of 29
Measurement of initial-state
final-state radiation interference in the
processes
e
þ
e
μ
þ
μ
γ
and
e
þ
e
π
þ
π
γ
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,9b,9c
A. R. Buzykaev,
9a
V. P. Druzhinin,
9a,9b
V. B. Golubev,
9a,9b
E. A. Kravchenko,
9a,9b
A. P. Onuchin,
9a,9b,9c
S. I. Serednyakov,
9a,9b
Yu. I. Skovpen,
9a,9b
E. P. Solodov,
9a,9b
K. Yu. Todyshev,
9a,9b
A. J. Lankford,
10
B. Dey,
11
J. W. Gary,
11
O. Long,
11
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
W. Panduro Vazquez,
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
J. Kim,
14
T. S. Miyashita,
14
P. Ongmongkolkul,
14
F. C. Porter,
14
M. Röhrken,
14
R. Andreassen,
15
Z. Huard,
15
B. T. Meadows,
15
B. G. Pushpawela,
15
M. D. Sokoloff,
15
L. Sun,
15
W. T. Ford,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17
,
W. H. Toki,
17
B. Spaan,
18
D. Bernard,
19
M. Verderi,
19
S. Playfer,
20
D. Bettoni,
21a
C. Bozzi,
21a
R. Calabrese,
21a,21b
G. Cibinetto,
21a,21b
E. Fioravanti,
21a,21b
I. Garzia,
21a,21b
E. Luppi,
21a,21b
L. Piemontese,
21a
V. Santoro,
21a
A. Calcaterra,
22
R. de Sangro,
22
G. Finocchiaro,
22
S. Martellotti,
22
P. Patteri,
22
I. M. Peruzzi,
22
M. Piccolo,
22
A. Zallo,
22
R. Contri,
23a,23b
M. R. Monge,
23a,23b
S. Passaggio,
23a
C. Patrignani,
23a,23b
B. Bhuyan,
24
V. Prasad,
24
A. Adametz,
25
U. Uwer,
25
H. M. Lacker,
26
U. Mallik,
27
C. Chen,
28
J. Cochran,
28
S. Prell,
28
H. Ahmed,
29
A. V. Gritsan,
30
N. Arnaud,
31
M. Davier,
31
D. Derkach,
31
G. Grosdidier,
31
F. Le Diberder,
31
A. M. Lutz,
31
B. Malaescu,
31
,
P. Roudeau,
31
A. Stocchi,
31
L. L. Wang,
31
G. Wormser,
31
D. J. Lange,
32
D. M. Wright,
32
J. P. Coleman,
33
J. R. Fry,
33
E. Gabathuler,
33
D. E. Hutchcroft,
33
D. J. Payne,
33
C. Touramanis,
33
A. J. Bevan,
34
F. Di Lodovico,
34
R. Sacco,
34
G. Cowan,
35
D. N. Brown,
36
C. L. Davis,
36
A. G. Denig,
37
M. Fritsch,
37
W. Gradl,
37
K. Griessinger,
37
A. Hafner,
37
K. R. Schubert,
37
R. J. Barlow,
38
G. D. Lafferty,
38
R. Cenci,
39
B. Hamilton,
39
A. Jawahery,
39
D. A. Roberts,
39
R. Cowan,
40
R. Cheaib,
41
P. M. Patel,
41
,*
S. H. Robertson,
41
N. Neri,
42a
F. Palombo,
42a,42b
L. Cremaldi,
43
R. Godang,
43
,**
D. J. Summers,
43
M. Simard,
44
P. Taras,
44
G. De Nardo,
45a,45b
G. Onorato,
45a,45b
C. Sciacca,
45a,45b
G. Raven,
46
C. P. Jessop,
47
J. M. LoSecco,
47
K. Honscheid,
48
R. Kass,
48
M. Margoni,
49a,49b
M. Morandin,
49a
M. Posocco,
49a
M. Rotondo,
49a
G. Simi,
49a,49b
F. Simonetto,
49a,49b
R. Stroili,
49a,49b
S. Akar,
50
E. Ben-Haim,
50
M. Bomben,
50
G. R. Bonneaud,
50
H. Briand,
50
G. Calderini,
50
J. Chauveau,
50
Ph. Leruste,
50
G. Marchiori,
50
J. Ocariz,
50
M. Biasini,
51a,51b
E. Manoni,
51a
A. Rossi,
51a
C. Angelini,
52a,52b
G. Batignani,
52a,52b
S. Bettarini,
52a,52b
M. Carpinelli,
52a,52b
,
††
G. Casarosa,
52a,52b
M. Chrzaszcz,
52a
F. Forti,
52a,52b
M. A. Giorgi,
52a,52b
A. Lusiani,
52a,52c
B. Oberhof,
52a,52b
E. Paoloni,
52a,52b
M. Rama,
52a
G. Rizzo,
52a,52b
J. J. Walsh,
52a
D. Lopes Pegna,
53
J. Olsen,
53
A. J. S. Smith,
53
F. Anulli,
54a
R. Faccini,
54a,54b
F. Ferrarotto,
54a
F. Ferroni,
54a,54b
M. Gaspero,
54a,54b
A. Pilloni,
54a,54b
G. Piredda,
54a
C. Bünger,
55
S. Dittrich,
55
O. Grünberg,
55
M. Hess,
55
T. Leddig,
55
C. Voß,
55
R. Waldi,
55
T. Adye,
56
E. O. Olaiya,
56
F. F. Wilson,
56
S. Emery,
57
G. Vasseur,
57
D. Aston,
58
D. J. Bard,
58
C. Cartaro,
58
M. R. Convery,
58
J. Dorfan,
58
G. P. Dubois-Felsmann,
58
W. Dunwoodie,
58
M. Ebert,
58
R. C. Field,
58
B. G. Fulsom,
58
M. T. Graham,
58
C. Hast,
58
W. R. Innes,
58
P. Kim,
58
D. W. G. S. Leith,
58
S. Luitz,
58
V. Luth,
58
D. B. MacFarlane,
58
D. R. Muller,
58
H. Neal,
58
T. Pulliam,
58
B. N. Ratcliff,
58
A. Roodman,
58
R. H. Schindler,
58
A. Snyder,
58
D. Su,
58
M. K. Sullivan,
58
J. Va
vra,
58
W. J. Wisniewski,
58
H. W. Wulsin,
58
M. V. Purohit,
59
J. R. Wilson,
59
A. Randle-Conde,
60
S. J. Sekula,
60
M. Bellis,
61
P. R. Burchat,
61
E. M. T. Puccio,
61
M. S. Alam,
62
J. A. Ernst,
62
R. Gorodeisky,
63
N. Guttman,
63
D. R. Peimer,
63
A. Soffer,
63
S. M. Spanier,
64
J. L. Ritchie,
65
R. F. Schwitters,
65
J. M. Izen,
66
X. C. Lou,
66
F. Bianchi,
67a,67b
F. De Mori,
67a,67b
A. Filippi,
67a
D. Gamba,
67a,67b
L. Lanceri,
68a,68b
L. Vitale,
68a,68b
F. Martinez-Vidal,
69
A. Oyanguren,
69
J. Albert,
70
Sw. Banerjee,
70
A. Beaulieu,
70
F. U. Bernlochner,
70
H. H. F. Choi,
70
G. J. King,
70
R. Kowalewski,
70
M. J. Lewczuk,
70
T. Lueck,
70
I. M. Nugent,
70
J. M. Roney,
70
R. J. Sobie,
70
N. Tasneem,
70
T. J. Gershon,
71
P. F. Harrison,
71
T. E. Latham,
71
H. R. Band,
72
S. Dasu,
72
Y. Pan,
72
R. Prepost,
72
and S. L. Wu
72
(The
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
3a
INFN Sezione di Bari, I-70126 Bari, Italy
3b
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, Canada V6T 1Z1
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
PHYSICAL REVIEW D
92,
072015 (2015)
1550-7998
=
2015
=
92(7)
=
072015(29)
072015-1
© 2015 American Physical Society
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
Technische Universität Dortmund, Fakultät Physik, D-44221 Dortmund, Germany
19
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
20
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
21a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
21b
Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy
22
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
23a
INFN Sezione di Genova, I-16146 Genova, Italy
23b
Dipartimento di Fisica, Università di Genova, I-16146 Genova, Italy
24
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
25
Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
26
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
27
University of Iowa, Iowa City, Iowa 52242, USA
28
Iowa State University, Ames, Iowa 50011-3160, USA
29
Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia
30
Johns Hopkins University, Baltimore, Maryland 21218, USA
31
Laboratoire de l
Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,
Centre Scientifique d
Orsay, F-91898 Orsay Cedex, France
32
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
33
University of Liverpool, Liverpool L69 7ZE, United Kingdom
34
Queen Mary, University of London, London E1 4NS, United Kingdom
35
University of London, Royal Holloway and Bedford New College, Egham,
Surrey TW20 0EX, United Kingdom
36
University of Louisville, Louisville, Kentucky 40292, USA
37
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
38
University of Manchester, Manchester M13 9PL, United Kingdom
39
University of Maryland, College Park, Maryland 20742, USA
40
Massachusetts Institute of Technology, Laboratory for Nuclear Science,
Cambridge, Massachusetts 02139, USA
41
McGill University, Montréal, Québec, Canada H3A 2T8
42a
INFN Sezione di Milano, I-20133 Milano, Italy
42b
Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy
43
University of Mississippi, University, Mississippi 38677, USA
44
Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7
45a
INFN Sezione di Napoli, I-80126 Napoli, Italy
45b
Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy
46
NIKHEF, National Institute for Nuclear Physics and High Energy Physics,
NL-1009 DB Amsterdam, The Netherlands
47
University of Notre Dame, Notre Dame, Indiana 46556, USA
48
