Search for Darkonium in
e
+
e
−
Collisions
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3
G. Eigen,
4
D. N. Brown,
5
Yu. G. Kolomensky,
5
M. Fritsch,
6
H. Koch,
6
T. Schroeder,
6
R. Cheaib,
7b
C. Hearty,
7a,7b
T. S. Mattison,
7b
J. A. McKenna,
7b
R. Y. So,
7b
V. E. Blinov,
8a,8b,8c
A. R. Buzykaev,
8a
V. P. Druzhinin,
8a,8b
V. B. Golubev,
8a,8b
E. A. Kozyrev,
8a,8b
E. A. Kravchenko,
8a,8b
A. P. Onuchin,
8a,8b,8c
,*
S. I. Serednyakov,
8a,8b
Yu. I. Skovpen,
8a,8b
E. P. Solodov,
8a,8b
K. Yu. Todyshev,
8a,8b
A. J. Lankford,
9
B. Dey,
10
J. W. Gary,
10
O. Long,
10
A. M. Eisner,
11
W. S. Lockman,
11
W. Panduro Vazquez,
11
D. S. Chao,
12
C. H. Cheng,
12
B. Echenard,
12
K. T. Flood,
12
D. G. Hitlin,
12
J. Kim,
12
Y. Li ,
12
D. X. Lin,
12
T. S. Miyashita,
12
P. Ongmongkolkul,
12
J. Oyang,
12
F. C. Porter,
12
M. Röhrken,
12
Z. Huard,
13
B. T. Meadows,
13
B. G. Pushpawela,
13
M. D. Sokoloff,
13
L. Sun,
13
,
†
J. G. Smith,
14
S. R. Wagner,
14
D. Bernard,
15
M. Verderi,
15
D. Bettoni,
16a
C. Bozzi,
16a
R. Calabrese,
16a,16b
G. Cibinetto,
16a,16b
E. Fioravanti,
16a,16b
I. Garzia,
16a,16b
E. Luppi,
16a,16b
V. Santoro,
16a
A. Calcaterra,
17
R. de Sangro,
17
G. Finocchiaro,
17
S. Martellotti,
17
P. Patteri,
17
I. M. Peruzzi,
17
M. Piccolo,
17
M. Rotondo,
17
A. Zallo,
17
S. Passaggio,
18
C. Patrignani,
18
,
‡
B. J. Shuve,
19
H. M. Lacker,
20
B. Bhuyan,
21
U. Mallik,
22
C. Chen,
23
J. Cochran,
23
S. Prell,
23
A. V. Gritsan,
24
N. Arnaud,
25
M. Davier,
25
F. Le Diberder,
25
A. M. Lutz,
25
G. Wormser,
25
D. J. Lange,
26
D. M. Wright,
26
J. P. Coleman,
27
E. Gabathuler,
27
,*
D. E. Hutchcroft,
27
D. J. Payne,
27
C. Touramanis,
27
A. J. Bevan,
28
F. Di Lodovico,
28
,§
R. Sacco,
28
G. Cowan,
29
Sw. Banerjee,
30
D. N. Brown,
30
C. L. Davis,
30
A. G. Denig,
31
W. Gradl,
31
K. Griessinger,
31
A. Hafner,
31
K. R. Schubert,
31
R. J. Barlow,
32
,
∥
G. D. Lafferty,
32
R. Cenci,
33
A. Jawahery,
33
D. A. Roberts,
33
R. Cowan,
34
S. H. Robertson,
35a,35b
R. M. Seddon,
35b
N. Neri,
36a
F. Palombo,
36a,36b
L. Cremaldi,
37
R. Godang,
37
,¶
D. J. Summers,
37
,*
P. Taras,
38
G. De Nardo,
39
C. Sciacca,
39
G. Raven,
40
C. P. Jessop,
41
J. M. LoSecco,
41
K. Honscheid,
42
R. Kass,
42
A. Gaz,
43a
M. Margoni,
43a,43b
M. Posocco,
43a
G. Simi,
43a,43b
F. Simonetto,
43a,43b
R. Stroili,
43a,43b
S. Akar,
44
E. Ben-Haim,
44
M. Bomben,
44
G. R. Bonneaud,
44
G. Calderini,
44
J. Chauveau,
44
G. Marchiori,
44
J. Ocariz,
44
M. Biasini,
45a,45b
E. Manoni,
45a
A. Rossi,
45a
G. Batignani,
46a,46b
S. Bettarini,
46a,46b
M. Carpinelli,
46a,46b
,**
G. Casarosa,
46a,46b
M. Chrzaszcz,
46a
F. Forti,
46a,46b
M. A. Giorgi,
46a,46b
A. Lusiani,
46a,46c
B. Oberhof,
46a,46b
E. Paoloni,
46a,46b
M. Rama,
46a
G. Rizzo,
46a,46b
J. J. Walsh,
46a
L. Zani,
46a,46b
A. J. S. Smith,
47
F. Anulli,
48a
R. Faccini,
48a,48b
F. Ferrarotto,
48a
F. Ferroni,
48a
,
††
A. Pilloni,
48a,48b
G. Piredda,
48a
,*
C. Bünger,
49
S. Dittrich,
49
O. Grünberg,
49
M. Heß,
49
T. Leddig,
49
C. Voß,
49
R. Waldi,
49
T. Adye,
50
F. F. Wilson,
50
S. Emery,
51
G. Vasseur,
51
D. Aston,
52
C. Cartaro,
52
M. R. Convery,
52
J. Dorfan,
52
W. Dunwoodie,
52
M. Ebert,
52
R. C. Field,
52
B. G. Fulsom,
52
M. T. Graham,
52
C. Hast,
52
W. R. Innes,
52
,*
P. Kim,
52
D. W. G. S. Leith,
52
,*
S. Luitz,
52
D. B. MacFarlane,
