# Search for a massless dark photon in $\Lambda^{+}_{c}\to p\gamma^{\prime}$
decay

M. Ablikim1, M. N. Achasov11,b, P. Adlarson70, M. Albrecht4, R. Aliberti31, A.
Amoroso69A,69C, M. R. An35, Q. An66,53, X. H. Bai61, Y. Bai52, O. Bakina32, R.
Baldini Ferroli26A, I. Balossino27A, Y. Ban42,g, V. Batozskaya1,40, D.
Becker31, K. Begzsuren29, N. Berger31, M. Bertani26A, D. Bettoni27A, F.
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Brueggemann63, H. Cai71, X. Cai1,53, A. Calcaterra26A, G. F. Cao1,58, N.
Cao1,58, S. A. Cetin57A, J. F. Chang1,53, W. L. Chang1,58, G. Chelkov32,a, C.
Chen39, Chao Chen50, G. Chen1, H. S. Chen1,58, M. L. Chen1,53, S. J. Chen38,
S. M. Chen56, T. Chen1, X. R. Chen28,58, X. T. Chen1, Y. B. Chen1,53, Z. J.
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Boer4, D. Dedovich32, Z. Y. Deng1, A. Denig31, I. Denysenko32, M.
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M. Y. Dong1,53,58, X. Dong71, S. X. Du75, P. Egorov32,a, Y. L. Fan71, J.
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Yang Gao66,53, S. Garbolino69C, I. Garzia27A,27B, P. T. Ge71, Z. W. Ge38, C.
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C. Y Guan1,58, A. Q. Guo28,58, L. B. Guo37, R. P. Guo44, Y. P. Guo10,f, A.
Guskov32,a, T. T. Han45, W. Y. Han35, X. Q. Hao18, F. A. Harris60, K. K. He50,
K. L. He1,58, F. H. Heinsius4, C. H. Heinz31, Y. K. Heng1,53,58, C. Herold55,
M. Himmelreich12,d, M. Himmelreich31,d, G. Y. Hou1,58, Y. R. Hou58, Z. L.
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R. Kliemt12, L. Koch33, O. B. Kolcu57A, B. Kopf4, M. Kuemmel4, M. Kuessner4,
A. Kupsc40,70, W. Kühn33, J. J. Lane62, J. S. Lange33, P. Larin17, A.
Lavania24, L. Lavezzi69A,69C, Z. H. Lei66,53, H. Leithoff31, M. Lellmann31, T.
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J. S. Li54, J. W. Li45, Ke Li1, L. J Li1, L. K. Li1, Lei Li3, M. H. Li39, P.
R. Li34,j,k, S. X. Li10, S. Y. Li56, T. Li45, W. D. Li1,58, W. G. Li1, X. H.
Li66,53, X. L. Li45, Xiaoyu Li1,58, Z. X. Li14, H. Liang30, H. Liang1,58, H.
Liang66,53, Y. F. Liang49, Y. T. Liang28,58, G. R. Liao13, L. Z. Liao45, J.
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X. Liu1, D. Liu17,66, F. H. Liu48, Fang Liu1, Feng Liu6, G. M. Liu51,i, H.
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Liu66,53, J. L. Liu67, J. Y. Liu1,58, K. Liu1, K. Y. Liu36, Ke Liu20, L.
Liu66,53, Lu Liu39, M. H. Liu10,f, P. L. Liu1, Q. Liu58, S. B. Liu66,53, T.
Liu10,f, W. K. Liu39, W. M. Liu66,53, X. Liu34,j,k, Y. Liu34,j,k, Y. B. Liu39,
Z. A. Liu1,53,58, Z. Q. Liu45, X. C. Lou1,53,58, F. X. Lu54, H. J. Lu21, J. G.
Lu1,53, X. L. Lu1, Y. Lu7, Y. P. Lu1,53, Z. H. Lu1, C. L. Luo37, M. X. Luo74,
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L. Ma45, M. M. Ma1,58, Q. M. Ma1, R. Q. Ma1,58, R. T. Ma58, X. Y. Ma1,53, Y.
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Meng61, G. Mezzadri27A, H. Miao1, T. J. Min38, R. E. Mitchell25, X. H.
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Qin67, L. Q. Qin13, X. P. Qin10,f, X. S. Qin45, Z. H. Qin1,53, J. F. Qiu1, S.
Q. Qu56, K. H. Rashid68, C. F. Redmer31, K. J. Ren35, A. Rivetti69C, V.
Rodin59, M. Rolo69C, G. Rong1,58, Ch. Rosner17, S. N. Ruan39, H. S. Sang66, A.
Sarantsev32,c, Y. Schelhaas31, C. Schnier4, K. Schoenning70, M.
Scodeggio27A,27B, K. Y. Shan10,f, W. Shan22, X. Y. Shan66,53, J. F.
Shangguan50, L. G. Shao1,58, M. Shao66,53, C. P. Shen10,f, H. F. Shen1,58, X.
Y. Shen1,58, B. A. Shi58, H. C. Shi66,53, J. Y. Shi1, q. q. Shi50, R. S.
Shi1,58, X. Shi1,53, X. D Shi66,53, J. J. Song18, W. M. Song30,1, Y. X.
Song42,g, S. Sosio69A,69C, S. Spataro69A,69C, F. Stieler31, K. X. Su71, P. P.
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Wang42,g, F. Wang67, H. J. Wang34,j,k, H. P. Wang1,58, K. Wang1,53, L. L.
Wang1, M. Wang45, M. Z. Wang42,g, Meng Wang1,58, S. Wang13, S. Wang10,f, T.
Wang10,f, T. J. Wang39, W. Wang54, W. H. Wang71, W. P. Wang66,53, X. Wang42,g,
X. F. Wang34,j,k, X. L. Wang10,f, Y. Wang56, Y. D. Wang41, Y. F. Wang1,53,58,
Y. H. Wang43, Y. Q. Wang1, Yaqian Wang16,1, Z. Wang1,53, Z. Y. Wang1,58, Ziyi
Wang58, D. H. Wei13, F. Weidner63, S. P. Wen1, D. J. White62, U. Wiedner4, G.
Wilkinson64, M. Wolke70, L. Wollenberg4, J. F. Wu1,58, L. H. Wu1, L. J.
Wu1,58, X. Wu10,f, X. H. Wu30, Y. Wu66, Z. Wu1,53, L. Xia66,53, T. Xiang42,g,
D. Xiao34,j,k, G. Y. Xiao38, H. Xiao10,f, S. Y. Xiao1, Y. L. Xiao10,f, Z. J.
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Xie66,53, T. Y. Xing1,58, C. F. Xu1, C. J. Xu54, G. F. Xu1, H. Y. Xu61, Q. J.
Xu15, X. P. Xu50, Y. C. Xu58, Z. P. Xu38, F. Yan10,f, L. Yan10,f, W. B.
Yan66,53, W. C. Yan75, H. J. Yang46,e, H. L. Yang30, H. X. Yang1, L. Yang47,
S. L. Yang58, Tao Yang1, Y. F. Yang39, Y. X. Yang1,58, Yifan Yang1,58, M.
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Yu1,58, T. Yu67, X. D. Yu42,g, C. Z. Yuan1,58, L. Yuan2, S. C. Yuan1, X. Q.
Yuan1, Y. Yuan1,58, Z. Y. Yuan54, C. X. Yue35, A. A. Zafar68, F. R. Zeng45, X.
Zeng6, Y. Zeng23,h, Y. H. Zhan54, A. Q. Zhang1, B. L. Zhang1, B. X. Zhang1, D.
H. Zhang39, G. Y. Zhang18, H. Zhang66, H. H. Zhang54, H. H. Zhang30, H. Y.
Zhang1,53, J. L. Zhang72, J. Q. Zhang37, J. W. Zhang1,53,58, J. X.
Zhang34,j,k, J. Y. Zhang1, J. Z. Zhang1,58, Jianyu Zhang1,58, Jiawei
Zhang1,58, L. M. Zhang56, L. Q. Zhang54, Lei Zhang38, P. Zhang1, Q. Y.
Zhang35,75, Shuihan Zhang1,58, Shulei Zhang23,h, X. D. Zhang41, X. M. Zhang1,
X. Y. Zhang50, X. Y. Zhang45, Y. Zhang64, Y. T. Zhang75, Y. H. Zhang1,53, Yan
Zhang66,53, Yao Zhang1, Z. H. Zhang1, Z. Y. Zhang39, Z. Y. Zhang71, G. Zhao1,
J. Zhao35, J. Y. Zhao1,58, J. Z. Zhao1,53, Lei Zhao66,53, Ling Zhao1, M. G.
Zhao39, Q. Zhao1, S. J. Zhao75, Y. B. Zhao1,53, Y. X. Zhao28,58, Z. G.
Zhao66,53, A. Zhemchugov32,a, B. Zheng67, J. P. Zheng1,53, Y. H. Zheng58, B.
Zhong37, C. Zhong67, X. Zhong54, H. Zhou45, L. P. Zhou1,58, X. Zhou71, X. K.
Zhou58, X. R. Zhou66,53, X. Y. Zhou35, Y. Z. Zhou10,f, J. Zhu39, K. Zhu1, K.
J. Zhu1,53,58, L. X. Zhu58, S. H. Zhu65, S. Q. Zhu38, T. J. Zhu72, W. J.
Zhu10,f, Y. C. Zhu66,53, Z. A. Zhu1,58, B. S. Zou1, J. H. Zou1 (BESIII
Collaboration) 1 Institute of High Energy Physics, Beijing 100049, People’s
Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
10 Fudan University, Shanghai 200433, People’s Republic of China
11 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
12 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
13 Guangxi Normal University, Guilin 541004, People’s Republic of China
14 Guangxi University, Nanning 530004, People’s Republic of China
15 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
16 Hebei University, Baoding 071002, People’s Republic of China
17 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
18 Henan Normal University, Xinxiang 453007, People’s Republic of China
19 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
20 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
21 Huangshan College, Huangshan 245000, People’s Republic of China
22 Hunan Normal University, Changsha 410081, People’s Republic of China
23 Hunan University, Changsha 410082, People’s Republic of China
24 Indian Institute of Technology Madras, Chennai 600036, India
25 Indiana University, Bloomington, Indiana 47405, USA
26 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
27 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
28 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
29 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
30 Jilin University, Changchun 130012, People’s Republic of China
31 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
32 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
33 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
34 Lanzhou University, Lanzhou 730000, People’s Republic of China
35 Liaoning Normal University, Dalian 116029, People’s Republic of China
36 Liaoning University, Shenyang 110036, People’s Republic of China
37 Nanjing Normal University, Nanjing 210023, People’s Republic of China
38 Nanjing University, Nanjing 210093, People’s Republic of China
39 Nankai University, Tianjin 300071, People’s Republic of China
40 National Centre for Nuclear Research, Warsaw 02-093, Poland
41 North China Electric Power University, Beijing 102206, People’s Republic of
China
42 Peking University, Beijing 100871, People’s Republic of China
43 Qufu Normal University, Qufu 273165, People’s Republic of China
44 Shandong Normal University, Jinan 250014, People’s Republic of China
45 Shandong University, Jinan 250100, People’s Republic of China
46 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
47 Shanxi Normal University, Linfen 041004, People’s Republic of China
48 Shanxi University, Taiyuan 030006, People’s Republic of China
49 Sichuan University, Chengdu 610064, People’s Republic of China
50 Soochow University, Suzhou 215006, People’s Republic of China
51 South China Normal University, Guangzhou 510006, People’s Republic of China
52 Southeast University, Nanjing 211100, People’s Republic of China
53 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
54 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
55 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
56 Tsinghua University, Beijing 100084, People’s Republic of China
57 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
58 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
59 University of Groningen, NL-9747 AA Groningen, The Netherlands
60 University of Hawaii, Honolulu, Hawaii 96822, USA
61 University of Jinan, Jinan 250022, People’s Republic of China
62 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
63 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
64 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
65 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
66 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
67 University of South China, Hengyang 421001, People’s Republic of China
68 University of the Punjab, Lahore-54590, Pakistan
69 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
70 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
71 Wuhan University, Wuhan 430072, People’s Republic of China
72 Xinyang Normal University, Xinyang 464000, People’s Republic of China
73 Yunnan University, Kunming 650500, People’s Republic of China
74 Zhejiang University, Hangzhou 310027, People’s Republic of China
75 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan

