SLAC-PUB-15700
B
A
B
AR
-PUB-13/012
Measurement of Collins asymmetries in inclusive production of charged
pion pairs in
e
+
e
−
annihilation at
B
A
B
AR
J. P. Lees, V. Poireau, and V. Tisserand
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),
Universit ́e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
A. Palano
ab
INFN Sezione di Bari
a
; Dipartimento di Fisica, Universit`a di Bari
b
, I-70126 Bari, Italy
G. Eigen and B. Stugu
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
D. N. Brown, L. T. Kerth, Yu. G. Kolomensky, M. J. Lee, and G. Lynch
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
H. Koch and T. Schroeder
Ruhr Universit ̈at Bochum, Institut f ̈ur Experimentalphysik 1, D-44780 Bochum, Germany
C. Hearty, T. S. Mattison, J. A. McKenna, and R. Y. So
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A. Khan
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
V. E. Blinov
ac
, A. R. Buzykaev
a
, V. P. Druzhinin
ab
, V. B. Golubev
ab
, E. A. Kravchenko
ab
, A. P. Onuchin
ac
,
S. I. Serednyakov
ab
, Yu. I. Skovpen
ab
, E. P. Solodov
ab
, K. Yu. Todyshev
ab
, and A. N. Yushkov
a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
a
,
Novosibirsk State University, Novosibirsk 630090
b
,
Novosibirsk State Technical University, Novosibirsk 630092
c
, Russia
D. Kirkby, A. J. Lankford, and M. Mandelkern
University of California at Irvine, Irvine, California 92697, USA
B. Dey, J. W. Gary, O. Long, and G. M. Vitug
University of California at Riverside, Riverside, California 92521, USA
C. Campagnari, M. Franco Sevilla, T. M. Hong, D. Kovalskyi, J. D. Richman, and C. A. West
University of California at Santa Barbara, Santa Barbara, California 93106, USA
A. M. Eisner, W. S. Lockman, B. A. Schumm, and A. Seiden
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
D. S. Chao, C. H. Cheng, B. Echenard, K. T. Flood, D. G. Hitlin, P. Ongmongkolkul, and F. C. Porter
California Institute of Technology, Pasadena, California 91125, USA
R. Andreassen, Z. Huard, B. T. Meadows, B. G. Pushpawela, M. D. Sokoloff, and L. Sun
University of Cincinnati, Cincinnati, Ohio 45221, USA
P. C. Bloom, W. T. Ford, A. Gaz, U. Nauenberg, J. G. Smith, and S. R. Wagner
University of Colorado, Boulder, Colorado 80309, USA
arXiv:1309.5278v2 [hep-ex] 24 Sep 2014
2
R. Ayad
∗
and W. H. Toki
Colorado State University, Fort Collins, Colorado 80523, USA
B. Spaan
Technische Universit ̈at Dortmund, Fakult ̈at Physik, D-44221 Dortmund, Germany
R. Schwierz
Technische Universit ̈at Dresden, Institut f ̈ur Kern- und Teilchenphysik, D-01062 Dresden, Germany
D. Bernard and M. Verderi
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
S. Playfer
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
D. Bettoni
a
, C. Bozzi
a
, R. Calabrese
ab
, G. Cibinetto
ab
, E. Fioravanti
ab
,
I. Garzia
ab
, E. Luppi
ab
, L. Piemontese
a
, and V. Santoro
a
INFN Sezione di Ferrara
a
; Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara
b
, I-44122 Ferrara, Italy
R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro,
S. Martellotti, P. Patteri, I. M. Peruzzi,
†
M. Piccolo, M. Rama, and A. Zallo
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
R. Contri
ab
, E. Guido
ab
, M. Lo Vetere
ab
, M. R. Monge
ab
, S. Passaggio
a
, C. Patrignani
ab
, and E. Robutti
a
INFN Sezione di Genova
a
; Dipartimento di Fisica, Universit`a di Genova
b
, I-16146 Genova, Italy
B. Bhuyan and V. Prasad