52
D. R. Muller,
52
H. Neal,
52
B. N. Ratcliff,
52
A. Roodman,
52
M. K. Sullivan,
52
J. Va
’
vra,
52
W. J. Wisniewski,
52
M. V. Purohit,
53
J. R. Wilson,
53
A. Randle-Conde,
54
S. J. Sekula,
54
H. Ahmed,
55
M. Bellis,
56
P. R. Burchat,
56
E. M. T. Puccio,
56
M. S. Alam,
57
J. A. Ernst,
57
R. Gorodeisky,
58
N. Guttman,
58
D. R. Peimer,
58
A. Soffer,
58
S. M. Spanier,
59
J. L. Ritchie,
60
R. F. Schwitters,
60
J. M. Izen,
61
X. C. Lou,
61
F. Bianchi,
62a,62b
F. De Mori,
62a,62b
A. Filippi,
62a
D. Gamba,
62a,62b
L. Lanceri,
63
L. Vitale,
63
F. Martinez-Vidal,
64
A. Oyanguren,
64
J. Albert,
65b
A. Beaulieu,
65b
F. U. Bernlochner,
65b
G. J. King,
65b
R. Kowalewski,
65b
T. Lueck,
65b
I. M. Nugent,
65b
J. M. Roney,
65b
R. J. Sobie,
65a,65b
N. Tasneem,
65b
T. J. Gershon,
66
P. F. Harrison,
66
T. E. Latham,
66
R. Prepost,
67
and S. L. Wu
67
(
B
A
B
AR
Collaboration)
1
Laboratoire d
’
Annecy-le-Vieux de Physique des Particules (LAPP), Universit ́
e de Savoie,
CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
3
INFN Sezione di Bari, 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
7a
Institute of Particle Physics, Vancouver, British Columbia V6T 1Z1, Canada
7b
University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
8a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
8b
Novosibirsk State University, Novosibirsk 630090, Russia
8c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
9
University of California at Irvine, Irvine, California 92697, USA
10
University of California at Riverside, Riverside, California 92521, USA
11
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
PHYSICAL REVIEW LETTERS
128,
021802 (2022)
0031-9007
=
22
=
128(2)
=
021802(7)
021802-1
Published by the American Physical Society
12
California Institute of Technology, Pasadena, California 91125, USA
13
University of Cincinnati, Cincinnati, Ohio 45221, USA
14
University of Colorado, Boulder, Colorado 80309, USA
15
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
16a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
16b
Dipartimento di Fisica e Scienze della Terra, Universit`
a di Ferrara, I-44122 Ferrara, Italy
17
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
18
INFN Sezione di Genova, I-16146 Genova, Italy
19
Harvey Mudd College, Claremont, California 91711, USA
20
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
21
Indian Institute of Technology Guwahati, Guwahati, Assam 781 039, India
22
University of Iowa, Iowa City, Iowa 52242, USA
23
Iowa State University, Ames, Iowa 50011, USA
24
Johns Hopkins University, Baltimore, Maryland 21218, USA
25
Universit ́
e Paris-Saclay, CNRS/IN2P3, IJCLab, F-91405 Orsay, France
26
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
27
University of Liverpool, Liverpool L69 7ZE, United Kingdom
28
Queen Mary, University of London, London, E1 4NS, United Kingdom
29
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
30
University of Louisville, Louisville, Kentucky 40292, USA
31
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
32
University of Manchester, Manchester M13 9PL, United Kingdom
33
University of Maryland, College Park, Maryland 20742, USA
34
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
35a
Institute of Particle Physics, Montr ́
eal, Qu ́