Ohio State University, Columbus, Ohio 43210, USA
49a
INFN Sezione di Padova, I-35131 Padova, Italy
49b
Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy
50
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS,
Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France
51a
INFN Sezione di Perugia, I-06123 Perugia, Italy
51b
Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy
52a
INFN Sezione di Pisa, I-56127 Pisa, Italy
52b
Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy
52c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
53
Princeton University, Princeton, New Jersey 08544, USA
54a
INFN Sezione di Roma, I-00185 Roma, Italy
54b
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy
55
Universität Rostock, D-18051 Rostock, Germany
J. P. LEES
et al.
PHYSICAL REVIEW D
92,
072015 (2015)
072015-2
56
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
57
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
58
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
59
University of South Carolina, Columbia, South Carolina 29208, USA
60
Southern Methodist University, Dallas, Texas 75275, USA
61
Stanford University, Stanford, California 94305-4060, USA
62
State University of New York, Albany, New York 12222, USA
63
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
64
University of Tennessee, Knoxville, Tennessee 37996, USA
65
University of Texas at Austin, Austin, Texas 78712, USA
66
University of Texas at Dallas, Richardson, Texas 75083, USA
67a
INFN Sezione di Torino, I-10125 Torino, Italy
67b
Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy
68a
INFN Sezione di Trieste, I-34127 Trieste, Italy
68b
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy
69
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
70
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
71
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
72
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 1 September 2015; published 28 October 2015)
Charge asymmetry in the processes
e
þ
e
μ
þ
μ
γ
and
e
þ
e
π
þ
π
γ
is measured using
232
fb
1
of
data collected with the
BABAR
detector at
e
þ
e
center-of-mass energies near 10.58 GeV. An observable is
introduced and shown to be very robust against detector asymmetries while keeping a large sensitivity to
the physical charge asymmetry that results from the interference between initial- and final-state radiation
(FSR). The asymmetry is determined as a function of the invariant mass of the final-state tracks from
production threshold to a few GeV
=c
2
. It is compared to the expectation from QED for
e
þ
e
μ
þ
μ
γ
, and
from theoretical models for
e
þ
e
π
þ
π
γ
. A clear interference pattern is observed in
e
þ
e
π
þ
π
γ
,
particularly in the vicinity of the
f
2
ð
1270
Þ
resonance. The inferred rate of lowest-order FSR production is
consistent with the QED expectation for
e
þ
e
μ
þ
μ
γ
, and is negligibly small for
e
þ
e
π
þ
π
γ
.
DOI:
10.1103/PhysRevD.92.072015
PACS numbers: 13.60.Hb, 13.66.Bc, 13.66.Jn
I. INTRODUCTION
The radiative processes
e
þ
e
X
γ
ð
1
Þ
have been extensively studied by several
e
þ
e
experiments
and the cross sections for
e
þ
e
X
have been measured
using the initial-state radiation (ISR) method
[1
4]
.At
BABAR
[5]
, the cross sections have thus been determined in
large energy ranges below the total
e
þ
e
center-of-mass
(c.m.) energy
ffiffiffi
s
p
10
.
58
GeV available at the SLAC
PEP-II collider. The state
X
can be either fully described
by quantum electrodynamics (QED) such as
μ
þ
μ
, or any
hadronic state with
J
PC
¼
1
−−
.
In reaction
(1)
at lowest order (LO) the photon can be
emitted from either the incoming electron or positron, or
from the final state (final-state radiation, or FSR). At
BABAR
, the kinematic conditions are such that the process
is dominated by ISR photons, which justifies the ISR
method. The LO FSR contribution to the hadronic radiative
process is neglected, as its theoretical estimates are well
below the systematic uncertainties of the cross section
measurement. This is due to the fact that the available
e
þ
e
c.m. energy is far beyond the domain of the hadronic
resonances that dominate the cross section, so that hadronic
form factors considerably reduce the probability that the
photon is emitted from the final state. However, the
theoretical estimations are model dependent, and it is thus
important to have a direct experimental proof of the
smallness of the FSR contribution to the hadronic cross
sections when high precision is at stake, as for the
determination of the hadronic contribution to the
g
2
value of the muon
[6]
. Because of the point-like nature of
the muon, the FSR reduction does not occur for the
e
þ
e
μ
þ
μ
γ
process. The LO FSR contribution to the cross
section is expected to vanish at threshold and to increase
*
Deceased.
Now at: University of Tabuk, Tabuk 71491, Saudi Arabia.
Now at: Laboratoire de Physique Nucléaire et de Hautes
Energies, IN2P3/CNRS, F-75252 Paris, France.
§
Now at: Institute of High Energy Physics, Beijing 100049,
China.
Now at: University of Huddersfield, Huddersfield HD1 3DH,
United Kingdom.
**
Now at: University of South Alabama, Mobile, Alabama
36688, USA.
††
Also at: Università di Sassari, I-07100 Sassari, Italy.
MEASUREMENT OF INITIAL-STATE
FINAL-STATE
...
PHYSICAL REVIEW D
92,
072015 (2015)
072015-3
with the invariant mass of the muon pair (
m
μμ
). Still, the
FSR fraction remains small for low di-muon mass (less than
1% for
m
μμ
<
1
GeV
=c
2
). For the
e
þ
e
μ
þ
μ
γ
cross
section measurement, a correction is applied for the LO
FSR contribution as a function of
m
μμ
, which is so far
determined by turning off FSR in the Monte Carlo (MC)
generation.
While it is not possible to distinguish ISR from FSR
photons on an event-by-event basis, as the corresponding
amplitudes are both present and interfere, a measurement of
the interference provides a sensitive and quantitative
determination of their relative strength. Measurement of
the forward-backward asymmetry of the pions was first
proposed in Ref.
[3]
, as a test of the underlying model for
final-state radiation. In this paper, the ISR-FSR interference
for
e
þ
e
μ
þ
μ
γ
and
e
þ
e
π
þ
π
γ
is studied through
the charge asymmetry of the production of these events at
various decay plane angles. The comparison between the
QED prediction and the measurement is done for the
charge asymmetry in
e
þ
e
μ
þ
μ
γ
. Various FSR models
are discussed for
e
þ
e
π
þ
π
γ
, and the most realistic
quark-FSR model is compared to the measurement of the
charge asymmetry in that channel.
This paper reports the first measurement of charge
asymmetry in the
e
þ
e
μ
þ
μ
γ
process. For
e
þ
e
π
þ
π
γ
, a preliminary measurement
[7]
of the forward-
backward asymmetry has been reported at low energies
(
ffiffiffi
s
p
1
GeV). No previous result exists at high energies.
II. ISR-FSR INTERFERENCE AND
CHARGE ASYMMETRY
A. Charge asymmetry
The Feynman diagrams for the LO ISR and LO FSR
emission in the process
e
þ
e
x
þ
x
γ
(where
x
¼
μ
or
π
),
are illustrated in Fig.
1
. The total LO amplitude
M
is the
sum of the corresponding amplitudes
M
ISR
and
M
FSR
, and
the cross section for
e
þ
e
x
þ
x
γ
is
σ
j
M
j
2
¼j
M
ISR
j
2
þj
M
FSR
j
2
þ
2
R
e
ð
M
ISR
M

