of 34
Cross sections for the reactions
e
þ
e

!
K
þ
K


þ


,
K
þ
K


0

0
,
and
K
þ
K

K
þ
K

measured using initial-state radiation events
J. P. Lees,
1
V. Poireau,
1
E. Prencipe,
1
V. Tisserand,
1
J. Garra Tico,
2
E. Grauges,
2
M. Martinelli,
3a,3b
D. A. Milanes,
3a,3b
A. Palano,
3a,3b
M. Pappagallo,
3a,3b
G. Eigen,
4
B. Stugu,
4
L. Sun,
4
D. N. Brown,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
G. Lynch,
5
H. Koch,
6
T. Schroeder,
6
D. J. Asgeirsson,
7
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
A. Khan,
8
V. E. Blinov,
9
A. R. Buzykaev,
9
V. P. Druzhinin,
9
V. B. Golubev,
9
E. A. Kravchenko,
9
A. P. Onuchin,
9
S. I. Serednyakov,
9
Yu. I. Skovpen,
9
E. P. Solodov,
9
K. Yu. Todyshev,
9
A. N. Yushkov,
9
M. Bondioli,
10
S. Curry,
10
D. Kirkby,
10
A. J. Lankford,
10
M. Mandelkern,
10
D. P. Stoker,
10
H. Atmacan,
11
J. W. Gary,
11
F. Liu,
11
O. Long,
11
G. M. Vitug,
11
C. Campagnari,
12
T. M. Hong,
12
D. Kovalskyi,
12
J. D. Richman,
12
C. A. West,
12
A. M. Eisner,
13
J. Kroseberg,
13
W. S. Lockman,
13
A. J. Martinez,
13
T. Schalk,
13
B. A. Schumm,
13
A. Seiden,
13
C. H. Cheng,
14
D. A. Doll,
14
B. Echenard,
14
K. T. Flood,
14
D. G. Hitlin,
14
P. Ongmongkolkul,
14
F. C. Porter,
14
A. Y. Rakitin,
14
R. Andreassen,
15
M. S. Dubrovin,
15
B. T. Meadows,
15
M. D. Sokoloff,
15
P. C. Bloom,
16
W. T. Ford,
16
A. Gaz,
16
M. Nagel,
16
U. Nauenberg,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17,
*
W. H. Toki,
17
B. Spaan,
18
M. J. Kobel,
19
K. R. Schubert,
19
R. Schwierz,
19
D. Bernard,
20
M. Verderi,
20
P. J. Clark,
21
S. Playfer,
21
J. E. Watson,
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
M.Munerato,
22a,22b
M.Negrini,
22a,22b
L.Piemontese,
22a
R.Baldini-Ferroli,
23
A.Calcaterra,
23
R.deSangro,
23
G. Finocchiaro,
23
M. Nicolaci,
23
S. Pacetti,
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
C. L.Lee,
26
M.Morii,
26
A. J. Edwards,
27
A.Adametz,
28
J. Marks,
28
U.Uwer,
28
F. U.Bernlochner,
29
M. Ebert,
29
H. M. Lacker,
29
T. Lueck,
29
P. D. Dauncey,
30
M. Tibbetts,
30
P. K. Behera,
31
U. Mallik,
31
C. Chen,
32
J. Cochran,
32
H. B. Crawley,
32
W. T. Meyer,
32
S. Prell,
32
E. I. Rosenberg,
32
A. E. Rubin,
32
A. V. Gritsan,
33
Z. J. Guo,
33
N. Arnaud,
34
M. Davier,
34
D. Derkach,
34
G. Grosdidier,
34
F. Le Diberder,
34
A. M. Lutz,
34
B. Malaescu,
34
P. Roudeau,
34
M. H. Schune,
34
A. Stocchi,
34
G. Wormser,
34
D. J. Lange,
35
D. M. Wright,
35
I. Bingham,
36
C. A. Chavez,
36
J. P. Coleman,
36
J. R. Fry,
36
E. Gabathuler,
36
D. E. Hutchcroft,
36
D. J. Payne,
36
C. Touramanis,
36
A. J. Bevan,
37
F. Di Lodovico,
37
R. Sacco,
37
M. Sigamani,
37
G. Cowan,
38
S. Paramesvaran,
38
D. N. Brown,
39
C. L. Davis,
39
A. G. Denig,
40
M. Fritsch,
40
W. Gradl,
40
A. Hafner,
40
K. E. Alwyn,
41
D. Bailey,
41
R. J. Barlow,
41
G. Jackson,
41
G. D. Lafferty,
41
R. Cenci,
42
B. Hamilton,
42
A. Jawahery,
42
D. A. Roberts,
42
G. Simi,
42
C. Dallapiccola,
43
E. Salvati,
43
R. Cowan,
44
D. Dujmic,
44
G. Sciolla,
44
D. Lindemann,
45
P. M. Patel,
45
S. H. Robertson,
45
M. Schram,
45
P. Biassoni,
46a,46b
A. Lazzaro,
46a,46b
V. Lombardo,
46a
F. Palombo,
46a,46b
S. Stracka,
46a,46b
L. Cremaldi,
47
R. Godang,
47,
R. Kroeger,
47
P. Sonnek,
47
D. J. Summers,
47
X. Nguyen,
48
P. Taras,
48
G. De Nardo,
49a,49b
D. Monorchio,
49a,49b
G. Onorato,
49a,49b
C. Sciacca,
49a,49b
G. Raven,
50
H. L. Snoek,
50
C. P. Jessop,
51
K. J. Knoepfel,
51
J. M. LoSecco,
51
W. F. Wang,
51
K. Honscheid,
52
R. Kass,
52
J. Brau,
53
R. Frey,
53
N. B. Sinev,
53
D. Strom,
53
E. Torrence,
53
E. Feltresi,
54a,54b
N. Gagliardi,
54a,54b
M. Margoni,
54a,54b
M. Morandin,
54a
M. Posocco,
54a
M. Rotondo,
54a
F. Simonetto,
54a,54b
R. Stroili,
54a,54b
E. Ben-Haim,
55
M. Bomben,
55
G. R. Bonneaud,
55
H. Briand,
55
G. Calderini,
55
J. Chauveau,
55
O. Hamon,
55
Ph. Leruste,
55
G. Marchiori,
55
J. Ocariz,
55
S. Sitt,
55
M. Biasini,
56a,56b
E. Manoni,
56a,56b
A. Rossi,
56a,56b
C. Angelini,
57a,57b
G. Batignani,
57a,57b
S. Bettarini,
57a,57b
M. Carpinelli,
57a,57b,
§
G. Casarosa,
57a,57b
A. Cervelli,
57a,57b
F. Forti,
57a,57b
M. A. Giorgi,
57a,57b
A. Lusiani,
57a,57c
N. Neri,
57a,57b
B. Oberhof,
57a,57b
E. Paoloni,
57a,57b
A. Perez,
57a
G. Rizzo,
57a,57b
J. J. Walsh,
57a
D. Lopes Pegna,
58
C. Lu,
58
J. Olsen,
58
A. J. S. Smith,
58
A. V. Telnov,
58
F. Anulli,
59a
G. Cavoto,
59a
R. Faccini,
59a,59b
F. Ferrarotto,
59a
F. Ferroni,
59a,59b
M. Gaspero,
59a,59b
L. Li Gioi,
59a
M. A. Mazzoni,
59a
G. Piredda,
59a
C. Bu
̈
nger,
60
T. Hartmann,
60
T. Leddig,
60
H. Schro
̈
der,
60
R. Waldi,
60
T. Adye,
61
E. O. Olaiya,
61
F. F. Wilson,
61
S. Emery,
62
G. Hamel de Monchenault,
62
G. Vasseur,
62
Ch. Ye
`
che,
62
D. Aston,
63
D. J. Bard,
63
R. Bartoldus,
63
J. F. Benitez,
63
C. Cartaro,
63
M. R. Convery,
63
J. Dorfan,
63
G. P. Dubois-Felsmann,
63
W. Dunwoodie,
63
R. C. Field,
63
M. Franco Sevilla,
63
B. G. Fulsom,
63
A. M. Gabareen,
63
M. T. Graham,
63
P. Grenier,
63
C. Hast,
63
W. R. Innes,
63
M. H. Kelsey,
63
H. Kim,
63
P. Kim,
63
M. L. Kocian,
63
D. W. G. S. Leith,
63
P. Lewis,
63
S. Li,
63
B. Lindquist,
63
S. Luitz,
63
V. Luth,
63
H. L. Lynch,
63
D. B. MacFarlane,
63
D. R. Muller,
63
H. Neal,
63
S. Nelson,
63
I. Ofte,
63
M. Perl,
63
T. Pulliam,
63
B. N. Ratcliff,
63
A. Roodman,
63
A. A. Salnikov,
63
V. Santoro,
63
R. H. Schindler,
63
A. Snyder,
63
D. Su,
63
M. K. Sullivan,
63
J. Va’vra,
63
A. P. Wagner,
63
M. Weaver,
63
W. J. Wisniewski,
63
M. Wittgen,
63
D. H. Wright,
63
H. W. Wulsin,
63
A. K. Yarritu,
63
C. C. Young,
63
V. Ziegler,
63
W. Park,
64
M. V. Purohit,
64
R. M. White,
64
J. R. Wilson,
64
A. Randle-Conde,
65
S. J. Sekula,
65
M. Bellis,
66
P. R. Burchat,
66
T. S. Miyashita,
66
