating the Collins effect
from asymmetries produced by detector acceptances and
radiative effects, we exploit the fact that most of the
(rad)
2
φ
+
1
φ
-3
-2
-1
0123
12
R
0.9
0.95
1
1.05
1.1
(a)
(rad)
2
φ
+
1
φ
-3
-2
-1
0123
12
R
0.9
0.95
1
1.05
1.1
(b)
FIG. 6 (color online). Normalized azimuthal distributions for
like-sign (
R
L
, full circles) and unlike-sign (
R
U
, open circles) pion
pairs, for (a) MC simulation and (b) data, in RF12.
(rad)
0
φ
2
-3
-2
-1
0
1
2
3
R
0.6
0.8
1
1.2
1.4
1.6
U
0
(rad)
0
φ
2
-3
-2
-1
0
1
2
3
L
0
(a)
R
(b)
R
FIG. 7 (color online). Normalized azimuthal distributions for
different intervals of cos
θ
th
measured in the RF0 frame for
unlike-sign (a) and like-sign (b) pion pairs. The cos
θ
th
intervals
are as follows:
0
.
8
<
cos
θ
th
<
0
.
9
for circles,
0
.
5
<
cos
θ
th
<
0
.
7
for squares,
0
.
3
<
cos
θ
th
<
0
.
5
for crosses,
0
<
cos
θ
th
<
0
.
3
for triangles.
MEASUREMENT OF COLLINS ASYMMETRIES IN
...
PHYSICAL REVIEW D
90,
052003 (2014)
052003-9
instrumental effects should largely cancel in ratios of
asymmetries, as for example, the ratio of unlike-sign over
like-sign asymmetries,
R
U
12
R
L
12
1
þh
sin
2
θ
th
1
þ
cos
2
θ
th
i
G
U
cos
ð
φ
1
þ
φ
2
Þ
1
þh
sin
2
θ
th
1
þ
cos
2
θ
th
i
G
L
cos
ð
φ
1
þ
φ
2
Þ
1
þ
sin
2
θ
th
1
þ
cos
2
θ
th
f
G
U
G
L
g
cos
ð
φ
1
þ
φ
2
Þ
:
ð
15
Þ
Here,
G
L
and
G
U
are, respectively,
G
U
5
H
fav
1
H
fav
1
þ
7
H
dis
1
H
dis
1
5
D
fav
1
D
fav
1
þ
7
D
dis
1
D
dis
1
;
G
L
5
H
fav
1
H
dis
1
þ
5
H
dis
1
H
fav
1
þ
2
H
dis
1
H
dis
1
5
D
fav
1
D
dis
1
þ
5
D
dis
1
D
fav
1
þ
2
D
dis
1
D
dis
1
;
ð
16
Þ
where we omitted the
z
and
p
t
dependence in order to
simplify the notation. The double ratio (DR) is performed
after the integration over the polar angle
θ
th
, so that
the average values of the quantity sin
2
θ
th
=
ð
1
þ
cos
2
θ
th
Þ
appear. These average values do not differ for like-, unlike-,
and all charged pion pairs. In Eq.
(15)
we assume that the
detector acceptance effects do not depend on the charge
combination of the pion pairs, that is
a
ε
;L
ð
θ
th
Þ¼
a
ε
;U
ð
θ
th
Þ
.
We also neglect the extra term proportional to
cos
2
ð
φ
1
þ
φ
2
Þ
, which couple the detector acceptance to
the true Collins asymmetries, and stop the series expansion
at the first order in cos
ð
φ
1
þ
φ
2
Þ
. We have checked for the
presence of these and other terms in addition to the
cos
ð
φ
1
þ
φ
2
Þ
modulation and found them negligible.
Also the assumption of acceptance effects independent
on the charge combination of the pion pairs seems to hold,
and noting that also the asymmetries produced by gluon
radiation do not depend on the charge combination, the
asymmetry amplitudes resulting from the double ratio
should mainly depend on a different combination of
favored and disfavored fragmentation functions (see also
discussion in Sec.
IX
).