FSR
Þ
:
ð
2
Þ
If the photon is emitted from the initial (final) state, the
x
þ
x
pair is produced with charge parity
C
¼
1
ðþ
1
Þ
,
which implies that the interference term changes sign if one
interchanges
x
þ
and
x
. While the contribution of the
interference term to the total cross section vanishes when
one integrates over the kinematic variables of the final state,
that term induces a significant observable charge asymme-
try in the differential cross section.
Charge asymmetry is defined as
A
¼
j
M
j
2
j
M
x
þ
x
j
2
j
M
j
2
þj
M
x
þ
x
j
2
¼
2
R
e
ð
M
ISR
M

FSR
Þ
j
M
ISR
j
2
þj
M
FSR
j
2
;
ð
3
Þ
where
x
þ
x
means that
x
þ
and
x
are interchanged.
Although it is not possible to reconstruct
M
ISR
or
M
FSR
from the charge asymmetry and the cross section, as the
relative phase between them remains unknown, informa-
tion on the ratio
j
M
FSR
=
M
ISR
j
can be derived within the
framework of specific models.
B. Choice of kinematic variables
Aside from an overall azimuthal rotation about the beam
axis, the kinematic topology of the
x
þ
x
γ
final state (where
x
¼
μ
or
π
) is described by four variables, which are the
muon-pair (pion-pair) invariant mass
m
xx
(or equivalently
E