###### Abstract

A search for a massless dark photon $\gamma^{\prime}$ is conducted using 4.5
$\mathrm{fb}^{-1}$ of $e^{+}e^{-}$ collision data collected at center-of-mass
energies between 4.600 and 4.699 $\ \mathrm{GeV}$ with the BESIII detector at
BEPCII. No significant signal is observed, and the upper limit on the
branching fraction $\mathcal{B}(\Lambda^{+}_{c}\to p\gamma^{\prime})$ is
determined to be $8.0\times 10^{-5}$ at 90% confidence level.

##  I Introduction

The flavor changing neutral current (FCNC) transitions of the charmed baryon
$\Lambda^{+}_{c}$ are of great interest since they can provide indications for
physics beyond the Standard Model (SM). In the framework of the SM, FCNC
transitions are strongly suppressed by the Glashow, Iliopoulos and Maiani
(GIM) mechanism Glashow:fcnc  in the charm sector. The SM predictions for the
branching fractions (BFs) of FCNC decays in the charm sector are less than
$10^{-9}$ Tanabashi:charmFCNC . The minimal super-symmetric SM with R-parity
violation Aulakh:model1  and the two-Higgs-doublet model Glashow:model2
predict the BFs of the same FCNC decays to be two to three orders of magnitude
larger. Observation of a FCNC decay with the current experimental sensitivity
would imply new physics beyond the SM.

Models of new physics beyond the SM may have a dark sector containing an extra
Abelian gauge group, $U(1)_{D}$, under which all the SM fields are singlets.
This symmetry may be broken spontaneously or may remain unbroken, causing the
associated gauge boson, the dark photon, to acquire a mass or remain massless.
These possibilities have received a great deal of attention in recent decades
dark:mode1 ; Dobrescu:dark1 ; Dobrescu:dark2 ; dark:mode2 ; Ying:darkphoton ;
Ackerman:intre1 ; Pan:intre2 ; Barger:intre3 ; Chiang:intre4 ; Batley:intre5 ;
Pospelov:intre6 . If $U(1)_{D}$ remains unbroken, there is always a linear
combination of the dark and SM Abelian gauge fields which does not have
renormalizable couplings to SM members and which can be identified with the
massless dark photon ($\gamma^{\prime}$) dark:mode1 ; Dobrescu:dark1 . While
it has no direct interactions with SM fermions, the $\gamma^{\prime}$ can
still exert influence on the SM via higher-dimensional operators generated by
loop diagrams involving particles that are charged under $U(1)_{D}$ and also
coupled to SM fields Dobrescu:dark1 ; Dobrescu:dark2 ; dark:mode2 .