Similarly, the DR of the normalized distributions of
unlike-sign and charged pion pairs is given by
R
U
12
R
C
12
1
þ
sin
2
θ
th
1
þ
cos
2
θ
th
f
G
U
G
C
g
cos
ð
φ
1
þ
φ
2
Þ
;
ð
17
Þ
with
G
C
5
ð
H
fav
1
þ
H
dis
1
Þð
H
fav
1
þ
H
dis
1
Þþ
4
H
dis
1
H
dis
1
5
ð
D
fav
1
þ
D
dis
1
Þð
D
fav
1
þ
D
dis
1
Þþ
4
D
dis
1
D
dis
1
:
ð
18
Þ
The measured
U=L
and
U=C
double ratios can be used
to derive information about the relative sign and magnitude
of favored and disfavored fragmentation functions
[31]
.
Analogous expressions can be obtained in the RF0 refer-
ence frame, with modulations in cos
ð
2
φ
0
Þ
instead
of cos
ð
φ
1
þ
φ
2
Þ
.
The DRs are still parametrized by a cosine function
R
i
α
R
j
α
¼
B
ij
α
þ
A
ij
α
· cos
ð
β
α
Þ
;
ð
19
Þ
where
B
and
A
are free parameters. The constant term
B
should be consistent with unity and the parameter
A
,
which depends on
z
, pt, and the average value of
sin
2
θ
=
ð
1
þ
cos
2
θ
Þ
, should mainly contain the Collins
effect.
Figure
8
shows the DR of unlike- to like-sign pion pairs
for samples of simulated and data events. The distribution
for the MC sample is now essentially flat as expected;
however, a slight deviation from zero asymmetry, of the
order of 0.2%, is measured. The origin and the effect of this
bias will be discussed in Sec.
IX A
. A clear cosine modu-
lation is instead visible in the data sample [Fig.
8(b)
], which
can be attributed to the Collins effect.
(rad)
2
φ
+
1
φ
-3
-2
-1
0123
L
12
/R
U
12
R
0.96
0.98
1
1.02
1.04
(a)
(rad)
2
φ
+
1
φ
-3
-2
-1
0123
L
12
/R
U
12
R
0.96
0.98
1
1.02
1.04
(b)
FIG. 8 (color online). Double ratio of azimuthal distributions of
unlike- over like- sign pion pairs for Monte Carlo (a) and data (b)
samples, in the RF12 system. The solid lines are the result of the
fits with the function reported in Eq.
(19)
.
J. P. LEES
et al.
PHYSICAL REVIEW D
90,
052003 (2014)
052003-10
Thanks to the large amount of data, we can study the
dependence of the asymmetry as a function of fractional
energies (
z
1
and
z
2
) and transverse momenta (
p
t
1
and
p
t
2
,
and
p
t
0
) of the selected pions. We choose
6
×
6
(
z
1
;z
2
)
bins, with the following
z
intervals: [0.15
0.2], [0.2
0.3],
[0.3
0.4], [0.4
0.5], [0.5
0.7], [0.7
0.9]; we use
4
×
4
(
p
t
1
;p
t
2
) bins in the RF12 frame, and 9
p
t
0
bins in the RF0
frame. The
p
t
intervals are defined in Table
III
.
Figure
9
shows the asymmetries obtained from fits to the
UL double ratio (
A
UL
) in the RF12 frame for data and MC
samples in every
ð
z
1
;z
2
Þ
bin. The asymmetries are not
corrected for the effects described in the next three sections,
and report only the statistical errors. Similar results are
obtained for the asymmetries measured in the RF0 frame,
and as a function of pion transverse momenta.
VII. CONTRIBUTION OF BACKGROUND EVENTS
TO THE ASYMMETRIES
The presence of background processes modifies the
measured asymmetry
A
meas
α
. This is obtained by fitting
the double ratio of the selected sample, and can be written
as
A
meas
α
¼

1
X
i
F
i

·
A
α
þ
X
i
F
i
·
A
i
α
:
ð
20
Þ
Here,
A
α
is the true Collins asymmetry produced by the
fragmentation of light quarks, while
A
i
α
and
F
i
are,
respectively, the asymmetry and the fraction of pion pairs
in the selected sample due to the
i
th background
component.