γ
, the energy of the radiated photon in the
e
þ
e
c.m.) and
three angular variables. At a given
m
xx
mass, the distribu-
tion of the three angular variables contains all the available
information on the ISR/FSR amplitudes.
At variance with the definition of forward-backward
asymmetry used in Ref.
[8]
, which refers to the polar angle
of
x
with respect to the incoming electron in the
e
þ
e
c.m.
system (c.m.s.), this analysis introduces the set of angular
variables illustrated in Fig.
2
. These are found to be more
sensitive observables to measure the ISR-FSR interference:
(i)
θ

γ
polar angle of the radiated photon in the
e
þ
e
c.m.s. (with respect to the
e
þ
e
axis);
(ii)
θ

polar angle of
x
with respect to the photon axis
in the
x
þ
x
c.m.s.;
(iii)
φ

azimuthal angle of
x
with respect to the
γ
e
þ
e
plane in the
x
þ
x
c.m.s. (or the
e
þ
e
c.m.s.).
Since
x
þ
x
interchange means reversal of the
x
direction to its opposite in the
x
þ
x
c.m.s., the charge
asymmetry, for fixed
m
xx
and
θ

γ
, is equal to
FIG. 1. Feynman diagrams for
e
þ
e
x
þ
x
γ
(
x
¼
μ
,
π
), where the photon is from LO ISR (left) or LO FSR (right).
J. P. LEES
et al.
PHYSICAL REVIEW D
92,
072015 (2015)
072015-4
A
ð
θ

;
φ

Þ¼
σ
ð
θ

;
φ

Þ
σ
ð
π
θ

;
π
þ
φ

Þ
σ
ð
θ

;
φ

Þþ
σ
ð
π
θ

;
π
þ
φ

Þ
:
ð
4
Þ
For the
e
þ
e
μ
þ
μ
γ
process, the charge asymmetry
as a function of cos
θ

and
φ

, studied with the
AFKQED
generator (see Sec.
IV B
), is shown in Fig.
3
. The FSR
amplitude is dominant at
j
cos
θ

j
1
, when one of the
charged-particle tracks is very close to the radiated photon.
However, Fig.
3
shows that
φ

is a more sensitive variable
to measure the ISR/FSR content over the full phase space,
with sign reversal of the charge asymmetry. After integra-
tion over cos
θ

γ
and integration over symmetrical cos
θ

intervals, the distribution of the integrated charge asym-
metry
A
ð
cos
φ

Þ
suggests a simple linear dependence
A
ð
cos
φ

Þ¼
A
0
cos
φ

:
ð
5
Þ
From the expressions of the differential cross section
detailed in the next section, it results that the slope
A
0
is
an estimator of the ISR-FSR interference, sensitive to the
ratio
j
M
FSR
=
M
ISR
j
in each
m
μμ
interval. Moreover, it will
be shown in Sec.
V
that the measurement of
A
0
is barely
affected by detector charge asymmetries.
III. THEORETICAL PREDICTIONS FOR THE
CHARGE ASYMMETRY
A. QED prediction for the
e
þ
e
μ
þ
μ
γ
process
In the massless limit
[9]
, the differential cross section of
the QED
e
þ
e
μ
þ
μ
γ
process, written as a function of
the four kinematic variables defined above (Sec.
II B
),
implies that the differential charge asymmetry is propor-
tional to cos
φ