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
M. Morii
Harvard University, Cambridge, Massachusetts 02138, USA
A. Adametz and U. Uwer
Universit ̈at Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
H. M. Lacker
Humboldt-Universit ̈at zu Berlin, Institut f ̈ur Physik, D-12489 Berlin, Germany
P. D. Dauncey
Imperial College London, London, SW7 2AZ, United Kingdom
U. Mallik
University of Iowa, Iowa City, Iowa 52242, USA
C. Chen, J. Cochran, W. T. Meyer, and S. Prell
Iowa State University, Ames, Iowa 50011-3160, USA
A. V. Gritsan
Johns Hopkins University, Baltimore, Maryland 21218, USA
N. Arnaud, M. Davier, D. Derkach, G. Grosdidier, F. Le Diberder,
A. M. Lutz, B. Malaescu,
‡
P. Roudeau, A. Stocchi, and G. Wormser
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
D. J. Lange and D. M. Wright
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
3
J. P. Coleman, J. R. Fry, E. Gabathuler, D. E. Hutchcroft, D. J. Payne, and C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom
A. J. Bevan, F. Di Lodovico, and R. Sacco
Queen Mary, University of London, London, E1 4NS, United Kingdom
G. Cowan
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
J. Bougher, D. N. Brown, and C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
A. G. Denig, M. Fritsch, W. Gradl, K. Griessinger, A. Hafner, E. Prencipe, and K. R. Schubert
Johannes Gutenberg-Universit ̈at Mainz, Institut f ̈ur Kernphysik, D-55099 Mainz, Germany
R. J. Barlow
§
and G. D. Lafferty
University of Manchester, Manchester M13 9PL, United Kingdom
E. Behn, R. Cenci, B. Hamilton, A. Jawahery, and D. A. Roberts
University of Maryland, College Park, Maryland 20742, USA
R. Cowan, D. Dujmic, and G. Sciolla
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
R. Cheaib, P. M. Patel,
¶
and S. H. Robertson
McGill University, Montr ́eal, Qu ́ebec, Canada H3A 2T8
P. Biassoni
ab
, N. Neri
a
, and F. Palombo
ab
INFN Sezione di Milano
a
; Dipartimento di Fisica, Universit`a di Milano
b
, I-20133 Milano, Italy
L. Cremaldi, R. Godang,
∗∗
P. Sonnek, and D. J. Summers
University of Mississippi, University, Mississippi 38677, USA
M. Simard and P. Taras
Universit ́e de Montr ́eal, Physique des Particules, Montr ́eal, Qu ́ebec, Canada H3C 3J7
G. De Nardo
ab
, D. Monorchio
ab
, G. Onorato
ab
, and C. Sciacca
ab
INFN Sezione di Napoli
a
; Dipartimento di Scienze Fisiche,
Universit`a di Napoli Federico II
b
, I-80126 Napoli, Italy
M. Martinelli and G. Raven
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop and J. M. LoSecco
University of Notre Dame, Notre Dame, Indiana 46556, USA
K. Honscheid and R. Kass
Ohio State University, Columbus, Ohio 43210, USA
J. Brau, R. Frey, N. B. Sinev, D. Strom, and E. Torrence
University of Oregon, Eugene, Oregon 97403, USA
E. Feltresi
ab
, M. Margoni
ab
, M. Morandin
a
, M. Posocco
a
, M. Rotondo
a
, G. Simi
a
, F. Simonetto
ab
, and R. Stroili
ab
INFN Sezione di Padova
a
; Dipartimento di Fisica, Universit`a di Padova
b
, I-35131 Padova, Italy
S. Akar, E. Ben-Haim, M. Bomben, G. R. Bonneaud, H. Briand,
G. Calderini, J. Chauveau, Ph. Leruste, G. Marchiori, J. Ocariz, and S. Sitt
Laboratoire de Physique Nucl ́eaire et de Hautes Energies,
4
IN2P3/CNRS, Universit ́e Pierre et Marie Curie-Paris6,
Universit ́e Denis Diderot-Paris7, F-75252 Paris, France
M. Biasini
ab
, E. Manoni
a
, S. Pacetti
ab
, and A. Rossi
a
INFN Sezione di Perugia
a
; Dipartimento di Fisica, Universit`a di Perugia
b
, I-06123 Perugia, Italy