ebec, Canada H3A 2T8
35b
McGill University, Montr ́
eal, Qu ́
ebec, Canada H3A 2T8
36a
INFN Sezione di Milano, I-20133 Milano, Italy
36b
Dipartimento di Fisica, Universit`
a di Milano, I-20133 Milano, Italy
37
University of Mississippi, University, Mississippi 38677, USA
38
Universit ́
e de Montr ́
eal, Physique des Particules, Montr ́
eal, Qu ́
ebec H3C 3J7, Canada
39
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Universit`
a di Napoli Federico II, I-80126 Napoli, Italy
40
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, Netherlands
41
University of Notre Dame, Notre Dame, Indiana 46556, USA
42
The Ohio State University, Columbus, Ohio 43210, USA
43a
INFN Sezione di Padova, I-35131 Padova, Italy
43b
Dipartimento di Fisica, Universit`
a di Padova, I-35131 Padova, Italy
44
Laboratoire de Physique Nucl ́
eaire et de Hautes Energies, Sorbonne Universit ́
e, Paris Diderot Sorbonne Paris Cit ́
e,
CNRS/IN2P3, F-75252 Paris, France
45a
INFN Sezione di Perugia, I-06123 Perugia, Italy
45b
Dipartimento di Fisica, Universit`
a di Perugia, I-06123 Perugia, Italy
46a
INFN Sezione di Pisa, I-56127 Pisa, Italy
46b
Dipartimento di Fisica, Universit`
a di Pisa, I-56127 Pisa, Italy
46c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
47
Princeton University, Princeton, New Jersey 08544, USA
48a
INFN Sezione di Roma, I-00185 Roma, Italy
48b
Dipartimento di Fisica, Universit`
a di Roma La Sapienza, I-00185 Roma, Italy
49
Universität Rostock, D-18051 Rostock, Germany
50
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
51
IRFU, CEA, Universit ́
e Paris-Saclay, F-91191 Gif-sur-Yvette, France
52
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
53
University of South Carolina, Columbia, South Carolina 29208, USA
54
Southern Methodist University, Dallas, Texas 75275, USA
55
St. Francis Xavier University, Antigonish, Nova Scotia B2G 2W5, Canada
56
Stanford University, Stanford, California 94305, USA
57
State University of New York, Albany, New York 12222, USA
58
Tel Aviv University, School of Physics and Astronomy, Tel Aviv 69978, Israel
59
University of Tennessee, Knoxville, Tennessee 37996, USA
60
University of Texas at Austin, Austin, Texas 78712, USA
61
University of Texas at Dallas, Richardson, Texas 75083, USA
62a
INFN Sezione di Torino, I-10125 Torino, Italy
PHYSICAL REVIEW LETTERS
128,
021802 (2022)
021802-2
62b
Dipartimento di Fisica, Universit`
a di Torino, I-10125 Torino, Italy
63
INFN Sezione di Trieste and Dipartimento di Fisica, Universit`
a di Trieste, I-34127 Trieste, Italy
64
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
65a
Institute of Particle Physics, Victoria, British Columbia V8W 3P6, Canada
65b
University of Victoria, Victoria, British Columbia V8W 3P6, Canada
66
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
67
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 20 June 2021; revised 17 September 2021; accepted 13 December 2021; published 11 January 2022)
Collider searches for dark sectors, new particles interacting only feebly with ordinary matter, have
largely focused on identifying signatures of new mediators, leaving much of dark sector structures
unexplored. In particular, the existence of dark matter bound states (darkonia) remains to be investigated.