:
A
e
þ
e
μ
þ
μ
γ
ð
m
μμ
;
θ

γ
;
θ

;
φ

Þ
¼
2
ffiffiffi
s
p
m
μμ
sin
θ

γ
sin
θ

cos
φ

s
sin
2
θ

þ
m
2
μμ
sin
2
θ

γ
:
ð
6
Þ
When the masses are taken into account, the effect from
the electron/positron mass is found to be negligible for
radiated photons away from the beams. The effect from the
muon mass is sizable, especially at large
m
μμ
when the
radiated photon is close to one of the muons. Predictions for
the charge asymmetry in the massive case are obtained by
numerical integration of several variants of the QED
differential cross section
[9
11]
. The phase space consid-
ered in those calculations is limited to the experimental
acceptance
20
°
<
θ

γ
<
160
°, and the results are shown in
FIG. 2. Definition of the angular variables describing the kinematic topology of the final states of the process
e
þ
e
x
þ
x
γ
ð
x
¼
μ
;
π
Þ
at a given
x
þ
x
invariant mass (left) in the
e
þ
e
c.m.s., and (right) in the
x
þ
x
c.m.s.
-1
-0.5
0
0.5
1
-2
0
2
-0.5
0
0.5
cos
θ
φ
*
(rad
)
Asymmetry
-1
-0.5
0
0.5
1
-2
0
2
-0.5
0
0.5
cos
θ
φ
*
(rad
)
Asymmetry
-1
-0.5
0
0.5
1
-2
0
2
-0.5
0
0.5
cos
θ
φ
*
(rad
)
Asymmetry
FIG. 3. Charge asymmetry at generator level in
e
þ
e
μ
þ
μ
γ
simulation, as a function of cos
θ

and
φ

for the same
m
μμ
interval
(
6
.
5
<m
μμ
<
7
.
0
GeV
=c
2
) and various cos
θ

γ
ranges: (left)
1
<
cos
θ

γ
<
0
.
6
, (middle)
0
.
6
<
cos
θ

γ
<
0
.
4
, (right)
0
.
4
<
cos
θ

γ
<
0
.
MEASUREMENT OF INITIAL-STATE
FINAL-STATE
...
PHYSICAL REVIEW D
92,
072015 (2015)
072015-5
Fig.
4
as a function of
m
μμ
. Predictions differ at the physical
threshold (
m
μμ
¼
2
m
μ
), where only the charge asymmetry
based on Ref.
[11]
extrapolates to zero as expected,
suggesting that the validity of formulas in Refs.
[9,10]
does not extend to small
m
μμ
. At large mass (
m
μμ
>
3
GeV
=c
2
), the prediction from Ref.
[9]
differs from the
others by up to a few percent. The formula of the differ-
ential LO cross section implemented in the
AFKQED
generator, which is used in this analysis for simulation
(see Sec.
IV B
), is the one by Arbuzov
et al.
[11]
, which has
the most reliable behavior over the full
m
μμ
range.
B. FSR models for the
e
þ
e
π
þ
π
γ
process
As in the
e
þ
e
μ
þ
μ
γ
process, ISR and FSR con-
tribute to
e
þ
e
π
þ
π
γ
(Fig.
5
). However, the charge
asymmetry is expected to be much smaller in the latter
process because the FSR contribution is strongly reduced
by the pion form factor at large
ffiffiffi
s
p
. In addition, its estimate
is model dependent.
1. FSR from point-like pions (model 1)
In the FSR model shown in Fig.
5(b)
, the photon is
emitted from one of the final-state pions, where the pion is
treated as a point-like particle. In this hypothesis, the FSR
amplitude
M
FSR
is proportional to the pion form factor at
the collision energy squared
s
, namely
F
π
ð
s
Þ
. The ISR
amplitude
M
ISR
shown in Fig.
5(a)
is proportional to the
pion form factor
F
π
ð
s
0
Þ
at a reduced energy squared
s
0
¼
s
ð
1
2
E

γ
=
ffiffiffi
s
p
Þ
. According to this FSR model, the
charge asymmetry to be measured at
BABAR
reflects the
relative magnitude of the pion form factor at
ffiffiffi
s
p
¼
10
.
58
GeV and at low energy. It is consequently negligibly
small, since
F
π
ð
s
0
Þ
, dominated by the
ρ
resonance in the
s
0
¼
m
2
ππ
domain accessible to the experiment, is 3 orders
of magnitude larger than
j
F
π
ð
10
.
58
2
GeV
2
Þj
0
.
01
,as
estimated from an extrapolation of existing data
[6,12]
using a
1
=s
dependence. This model is studied with the
PHOKHARA
4.0
[13]
generator, in which the FSR current has
a point-like Lorentz structure, including a contact term,
m
μμ
(GeV/c
2
)
A
0
GW
BK
AF
-1
-0.8
-0.6
-0.4
-0.2
0
m
μμ
(GeV/c
2
)
Δ
A
0
= A(
φ
*
=0) - (A
0
)
AfkQed
(LO)
GW
BK
AF
-0.1
-0.05
0
0.05
0.1
0246
0246
FIG. 4. (Left) Charge asymmetry at
φ