C. Angelini
ab
, G. Batignani
ab
, S. Bettarini
ab
, M. Carpinelli
ab
,
††
G. Casarosa
ab
, A. Cervelli
ab
, F. Forti
ab
,
M. A. Giorgi
ab
, A. Lusiani
ac
, B. Oberhof
ab
, E. Paoloni
ab
, A. Perez
a
, G. Rizzo
ab
, and J. J. Walsh
a
INFN Sezione di Pisa
a
; Dipartimento di Fisica, Universit`a di Pisa
b
; Scuola Normale Superiore di Pisa
c
, I-56127 Pisa, Italy
D. Lopes Pegna, J. Olsen, and A. J. S. Smith
Princeton University, Princeton, New Jersey 08544, USA
R. Faccini
ab
, F. Ferrarotto
a
, F. Ferroni
ab
, M. Gaspero
ab
, L. Li Gioi
a
, and G. Piredda
a
INFN Sezione di Roma
a
; Dipartimento di Fisica,
Universit`a di Roma La Sapienza
b
, I-00185 Roma, Italy
C. B ̈unger, O. Gr ̈unberg, T. Hartmann, T. Leddig, C. Voß, and R. Waldi
Universit ̈at Rostock, D-18051 Rostock, Germany
T. Adye, E. O. Olaiya, and F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
S. Emery, G. Hamel de Monchenault, G. Vasseur, and Ch. Y`eche
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
F. Anulli,
‡‡
D. Aston, D. J. Bard, J. F. Benitez, C. Cartaro, M. R. Convery, J. Dorfan, G. P. Dubois-Felsmann,
W. Dunwoodie, M. Ebert, R. C. Field, B. G. Fulsom, A. M. Gabareen, M. T. Graham, C. Hast,
W. R. Innes, P. Kim, M. L. Kocian, D. W. G. S. Leith, P. Lewis, D. Lindemann, B. Lindquist, S. Luitz,
V. Luth, H. L. Lynch, D. B. MacFarlane, D. R. Muller, H. Neal, S. Nelson, M. Perl, T. Pulliam,
B. N. Ratcliff, A. Roodman, A. A. Salnikov, R. H. Schindler, A. Snyder, D. Su, M. K. Sullivan, J. Va’vra,
A. P. Wagner, W. F. Wang, W. J. Wisniewski, M. Wittgen, D. H. Wright, H. W. Wulsin, and V. Ziegler
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
W. Park, M. V. Purohit, R. M. White,
§§
and J. R. Wilson
University of South Carolina, Columbia, South Carolina 29208, USA
A. Randle-Conde and S. J. Sekula
Southern Methodist University, Dallas, Texas 75275, USA
M. Bellis, P. R. Burchat, T. S. Miyashita, and E. M. T. Puccio
Stanford University, Stanford, California 94305-4060, USA
M. S. Alam and J. A. Ernst
State University of New York, Albany, New York 12222, USA
R. Gorodeisky, N. Guttman, D. R. Peimer, and A. Soffer
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
J. L. Ritchie, A. M. Ruland, R. F. Schwitters, and B. C. Wray
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen and X. C. Lou
University of Texas at Dallas, Richardson, Texas 75083, USA
5
F. Bianchi
ab
, F. De Mori
ab
, A. Filippi
a
, D. Gamba
ab
, and S. Zambito
ab
INFN Sezione di Torino
a
; Dipartimento di Fisica, Universit`a di Torino
b
, I-10125 Torino, Italy
L. Lanceri
ab
and L. Vitale
ab
INFN Sezione di Trieste
a
; Dipartimento di Fisica, Universit`a di Trieste
b
, I-34127 Trieste, Italy
F. Martinez-Vidal, A. Oyanguren, and P. Villanueva-Perez
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
H. Ahmed, J. Albert, Sw. Banerjee, F. U. Bernlochner, H. H. F. Choi, G. J. King, R. Kowalewski,
M. J. Lewczuk, T. Lueck, I. M. Nugent, J. M. Roney, R. J. Sobie, and N. Tasneem
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
T. J. Gershon, P. F. Harrison, and T. E. Latham
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
H. R. Band, S. Dasu, Y. Pan, R. Prepost, and S. L. Wu
University of Wisconsin, Madison, Wisconsin 53706, USA
We present measurements of Collins asymmetries in the inclusive process
e
+
e
−
→
ππX
, 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
B
A
B
AR
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.