This possibility could arise in a simple model in which a dark photon (
A
0
) is light enough to generate an
attractive force between dark fermions. We report herein a search for a
J
PC
¼
1
−−
darkonium state, the
Υ
D
,
produced in the reaction
e
þ
e
−
→
γ
Υ
D
,
Υ
D
→
A
0
A
0
A
0
, where the dark photons subsequently decay into
pairs of leptons or pions, using
514
fb
−
1
of data collected with the
BABAR
detector. No significant signal is
observed, and we set bounds on the
γ
−
A
0
kinetic mixing as a function of the dark sector coupling constant
for
0
.
001
<m
A
0
<
3
.
16
GeV and
0
.
05
<m
Υ
D
<
9
.
5
GeV.
DOI:
10.1103/PhysRevLett.128.021802
The possibility of dark sectors, new quantum fields
neutral under all standard model (SM) forces, has emerged
as an intriguing framework to explain the presence of dark
matter in the Universe
[1,2]
. While these particles do not
couple directly to ordinary matter, indirect interactions
through low-dimensional operators called
“
portals
”
are
possible
[3]
. The physics of these dark sectors could
involve an arbitrary number of fields and interactions,
including the possibility of self-interacting dark matter.
This scenario can be realized in a minimal dark sector
model containing a single Dirac fermion (
χ
) charged under
a new U(1) gauge group with a coupling constant
g
D
[4]
.
The corresponding force carrier is conventionally referred
to as a dark photon (
A
0
), and couples to the SM photon via
kinetic mixing with strength
ε
[5,6]
. A low-mass dark
photon would give rise to an attractive force between the
χ
and
̄
χ
particles, resulting in the formation of bound states
(darkonia) when
1
.
68
m
A
0
<
α
D
m
χ
for
α
D
¼
g
2
D
=
4
π
[4,7]
.
The two lowest energy bound states in this model are
denoted
η
D
(
J
PC
¼
0
−
þ
) and
Υ
D
(
J
PC
¼
1
−−
), in analogy
with similar SM states. The quantum numbers predict the
following production and decay mechanisms at
e
þ
e
−
colliders:
e
þ
e
−
→
η
D
þ
A
0
,
η
D
→
A
0
A
0
and initial-state
radiation (ISR) process
e
þ
e
−
→
Υ
D
þ
γ
ISR
,
Υ
D
→
A
0
A
0
A
0
. In the regime
m
A
0
<
2
m
χ
, the dark photon decays
visibly into a pair of SM fermions with a decay width
proportional to the product
m
A
0
ε
2
. Depending on the value
of these parameters, the decays can be prompt or signifi-
cantly displaced from the
e
þ
e
−
interaction point. Current
constraints on visible
A
0
decays
[8
–
18]
exclude values of
ε
≳
10
−
3
over a wide range of masses from the dielectron
threshold up to tens of GeV
[19]
.
We report herein a search for darkonium in the ISR
process
e
þ
e
−
→
γ
ISR
Υ
D
,
Υ
D
→
A
0
A
0
A
0
, where the dark
photons subsequently decay into pairs of electrons, muons,
or pions. The cross section is determined for prompt
A
0
decays in the region
0
.
001
GeV
<m
A
0
<
3
.
16
GeV and
0
.
05
GeV
<m
Υ
D
<
9
.
5
GeV. For
m
A
0
≥
0
.
2
GeV, the
flight distance in the detector is smaller than 0.01 mm,
effectively indistinguishable from prompt decays. For
m
A
0
<
0
.