M. S. Alam,
67
J. A. Ernst,
67
R. Gorodeisky,
68
N. Guttman,
68
D. R. Peimer,
68
A. Soffer,
68
P. Lund,
69
S. M. Spanier,
69
R. Eckmann,
70
J. L. Ritchie,
70
A. M. Ruland,
70
C. J. Schilling,
70
R. F. Schwitters,
70
B. C. Wray,
70
J. M. Izen,
71
X. C. Lou,
71
F. Bianchi,
72a,72b
D. Gamba,
72a,72b
L. Lanceri,
73a,73b
L. Vitale,
73a,73b
N. Lopez-March,
74
F. Martinez-Vidal,
74
A. Oyanguren,
74
H. Ahmed,
75
J. Albert,
75
Sw. Banerjee,
75
H. H. F. Choi,
75
PHYSICAL REVIEW D
86,
012008 (2012)
1550-7998
=
2012
=
86(1)
=
012008(34)
012008-1
Ó
2012 American Physical Society
G. J. King,
75
R. Kowalewski,
75
M. J. Lewczuk,
75
C. Lindsay,
75
I. M. Nugent,
75
J. M. Roney,
75
R. J. Sobie,
75
T. J. Gershon,
76
P. F. Harrison,
76
T. E. Latham,
76
E. M. T. Puccio,
76
H. R. Band,
77
S. Dasu,
77
Y. Pan,
77
R. Prepost,
77
C. O. Vuosalo,
77
and S. L. Wu
77
(
B
A
B
AR
Collaboration)
1
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Universite
́
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, Universita
`
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 Universita
̈
t Bochum, Institut fu
̈
r 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
9
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, 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
Technische Universita
̈
t Dortmund, Fakulta
̈
t Physik, D-44221 Dortmund, Germany
19
Technische Universita
̈
t Dresden, Institut fu
̈
r Kern- und Teilchenphysik, D-01062 Dresden, Germany
20
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
21
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
22a
INFN Sezione di Ferrara, I-44100 Ferrara, Italy
22b
Dipartimento di Fisica, Universita
`
di Ferrara, I-44100 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, Universita
`
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
Harvey Mudd College, Claremont, California 91711, USA
28
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
29
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstrasse 15, D-12489 Berlin, Germany
30
Imperial College London, London, SW7 2AZ, United Kingdom
31
University of Iowa, Iowa City, Iowa 52242, USA
32
Iowa State University, Ames, Iowa 50011-3160, USA
33
Johns Hopkins University, Baltimore, Maryland 21218, USA
34
Laboratoire de l’Acce
́
le
́
rateur Line
́
aire, IN2P3/CNRS et Universite
́
Paris-Sud 11, Centre Scientifique d’Orsay,
B. P. 34, F-91898 Orsay Cedex, France
35
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
36
University of Liverpool, Liverpool L69 7ZE, United Kingdom
37
Queen Mary, University of London, London, E1 4NS, United Kingdom
38
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
39
University of Louisville, Louisville, Kentucky 40292, USA
40
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
41
University of Manchester, Manchester M13 9PL, United Kingdom
42
University of Maryland, College Park, Maryland 20742, USA
43
University of Massachusetts, Amherst, Massachusetts 01003, USA
44
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
45
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
46a
INFN Sezione di Milano, I-20133 Milano, Italy
46b
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
47
University of Mississippi, University, Mississippi 38677, USA
48
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
J. P. LEES
et al.
PHYSICAL REVIEW D
86,
012008 (2012)
012008-2
49a
INFN Sezione di Napoli, I-80126 Napoli, Italy
49b
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
50
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, Netherlands
51
University of Notre Dame, Notre Dame, Indiana 46556, USA
52
The Ohio State University, Columbus, Ohio 43210, USA
53
University of Oregon, Eugene, Oregon 97403, USA
54a
INFN Sezione di Padova, I-35131 Padova, Italy
54b
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
55
Laboratoire de Physique Nucle
́
aire et de Hautes Energies, IN2P3/CNRS, Universite
́
Pierre et Marie Curie-Paris6,
Universite
́
Denis Diderot-Paris7, F-75252 Paris, France
56a
INFN Sezione di Perugia, I-06100 Perugia, Italy
56b
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
57a
INFN Sezione di Pisa, I-56127 Pisa, Italy
57b
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy
57c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
58
Princeton University, Princeton, New Jersey 08544, USA
59a
INFN Sezione di Roma, I-00185 Roma, Italy
59b
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
60
Universita
̈
t Rostock, D-18051 Rostock, Germany
61
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
62
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
63
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
64
University of South Carolina, Columbia, South Carolina 29208, USA
65
Southern Methodist University, Dallas, Texas 75275, USA
66
Stanford University, Stanford, California 94305-4060, USA
67
State University of New York, Albany, New York 12222, USA
68
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
69
University of Tennessee, Knoxville, Tennessee 37996, USA
70
University of Texas at Austin, Austin, Texas 78712, USA
71
University of Texas at Dallas, Richardson, Texas 75083, USA
72a
INFN Sezione di Torino, I-10125 Torino, Italy
72b
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
73a
INFN Sezione di Trieste, I-34127 Trieste, Italy
73b
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
74
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
75
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
76
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
77
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 15 March 2011; published 26 July 2012)
We study the processes
e
þ
e