PACS numbers: 13.66.Bc, 13.87.Fh, 13.88.+e, 14.65.-q
I. INTRODUCTION
Parton fragmentation functions describe the probabil-
ity for a parton to fragment into a hadron carrying a
certain fraction
z
of the parton momentum. These func-
tions are denoted
D
i
h
(
z
), where
i
represents the frag-
menting parton (
g
,
u
, ̄
u
,
d
,
̄
d
,...), and
h
is the produced
hadron. Since the
D
i
h
(
z
) incorporate the long distance,
non-perturbative physics of the hadronization processes,
they cannot be calculated in perturbative QCD, but can
be evolved from a starting distribution at a defined en-
ergy scale. Fragmentation processes have been studied in
lepton-hadron and hadron-hadron scattering, as well as
in
e
+
e
−
annihilation, which provides the cleanest envi-
∗
Now at the University of Tabuk, Tabuk 71491, Saudi Arabia
†
Also with Universit`a di Perugia, Dipartimento di Fisica, Perugia,
Italy
‡
Now at Laboratoire de Physique Nucl ́aire et de Hautes Energies,
IN2P3/CNRS, Paris, France
§
Now at the University of Huddersfield, Huddersfield HD1 3DH,
UK
¶
Deceased
∗∗
Now at University of South Alabama, Mobile, Alabama 36688,
USA
††
Also with Universit`a di Sassari, Sassari, Italy
‡‡
Also with INFN Sezione di Roma, Roma, Italy
§§
Now at Universidad T ́ecnica Federico Santa Maria, Valparaiso,
Chile 2390123
ronment since no hadrons are present in the initial state.
Due to the large amount of experimental data collected
at several
e
+
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
√
s
∼
10 GeV, the unpolar-
ized functions are presently well known.
Transverse spin-dependent effects in fragmentation
processes were first proposed by Collins [9, 10], who in-
troduced the chiral-odd polarized Collins fragmentation
function
H
⊥
1
. It describes the relation between the trans-
verse 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 frac-
tional energy
z
= 2
E
h
/
√
s
, where
E
h
and
√
s/
2 are, re-
spectively, 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 un-
polarized
D
q
1
, and Collins
H
⊥
q
1
fragmentation functions,
as [11]
D
q
↑
h
(
z,
P
⊥
h
) =
D
q
1
(
z,P
⊥
h
2
) +
H
⊥
q
1
(
z,P
⊥
h
2
)
(
ˆ
k
×
P
⊥
h
)
·
S
q
zM
h
.
(1)
The term containing the Collins function depends on
6
the spin vector of the quark
S
q
, and introduces an az-
imuthal asymmetry in the distribution of hadrons around
the quark three-momentum direction
ˆ
k
. The triple pro-
duct of Eq. (1), in fact, produces a cos
φ
modulation,
where
φ
is the azimuthal angle between the plane per-
pendicular 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 asym-
metry or the Collins effect.
φ
P
h
S
q
k
P
h
⊥
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
.
Experimental evidence for a non-zero Collins func-
tion was found by the HERMES [12, 13] and COM-
PASS [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 convolution of the Collins function and the chiral-
odd transversity 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
measurement of at least one of them is needed.