In experiment, LHCb reported the evidence for the breaking of lepton
universality in bottom-quark FCNC decays to charged lepton pairs with a
significance of $3.1\sigma$ LHCb:lepvio . As a complementary study, we
concentrate on FCNC effects arising from the dark photon with the $c$ and $u$
quarks, where the missing energy due to the dark photon is the feature of the
signal processes. BESIII has searched for the invisible signals within various
hadron decays, including $\eta/\eta^{\prime}\to\rm{invisible}$ BESIII:etainv ,
$\omega/\phi\to\rm{invisible}$ BESIII:omegainv , $\Lambda\to\rm{invisible}$
BESIII:Laminv  and $J/\psi\to\gamma+\rm{invisible}$ BESIII:Jpsiinv , and no
significant signals are observed. However, this has never been probed in the
charmed baryon sector. The two-body charmed baryon decay potentially offers a
competitive window to access $c\to u\gamma^{\prime}$, which gives rise to the
FCNC decays of charmed baryon into a lighter baryon plus missing energy
carried away by the massless dark photon. Figure 1 presents a typical Feynman
diagram of $\Lambda^{+}_{c}\to p\gamma^{\prime}$. It is found that the BFs of
some charmed baryon decays are allowed to be as high as a few times $10^{-5}$
Tandean:cFCNC  . Such BFs are likely to be within the sensitivity reaches of
some ongoing experiments like BESIII.

Fig. 1:  Feynman diagram for $\Lambda^{+}_{c}\to p\gamma^{\prime}$.

This paper presents an experimental search for a massless dark photon in
$\Lambda^{+}_{c}\to p\gamma^{\prime}$ decay using 4.5 $\mathrm{fb}^{-1}$ of
$e^{+}e^{-}$ collision data collected with the BESIII detector at seven
center-of-mass (CM) energies between 4.600 and 4.699 GeV. The CM energies and
the integrated luminosities for each energy point are listed in Table 1
BESIII:lumi0 ; BESIII:lumi1 ; BESIII:lumi2 . Taking advantage of the
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ production just above the mass
threshold 4572.92 MeV, a double-tag (DT) approach MarkIII:DT  is implemented.
Throughout the text, the charge conjugate states are always implied.

Table 1: The CM energies and the integrated luminosities ($\mathcal{L}_{int}$) for each energy point. The first and the second uncertainties are statistical and systematic, respectively. $\sqrt{s}$ (MeV) |  $\mathcal{L}_{int}$ (pb-1)
---|---
4599.53 $\pm$ 0.07 $\pm$ 0.74 |  586.90 $\pm$ 0.10 $\pm$ 3.90
4611.86 $\pm$ 0.12 $\pm$ 0.32 |  103.83 $\pm$ 0.05 $\pm$ 0.55
4628.00 $\pm$ 0.06 $\pm$ 0.32 |  521.52 $\pm$ 0.11 $\pm$ 2.76
4640.91 $\pm$ 0.06 $\pm$ 0.38 |  552.41 $\pm$ 0.12 $\pm$ 2.93
4661.24 $\pm$ 0.06 $\pm$ 0.29 |  529.63 $\pm$ 0.12 $\pm$ 2.81
4681.92 $\pm$ 0.08 $\pm$ 0.29 |  1669.31 $\pm$ 0.21 $\pm$ 8.85
4698.82 $\pm$ 0.10 $\pm$ 0.39 |  536.45 $\pm$ 0.12 $\pm$ 2.84

##  II BESIII Detector and Monte Carlo simulation

The BESIII detector Ablikim:2009aa  records symmetric $e^{+}e^{-}$ collisions
provided by the BEPCII storage ring, which operates with a peak luminosity of
$1\times 10^{33}$ cm-2s-1 in the CM energy range from 2.0 GeV to 4.9 GeV.
BESIII has collected large data samples at these energy regions
Ablikim:2019hff . The cylindrical core of the BESIII detector covers 93% of
the full solid angle and consists of a helium-based multilayer drift chamber
(MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl)
electromagnetic calorimeter (EMC), which are all enclosed in a superconducting
solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported
by an octagonal flux-return yoke with resistive plate counter based muon
identification modules interleaved with steel. The charged-particle momentum
resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and resolution of the ionization
energy loss in the MDC ($\mathrm{d}E/\mathrm{d}x$) is $6\%$ for electrons from
Bhabha scattering. The EMC measures photon energies with a resolution of
$2.5\%$ ($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution
in the TOF barrel region is 68 ps, while that in the end cap region is 110 ps.
The end cap TOF system was upgraded in 2015 using multi-gap resistive plate
chamber technology, providing a time resolution of 60 ps LiGuo:MRPC . About
13% of the data (the 4.600 GeV sample) used in the current analysis predates
this upgrade.

Monte Carlo (MC) simulated data samples are produced with a geant4-based
Agostinelli:2002hh  MC package, which includes the geometric description and
response of the BESIII detector. The signal MC samples of
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ with
$\bar{\Lambda}_{c}^{-}$ decaying into ten specific tag modes (as described
below and listed in Table 2) and $\Lambda^{+}_{c}\to p\gamma^{\prime}$, which
are used to determine the detection efficiencies, are generated for each CM
energy using the generator kkmc Jadach:2000ir  incorporating initial-state
radiation (ISR) effects and the beam energy spread. The inclusive MC samples,
which consist of $\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ events, charmed meson
$D_{(s)}^{(\ast)}$ pair production, ISR return to the charmonium(-like) $\psi$
states at lower masses, and continuum processes $e^{+}e^{-}\rightarrow
q\bar{q}$ ($q=u,d,s$), are generated to estimate the potential background.
Decay modes as specified in the Particle Data Group (PDG) PDG:2020  are
modeled with evtgen Lange:2001uf ; Ping:2008zz , and the remaining unknown
decays are modeled with lundcharm Chen:2000tv ; YANG:2014 . Final state
radiation (FSR) from charged final state particles is incorporated using
photos Richter-Was:1992hxq .

##  III METHODOLOGY

A DT approach MarkIII:DT  is implemented to search for $\Lambda^{+}_{c}\to
p\gamma^{\prime}$. A data sample of $\bar{\Lambda}_{c}^{-}$ baryon, referred
to as the single-tag (ST) sample, is reconstructed with ten exclusive hadronic
decay modes, as listed in Table 2. The subset of those events in which a
signal decay $\Lambda^{+}_{c}\to p\gamma^{\prime}$ candidate is reconstructed
in the system recoiling against the $\bar{\Lambda}_{c}^{-}$ candidate are
denoted as DT candidates. The $\Lambda^{+}_{c}$ decay BF is determined as

$\mathcal{B}(\Lambda^{+}_{c}\to
p\gamma^{\prime})=\frac{N_{\mathrm{obs}}-N_{\mathrm{bkg}}}{\sum_{ij}N_{ij}^{\mathrm{ST}}\cdot(\epsilon_{ij}^{\mathrm{DT}}/\epsilon_{ij}^{\mathrm{ST}})},$
(1)

where the $N_{\mathrm{obs}}$ is the number of observed events in the signal
region from data, and $N_{\mathrm{bkg}}$ is the number of estimated background
as explained explicitly in Section VI. The subscripts $i$ and $j$ represent
the ST modes and the data samples at different CM energies, respectively. The
parameters $N_{ij}^{\mathrm{ST}}$, $\epsilon_{ij}^{\mathrm{ST}}$ and
$\epsilon_{ij}^{\mathrm{DT}}$ are the ST yields, ST and DT detection
efficiencies, respectively.

##  IV ST Event Selections

Charged tracks detected in the MDC are required to be within a polar angle
($\theta$) range of $|\\!\cos\theta|<0.93$, where $\theta$ is defined with
respect to the positron beam direction. For prompt tracks not from
$K_{\mathrm{S}}^{0}$ and $\bar{\Lambda}$ decays, the distances of the closest
approach to the interaction point (IP) are required to be within $\pm$10 cm
along the beam direction and 1 cm in the plane perpendicular to the beam
(referred to as tight track hereafter). The particle identification (PID) is
implemented by combining measurements of the $\mathrm{d}E/\mathrm{d}x$ and the
flight time in the TOF. Every charged track is assigned a particle type of
pion, kaon or proton, by choosing the type with the highest probability.