¼
0
,
A
0
, as a function of
m
μμ
, obtained by numerical integration according to three different
theoretical predictions (see text), with the condition
20
°
<
θ

γ
<
160
° applied. (Right) The difference between the prediction and the
AFKQED
LO value. Results labeled GW, BK, AF are obtained from Refs.
[9
11]
, respectively.
(a)
(b)
(c)
FIG. 5. Feynman diagrams for
e
þ
e
π
þ
π
γ
. (a) Initial state radiation, (b) Final state radiation with pions treated as point-like
particles (FSR model 1), (c) Final state radiation at quark level (FSR model 2).
J. P. LEES
et al.
PHYSICAL REVIEW D
92,
072015 (2015)
072015-6
globally multiplied by the pion form factor. In this model,
the
A
0
ð
m
ππ
Þ
distribution is expected to increase quadrati-
cally with mass on the
ρ
resonance, with a change of sign at
the
ρ
mass
A
0
2
×
10
3
ð
m
2
ππ
m
2
ρ
Þ
;
ð
7
Þ
with values well below the sensitivity of this analysis
because of the large pion form factor suppression at
10.58 GeV.
2. FSR from quarks (model 2)
In the
a priori
more realistic FSR model for
e
þ
e
π
þ
π
γ
depicted in Fig.
5(c)
, the FSR photon is emitted
from the quarks, which subsequently hadronize into a pion
pair
[14]
. The dominant ISR
1
and FSR contributions, and
their interference, are written in terms of the variables
defined in Sec.
II B
:
d
σ
ISR
e
þ
e
π
þ
π
γ
dm
2
ππ
d
cos
θ

γ
d
cos
θ

d
φ

¼
α
3
β
3
16
π
s
2
m
2
ππ
ð
s
m
2
ππ
Þ
j
F
π
ð
m
2
ππ
Þj
2
×

ð
s
2
þ
m
4
ππ
Þ
1
þ
cos
2
θ

γ
sin
2
θ

γ
sin
2
θ

þ
4
sm
2
ππ
cos
2
θ

2
ffiffiffi
s
p
m
ππ
ð
s
þ
m
2
ππ
Þð
tan
θ

γ
Þ
1
sin
2
θ

cos
φ

2
sm
2
ππ
sin
2
θ

cos
2
φ


;
ð
8
Þ
where
α
and
β
are the QED fine-structure constant and the
pion velocity
β
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4
m
2
π
=m
2
ππ
p
, respectively. The FSR
contribution is
d
σ
FSR
e
þ
e
π
þ
π
γ
dm
2
ππ
d
cos
θ

γ
d
cos
θ

d
φ

¼
α
3
β
ð
s
m
2
ππ
Þ
64
π
s
3
ð
1
þ
cos
2
θ

γ
Þj
V
ð
m
2
ππ
;
θ

Þj
2
;
ð
9
Þ
and the interference term
d
σ
I
e
þ
e
π
þ
π
γ
dm
2
ππ
d
cos
θ

γ
d
cos
θ

d
φ

¼
α
3
β
2
16
π
s
2
ffiffiffi
s
p
m
ππ
Re
f
F

π
ð
m
2
ππ
Þ
V
ð
m
2
ππ
;
θ

Þg
×

ffiffiffi
s
p
m
ππ
cos
θ

γ
cos
θ

þ½ð
1
þ
cos
2
θ

γ
Þ
s
þ
m
2
ππ
sin
2
θ

γ

sin
θ

cos
φ

2
sin
θ

γ

;
ð
10
Þ
where
V
¼
X
q
e
2
q
V
q
¼
X
q
e
2
q
Z
1
0
dz
2
z
1
z
ð
1
z
Þ
Φ
þ
q
ð
z; m
2
ππ
;
cos
θ

Þ
ð
q
¼
u; d
Þ
;
ð
11
Þ
and
Φ
þ
q
ð
z; m
2
ππ
;
cos
θ

Þ
is the
C
-even part of the two-pion
generalized distribution amplitudes (GDA). The pion time-
like form factor
F
π
ð
m
2
ππ
Þ
is taken from a fit to
BABAR
data
[6]
with a vector dominance model.
So far, there is no implementation of this model in an
MC generator to describe the ISR-FSR interference in the
e
þ
e
π
þ
π
γ
process. In order to predict the charge
asymmetry numerically, we take the following GDA
model, which is a modified version of the model found
in Ref.
[16]
:
Φ
þ
u
ð
z; m
2
ππ
;
cos
θ

Þ
¼
Φ
þ
d
ð
z; m
2
ππ
;
cos
θ

Þ
¼
10
z
ð
1
z
Þð
2
z
1
Þ
×

c
0
3
β
2
2
e
i
δ
0
ð
m
ππ
Þ
þ
c
2
β
2
BW
ð
m
ππ
Þ
P
2
ð
cos
θ

Þ

;
ð
12
Þ
where
c
0
and
c
2
are the magnitudes of the S-wave and
D-wave contributions, respectively. As the scalar sector
is known to involve wide resonances, the S-wave contri-
bution is approximated by a constant amplitude with
a mass-dependent phase
δ
0
ð
m
ππ
Þ
taken from pion-pion
phase-shift analyses
[17]
in the region below
1
.
6
GeV
=c
2
.
This model incorporates the rapid phase variation across
the
f
0
ð
980
Þ
resonance. Using
c
0
¼
0
.
5
[16]
yields an
A
0
value of about
1%
near the
ρ
resonance and nearly flat
with mass. For the D-wave tensor contribution, we use a
Breit-Wigner (BW) form for the
f
2
ð
1270
Þ
resonance in
order to take properly into account the mass dependence of
the amplitude, the phase variation being given by the BW
form in agreement with the measured
δ
2
ð
m
ππ
Þ
values
[17]
.
The angular dependence in the
ππ
center of mass is given
by the Legendre polynomial
P
2
ð
cos
θ