The background processes giving a significant contri-
bution after the selection procedure are
e
þ
e
τ
þ
τ
,
e
þ
e
c
c
, and
e
þ
e
Υ
ð
4
S
Þ
B
B
. We refer to them
as the
τ
, charm, and bottom backgrounds, respectively. In
the former process, azimuthal asymmetries can arise from
parity violation in the weak decay of the heavy leptons.
For charm processes the Collins effect is expected to be
suppressed by the heavy mass of the fragmenting quarks.
The study of the azimuthal asymmetries for
c
c
processes
would be interesting on its own, but larger data samples and
an optimized analysis would be necessary to perform
precise measurements. No asymmetries arising from the
Collins effect are expected from
Υ
ð
4
S
Þ
B
B
decays.
The fractions
F
i
and the asymmetries
A
i
α
of the back-
ground components are determined using both MC and
data control samples specific to each process, and evaluated
for each bin of
z
and
p
t
.
A. The
e
þ
e
τ
þ
τ
background
In order to assess whether a significant asymmetry
is produced by
e
þ
e
τ
þ
τ
processes we study a
τ
-enhanced data sample, consisting of the events in the
lower-right side of the
E
vis
-vs-
T
distribution of Fig.
5
, and
rejected by the cut shown in the same picture. The purity of
this control sample is estimated to be about 75%; the fitted
asymmetries are very small and consistent with zero within
about two standard deviations. We also perform the
analysis on a sample of simulated
τ
þ
τ
events, applying
the same event selection as for the data, and obtain
asymmetries consistent with the small bias observed in
the
uds
MC sample.
The contribution of the
e
þ
e
τ
þ
τ
background
appears in Eq.
(20)
as the product of the asymmetry
A
τ
α
multiplied by the pion pairs fraction
F
τ
. We estimate
F
τ
from the number of pion pairs selected in a MC sample
of
τ
þ
τ
events scaled by the data/MC luminosity ratio,
independently for every
z
and
p
t
bin. The values of
F
τ
range from about 1% at low
z
i
, to more than 18% at high
z
i
,
and are around 2% independently of
p
t
.
Considering that the asymmetries measured in the
τ
-enhanced samples are consistent with zero or give only
very small deviations from zero, and that the contamination
from
τ
þ
τ
events is significant only at large
z
i
, where the
Collins effect from
uds
is large (see Fig.
9
), we set
A
τ
α
¼
0
everywhere.
B. The
e
þ
e
c
c
and
e
þ
e
B
B
backgrounds
The fraction of pion pairs due to
e
þ
e
c
c
events is
much larger than the
τ
þ
τ
component, because of the
higher production cross section and of event shapes similar
to those for light quark production. The fraction
F
c
,
estimated with a generic
c
c
-MC sample, amounts to about
25% for the whole data sample, roughly independent of
p
t
,
but ranging from about 30% for pairs with low fractional
energies down to less than one percent at the highest
z
i
values.
The
B
B
events are strongly suppressed by the event
selection, mainly because of the cut on the event thrust, and
the fractions
F
B
are estimated to be at most 2% for low
UL, raw
12
A
0
0.05
0.1
0.15
Data
MC
=[0.15-0.2]
1
z
2
z
0.2 0.4 0.6 0.8
1
UL, raw
12
A
0
0.05
0.1
0.15
=[0.4-0.5]
1
z
=[0.2-0.3]
1
z
2
z
0.2 0.4 0.6 0.8
1
=[0.5-0.7]
1
z
=[0.3-0.4]
1
z
2
z
0.2 0.4 0.6 0.8
1
=[0.7-0.9]
1
z
FIG. 9 (color online). Comparison of raw DR asymmetries as a
function of
6
×
6
(
z
1
;z
2
)-bin subdivision calculated in data
(triangles) and MC samples (squares). In each plot,
z
1
is fixed
following the interval subdivisions described in the text, and
z
2
ranges between 0.15 to 0.9.
MEASUREMENT OF COLLINS ASYMMETRIES IN
...
PHYSICAL REVIEW D
90,
052003 (2014)
052003-11