Photon candidates are selected from showers reconstructed in the EMC. The
deposited energy of each shower must be more than 25 MeV in the barrel region
($|\\!\cos\theta|\leq 0.80$) or more than 50 MeV in the end cap region
($0.86\leq|\\!\cos\theta|\leq 0.92$). To suppress electronic noise and showers
unrelated to the event, the difference between the EMC time and the event
start time is required to be within (0, 700) ns. The $\pi^{0}$ candidates are
reconstructed from photon pairs with an invariant mass in the range (0.115,
0.150) GeV/$c^{2}$. To improve the resolution, a kinematic fit is performed
constraining the invariant mass of the photon pair to the known $\pi^{0}$ mass
PDG:2020 . The corresponding $\chi^{2}$ of the fit must be less than 200. The
momenta updated by the kinematic fit are used in further analysis.

Candidates for $K_{\mathrm{S}}^{0}$ and $\bar{\Lambda}$ are reconstructed in
their decays to $\pi^{+}\pi^{-}$ and $\bar{p}\pi^{+}$, respectively. Each
charged track must have a distance of closest approach to the IP within
$\pm$20 cm along the beam direction (referred to as loose track hereafter). To
improve the signal purity, PID is applied to the proton candidates, but not
the pion candidates. A secondary vertex fit is performed to each
$K_{\mathrm{S}}^{0}$ or $\bar{\Lambda}$ candidate, and the momenta updated by
the fit are used in the further analysis. To keep a high signal efficiency, a
$K_{\mathrm{S}}^{0}$ or $\bar{\Lambda}$ candidate is accepted if the
$\chi^{2}$ of this fit is less than 100. Furthermore, the decay vertex is
required to be separated from the IP by a distance of at least twice the
fitted vertex resolution, and the invariant mass must be within (0.487, 0.511)
GeV/$c^{2}$ for $\pi^{+}\pi^{-}$ or (1.111, 1.121) GeV/$c^{2}$ for
$\bar{p}\pi^{+}$. The $\bar{\Sigma}^{0}$ and $\bar{\Sigma}^{-}$ candidates are
reconstructed with the $\gamma\bar{\Lambda}$ and $\bar{p}\pi^{0}$ final
states, requiring the invariant masses to lie within (1.179, 1.203)
GeV/$c^{2}$ and (1.176, 1.200) GeV/$c^{2}$, respectively.

The ST $\bar{\Lambda}_{c}^{-}$ candidates are identified using the beam
constrained invariant mass
$M_{\mathrm{BC}}=\sqrt{E_{\mathrm{beam}}^{2}/c^{4}-|\vec{p}_{\bar{\Lambda}^{-}_{c}}|^{2}/c^{2}}$
and energy difference $\Delta E=E_{\bar{\Lambda}^{-}_{c}}-E_{\mathrm{beam}}$,
where $E_{\mathrm{beam}}$ is the beam energy, $E_{\bar{\Lambda}^{-}_{c}}$ and
$\vec{p}_{\bar{\Lambda}^{-}_{c}}$ are the energy and momentum of the
$\bar{\Lambda}_{c}^{-}$ candidate in the $e^{+}e^{-}$ CM frame, respectively.
The $\bar{\Lambda}_{c}^{-}$ candidates are required to satisfy the tag-mode
dependent $\Delta E$ requirements, the asymmetric intervals of which take into
account the effects of ISR and correspond to three times the resolution around
the peak, as summarized in Table 2. If there are more than one candidate
satisfying the above requirements for a specific tag mode, the one with the
minimum $|\Delta E|$ is kept.

Table 2: $\Delta E$ requirement, the ST yield, and the ST detection efficiency of each tag mode for data sample at $\sqrt{s}=4.600$ GeV. The uncertainty in the ST yield is statistical only.  |  $\Delta E$ (MeV) | $N_{i}^{\mathrm{ST}}$ |  $\epsilon_{i}^{\mathrm{ST}}$(%)
---|---|---|---
$\bar{p}K^{+}\pi^{-}$ | $(-34,~{}20)$ |  $6705$ $\pm$ |  90 | 51.0
$\bar{p}K_{S}^{0}$ | $(-20,~{}20)$ |  $1268$ $\pm$ |  37 | 56.2
$\bar{\Lambda}\pi^{-}$ | $(-20,~{}20)$ |  $741$ $\pm$ |  28 | 47.7
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | $(-30,~{}20)$ |  $1539$ $\pm$ |  57 | 15.4
$\bar{p}K_{S}^{0}\pi^{0}$ | $(-30,~{}20)$ |  $485$ $\pm$ |  29 | 18.4
$\bar{\Lambda}\pi^{-}\pi^{0}$ | $(-30,~{}20)$ |  $1382$ $\pm$ |  49 | 16.6
$\bar{p}K_{S}^{0}\pi^{+}\pi^{-}$ | $(-20,~{}20)$ |  $512$ $\pm$ |  29 | 19.9
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | $(-20,~{}20)$ |  $646$ $\pm$ |  31 | 13.7
$\bar{\Sigma}^{0}\pi^{-}$ | $(-20,~{}20)$ |  $404$ $\pm$ |  22 | 22.5
$\bar{\Sigma}^{-}\pi^{+}\pi^{-}$ | $(-30,~{}20)$ |  $872$ $\pm$ |  38 | 18.1

For the $\bar{\Lambda}_{c}^{-}\to\bar{p}K_{S}^{0}\pi^{0}$ ST mode, candidate
events with $M_{\bar{p}\pi^{+}}\in(1.100,1.125)$ GeV/$c^{2}$ and
$M_{\bar{p}\pi^{0}}\in(1.170,1.200)$ GeV/$c^{2}$ are vetoed to avoid double
counting with the $\bar{\Lambda}_{c}^{-}\to\bar{\Lambda}\pi^{-}\pi^{0}$ or
$\bar{\Lambda}_{c}^{-}\to\bar{\Sigma}^{-}\pi^{+}\pi^{-}$ ST modes,
respectively. For the $\bar{\Lambda}_{c}^{-}\to\bar{\Sigma}^{-}\pi^{+}\pi^{-}$
ST mode, candidate events with $M_{\pi^{+}\pi^{-}}\in(0.490,0.510)$
GeV/$c^{2}$ and $M_{\bar{p}\pi^{+}}\in(1.110,1.120)$ GeV/$c^{2}$ are rejected
to avoid double counting with the
$\bar{\Lambda}_{c}^{-}\to\bar{p}K_{S}^{0}\pi^{0}$ or
$\bar{\Lambda}_{c}^{-}\to\bar{\Lambda}\pi^{-}\pi^{0}$ ST modes, respectively.
In the $\bar{\Lambda}_{c}^{-}\to\bar{p}K_{S}^{0}\pi^{+}\pi^{-}$ and
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ selections, candidate events with
$M_{\bar{p}\pi^{+}}\in(1.100,1.125)$ GeV/$c^{2}$ and
$M_{\pi^{+}\pi^{-}}\in(0.490,0.510)$ GeV/$c^{2}$ are rejected, respectively.

The $M_{\mathrm{BC}}$ distributions of candidates for the ten ST modes with
the data sample at $\sqrt{s}=4.600~{}\mathrm{GeV}$ are illustrated in Fig. 2,
where clear $\bar{\Lambda}_{c}^{-}$ signals are observed in each mode. No
peaking background is found using the inclusive MC samples. To obtain the ST
yields, unbinned maximum likelihood fits on these $M_{\mathrm{BC}}$
distributions are performed, where the signal shape is modeled with the MC-
simulated shape convolved with a Gaussian function representing for the
resolution difference between data and MC simulation, and the background shape
is described by the ARGUS function ARGUS:1990hfq . The candidates with
$M_{\mathrm{BC}}\in(2.275,2.310)$ GeV/$c^{2}$ are retained for further
analysis, and the signal yields for the individual ST modes are summarized in
Table 2. The same procedure is performed for the other six data samples at
different CM energies, the results can be found in Ref. Ablikim:2022SCS  and
its supplemental material. The sum of ST yields for all data samples at
different CM energies is $105244\pm 384$, where the uncertainty is
statistical.