!
K
þ
K


þ



,
K
þ
K


0

0

,and
K
þ
K

K
þ
K


, where the photon is
radiated from the initial state. About 84000, 8000, and 4200 fully reconstructed events, respectively, are
selected from
454 fb

1
of
BABAR
data. The invariant mass of the hadronic final state defines the
e
þ
e

center-
of-mass energy, so that the
K
þ
K


þ



data can be compared with direct measurements of the
e
þ
e

!
K
þ
K


þ


reaction. No direct measurements exist for the
e
þ
e

!
K
þ
K


0

0
or
e
þ
e

!
K
þ
K

K
þ
K

reactions,andwepresentanupdateofourpreviousresultbasedonadatasamplethatistwiceaslarge.Studying
the structure of these events, we find contributions from a number of intermediate states and extract their cross
sections. In particular, we perform a more detailed study of the
e
þ
e

!

ð
1020
Þ

reaction and confirm
the presence of the
Y
ð
2175
Þ
resonance in the

ð
1020
Þ
f
0
ð
980
Þ
and
K
þ
K

f
0
ð
980
Þ
modes. In the charmonium
region, we observe the
J=
c
in all three final states and in several intermediate states, as well as the
c
ð
2
S
Þ
in
some modes, and measure the corresponding products of branching fraction and electron width.
DOI:
10.1103/PhysRevD.86.012008
PACS numbers: 13.66.Bc, 13.25.Gv, 13.25.Jx, 14.40.