Þ
, which assumes the
dominance of helicity 0 for
f
2
ð
1270
Þ
production.
1
We thank Leonard Lesniak for pointing out a sign mistake in
the sin
2
θ

cos
φ

term of Eq.
(8)
as given in the erratum of
Ref.
[14]
. The correct sign has been checked with the formulas
given in Ref.
[15]
.
MEASUREMENT OF INITIAL-STATE
FINAL-STATE
...
PHYSICAL REVIEW D
92,
072015 (2015)
072015-7
C. Other sources of charge asymmetry
Next-to-leading-order (NLO) corrections including addi-
tional photons (soft and hard) and loops are expected to
affect the LO predictions for the charge asymmetry. For the
μ
þ
μ
γ
process these corrections have been computed
recently
[18]
and implemented in the
PHOKHARA
9.0
generator
[18]
. As discussed in Sec.
VI C
, the effects are
found to be small, at the percent level for the experimental
conditions of the present analysis, and to be well accounted
for by the simpler structure function approach implemented
in
AFKQED
. No exact NLO calculation is available for the
π
þ
π
γ
process. In this case, since the LO charge asym-
metry is expected to be small because the FSR amplitude is
suppressed, NLO corrections could play a relatively more
important role. The soft and virtual photon contributions to
the Born process
e
þ
e
π
þ
π
are known
[15,19]
to
generate an asymmetry of the pion production, with
asymmetry values at the percent level at a
ππ
mass of
1
GeV
=c
2
. However, it is unclear if the above result can be
used in the conditions of the present process
e
þ
e
π
þ
π
γ
, where one of the incoming electrons is highly off
shell after emission of a hard ISR photon. Furthermore,
such an asymmetry would vanish because of the symmet-
rical integration in cos
θ

. NLO corrections as implemented
in
AFKQED
have indeed no effect on the charge asymmetry.
No correction on the measured charge asymmetry is
therefore applied for the
π
þ
π
γ
process.
Another potential source of charge asymmetry comes
from
Z
exchange. This contribution is strongly suppressed
by the
Z
propagator, especially for the ISR diagrams where
m
2
xx
=M
2
Z
10
4
. Therefore one expects this effect to be
negligible for
π
þ
π
γ
. The contribution is larger for the FSR
diagrams for
μ
þ
μ
γ
since here the relevant ratio is
s=M
2
Z
¼
1
.
4%
. The contribution of
Z
exchange is studied
with the
KKMC
generator
[20]
. As reported in Sec.
VI C
, the
effect is at the level of a few per mille.
IV. EXPERIMENTAL ANALYSIS
A. The
BABAR
detector and data samples
The analysis is based on
232
fb
1
of data
[21]
collected
with the
BABAR
detector at the SLAC National
Accelerator Laboratory at the PEP-II asymmetric-energy
e
þ
e
collider operated at the
Υ
ð
4
S
Þ
resonance. About
10% of the data was collected 40 MeV below the
resonance. The
BABAR
detector is described in detail
elsewhere
[22]
. Charged-particle tracks are measured with
a five-layer double-sided silicon vertex tracker (SVT)
together with a 40-layer drift chamber (DCH), both inside
a 1.5 T superconducting solenoid. Photons are assumed to
originate from the primary vertex defined by the charged-
particle tracks of the event, and their energy and position
are measured in a CsI(Tl) electromagnetic calorimeter
(EMC). Charged-particle identification (PID) uses the
ionization energy loss d
E=
d
x
in the SVT and DCH, the
Cherenkov radiation detected in a ring-imaging device
(DIRC), the shower energy deposit (
E
cal
)intheEMC,and
the shower shape in the instrumented flux return (IFR) of
the magnet. The IFR system is made of modules of
resistive plate chambers interspaced with iron slabs,
arranged in a layout with a barrel and two end caps.
Collision events are recorded and reconstructed if they
pass three levels of trigger (hardware, online software, and
offline filter), each using complementary information
from the subdetectors.
B. Monte Carlo generators and simulation
Signal and background processes
e
þ
e
X
γ
are simu-
lated with the
AFKQED
event generator, which is based on
QED for
e
þ
e
μ
þ
μ
γ
and Ref.
[23]
for hadronic
production. LO ISR and FSR emission is simulated for
e
þ
e
μ
þ
μ
γ
, while LO FSR is neglected for hadronic
processes. The main photon (hereafter called the
ISR
photon) is emitted within the angular range
20
°
<
θ