Fig. 2:  The $M_{\mathrm{BC}}$ distributions of the ST modes for data sample
at $\sqrt{s}=4.600~{}\mathrm{GeV}$. The points with error bars represent data.
The red solid curves indicate the fit results and the blue dashed curves
describe the background shapes.

##  V Reconstruction of $p\gamma^{\prime}$ candidates

The decay $\Lambda^{+}_{c}\to p\gamma^{\prime}$ is searched for among the
remaining tracks and showers recoiling against the $\bar{\Lambda}_{c}^{-}$
candidates. Only one tight track is allowed, and it must satisfy the PID
criteria $\mathcal{L}$($p$) $>\mathcal{L}$($K$) and $\mathcal{L}$($p$)
$>\mathcal{L}$($\pi$). To suppress contamination from long-lifetime particles
in the final state, the candidate events are further required to be without
any unused loose tracks. The $\gamma^{\prime}$ does not interact directly with
SM fermions and thus it deposits no energy in the EMC. Backgrounds containing
a $\pi^{0}$ are vetoed with the requirement of $E_{\rm max}<0.3$ GeV and
$E_{\rm sum}<0.5$ GeV, where $E_{\rm max}$ and $E_{\rm sum}$ are the maximum
energy and the energy sum of the unused showers, respectively. The
$\gamma^{\prime}$ signal is selected using the square of the recoil mass,
$M^{2}_{\rm rec(\bar{\Lambda}_{c}^{-}p)}$, against the ST
$\bar{\Lambda}_{c}^{-}$ and $p$.

After imposing all selection conditions mentioned above, the distribution of
$M^{2}_{\rm rec(\bar{\Lambda}_{c}^{-}p)}$ of the accepted DT candidate events
from the combined seven data samples at different CM energies is shown in Fig.
3(a). There is a peaking structure at the $K_{L}^{0}$ mass position, from the
process $\Lambda^{+}_{c}\to pK_{L}^{0}$.

##  VI Background analysis

The potential background can be classified into two categories: those directly
originated from continuum hadron production in the $e^{+}e^{-}$ annihilation,
denoted as $q\bar{q}$ background, and those from the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ events, denoted as
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background. The distribution and
magnitude of $\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background are estimated
with the inclusive MC samples, where the peaking background
$\Lambda^{+}_{c}\to pK_{L}^{0}$ is extracted separately from the inclusive MC
samples. The $\Lambda^{+}_{c}\to pK_{L}^{0}$ rate is normalized to the known
BF PDG:2020  and the remaining backgrounds are normalized to
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ data yields. The $q\bar{q}$ background
is investigated with $M_{\mathrm{BC}}$ sideband region (2.21,2.26) GeV/$c^{2}$
of ST candidates in data, which is than reweighted to agree with those of the
data in signal region. The $q\bar{q}$ and
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ backgrounds are estimated to be $7.4\pm
0.4$ and $7.2\pm 1.4$, respectively. The resultant $M^{2}_{\rm
rec(\bar{\Lambda}_{c}^{-}p)}$ distribution is depicted in Fig. 3(a).

Fig. 3:  (a) The full spectrum of $M^{2}_{\rm rec(\bar{\Lambda}_{c}^{-}p)}$ of
the accepted DT candidate events from the combined seven data samples. (b) The
spectrum of $M^{2}_{\rm rec(\bar{\Lambda}_{c}^{-}p)}$ of the accepted DT
candidate events from the combined data in the signal region. The black points
with error bars are data; no events have been observed in the last bin. The
red hatched histogram indicates $pK_{L}^{0}$ background. The blue hatched
histogram is the $\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background excluding
$\Lambda^{+}_{c}\to pK_{L}^{0}$ process. The brown hatched histogram
represents the $q\bar{q}$ background. The yellow hatched histogram is the
$\Lambda^{+}_{c}\to p\gamma^{\prime}$ signal, which is normalized to data
luminosity with the upper limit of the BF.

##  VII upper limit setting

In order to extract the signal yield, a signal region is defined as (0.0, 0.1)
$\mathrm{GeV}^{2}/{c}^{4}$ in the $M^{2}_{\rm rec(\bar{\Lambda}_{c}^{-}p)}$
distribution, corresponding to a 96% signal detection efficiency after
imposing all the selection criteria. The distribution of $M^{2}_{\rm
rec(\bar{\Lambda}_{c}^{-}p)}$ in the signal region is shown in Fig. 3(b).
Thirteen candidate events are observed in the signal region, while the
background events are estimated to be $14.6\pm 1.5$, where the uncertainty is
statistical only. The $\Lambda^{+}_{c}$ decay BF is determined as in Eq. (1).
The detection efficiency $\epsilon_{ij}^{\mathrm{ST}}$ is obtained with the
same procedure as in Ref. Ablikim:2022SCS , and the
$\epsilon_{ij}^{\mathrm{DT}}$ is derived with the signal MC samples. The DT
efficiencies are summarized in Table 3.

Table 3:  The DT detection efficiencies in percentage for ten tag modes and seven data samples at different CM energies. The statistical uncertainties are lower than 0.3%. $\sqrt{s}$ (GeV) | $4.600$ | $4.612$ | $4.628$ | $4.641$ | $4.661$ | $4.682$ | $4.699$
---|---|---|---|---|---|---|---
$\bar{p}K^{+}\pi^{-}$ | 34.3 | 33.8 | 32.3 | 32.0 | 31.6 | 31.3 | 30.7
$\bar{p}K_{S}^{0}$ | 39.1 | 37.1 | 34.6 | 33.6 | 33.9 | 32.8 | 31.8
$\bar{\Lambda}\pi^{-}$ | 32.9 | 30.5 | 29.1 | 29.5 | 28.5 | 26.9 | 25.4
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | 12.9 | 12.4 | 12.3 | 12.6 | 11.9 | 11.9 | 11.5
$\bar{p}K_{S}^{0}\pi^{0}$ | 14.9 | 14.2 | 13.5 | 13.4 | 13.4 | 13.5 | 12.7
$\bar{\Lambda}\pi^{-}\pi^{0}$ | 13.6 | 12.7 | 12.2 | 12.3 | 12.0 | 11.6 | 11.2
$\bar{p}K_{S}^{0}\pi^{+}\pi^{-}$ | 15.4 | 14.5 | 13.7 | 13.6 | 13.4 | 13.3 | 13.2
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | 10.4 | 9.8 | 9.3 | 9.6 | 9.3 | 9.3 | 9.1
$\bar{\Sigma}^{0}\pi^{-}$ | 18.8 | 17.7 | 17.1 | 16.2 | 14.7 | 14.6 | 15.0
$\bar{\Sigma}^{-}\pi^{+}\pi^{-}$ | 17.5 | 16.5 | 16.3 | 16.2 | 15.8 | 15.0 | 14.9

Since no significant signal is observed, the profile-likelihood approach
Cowan:LH  is used to determine the upper limit on the BF of
$\Lambda^{+}_{c}\to p\gamma^{\prime}$. The likelihood function which depends
on the parameter of interest $\mathcal{B}(\Lambda^{+}_{c}\to
p\gamma^{\prime})$ and the nuisance parameters $\theta=(\epsilon_{\rm
eff},N_{\rm bkg})$ is defined as:

$\mathcal{L}(\mathcal{B}(\Lambda^{+}_{c}\to p\gamma^{\prime}),\theta)=P(N_{\rm
obs}|N_{\rm exp})\cdot G(\theta)$ (2)

where the observed events are assumed to follow a Poisson distribution ($P$).
The $N_{\rm exp}$ is the expected number of events; it is defined as the sum
of the number of background events and number of signal events estimated in
the signal region, corresponding to Eq. (1). The detection efficiency
$\epsilon_{\rm eff}$ and $N_{\rm bkg}$ follow Gaussian distributions ($G$).
The upper limit on the BF of $\Lambda^{+}_{c}\to p\gamma^{\prime}$ is
determined by scanning the parameter of interest. The resultant profile-
likelihood scan distribution is presented in Fig. 4. The upper limit is
calculated to be $\mathcal{B}(\Lambda^{+}_{c}\to p\gamma^{\prime})<8.0\times
10^{-5}$ at 90% confidence level (CL), where the statistical and systematic
uncertainties are all incorporated. The systematic uncertainties associated
with the detection efficiency and the background estimation are performed with
the two nuisance parameters in a profile-likelihood fit.