n
*
Present address: Temple University, Philadelphia, PA 19122, USA.
Also with Universita
`
di Perugia, Dipartimento di Fisica, Perugia, Italy.
Present address: University of South Alabama, Mobile, AL 36688, USA.
{
Also with Universita
`
di Sassari, Sassari, Italy.
§
Also with Universita
`
di Sassari, Sassari, Italy.
k
Also with Universita
`
della Basilicata, Potenza, Italy.
CROSS SECTIONS FOR THE REACTIONS
...
PHYSICAL REVIEW D
86,
012008 (2012)
012008-3
I. INTRODUCTION
Electron-positron annihilation at fixed center-of-mass
(c.m.) energies has long been a mainstay of research in
elementary particle physics. The idea of utilizing initial-
state radiation (ISR) to explore
e
þ
e

reactions below the
nominal c.m. energies was outlined in Ref. [
1
], and dis-
cussed in the context of high-luminosity

and
B
factories
in Refs. [
2
4
]. At high c.m. energies,
e
þ
e

annihilation is
dominated by quark-level processes producing two or more
hadronic jets. Low-multiplicity processes dominate below
or around 2 GeV, and the region near the charm threshold,
3.0–4.5 GeV, features a number of resonances [
5
]. Thus,
studies with ISR events allow us to probe a wealth of
physics topics, including cross sections, spectroscopy,
and form factors. Charmonium and other states with
J
PC
¼
1

can be observed, and intermediate states may
contribute to the final-state hadronic system. Measurements
of their decay modes and branching fractions are important
for an understanding of the nature of such states.
Of particular current interest (see Ref. [
6
]) is the
Y
ð
2175
Þ
state observed to decay to

ð
1020
Þ
f
0
ð
980
Þ
in
our previous study [
7
] and confirmed by the BES [
8
] and
Belle [
9
] Collaborations. With twice the integrated lumi-
nosity (compared to Ref. [
7
]) in the present analysis, we
perform a more detailed study of this structure.
The study of
e
þ
e

!
hadrons reactions in data is also
critical to hadronic-loop corrections to the muon magnetic
anomaly,
a

¼ð
g


2
Þ
=
2
. The theoretical predictions of
this anomaly rely on these measurements [
10
]. Improving
this prediction requires not only more precise measure-
ments but also measurements from threshold to the highest
c.m. energy possible. In addition, all the important subpro-
cesses should be studied in order to properly incorporate
possible acceptance effects. Events produced via ISR at
B
factories provide independent and contiguous measure-
ments of hadronic cross sections from the production
threshold to a c.m. energy of

5 GeV
. With more data
we also are able to reduce systematic uncertainties in the
cross section measurements.
The cross section for the radiation of a photon of energy
E

in the c.m. frame, followed by the production of a
particular hadronic final-state
f
, is related to the corre-
sponding direct
e
þ
e

!
f
cross section

f
ð
s
Þ
by
d
f
ð
s
0
;x
Þ
dx
¼
W
ð
s
0
;x
Þ

f
ð
s
0
ð
1

x
ÞÞ
;
(1)
where
ffiffiffiffiffi
s
0
p
is the nominal
e
þ
e

c.m. energy,
x
¼
2
E

=
ffiffiffiffiffi
s
0
p
is the fraction of the beam energy carried by the ISR
photon, and
E
c
:
m
:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
0
ð
1

x
Þ
p

ffiffiffi
s
p
is the effective
c.m. energy at which the final state
f
is produced. The
probability density function
W
ð
s
0
;x
Þ
for ISR-photon emis-
sion has been calculated with better than 1% precision (see,
e.g., Ref. [
4
]). It falls rapidly as
E

increases from zero, but
has a long tail, which in combination with the increasing

f
ð
s
0
ð
1

x
ÞÞ
produces a sizable event rate at very low
E
c
:
m
:
. The angular distribution of the ISR photon peaks
along the beam directions. For a typical
e
þ
e

detector,
around 10%–15% of the ISR photons fall within the
experimental acceptance [
4
].
Experimentally, the measured invariant mass of the had-
ronic final state defines
E
c
:
m
:
. An important feature of ISR
data is that a wide range of energies is scanned continu-
ously in a single experiment, so that no structure is missed,
and the relative normalization uncertainties in data from
different experiments are avoided. Furthermore, for large
values of
x
the hadronic system is collimated, reducing
acceptance issues and allowing measurements down to
production threshold. The mass resolution is not as good
as the typical beam energy spread used in direct measure-
ments, but resolution and absolute energy scale can be
monitored by means of the measured values of the width
and mass of well-known resonances, such as the
J=
c
produced in the reaction
e
þ
e

!
J=
c

. Backgrounds
from
e
þ
e

!
hadrons events at the nominal
ffiffiffiffiffi
s
0
p
and
from other ISR processes can be suppressed by a combi-
nation of particle identification and kinematic fitting
techniques. Studies of
e
þ
e

!

þ



and several multi-
hadron ISR processes using
BABAR
data have been per-
formed [
7
,
11
17
], demonstrating the viability of such
measurements. These analyses have led to improvements
in background reduction procedures for more rare ISR
processes.
The
K
þ
K


þ


final state has been measured directly
by the DM1 Collaboration [
18
] for
ffiffiffi
s
p
<
2
:
2 GeV
, and we
have previously published ISR measurements of the
K
þ
K


þ


and
K
þ
K

K
þ
K

final states [
13
] for
E
c
:
m
:
<
4
:
5 GeV
. Later we reported an updated measure-
ment of the
K
þ
K


þ


final state with a larger data
sample, together with the first measurement of the
K
þ
K


0

0
final state, in which we observed a structure
near threshold in the
f
0
intermediate state [
7
].
In this paper we present a more detailed study of these
two final states along with an updated measurement of the
K
þ
K