In
e
+
e
−
annihilation to a quark-antiquark pair, each
quark spin is unknown: the Collins asymmetry in a sin-
gle jet (
q
→
hX
) will be zero. However, the Collins
effect can be observed when the products of the quark
and antiquark fragmentation are simultaneously consi-
dered. Experimentally, this is achieved by studying the
process
e
+
e
−
→
q
q
→
h
1
h
2
X
, where
h
1
(
h
2
) is a spin-
less hadron (
π
or
K
) coming from the fragmenting quark
q
( ̄
q
). 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
+
e
−
annihilation, by studying
Z
0
→
2 jets events, were
performed by the DELPHI Collaboration [17], while the
first observation was obtained by the Belle Collabora-
tion [18–20], from a study of inclusive production of
charged pion pairs at a center-of-mass (c.m.) energy of
approximately 10.6 GeV. Assuming the universality of
the Collins function [21, 22], a global analysis of SIDIS
and
e
+
e
−
annihilation data has been carried out by the
authors of Refs. [23, 24], allowing the simultaneous ex-
traction of the transversity and Collins functions for the
pion system.
We report the measurements of the azimuthal modula-
tion due to the Collins effect (Collins asymmetries) in the
process
e
+
e
−
→
q
q
→
ππX
, 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 refer-
ence 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, de-
fined in Sec. II B. This choice follows the scheme out-
lined 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. Dilu-
tions of the asymmetries induced by background sources
and by detector effects not related to the Collins func-
tion 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 mani-
fests itself as an azimuthal modulation of two final state
pions around the fragmenting quark-antiquark momen-
tum. The
q
q
direction is not accessible to a direct mea-
surement and is approximated by the thrust axis of the
event [26]. The kinematics in the
e
+
e
−
c.m. system
corresponding to
e
+
e
−
→
π
1
π
2
X
, together with the defi-
nition of the two azimuthal angles, are schematically re-
presented 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 antiquark Collins functions in opposite
7
hemispheres gives a product of two modulations for the
azimuthal angles
φ
1
and
φ
2
, resulting in a cos(
φ
1
+
φ
2
)
modulation. The azimuthal angles are defined as
φ
1
,
2
= sign[
ˆ
n
·{
(
ˆ
u
×
ˆ
n
)
×
(
ˆ
n
×
ˆ
P
1
,
2
)
}
]
×
arccos
(
ˆ
u
×
ˆ
n
|
ˆ
u
×
ˆ
n
|
·
ˆ
n
×
P
1
,
2
|
ˆ
n
×
P
1
,
2
|
)
,
(2)
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
(
e
+
e
−
→
π
1
π
2
X
)
d
z
1
d
z
2
d
φ
1
d
φ
2
d cos
θ
th
=
∑
q,
q
3
α
2
s
e
2
q
4
z
2
1
z
2
2
×
{
(1 + cos
2
θ
th
)
D
q,
[0]
1
(
z
1
)
D
q,
[0]
1
(
z
2
) + sin
2
θ
th
cos(
φ
1
+
φ
2
)
H
⊥
q,
[1]
1
(
z
1
)
H
⊥
q,
[1]
1
(
z
2
)
}
,
(3)
where the summation runs over all quark flavors accessi-
ble at the c.m. energy
√
s
,
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
[
n
]
(
z
) =
∫
d
|
k
2
T
|
(
|
k
T
|
M
π
)
n
F
(
z,
k
2
T
)
,
(4)
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
=
z
k
T
, and
M
π
is the pion mass. The Collins asym-
metry can be extracted by measuring the cosine modula-
tion of the distribution of the quantity (
φ
1
+
φ
2
) on top
of the uniform distribution due to the unpolarized part
of the fragmentation function. Dividing the full (
φ
1
+
φ
2
)
range into one hundred intervals, we define the normal-
ized azimuthal distribution as
R
12
(
φ
1
+
φ
2
) =
N
(
φ
1
+
φ
2
)
〈
N
12
〉
,
(5)
with
N
(
φ
1
+
φ
2
) the di-pion yield in each (
φ
1
+
φ
2
) sub-
division, and
〈
N
12
〉
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
1
Throughout the paper we use the subscript
T
to denote the trans-
verse component of a vector to the dipion axis in the frame where
they are collinear, while the superscript
⊥
indicates the compo-
nent transverse to the
q
q
axis.