Fig. 4:  The profile-likelihood curve versus $\mathcal{B}(\Lambda^{+}_{c}\to
p\gamma^{\prime})$. The black solid curve is the scan result with systematic
uncertainties. The cross of the curve indicates the upper limit of the BF at
90% CL.

##  VIII Systematic Uncertainty

The systematic uncertainties for the BF measurement include those associated
with the ST yields ($N_{ij}^{\mathrm{ST}}$), reconstruction efficiencies of
the ST $\bar{\Lambda}_{c}^{-}$ ($\epsilon_{ij}^{\mathrm{ST}}$) and
reconstruction efficiencies of the DT ($\epsilon_{ij}^{\mathrm{DT}}$). As the
DT technique is adopted, the systematic uncertainties originating from
reconstructing the ST side largely cancel. Table 4 summarizes the possible
sources of systematic uncertainties. Each of them is evaluated relative to the
measured BF. The details are described below.

The uncertainties associated with the proton tracking and PID efficiencies are
determined with the control sample $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$
Uncer:pidTracking . The systematic uncertainties in proton tracking and PID
are assigned to be 1.0% each.

The uncertainty in the ST yields is 0.5%, which arises from the statistical
uncertainty and fitting $M_{\mathrm{BC}}$ distributions. The uncertainty in
the fitting procedure is evaluated by varying the ARGUS background parameter
and changing the Gaussian function to a sum of two Gaussian functions.

The systematic uncertainty for the requirements of $E_{\rm max}$ and $E_{\rm
sum}$ is studied using the control sample of $\Lambda_{c}^{+}\to
pK^{-}\pi^{+}$. The systematic uncertainty, 3.0%, is determined by comparing
the efficiencies between data and MC simulation.

According to Eq. (1), the uncertainty related to the ST efficiency is expected
to be canceled. However, due to the different multiplicities of tracks and
showers in the rest of the event, the ST efficiencies estimated with the
generic and the signal MC samples are expected to differ slightly. Thus, the
uncertainty associated with the ST efficiency is not canceled fully, which
results in a so called “tag bias” uncertainty. The difference of ST efficiency
between generic and the signal MC samples, 0.9%, is assigned as the
corresponding uncertainty.

The uncertainty in the $q\bar{q}$ background estimation is obtained by
widening the $M_{\mathrm{BC}}$ sideband range by 5 MeV/$c^{2}$ in comparison
to the nominal one, yielding 4.1%. In addition, the statistical uncertainty of
5.7% on the data yield in the $M_{\mathrm{BC}}$ sideband range (2.21,2.26)
GeV/$c^{2}$ is taken into account. The total $q\bar{q}$ background systematic
uncertainty of 7.0% is the quadratic sum of these two effects.

The main $\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background is the
$\Lambda^{+}_{c}\to pK_{L}^{0}$ process. The uncertainty of its BF quoted from
the PDG on ${\cal B}(\Lambda^{+}_{c}\to pK_{S}^{0})$ PDG:2020  is 5.0%, and
the fraction of $pK_{L}^{0}$ in the signal region is 36%. The net systematic
uncertainty in $\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background estimation is
thus 1.8%.

Other uncertainties are negligible. Assuming that all the sources of the
uncertainties are uncorrelated, the total systematic uncertainties associated
with the detection efficiency and the background estimation are 3.5% and 7.2%,
respectively.

Table 4: The systematic uncertainties in percentage for $\Lambda^{+}_{c}\to p\gamma^{\prime}$. mode | uncertainty (%)
---|---
$p$ tracking | 1.0
$p$ PID | 1.0
ST yield | 0.5
$E_{\rm max}$ and $E_{\rm sum}$ requirements | 3.0
Tag bias | 0.9
$q\bar{q}$ background estimation | 7.0
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background estimation | 1.8

##  IX Summary

In summary, with a sample of 4.5 $\mathrm{fb}^{-1}$ collected at CM energies
between 4.600 and 4.699 GeV with the BESIII detector, the first investigation
for a massless dark photon in $\Lambda^{+}_{c}\to p\gamma^{\prime}$ decay is
carried out. No significant signal is observed with respect to the expected
background. The upper limit on the BF of $\Lambda^{+}_{c}\to p\gamma^{\prime}$
is measured to be $\mathcal{B}(\Lambda^{+}_{c}\to p\gamma^{\prime})<8.0\times
10^{-5}$ at 90% CL. It is below the sensitivity of theory prediction in Ref.
Tandean:cFCNC , which predicts the BF to be $1.6\times 10^{-5}$ or $9.1\times
10^{-6}$ with different inputs of form factors. A more stringent constrain on
$\mathcal{B}(\Lambda^{+}_{c}\to p\gamma^{\prime})$ is expected in the near
future with larger $\Lambda_{c}^{+}$ samples at BESIII Ablikim:2019hff .

## ACKNOWLEDGEMENTS

The BESIII collaboration thanks the staff of BEPCII, the IHEP computing center
and the supercomputing center of the University of Science and Technology of
China (USTC) for their strong support. Authors are grateful to Jusak Tandean
for enlightening discussions. This work is supported in part by National Key
R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300;
National Natural Science Foundation of China (NSFC) under Contracts No.
11635010, No. 11735014, No. 11835012, No. 11935015, No. 11935016, No.
11935018, No. 11961141012, No. 12022510, No. 12025502, No. 12035009, No.
12035013, No. 12192260, No. 12192261, No. 12192262, No. 12192263, No.
12192264, No. 12192265, No. 12005311; the Fundamental Research Funds for the
Central Universities, Sun Yat-sen University, University of Science and
Technology of China; 100 Talents Program of Sun Yat-sen University; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program;
Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract
No. U1832207; 100 Talents Program of CAS; The Institute of Nuclear and
Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and
Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020
research and innovation programme under Marie Sklodowska-Curie grant agreement
under Contract No. 894790; German Research Foundation DFG under Contracts Nos.
443159800, Collaborative Research Center CRC 1044, GRK 2149; Istituto
Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; National Science and Technology fund; National
Science Research and Innovation Fund (NSRF) via the Program Management Unit
for Human Resources & Institutional Development, Research and Innovation under
Contract No. B16F640076; STFC (United Kingdom); Suranaree University of
Technology (SUT), Thailand Science Research and Innovation (TSRI), and
National Science Research and Innovation Fund (NSRF) under Contract No.
160355; The Royal Society, UK under Contracts Nos. DH140054, DH160214; The
Swedish Research Council; U. S. Department of Energy under Contract No. DE-
FG02-05ER41374.

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# Full Attention Bidirectional Deep Learning Structure for Single Channel
Speech Enhancement

###### Abstract

As the cornerstone of other important technologies, such as speech recognition
and speech synthesis, speech enhancement is a critical area in audio signal
processing. In this paper, a new deep learning structure for speech
enhancement is demonstrated. The model introduces a “full” attention mechanism
to a bidirectional sequence-to-sequence method to make use of latent
information after each focal frame. This is an extension of the previous
attention-based RNN method. The proposed bidirectional attention-based
architecture achieves better performance in terms of speech quality (PESQ),
compared with OM-LSA, CNN-LSTM, T-GSA and the unidirectional attention-based
LSTM baseline.