γ
<
160
° in the
e
þ
e
c.m. system, bracketing the photon
detection range with a margin for resolution. Additional
ISR photons are generated with the structure function
method
[24]
, and additional FSR photons with the
PHOTOS
[25]
program. Additional ISR photons are emitted
along the
e
þ
or
e
beam particle direction. A minimum
mass
m
X
γ
ISR
>
8
GeV
=c
2
is imposed at generation, which
puts an upper bound on the additional ISR photon energy.
Samples corresponding to 5 to 10 times the data are
generated for the signal
e
þ
e
μ
þ
μ
γ
and
e
þ
e
π
þ
π
γ
channels, as well as large samples of backgrounds
from the other two-prong and multihadron ISR processes.
Background processes
e
þ
e
q
̄
q
(
q
¼
u; d; s; c
) are gen-
erated with the
JETSET
[26]
generator, and
e
þ
e
τ
þ
τ
with the
KORALB
[27]
program. The response of the
BABAR
detector is simulated using the
GEANT
4
[28]
package.
C. Event selection
Event selection follows the same procedure as the
selection of two-charged particle ISR events used for cross
section measurements
[6]
. It requires a photon with energy
E

γ
>
3
GeV in the
e
þ
e
c.m. and laboratory polar angle
with respect to the
e
beam in the range [0.35
2.4] rad, and
exactly two tracks of opposite charge, each with momen-
tum
p>
1
GeV
=c
and within the angular range [0.40
2.45] rad. If more than one photon is detected, the candidate
with the highest
E

γ
is taken to be the
ISR
photon. To
ensure a rough momentum balance at an early stage of the
selection, the
ISR
photon is required to lie within 0.3 rad
of the missing momentum of the charged particles (or of the
tracks plus the other photons). The tracks are required to
have at least 15 hits in the DCH, to originate within 5 mm
of the collision axis and within 6 cm from the beam spot
along the beam direction, and to extrapolate to the DIRC
and IFR active areas in order to exclude low-efficiency
regions. Both tracks are required to be identified either as
J. P. LEES
et al.
PHYSICAL REVIEW D
92,
072015 (2015)
072015-8
0
2000
4000
6000
8000
-1
-0.5
0
0.5
1
2m
μ
0.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
2000
4000
6000
-1
-0.5
0
0.5
1
0.5
1GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
3000
-1
-0.5
0
0.5
1
1
1.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
-1
-0.5
0
0.5
1
1.5
2GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
500
1000
1500
2000
-1
-0.5
0
0.5
1
2
2.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
500
1000
1500
-1
-0.5
0
0.5
1
2.5
3GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
500
1000
1500
-1
-0.5
0
0.5
1
3
3.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
500
1000
1500
-1
-0.5
0
0.5
1
3.5
4GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
-1
-0.5
0
0.5
1
4
4.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
3000
-1
-0.5
0
0.5
1
4.5
5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
3000
-1
-0.5
0
0.5
1
5
5.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
3000
4000
-1
-0.5
0
0.5
1
5.5
6GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
3000
4000
-1
-0.5
0
0.5
1
6
6.5GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
0
1000
2000
3000
4000
-1
-0.5
0
0.5
1
6.5
7GeV/c
2
cos
φ
Events / 0.05
μ
-
μ
↔μ
+
FIG. 6 (color online). The cos
φ

distributions for
e
þ
e
μ
þ
μ
γ
data in
0
.
5
GeV
=c
2
m
μμ
intervals. The points labeled
μ
refer to
the configurations with
φ

½
0
;
π

, while the points labeled
μ
μ
þ
correspond to
φ

þ
½
0
;
π

.
MEASUREMENT OF INITIAL-STATE
FINAL-STATE
...
PHYSICAL REVIEW D
92,
072015 (2015)
072015-9
0
200
400
-1
-0.5
0
0.5
1
0.3
0.4GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
250
500
750
1000
-1
-0.5
0
0.5
1
0.4
0.5GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
500
1000
1500
2000
-1
-0.5
0
0.5
1
0.5
0.6GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
2000
4000
-1
-0.5
0
0.5
1
0.6
0.7GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
5000
10000
-1
-0.5
0
0.5
1
0.7
0.8GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
2000
4000
-1
-0.5
0
0.5
1
0.8
0.9GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
500
1000
-1
-0.5
0
0.5
1
0.9
1GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
200
400
-1
-0.5
0
0.5
1
1
1.1GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
100
200
300
-1
-0.5
0
0.5
1
1.1
1.2GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
50
100
150
-1
-0.5
0
0.5
1
1.2
1.3GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
25
50
75
100
-1
-0.5
0
0.5
1
1.3
1.4GeV/c
2
cos
φ
Events / 0.1
π
π
↔π
+
0
25
50
75
100
-1
-0.5
0
0.5
1
1.4
1.5GeV/c
2
cos
φ
Events / 0.5
π
π
↔π
+
0
20
40
60
-1
-0.5
0
0.5
1
1.5
1.6GeV/c
2
cos
φ
Events / 0.5
π
π
↔π
+
0
20
40
60
-1
-0.5
0
0.5
1
1.6
1.7GeV/c
2
cos
φ
Events / 0.5
π
π
↔π
+
0
25
50
75
100
-1
-0.5
0
0.5
1
1.7
1.8GeV/c
2
cos
φ
Events / 0.5
π
π
↔π
+
FIG. 7 (color online). The cos
φ

distributions for
e
þ
e
π
þ
π
γ
data in
0
.
1
GeV
=c
2
m
ππ
intervals. The points labeled
π
refer to
the configurations with
φ

½
0
;
π

, while the points labeled
π
π
þ
correspond to
φ

þ
½
0
;
π

.
J. P. LEES
et al.
PHYSICAL REVIEW D
92,
072015 (2015)
072015-10