2
GeV, the dark photon decay length becomes
significant for values of
ε
we expect to probe, and we
additionally report cross sections for lifetimes
τ
A
0
corre-
sponding to
c
τ
A
0
values of 0.1, 1, and 10 mm. This search is
based on
514
fb
−
1
of data collected with the
BABAR
detector at the SLAC PEP-II
e
þ
e
−
collider at the
Υ
ð
4
S
Þ
,
Υ
ð
3
S
Þ
, and
Υ
ð
2
S
Þ
resonances and their vicinities
[20]
. The
BABAR
detector is described in detail elsewhere
[21,22]
.
To avoid experimental bias, the data are not examined until
the selection procedure is finalized. The analysis is devel-
oped using simulated signal events and a small fraction of
real data for background studies.
Signal events are generated using
MADGRAPH
5
[23]
with
prompt dark photon decays for 119 different
A
0
and
Υ
D
mass hypotheses. For
m
A
0
<
0
.
2
GeV, we also simulate
samples with non-zero dark photon lifetimes corresponding
to proper decay lengths 0.1, 1, and 10 mm. The detector
acceptance and reconstruction efficiencies are estimated
with a simulation based on
GEANT
4
[24]
. Since the back-
ground is too complex to be accurately simulated, we use
5% of the data together with the simulated signal samples to
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW LETTERS
128,
021802 (2022)
021802-3
optimize the selection criteria, assuming that any signal
component has a negligible impact on this procedure. This
data set, referred to as the optimization sample, is discarded
from the final results.
The event selection for prompt
A
0
decays proceeds by
selecting events containing exactly six charged tracks, and
reconstructing dark photon candidates as pairs of oppo-
sitely charged tracks identified as electrons, muons, or
pions by particle identification algorithms. We require the
presence of at least one lepton pair of opposite charge with
the same flavor to limit the large accidental background.
We form
Υ
D
candidates by combining three dark photon
candidates, and fit them, constraining all tracks to originate
from a common point compatible with the beam interaction
region. For each
Υ
D
candidate, we additionally form same-
sign track combinations by swapping particles with iden-
tical flavor between reconstructed
A
0
pairs, such as
ð
e
þ
e
þ
Þð
e
−
e
−
Þð
μ
þ
μ
−
Þ
or
ð
π
þ
π
þ
Þð
π
−
π
−
Þð
e
þ
e
−
Þ
. For the
fully mixed final state
ð
μ
þ
μ
−
Þð
π
þ
π
−
Þð
e
þ
e
−
Þ
, we use the
same-sign combination
ð
μ
þ
π
þ
Þð
μ
−
π
−
Þð
e
þ
e
−
Þ
, since pions
are more easily misidentified as muons than electrons.
Because of the combinatoric nature of the background, the
distributions of the mass difference for same-sign and
opposite-sign pairs tend to be similar. By contrast, the
differences between these distributions tend to be larger for
signal events, effectively providing discrimination power.
The detection of the ISR photon accompanying
Υ
D
production is not explicitly required. Instead, we infer the
kinematics of the particle recoiling against the
Υ
D
candi-
dates, and we select the ISR photon candidate that is most
compatible with the photon hypothesis as follows. If the
recoil particle is determined to have been emitted inside
the electromagnetic calorimeter acceptance, we search for
the presence of a corresponding ISR photon candidate,
which is defined as a neutral cluster having an energy
within 10% of that of the recoiling particle, and an angle
compatible with the direction of the recoiling particle to
better than 0.1 rad.
To improve the signal purity, we train three multivariate
classifiers consisting of logistic regressions stacked on top
of random forest (RF) classifiers
[25]
. The following 13
variables are used as inputs to the RF: the
χ
2
of the
constrained fit to the
Υ
D
candidate; combined particle
identification information of the six tracks; the maximum
mass difference between any pair of
A
0
candidates; the
polar angle and the invariant mass of the particle recoiling
against the reconstructed
Υ
D
candidate; a categorical
feature indicating whether the recoiling particle is emitted
inside the calorimeter acceptance and if a corresponding
ISR photon candidate is found; the sum of neutral energy
deposited in the electromagnetic calorimeter, excluding the
ISR photon candidate; the average of the three dark photon
helicity angles
[26]
; the average of the angles between pairs
of dark photons in the
Υ
D
rest frame; the average of the
dihedral angles between pairs of dark photons; the average
of the three helicity angles of the tracks produced in the
A
0
decays; the average of the dark photon decay lengths,
defined as the distances between the primary interaction
point and the
A
0
decay vertices; and the maximum mass
difference between same-sign pairs.