Index Terms: Speech enhancement, bidirectional, full attention mechanism, long
short term memory (LSTM).

##  1 Introduction

Neural networks have seen great success in the field of speech enhancement and
noise suppression in the last decade [1, 2, 3, 4, 5, 6, 7, 8]. This approach
now outperforms most traditional model-based statistical approaches, such as
spectral subtraction [9], the minimum mean-square error log-spectral method
[8], Wiener filtering [10] or OM-LSA [11]. Recurrent neural network (RNN), has
been widely used [12, 13, 14].

Various RNN-based structures such as gated recurrent unit (GRU) networks, long
short term memory (LSTM) networks, have been explored due to their powerful
sequence learning ability, as well as combinations and variations [15], which
show superiority in the field of audio signal processing. The introduction of
an attention mechanism further enhances the potential of these seq-to-seq
methods [16]. The proposed approach makes use of Mel frequency features [17]
to extract features to represent the sound sequence and introduces a new
bidirectional attention-based structure in speech enhancement.

Our work is inspired by the recent success of unidirectional attention-RNN
models in various seq-to-seq learning tasks [18], which allow the clean speech
to be perceived with high attention while the noise and background have low
attention. We extend this idea to make full use of the information around the
focal frame as indicated by a high level of attention, and refer to this as a
_full_ as opposed to partial or half attention method typical of previous
approaches. The new architecture uses a bidirectional connection structure
combined with this full attention mechanism.

The rest of the paper is organized as follows. In Section 2, our method is
described in detail. The experiments and results are presented in Section 3.
Finally we draw conclusions in Section 4.

##  2 Method

As illustrated in Figure 1, the process is divided into three major parts:
feature extracting, deep learning and filtering. The input audio signal is
speech with noise and the output audio signal is expected to be clean. Feature
extraction calculates Mel filter bank (FBank) features, while the neural
network implements regression and generates clean estimates of the FBank [17]
for filtering.

Figure 1: System Overview.

###  2.1 Feature Extraction

Frequency features are extracted from the short-time Fourier transform (STFT)
of input audio signal. In contrast to typical methods that use all sampling
points in the spectrum as features [18], we choose to extract the features in
FBank to reduce the number of the weight to estimate in the system. We use the
input and output feature vectors to conduct spectrum suppression as described
in Section 2.3.

###  2.2 Attention-based Bidirectional Architecture

The deep learning architecture is shown in Figure 2. The input is
$\bm{X}=[\bm{x}_{1},\bm{x}_{2},...\bm{x}_{T}]$, where $\bm{x}_{t}$ is the
vector that represents the amplitude at specific frequency points of noisy
speech at frame $t$. $T$ represents the total number of frames.

Figure 2: Deep Learning Structure: The left and right sides respectively
represent the calculation process of the forward and reverse attention
parameters.

A dense layer serves as the encoder to give a high-level representation from
the input features.

$\bm{\hat{x}}_{t}=\tanh{(\bm{W}\bm{x}_{t}+\bm{b})}$ (1)

The number of output features requires fine adjustment to maintain balance
between representation and generalization. $\bm{\hat{x}}_{t}$ is used in both
the forward LSTM cell and the backward LSTM cell to generate the key and the
query vectors.

$\displaystyle\bm{h}_{t,f}^{K}$
$\displaystyle={\rm{LSTM_{1}}}({\rm{forward}}(\bm{\hat{x}}_{t}))$ (2)
$\displaystyle\bm{h}_{t,b}^{K}$
$\displaystyle={\rm{LSTM_{2}}}({\rm{backward}}(\bm{\hat{x}}_{t}))$ (3)
$\displaystyle\bm{h}_{t,f}^{Q}$
$\displaystyle={\rm{LSTM_{3}}}({\rm{forward}}(\bm{\hat{x}}_{t}))$ (4)
$\displaystyle\bm{h}_{t,b}^{Q}$
$\displaystyle={\rm{LSTM_{4}}}({\rm{backward}}(\bm{\hat{x}}_{t}))$ (5)

The key and query vectors are shown as $\bm{h}_{t,f}^{K}$, $\bm{h}_{t,b}^{K}$,
$\bm{h}_{t,f}^{Q}$ and $\bm{h}_{t,f}^{Q}$, where the backward or forward
direction is denoted by subscript $b$ or $f$, respectively. As shown above,
backward LSTM cell can be regarded as a normal LSTM cell with the reverse
sequence input.

This approach will make it possible to use latent information in the attention
mechanism. A dense layer is also added before generating the final query, but
this step is omitted here for the simplicity of explanation. In forward and
backward processing, the attention mechanism generates two attention weight
vectors.

$\displaystyle\bm{\alpha}_{t}$
$\displaystyle=[\alpha_{t,t-\omega},\alpha_{t,t-\omega+1},...,\alpha_{t,t}]$
(6) $\displaystyle\bm{\gamma}_{t}$
$\displaystyle=[\gamma_{t,t+\xi},\gamma_{t,t+\xi-1},...,\gamma_{t,t}]$ (7)

Following the correlation calculation in [18] and [19], normalized attention
weight $\alpha$ and $\gamma$ can be learned as:

$\displaystyle\alpha_{t,k}$
$\displaystyle=\frac{\exp{(\bm{h}_{k,f}^{K}\bm{W}\bm{h}_{t,f}^{Q})}}{\sum_{k=t-\omega}^{t}\exp{(\bm{h}_{k,f}^{K}\bm{W}\bm{h}_{t,f}^{Q}})}$
(8) $\displaystyle\gamma_{t,k}$
$\displaystyle=\frac{\exp{(\bm{h}_{k,b}^{K}\bm{W}\bm{h}_{t,b}^{Q})}}{\sum_{k=t}^{t+\xi}\exp{(\bm{h}_{k,b}^{K}\bm{W}\bm{h}_{t,b}^{Q})}}$
(9)

The attention weight vectors indicate the degree of relevance of neighboring
frames to the time frame $t$, which we refer to as a _focal frame_. If the
utterance is too long, the attention weights of distant frames may be nearly
zero. Based on human pronunciation patterns, it is reasonable to assume that
the length of backward effective attention sequences and the length of forward
effective attention sequences may be different. Based on thie, $\omega$ and
$\xi$ are implemented as separate constants.

If $t-\omega<0$, we can set $t-\omega=0$. If $t+\xi>N$, we can set $t+\xi=N$,
where $N$ is the total number of frames in the input sequence. The context
vector $\bm{c}_{t}$ containing the information of key vectors
$\bm{h}_{k,f}^{K}$ and $\bm{h}_{k,b}^{K}$ can be computed.

$\displaystyle\bm{c}_{t,f}$
$\displaystyle=\sum_{k=t-\omega}^{t}(\alpha_{t,k}\bm{h}_{k,f}^{K})$ (10)
$\displaystyle\bm{c}_{t,b}$
$\displaystyle=\sum_{k=t}^{t+\xi}(\gamma_{t,k}\bm{h}_{k,b}^{K})$ (11)
$\displaystyle\bm{c}_{t}$ $\displaystyle=[\bm{c}_{t,f};\bm{c}_{t,b}]$ (12)

where $[\cdot;\cdot]$ denotes the concatenation of two vectors. Another dense
layer severs as a decoder to make use of the context vector $\bm{c}_{t}$,
query vectors $\bm{h}_{t}^{Q}=[\bm{h}_{t,f}^{Q};\bm{h}_{t,b}^{Q}]$ and the
input feature $\bm{x}_{t}$. Similar to [18], the model generate an enhancement
vector $\bm{e}_{t}$ and the final gain $\bm{g}_{t}$ as:

$\displaystyle\bm{e}_{t}$
$\displaystyle=\tanh{(\bm{W}_{e}[\bm{c}_{t};\bm{h}_{t}^{Q}]+\bm{b}_{e})}$ (13)
$\displaystyle\bm{g}_{t}$
$\displaystyle={\rm{sigmoid}}(\bm{W}_{m}\bm{e}_{t}+\bm{b}_{m})$ (14)
$\displaystyle\bm{y}_{t}$ $\displaystyle=\bm{x}_{t}\odot\bm{g}_{t}$ (15)

###  2.3 Mel Filter Bank Generator

The model generates a FBank feature vector for every frame to use for noise
filtering. The gain $\bm{g}_{t}$ of each triangular filter at the peak
response is calculated from the input vector $\bm{x}_{t}$ and the output
vector $\bm{y}_{t}$. The amplitude of other points of the filter bank is then
adjusted proportionally according to the gain at the peak.