n
̂
e
-
e
+
φ
2
φ
1
θ
th
p
t2
P
1
P
2
p
t1
u
̂
FIG. 2: (color online). Thrust reference frame (RF12). The
azimuthal angles
φ
1
and
φ
2
are the angles between the scat-
tering plane and the transverse pion momenta
p
ti
around
the thrust axis. The polar angle
θ
th
is defined as the an-
gle between the beam axis and the thrust axis
ˆn
. The pion
transverse momenta
p
ti
used in the analysis differ from the
corresponding
P
⊥
h
, which refer to the true
q
q
direction.
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
+
e
−
c.m. system, but only one azimuthal angle,
φ
0
, is defined:
φ
0
= sign[
P
2
·{
(
ˆ
u
×
P
2
)
×
(
P
2
×
P
1
)
}
]
×
arccos
(
ˆ
u
×
P
2
|
ˆ
u
×
P
2
|
·
P
2
×
P
1
|
P
2
×
P
1
|
)
.
(6)
8
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
(
e
+
e
−
→
π
1
π
2
X
)
d
z
1
d
z
2
d
2
q
T
d cos(
θ
2
)
dφ
0
=
3
α
2
s
z
2
1
z
2
2
4
×
{
(1 + cos
2
θ
2
)
F
(
D
1
(
z
1
)
D
1
(
z
2
)) + sin
2
θ
2
cos(2
φ
0
)
×F
[
(2
ˆ
h
·
k
T
ˆ
h
·
p
T
−
k
T
·
p
T
)
H
⊥
1
(
z
1
)
H
⊥
1
(
z
2
)
M
2
π
]}
,
(7)
where
|
q
T
|
=
Q
t
is the transverse momentum of the vir-
tual photon from
e
+
e
−
annihilation in the frame where
P
1
and
P
2
are collinear [28].
F
is a convolution in-
tegral over the transverse momenta
P
⊥
1
=
z
1
k
T
and
P
⊥
2
=
z
2
p
T
, with
k
T
and
p
T
the transverse momenta
of the two fragmenting quarks:
F
(
X
X
) =
∑
a,
̄
a
e
2
a
∫
d
2
k
T
d
2
p
T
δ
2
(
p
T
+
k
T
−
q
T
)
X
(
z
1
,z
2
1
k
2
T
)
X
(
z
2
,z
2
2
p
2
T
)
,
(8)
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
(2
φ
0
) =
N
(2
φ
0
)
〈
N
0
〉
.
(9)
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 sec-
tion is obtained. A similar relation exists for the Collins
contributions. However, due to the additional convo-
lution 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
, as
is the unpolarized fragmentation function, with a differ-
ent 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 op-
posite charges allows sensitivity to “favored” and “dis-
favored” fragmentation processes. A favored fragmen-
tation 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
+
and
d
→
π
−
.
e
-
e
+
φ
0
θ
2
p
t0
P
1
P
2
u
̂
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 momen-
tum
P
2
, and the transverse momentum
p
t
0
of the first pion
around the second-pion direction. The polar angle
θ
2
is de-
fined as the angle between the beam axis and the momentum
P
2
of the second pion.