To improve the robustness of the predictions of the
classifiers, we group the final states into three categories
based on the number of pion pairs: zero (
C
0
), one (
C
1
), or
two (
C
2
) pion pairs. A classifier is trained for each category
with a sample of simulated events for different
Υ
D
and
A
0
masses and a fraction of the optimization sample to
describe the background. The classifier outputs are then
transformed into classifier scores using a logit function
[27]
, with higher scores indicating greater probabilities of
being signal events. The distribution of the classifier scores
for each category are shown in Fig.
1
. The optimal selection
criteria are determined by maximizing a figure of merit
averaged over a wide range of
Υ
D
and
A
0
masses. We adopt
a conservative approach and treat observed events as signal
candidates for the purposes of calculating the figure of
merit. If multiple
Υ
D
candidates are selected in an event, a
single one is chosen based on its final state according to the
following sequence of hypotheses:
6
e
,
4
e
2
μ
,
2
e
4
μ
,
6
μ
,
4
e
2
π
,
2
e
2
μ
2
π
,
4
μ
2
π
,
2
e
4
π
,
2
μ
4
π
. The sequence of
hypotheses is ordered according to the purity of the final
states to minimize misidentification between channels.
A total of 69 events pass all the selection criteria. The
corresponding
ð
m
Υ
D
;m
A
0
Þ
distribution is shown in Fig.
2
.
The events near
m
Υ
D
∼
0
.
1
GeV and
m
A
0
∼
0
.
05
GeV arise
from
e
þ
e
−
→
γγγ
events in which all three photons convert
to
e
þ
e
−
pairs.
The signal is extracted by combining all event cate-
gories into a single sample, and scanning the
ð
m
Υ
D
;m
A
0
Þ
plane in steps of the signal resolution. The signal region for
a given mass hypothesis is defined as the interval
½
m
Υ
D
−
4
σ
m
Υ
D
;
m
Υ
D
þ
4
σ
m
Υ
D
and
½
m
A
0
−
4
σ
m
A
0
;
m
A
0
þ
4
σ
m
A
0
, where
FIG. 1. The distribution of the classifier scores for each event
category for the data (markers) and signal Monte Carlo (solid
lines) samples. The MC simulations are arbitrarily normalized.
PHYSICAL REVIEW LETTERS
128,
021802 (2022)
021802-4
σ
m
Υ
D
(
σ
m
A
0
) denotes the corresponding
Υ
D
(
A
0
) mass
resolution. The resolutions are determined by fitting the
different signal Monte Carlo (MC) samples with a crystal
ball function
[28]
and interpolating the results throughout
the full mass range. The
Υ
D
(
A
0
) mass resolution varies
between 5
–
40 MeV (1
–
8 MeV); the detailed results are
available in Supplemental Material
[29]
. The number of
observed background events is estimated by averaging two
neighboring regions along the
m
Υ
D
axis:
½
m
Υ
D
−
8
σ
m
Υ
D
;
m
Υ
D
−
4
σ
m
Υ
D
and
½
m
Υ
D
þ
4
σ
m
Υ
D
;
m
Υ
D
þ
8
σ
m
Υ
D
. This
choice is motivated by the potential background contribu-
tion due to hadronic resonances or photon conversions,
which would be concentrated at similar values of dark
photon masses. The signal significance is assessed from
MC samples, using sideband data from the classifier score
distribution to model the
ð
m
Υ
D
;m
A
0
Þ
distribution of the
background. The most significant measurement contains
two events in the signal window, corresponding to a
p
value
of 30%, which is compatible with the null hypothesis.
In the absence of signal, we derive 90% confidence
level (C.L.) upper limits on the
e
þ
e
−
→
γ
Υ
D
cross section
using a profile likelihood method
[30]
. The probability of
observing
N
events in a given signal region is described by
the following model:
P
ð
N
j
n
þ
b
Þ
¼
e
−
n
n
N
N
!
e
−
b
b
B
B
!