K
þ
K

final state. In all cases we require the detec-
tion of the ISR photon and perform a set of kinematic fits.
We are able to suppress backgrounds sufficiently to study
these final states from their respective production thresh-
olds up to
E
c
:
m
:
¼
5 GeV
. In addition to measuring the
overall cross sections, we study the internal structure of the
final states and measure cross sections for a number of
intermediate states that contribute to them. We also study
the charmonium region, measure several
J=
c
and
c
ð
2
S
Þ
products of branching fraction and electron width, and set
limits on other states.
II. THE
BABAR
DETECTOR AND DATA SET
The data used in this analysis were collected with
the
BABAR
detector at the PEP-II asymmetric-energy
e
þ
e

storage rings at the SLAC National Accelerator
J. P. LEES
et al.
PHYSICAL REVIEW D
86,
012008 (2012)
012008-4
Laboratory. The total integrated luminosity used is
454
:
2fb

1
, which includes
413
:
1fb

1
collected at the

ð
4
S
Þ
peak,
ffiffiffiffiffi
s
0
p
¼
10
:
58 GeV
, and
41
:
1fb

1
collected
at about
ffiffiffiffiffi
s
0
p
¼
10
:
54 GeV
.
The
BABAR
detector is described elsewhere [
19
]. In the
present work, we use charged-particle tracks reconstructed
in the tracking system, which is composed of a five double-
sided-layer silicon vertex tracker (SVT) and a 40-layer drift
chamber (DCH) in a 1.5 T axial magnetic field. Separation
of charged pions, kaons, and protons is achieved using a
combination of Cherenkov angles measured in the detector
of internally reflected Cherenkov light (DIRC) and specific-
ionization measurements in the SVT and DCH. For the
present study we use a kaon identification algorithm that
provides 90%–95% efficiency, depending on momentum,
and pion and proton rejection factors in the 20–100 range.
Photon and electron energies are measured in a CsI(Tl)
electromagnetic calorimeter (EMC). We use muon identi-
fication provided by an instrumented flux return to select
the

þ



final state used for photon efficiency studies.
To study the detector acceptance and efficiency, we use a
simulation package developed for radiative processes.
The simulation of hadronic final states, including
K
þ
K


þ



,
K
þ
K


0

0

, and
K
þ
K

K
þ
K


,is
based on the approach suggested by Czyz
̇
and Ku
̈
hn [
20
].
Multiple soft-photon emission from the initial-state
charged particles is implemented with a structure-function
technique [
21
,
22
], and photon radiation from the final-state
particles (FSR) is simulated by the
PHOTOS
package [
23
].
The precision of the radiative corrections is about 1%
[
21
,
22
].
We simulate the two
K
þ
K


(

þ


,

0

0
) final
states uniformly in phase space, and also according to
models that include the

ð
1020
Þ!
K
þ
K

and/or
f
0
ð
980
Þ!

channels. The
K
þ
K

K
þ
K

final state is
simulated according to phase space, and also including the

!
K
þ
K

channel. The generated events are subjected
to a detailed detector simulation [
24
], and we reconstruct
them with the same software chain used for the experi-
mental data. Variations in detector and background con-
ditions over the course of the experiment are taken into
account.
We also generate a large number of potential
background processes, including the ISR reactions
e
þ
e

!

þ



þ



,
e
þ
e

!

þ



0

0

, and
e
þ
e

!
K
S
K
, which can contribute due to particle
misidentification. We also simulate
e
þ
e

!

,
e
þ
e

!

0

, and
e
þ
e

!

þ



0

, which have
larger cross sections and can contribute background via
missing or spurious tracks or photons. In addition, we study
non-ISR backgrounds resulting from
e
þ
e

!
q

q
(
q
¼
u
,
d
,
s
,
c
) generated using
JETSET
[
25
] and from
e
þ
e

!

þ


generated using
KORALB
[
26
]. The cross sections for
these processes are known to about 10% accuracy or better,
which is sufficiently precise for the purposes of the
measurements in this paper. The contribution from

ð
4
S
Þ
decays is found to be negligible.
III. EVENT SELECTION AND KINEMATIC FIT
In the selection of candidate events, we consider photon
candidates in the EMC with energy above 0.03 GeV, and
charged-particle tracks reconstructed in either or both of
the DCH and SVT, that extrapolate within 0.25 cm of the
collision axis in the transverse plane and within 3 cm of the
nominal collision point along this axis. We require a pho-
ton with c.m. energy
E

>
3 GeV
in each event and either
four charged-particle tracks with zero net charge and total
momentum roughly (within 0.3 radians) opposite to the
photon direction or two oppositely charged tracks that
combine with other photons to roughly balance the high-
energy photon momentum. We assume that the photon
with the largest value of
E

is the ISR photon. We fit the
set of charged-particle tracks to a common vertex and use
this as the point of origin in calculating the photon direc-
tion(s). If additional well-reconstructed tracks exist, the
nearest four (two) to the interaction region are chosen for
the four-track (two-track) analysis. Most events contain
additional soft photons due to machine background or
interactions in the detector material.
We subject each candidate event to a set of constrained
kinematic fits and use the fit results, along with charged-
particle identification, both to select the final states of
interest and to measure backgrounds from other processes.
The kinematic fits use the ISR-photon direction and energy
along with the four-momenta and covariance matrices of
the initial
e
þ
e

and the set of selected tracks and photons.
The ISR-photon energy and position are additionally
aligned and calibrated using the

þ



ISR process,
since the two well-identified muons predict precisely the
position and energy of the photon. This process is also used
to identify and measure data—Monte Carlo (MC) simula-
tion differences in the photon detection efficiency and
resolution. The fitted three-momentum for each charged-
particle track and the photon are used in further kinemati-
cal calculations.
For the four-track event candidates the fits have four
constraints (4C). We first fit to the