Vice versa, we refer to
u
→
π
−
and
d
→
π
+
as disfa-
vored fragmentation processes. Therefore, the produc-
tion of pion pairs with opposite charge from the frag-
mentation of a
u
u
pair (
e
+
e
−
→
u
u
→
π
±
π
∓
X
) can
proceed through two favored fragmentation processes
(
u
→
π
+
and
u
→
π
−
) or through two disfavored
fragmentation processes (
u
→
π
−
and
u
→
π
+
). Ne-
glecting heavy quark fragmentation and introducing the
favored functions
D
fav
(
z
) =
D
π
+
u
(
z
) =
D
π
−
d
(
z
), and
D
fav
(
z
) =
D
π
−
̄
u
(
z
) =
D
π
+
̄
d
(
z
), as well as the disfavored
functions
D
dis
(
z
) =
D
π
−
u
(
z
) =
D
π
+
d
(
z
) =
D
π
±
s
(
z
) and
D
dis
(
z
) =
D
π
+
̄
u
(
z
) =
D
π
−
̄
d
(
z
) =
D
π
±
̄
s
(
z
), the cross section
for charged pion pair production can be written as [31]:
9
N
U
(
φ
) =
d
σ
(
e
+
e
−
→
π
±
π
∓
X
)
dΩd
z
1
d
z
2
∝
5
9
D
fav
(
z
1
)
D
fav
(
z
2
) +
7
9
D
dis
(
z
1
)
D
dis
(
z
2
)
N
L
(
φ
) =
d
σ
(
e
+
e
−
→
π
±
π
±
X
)
dΩd
z
1
d
z
2
∝
5
9
D
fav
(
z
1
)
D
dis
(
z
2
) +
5
9
D
dis
(
z
1
)
D
fav
(
z
2
) +
2
9
D
dis
(
z
1
)
D
dis
(
z
2
)
N
C
(
φ
) =
d
σ
(
e
+
e
−
→
ππX
)
dΩd
z
1
d
z
2
=
N
U
(
φ
) +
N
L
(
φ
)
∝
5
9
[
D
fav
(
z
1
) +
D
dis
(
z
1
)][
D
fav
(
z
2
) +
D
dis
(
z
2
)] +
4
9
D
dis
(
z
1
)
D
dis
(
z
2
)
(10)
where
π
stands for a generic charged pion,
φ
is the az-
imuthal angle
φ
1
+
φ
2
in RF12 or
φ
0
in RF0, dΩ =
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.
B
A
B
AR
EXPERIMENT AND DATA SAMPLE
The results presented here are based on a sample of
data collected with the
B
A
B
AR
detector at the PEP-II
asymmetric-energy
e
+
e
−
collider, at the SLAC National
Accelerator Laboratory. A total integrated luminosity
of about 468 fb
−
1
[32] is used, consisting of 424 fb
−
1
collected at the peak of the
Υ
(4
S
) resonance, and about
44 fb
−
1
collected 40 MeV below the peak.
The
B
A
B
AR
detector is described in detail in refer-
ences [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 mag-
net. 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
B
A
B
AR
detector and has a 95% (91%) identification effi-
ciency 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 elec-
tromagnetic calorimeter (EMC). Muons are identified in
the instrumented flux return (IFR), 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
Jetset
[35]
is used to generate hadronic events in non-resonant
e
+
e
−
annihilation. Separate MC samples are generated for
2
Natural units are used throughout this article
light quarks,
e
+
e
−
→
q
q
(
q
=
u, d, s
), called generic
uds
MC, and charm quarks,
e
+
e
−
→
c
c
. Samples of
B
B
events with generic
B
decays are generated with
the
EvtGen
[36] package. Finally,
τ
+
τ
−
and
μ
+
μ
−
event
samples are produced with the
KK2F
[37] generator, and
μ
+
μ
−
γ
events with
AfkQed
[38]. The generated events
undergo a full detector simulation based on
Geant4
[39]
and are analyzed in the same way as the experimental
data. No transverse spin effects are implemented in the
MC generation, so it can be used to evaluate detector bi-
ases. 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
correlation between the fragmenting quark and antiquark
may be lost for heavy quarks. In this section, we sum-
marize the event and track selection requirements.
Multi-hadronic events are selected by requiring at least
three reconstructed charged particles and the value of the
2nd divided by the 0th Fox-Wolfram moment [40], cal-
culated from charged tracks only,
R
′
2
<
0
.
98. To sup-
press backgrounds from
e
+
e
−
→
τ
+
τ
−
,
γγ
processes,
and events characterized by emission of a very energetic
photon via initial state radiation, we require the visi-
ble 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. Ev-
ery 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
|
d
Z
|
<
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
|
d
Z
|
<
10 cm, as well