1
2
πσ
Z
σ
L
e
½
−
ð
z
−
Z
Þ
2
=
2
σ
2
Z
e
½
−
ð
l
−
L
Þ
2
=
2
σ
2
L
;
where
b
(
B
) is the expected (estimated) number of back-
ground events, and
n
¼
lz
σ
ð
e
þ
e
−
→
γ
Υ
D
Þ
is the expected
number of signal events given by the product of the
integrated luminosity
l
, the
e
þ
e
−
→
γ
Υ
D
cross section,
and the signal efficiency
z
. The measured luminosity, signal
efficiency, and their uncertainties are denoted by
L
,
Z
,
σ
L
,
and
σ
Z
, respectively. The signal efficiency includes the
dark photon branching fractions, taken from Ref.
[31]
.
The efficiency is determined for each simulated sample
and interpolated to the full parameter space, ranging
from 0.1% for
m
Υ
D
∼
0
.
15
GeV,
m
A
0
∼
0
.
05
GeV to
34% for
m
Υ
D
∼
8
.
0
GeV,
m
A
0
∼
1
.
0
GeV. The uncertainty
in the signal efficiency arises mainly from particle iden-
tification algorithms, assessed with high-purity samples of
leptons and pions. This source of uncertainty varies
between 9% and 11%. The uncertainty associated with
the efficiency extrapolation procedure ranges from 4% to
7%, depending on the
m
Υ
D
and
m
A
0
. Other uncertainties
include the tracking efficiency (1.2%) and the limited
statistical precision of the simulated sample (1%
–
5%).
The uncertainty in the dark photon branching fraction
[31]
ranges from parts per mille to 1%. The uncertainty in
the luminosity is determined to be 0.6%
[20]
. The cross
section at 90% C.L. upper limits are displayed in Fig.
3
.
The dark photon decays predominantly into
π
þ
π
−
π
0
(
K
þ
K
−
) near the
ω
(
φ
) resonance which are not considered
FIG. 2. The
ð
m
Υ
D
;m
A
0
Þ
distribution for events passing all
selection criteria for prompt dark photon decays.
FIG. 3. The 90% C.L. upper limits on the
e
þ
e
−
→
γ
Υ
D
cross
section for prompt dark photon decays.
FIG. 4. The
ð
m
Υ
D
;m
A
0
Þ
mass distribution of event candidates
passing all selection criteria for the datasets optimized for each
dark photon lifetime.
PHYSICAL REVIEW LETTERS
128,
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021802-5
in this analysis, resulting in much looser bounds around
m
A
0
∼
0
.
8
GeV (
m
A
0
∼
1
GeV).
We follow a similar procedure to determine the
e
þ
e
−
→
γ
Υ
D
cross section for each dark photon lifetime hypothesis.
The measurement is performed for
m
A
0
<
0
.
2
GeV. In this
mass range, the
A
0
decays almost exclusively to an
e
þ
e
−
pair. The event selection is analogous to that previously
described, except that we constrain the momentum vector
of the
A
0
candidates to point back to the beam interaction
region instead of requiring the tracks to originate from this
location when performing the
Υ
D
kinematic fit. To further
suppress photon conversions in the detector material, we
add the following variables to the RF classifier, averaged
over the three dark photon candidates: the
χ
2
of a fit of the
A
0
candidate; the angle between the secondary vertex flight
direction and the
A
0
momentum; and the ratio between the
flight length and its uncertainty. We train a classifier for
each
c
τ
A
0
value to improve the search sensitivity. A total of
56, 33, and 31 events are selected for the
c
τ
A
0
¼
0
.
1
, 1, and
10 mm data sample, respectively. The resulting mass
distributions are shown in Fig.
4
. The signal extraction
procedure described above is applied to each selected
sample separately. No significant signal is observed for
any
A
0
lifetime hypothesis, and limits on the cross section
for each value of
c
τ
A
0
are extracted. The classifier score
distributions and the cross section at 90% C.L. upper limits
are shown in Supplemental Material
[29]
.