þ



þ


hypothe-
sis, obtaining the chi-squared value

2
4

. If the four tracks
include one identified
K
þ
and one identified
K

,wefitto
the
K
þ
K


þ


hypothesis and retain the event as a
K
þ
K


þ


candidate. For events with one identified
kaon, we perform fits with each of the two oppositely
charged tracks given the kaon hypothesis, and the combi-
nation with the lower

2
2
K
2

is retained if its value is less
than

2
4

. If the event contains three or four identified
K

,
we fit to the
K
þ
K

K
þ
K

hypothesis and retain the event
as a
K
þ
K

K
þ
K

candidate with chi-squared value

2
4
K
.
For the events with two charged-particle tracks and five
or more photon candidates, we require that both tracks be
identified as kaons to suppress background from ISR
CROSS SECTIONS FOR THE REACTIONS
...
PHYSICAL REVIEW D
86,
012008 (2012)
012008-5

þ



0

0
and
K

K
0
S


events. We then pair all non-
ISR photon candidates and consider combinations with
invariant mass within

30 MeV
=c
2
of the

0
mass [
5
]
as

0
candidates. We perform a six-constraint (6C) fit to
each set of two nonoverlapping

0
candidates, the ISR
photon, the two charged-particle tracks, and the beam
particles. Both

0
candidates are constrained to the

0
mass, and we retain the combination with the lowest
chi-squared value,

2
2
K
2

0
.
IV. THE
K
þ
K


þ


FINAL STATE
A. Final selection and backgrounds
The

2
2
K
2

distribution in data for the
K
þ
K


þ


candidates is shown in Fig.
1
(points); the open histogram
is the distribution for the simulated
K
þ
K


þ


events.
The distributions are broader than those for a typical 4C

2
distribution due to higher order ISR, and the experimental
distribution has contributions from background processes.
The simulated distribution is normalized to the data in the
region

2
2
K
2

<
10
where the contributions of the back-
grounds and radiative corrections do not exceed 10%.
The shaded histogram in Fig.
1
represents the back-
ground from non-ISR
e
þ
e

!
q

q
events obtained from
the
JETSET
simulation. It is dominated by events with a hard

0
that results in a fake ISR photon. These events other-
wise have kinematics similar to the signal, resulting in the
peaking structure at low values of

2
2
K
2

. We evaluate this
background in a number of
E
c
:
m
:
ranges by combining the
ISR-photon candidate with another photon candidate in
both data and simulated events, and comparing the

0
signals in the resulting

invariant-mass distributions.
The simulation gives an
E
c
:
m
:
-dependence consistent with
the data, so we normalize it using an overall factor. The
cross-hatched region in Fig.
1
represents
e
þ
e

!
K
S
K
events with
K
S
!

þ


decays close to the interaction
region and one pion misidentified as a kaon. The process
has similar kinematics to the signal process, and a
contribution of about 1% is estimated using the cross
section measured in our previous study [
16
]. The hatched
region represents the contribution from ISR
e
þ
e

!

þ



þ


events with one or two misidentified pions;
this process contributes mainly at low

2
values. We
estimate the contribution as a function of
E
c
:
m
:
from a
simulation using the cross section value and shape from
our previous study [
13
].
All remaining background sources either are negligible
or give a

2
2
K
2

distribution that is nearly uniform over the
range shown in Fig.
1
. We define the signal region by
requiring

2
2
K
2

<
30
and estimate the sum of the remain-
ing backgrounds from the difference between the number
of data and simulated entries in the control region,
30
<

2
2
K
2

<
60
, as shown in Fig.
1
. The background contribu-
tion to any distribution other than

2
is estimated as the
difference between the distributions in the relevant quantity
for data and MC events from the control region of Fig.
1
,
normalized to the difference between the number of data
and MC events in the signal region. The non-ISR back-
ground is subtracted separately. The signal region contains
85 598 data and 63 784 simulated events; the control region
contains 9684 data and 4315 simulated events.
Figure
2
shows the
K
þ
K


þ


invariant-mass distri-
bution from threshold up to
5
:
0 GeV
=c
2
for events in the
signal region. Narrow peaks are apparent at the
J=
c
and
c
ð
2
S
Þ
masses. The shaded histogram represents the
q

q
background, which is negligible at low mass but dominates
at higher masses. The cross-hatched region represents the
background from the
K
S
K
channel [which exhibits a

ð
1680
Þ
peak [
16
] ] and from the

2
control region. The
hatched region represents the contribution from misidenti-
fied ISR

þ



þ


and is dominant for masses below
3
:
0 GeV
=c
2
. The total background is 6%–8% at low mass,
but accounts for 20%–25% of the observed distribution
near
4 GeV
=c
2
and increases further for higher masses.
We subtract the sum of backgrounds in each mass inter-
val to obtain the number of signal events. Considering
uncertainties in the cross sections for the background
processes, the normalization of events in the control re-
gion, and the simulation statistics, we estimate a systematic
uncertainty on the signal yield that is 2% or less in the
1
:
6
3
:
3 GeV
=c
2
mass region, but increases linearly to 10%
in the
3
:
3
5
:
0 GeV
=c
2
region, and is about 20% for the
masses below
1
:
6 GeV
=c
2
.
10
2
10
3
10
4
0204060
χ
2
2K2
π
Events/unit
χ
2
FIG. 1 (color online). Distribution of