The 90% C.L. upper limits on the kinetic mixing
parameter are extracted by an iterative procedure taking
into account the effect of the potentially long dark photon
lifetime. At each step, we estimate the dark photon lifetime
given the current value of the kinetic mixing, compare the
limit on the production cross section interpolated at that
lifetime, update the kinetic mixing, and repeat the pro-
cedure until convergence is obtained. Since the dark photon
lifetime is independent of the dark sector coupling constant,
we derive separate limits for
α
D
values set to 0.1, 0.3, 0.5,
0.7, 0.9, and 1.1. The results are shown in Fig.
5
for
α
D
¼
0
.
5
, and in Supplemental Material
[29]
for the
remaining values. Bounds on the mixing strength
ε
down
to
5
×
10
−
5
−
10
−
3
are set for a large fraction of the
parameter space. Constraints for different values of
α
D
,
m
A
0
and
m
Υ
D
are also shown in Fig.
6
.
In summary, we report the first search for a dark sector
bound state decaying into three dark photons in the range
0
.
001
GeV
<m
A
0
<
3
.
16
GeV and
0
.
05
GeV
<m
Υ
D
<
9
.
5
GeV. No significant signal is seen, and we derive
limits on the
γ
−
A
0
kinetic mixing
ε
at the level of
5
×
10
−
5
−
10
−
3
, depending on the values of the model
parameters. These measurements improve upon existing
constraints over a significant fraction of dark photon
masses below 1 GeV for large values of the dark sector
coupling constant. Were the
η
D
bound state to be included
in the search, the upper limits on the cross section (in the
absence of a signal) could be improved by around a factor
of 2, leading to an improvement on the constraints on the
kinetic mixing strength by about a factor of
ffiffiffi
2
p
.
The authors wish to thank Haipeng An and Yue Zhang
for useful discussions and for providing us with
MADGRAPH
FIG. 5. The 90% C.L. upper limits on the kinetic mixing
ε
2
as a
function of the
Υ
D
mass
m
Υ
D
and dark photon mass
m
A
0
,
assuming
α
D
¼
0
.
5
.
FIG. 6. The 90% C.L. upper limits on the kinetic mixing
ε
for
(top) various
Υ
D
masses assuming
α
D
¼
0
.
5
and (bottom)
various
α
D
values assuming
m
Υ
D
¼
9
GeV together with current
constraints (gray area)
[8
–
18]
.
PHYSICAL REVIEW LETTERS
128,
021802 (2022)
021802-6
simulations of self-interacting dark matter processes. We
also thank Gaia Lanfranchi for providing us constraints
from existing experiments. We are grateful for the extraor-
dinary contributions of our PEP-II colleagues in achieving
the excellent luminosity and machine conditions that
have made this work possible. The success of this project
also relies critically on the expertise and dedication of the
computing organizations that support
BABAR
. The col-
laborating institutions wish to thank SLAC for its support
and the kind hospitality extended to them. This work
is supported by the U.S. Department of Energy and
National Science Foundation, the Natural Sciences and
Engineering Research Council (Canada), the Commissariat
`
al
’
Energie Atomique and Institut National de Physique
Nucl ́
eaire et de Physique des Particules (France), the
Bundesministerium für Bildung und Forschung and
Deutsche Forschungsgemeinschaft (Germany), the Istituto
Nazionale di Fisica Nucleare (Italy), the Foundation for
Fundamental Research on Matter (Netherlands), the
Research Council of Norway, the Ministry of Education
and Science of the Russian Federation, Ministerio de
Economía y Competitividad (Spain), the Science and
Technology Facilities Council (U.K.), and the Binational
Science Foundation (U.S.-Israel). Individuals have received
support from the Marie-Curie IEF program (European
Union) and the A. P. Sloan Foundation (U.S.).
*
Deceased.
†
Present address: Wuhan University, Wuhan 430072, China.
‡
Present address: Universit`
a di Bologna and INFN Sezione
di Bologna, I-47921 Rimini, Italy.
§
Present address: King
’
s College, London WC2R 2LS,
United Kingdom.
∥
Present address: University of Huddersfield, Huddersfield
HD1 3DH, United Kingdom.
¶
Present address: University of South Alabama, Mobile,
Alabama 36688, USA.
**
Also at Universit`
a di Sassari, I-07100 Sassari, Italy.
††
Also at Gran Sasso Science Institute, I-67100 LAquila,
Italy.
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128,
021802 (2022)
021802-7