2
from the four-
constraint fit for
K
þ
K


þ


candidates in the data (points).
The open histogram is the distribution for simulated signal
events, normalized as described in the text. The shaded, cross-
hatched, and hatched regions represent, respectively, the back-
ground from non-ISR events, from the ISR
K
S
K
process, and
backgrounds with dominant contribution from misidentified ISR
4

events. Signal and control regions are indicated.
J. P. LEES
et al.
PHYSICAL REVIEW D
86,
012008 (2012)
012008-6
B. Selection efficiency
The selection procedure applied to the data is also applied
to the simulated signal samples. The resulting
K
þ
K


þ


invariant-mass distributions in the signal and control re-
gions are shown in Fig.
3(a)
for the uniform phase space
simulation. This model reproduces the observed distribu-
tions of kaon and pion momenta and polar angles. A broad,
smooth mass distribution is chosen to facilitate the estima-
tion of the efficiency as a function of mass. We divide the
number of reconstructed simulated events in each mass
interval by the number generated in that interval to obtain
the efficiency shown by the points in Fig.
3(b)
. The result of
fitting a third-order polynomial to the points is used for
further calculations. We simulate events with the ISR pho-
ton confined to the angular range 20

–160

with respect to
the electron beam in the
e
þ
e

c.m. frame; this angular
range is wider than the actual EMC acceptance. The calcu-
lated efficiency is for this fiducial region, and includes the
acceptance for the final-state hadrons, the inefficiencies of
the detector subsystems, and the event loss due to additional
soft-photon emission.
The simulations including the

ð
1020
Þ

þ


and/or
K
þ
K

f
0
ð
980
Þ
channels give very different mass and angu-
lar distributions in the
K
þ
K


þ


rest frame. However,
the angular acceptance is quite uniform for ISR events
(see Ref. [
13
]), and the efficiencies are within 1% of those
from the uniform phase space simulation, as shown by the
dashed curve in Fig.
3(b)
for the

ð
1020
Þ

þ


final state.
To study possible mismodeling of the acceptance, we
repeat the analysis with tighter requirements. All charged
tracks are required to lie within the DIRC acceptance,
0
:
45
<
ch
<
2
:
4
radians, and the ISR photon must not
appear near the edges of the EMC,
0
:
35
<
ISR
<
2
:
4
radians. The fraction of selected data events satisfying
the tighter requirements differs from the simulated ratio
by 1.5%. We take the sum in quadrature of this variation
and the 1% model variation (2% total) as the systematic
uncertainty due to acceptance and model dependence.
Our data sample contains about 3000 events in the
J=
c
peak. Comparing this number with and without selection on

2
2
K
2

we find less than a 1% difference between data and
MCsimulation due tomismodeling oftheshapeofthe

2
2
K
2

distribution. This value is taken as an estimate of the system-
atic uncertainty associated with the

2
2
K
2

selection crite-
rion. To measure tracking efficiency, we consider data and
simulated events that contain a high-energy photon and
exactly three charged-particle tracks, which satisfy a set of
kinematicalcriteria,includinga good

2
fromakinematicfit
to the

þ



þ


hypothesis, assuming one missing pion
track in the event. We find that the simulated track-finding
efficiency is overestimated by
ð
0
:
75

0
:
25
Þ
%
per track, so
we apply a correction of
þð
3

1
Þ
%
to the signal yield.
The kaon identification efficiency is studied in
BABAR
using many different test processes [e.g.
e
þ
e

!

ð
1020
Þ

!
K
þ
K


], and we conservatively estimate a
systematic uncertainty of

1
:
0%
per kaon due to data-MC
differences in our kaon momentum range.
The data-MC simulation correction due to ISR-photon-
detection efficiency was studied with a sample of
e
þ
e

!

þ



events and was found to be
þð
1
:
0

0
:
5
Þ
%
.
0
500
1000
1234
m(K
+
K
-
π
+
π
-
) (GeV/c
2
)
Events/0.1 GeV/c
2
0
0.1
0.2
0.3
1234
m(K
+
K
-
π
+
π
-
) (GeV/c
2
)
Eff./0.1 GeV/c
2
FIG. 3 (color online). (a) The invariant-mass distributions for
K
þ
K


þ


MC events that are simulated uniformly in phase
space, reconstructed in the signal (open) and control (hatched)
regions of Fig.
1
; (b) net reconstruction and selection efficiency
as a function of mass obtained from this simulation (the curve
represents a third-order polynomial fit). The dashed curve is
obtained for the

ð
1020
Þ

þ


final state.
1
10
10
2
10
3
12345
m(K
+
K
-
π
+
π
-
) (GeV/c
2
)
Events/0.025 GeV/c
2
FIG. 2. The invariant-mass distribution for
K
þ
K


þ


can-
didates in the data (points): the shaded, cross-hatched, and hatched
regions show, respectively, the non-ISR background from
JETSET
simulation, the
K
S
K
background with a small contribution from
the control region of Fig.
1
, and the dominant contribution result-
ing from ISR misidentified

þ



þ


events.
CROSS SECTIONS FOR THE REACTIONS
...
PHYSICAL REVIEW D
86,
012008 (2012)
012008-7
C. Cross section for
e
þ
e

!
K
þ
K


þ


We calculate the
e
þ
e

!
K
þ
K


þ


cross section as
a function of the effective c.m. energy from

2
K
2

ð
E
c
:
m
:
Þ¼
dN
2
K
2

ð
E
c
:
m
:
Þ
d
L
ð
E
c
:
m
:
Þ
2
K
2

ð
E
c
:
m
:
Þ
R
ð
E
c
:
m
:
Þ
;
(2)
where
E
c
:
m
:

m
2
K
2

c
2
with
m
2
K
2

the measured invariant
mass of the
K
þ
K


þ


system,
dN
2
K
2

the number of
selected events after background subtraction in the interval
d
E
c
:
m
:
,
2
K
2

ð
E
c
:
m
:
Þ
the corrected detection efficiency, and
R
a radiative correction.
We calculate the differential luminosity
d
L
ð
E
c
:
m
:
Þ
in
each interval
d
E
c
:
m
:
, with the photon in the same fiducial
range as that used for the simulation, using the simple
leading order formula described in Ref. [
12
]. From the
mass spectra, obtained from the MC simulation with and
without extra-soft-photon (ISR and FSR) radiation, we
extract
R
ð
E
c
:
m
:
Þ
, which gives a correction less than 1%.
Our data, calculated according to Eq. (
2
), include vacuum
polarization (VP) and exclude any radiative effects, as is
conventional for the reporting of
e
þ
e

cross sections. Note
that VP should be excluded and FSR included for calcu-
lations of
a

. From data-simulation comparisons for the
e
þ
e

!

þ



events we estimate a systematic uncer-
tainty on
d
L
of 1% [
17
].
We show the cross section as a function of
E
c
:
m
:
in Fig.
4
with statistical errors only in comparison with the direct
measurements from DM1 [
18
], and list our results in
Table
I
. The results are consistent with our previous
measurements for this reaction [
7
,
13
] but have increased
statistical precision. Our data lie systematically below the
DM1 data for
E
c
:
m
:
above 1.9 GeV. The systematic uncer-
tainties, summarized in Table
II
, affect the normalization
but have little effect on the energy dependence.
The cross section rises from threshold to a peak value of
about 4.6 nb near 1.86 GeV and then generally decreases
with increasing energy. In addition to narrow peaks at the
J=
c
and
c
ð
2
S
Þ
mass values, there are several possible
wider structures in the 1.8–2.8 GeV region. Such structures
might be due to thresholds for intermediate resonant states,
such as
f
0
ð
980
Þ
near 2 GeV. Gaussian fits to the distri-
butions of the mass difference between generated and
reconstructed MC data yield
K
þ
K


þ


mass resolution
values that vary from
4
:
2 MeV
=c
2
in the
1
:
5
2
:
5 GeV
=c
2
region to
5
:
5 MeV
=c
2
in the
2
:
5
3
:
5 GeV
=c
2
region. The
resolution functions are not purely Gaussian due to soft-
photon radiation, but less than 10% of the signal is outside
the
0
:
025 GeV
=c
2
mass interval used in Fig.
4
. Since the
cross section has no sharp structure other than the
J=
c
and
c
ð
2
S
Þ
peaks discussed in Sec.
IX
below, we apply no
correction for mass resolution.
D. Substructures in the
K
þ
K


þ


final state
Our previous study [
7
,
13
] showed evidence for many
intermediate resonances in the
K
þ
K


þ


final state.
With the larger data sample used here, these can be
seen more clearly and, in some cases, studied in detail.
Figure
5(a)
shows a plot of the invariant mass of the
K


þ
pair versus that of the
K
þ


pair. Signal for the
K

ð
892
Þ
0
is clearly visible. Figure
5(b)
shows the
K



mass
distribution (two entries per event) for all selected
K
þ
K


þ


events. As we show in our previous study
[
7
], the signal at about
1400 GeV
=c
2
has parameters con-
sistent with
K

2
ð
1430
Þ
0
. Therefore, we perform a fit to this
distribution using P- and D-wave Breit–Wigner (BW)
functions for the
K

0
and
K

0
2
signals, respectively, and a
third-order polynomial function for the remainder of the
distribution, taking into account the
K
threshold. The fit
result is shown by the curves in Fig.
5(b)
. The fit yields a
K

0
signal of
53 997

526
events with
m
ð
K

0
Þ¼
0
:
8932

0
:
0002GeV
=c
2
and

ð
K

0
Þ¼
0
:
0521

0
:
0007GeV
, and a
K

0
2
signal of
4361

235
events with
m
ð
K

0
2
Þ¼
1
:
4274

0
:
0019GeV
=c
2
and

ð
K

0
2
Þ¼
0
:
0902

0
:
0056GeV
. These
values are consistent with current world averages for
K

ð
892
Þ
0
and
K

2
ð
1430
Þ
0
[
5
], and the fit describes the
data well, indicating that contributions from other reso-
nances decaying into
K



, like
K

ð
1410
Þ
0
and/or
K

0
ð
1430
Þ
0
, are small.
We combine
K

0
=

K

0
candidates within the lines in
Fig.
5(a)
with the remaining pion and kaon to obtain the
K

ð
892
Þ
0


invariant-mass distribution shown in Fig.
6(b)
,
and the
K

ð
892
Þ
0


versus
K

ð
892
Þ
0
K

mass plot in
Fig.
6(a)
. The bulk of Fig.
6(a)
shows a strong positive
correlation, characteristic of
K

0
K
final states with no
higher resonances. The horizontal bands in Fig.
6(a)
corre-
spond to the peak regions of the projection plot of Fig.
6(b)
0
2
4
6
12345
- DM1
- BaBar ISR
E
c.m.
(GeV)
σ
(K
+
K
-
π
+
π
-
) (nb)
FIG. 4 (color online). The
e
þ
e

!
K
þ
K


þ


cross sec-
tion as a function of
e
þ
e

c.m. energy measured with ISR data at
BABAR
(dots). The direct measurements from DM1 [
18
] are
shown as the open circles. Only statistical errors are shown.
J. P. LEES
et al.
PHYSICAL REVIEW D
86,
012008 (2012)
012008-8