##  3 Results

###  3.1 Datasets

We tested the quality of the speech enhancement on two free speech databases,
THCHS-30 [20] and QUT-NOISE-TIMIT [21].

THCHS-30 is a free Chinese speech database, which involves more than 30 hours
of clean speech signals recorded by a single microphone in silence and with
white noise, car noise and cafeteria noise. We extend the noise conditions
further by creating a mix of car noise and cafeteria noise, implemented along
with the white noise condition. Signal-to-noise ratio (SNR) ranges from $-10$
dB to 10 dB.

QUT-NOISE-TIMIT is synthesized by mixing 5 different background noise sources
with TIMIT [22]. For the training set, $-5$ and 5 dB SNR data were used, but
the evaluation set contains SNR ranging from $-10$ dB to 15 dB. The total
length of train and test data corresponds to 25 hours and 12 hours,
respectively.

###  3.2 Experiment Setup

The sampling rate of the audio files is 16 kHz, with 512 point Hanning-
windowed frames and 128 points overlap.

In both datasets, we randomly divide the data into training set and test set
in a 4:1 ratio. The result is verified using 5-fold cross-validation. The
input and the output are both 42-dimensional FBank vectors. The batch size is
set to 96, the number of LSTM cells is set to 350, and the value of $\omega$
and $\xi$ are set to 15 and 5, respectively. The loss function is mean square
error (MSE), and dropout regularization is used. The setting of the learning
rate is dynamically adjusted according to the SNR as follows:

$\rm LearningRate=\left\\{\begin{array}[]{rcl}1\times 10^{-6}&&{\rm{SNR}\geq
10dB}\\\ 1\times 10^{-5}&&{\rm 5dB\leq\rm{SNR}<10dB}\\\ 5\times 10^{-5}&&{\rm
0dB\leq\rm{SNR}<5dB}\\\ 1\times 10^{-4}&&{\rm-5dB\leq\rm{SNR}<0dB}\\\ 5\times
10^{-4}&&{\rm{SNR}<-5dB}\end{array}\right.$

In the experiments on the THCHS-30 dataset, we compared our approach with OM-
LSA, LSTM-RNN and LSTM with attention(LSTM-Att) [18]. It is worth mentioning
that the LSTM-Att method is unidirectional, which can make the comparision
more convincing.

In the experiments on the QUT-NOISE-TIMIT dataset, we choose CNN-LSTM, O-T,
T-AB, T-GSA, C-T-GSA to be the baseline [23], in order to compare our model
with other end-to-end sequence structures. The perceptual evaluation of speech
quality (PESQ) [24] is used as the evaluation criteria.

###  3.3 Main Results

Results for THCHS-30 are shown in Table 1. The new approach outperforms the
three baselines in most instances, especially when the noise pattern is
irregular and SNR is relatively low. The bidirectional model performs better
than the unidirectional model or the model without an attention mechanism.
This sheds light on the effectiveness of using additional context when
implementing the attention mechanism.

Table 1: PESQ results on THCHS-30: Test set consists of 5 SNR ranges:-10, -5, 0, 5, 10dB. Bold indicates best results. Method | Raw | OM-LSA | LSTM-RNN | LSTM-Att | Bi-Att
---|---|---|---|---|---
cafe&car | white | cafe&car | white | cafe&car | white | cafe&car | white | cafe&car | white
$-10$ dB | 0.6424 | 0.0310 | 0.8337 | 0.8258 | 1.3532 | 1.3750 | 1.3786 | 1.3756 | $\bm{1.5615}$ | $\bm{1.5044}$
$-5$ dB | 1.1098 | 0.4169 | 1.2895 | 1.4132 | 1.7820 | 1.6757 | 1.8277 | 1.7512 | $\bm{1.8317}$ | $\bm{1.7986}$
0 dB | 1.5216 | 0.9977 | 2.1160 | 1.9523 | 2.2624 | 2.2322 | 2.4584 | 2.2324 | $\bm{2.5195}$ | $\bm{2.2745}$
5 dB | 2.1597 | 1.6791 | 2.2753 | $\bm{2.4582}$ | 2.4723 | 2.4023 | 2.6439 | 2.4330 | $\bm{2.6675}$ | 2.4468
10 dB | 2.6210 | 2.3163 | 2.6620 | $\bm{2.8326}$ | 2.7027 | 2.4732 | 2.8659 | 2.4942 | $\bm{2.9185}$ | 2.5165

Results for QUT-NOISE-TIMIT are shown in Table 2. The bidirectional attention
model outperforms the selected baseline models in most SNR situation. The area
where the improvement is most significant is in the SNR range from $-5$ to 5
dB. Our hypothesis is that the FBank structure can better suppress the full
frequency band under high noise conditions, but that in the case of low noise
energy, insufficiently fine point-to-point suppression may reduce the quality
of the entire speech signal.

Table 2: PESQ results on QUT-NOISE-TIMIT: Test set consists of 6 SNR ranges: $-10$, $-5$, 0, 5, 10, 15 dB, according to result in [23]. SNR (dB) | $-10$ | $-5$ | 0 | 5 | 10 | 15
---|---|---|---|---|---|---
Raw | 1.07 | 1.08 | 1.13 | 1.26 | 1.44 | 1.72
CNNLSTM | 1.43 | 1.65 | 1.89 | 2.16 | 2.35 | 2.54
O-T | 1.29 | 1.45 | 1.63 | 1.87 | 2.07 | 2.29
T-AB | 1.49 | 1.67 | 1.85 | 2.01 | 2.28 | 2.50
T-GSA | 1.54 | 1.76 | 2.00 | 2.28 | $\bm{2.51}$ | 2.74
C-T-GSA | 1.43 | 1.64 | 1.88 | 2.17 | 2.40 | 2.67
Bi-Att | $\bm{1.56}$ | $\bm{1.84}$ | $\bm{2.09}$ | $\bm{2.35}$ | 2.47 | $\bm{2.79}$

Based on the asymmetry of speech in the time domain, it is reasonable to
assume that the attention may vary in different ways in forward and backward
directions. The results of having separate attention sequence length
constants, i.e. $\omega$ and $\xi$, are shown in Figure 3, for the THCHS-30
dataset in mixed car and cafe noise at an SNR of 0 dB. The resulting PESQ
changes slightly as $\omega$ and $\xi$ vary. We notice that bigger values of
$\omega$ and $\xi$ result in no obvious improvement. The model performs best
when the value of $\omega$ and $\xi$ are both in the range of 5 to 20 dB.

Figure 3: PESQ at different values of $\omega$ and $\xi$ and visualization of
results: darker colors represent larger PESQ. $\omega$ and $\xi$ represents
the length of forward and backward attention sequence respectively.

The impact of the attention weight is illustrated in Figure 4. We set $\omega$
as $15$ and $\xi$ as $5$ on a test example on THCHS-30 (251 frames). In the
left figure, the $x$ and $y$ axis represents $\bm{h}_{t,f}^{K}$ and
$\bm{h}_{t,f}^{Q}$. On the right, the $x$ and $y$ axis represents
$\bm{h}_{t,b}^{K}$ and $\bm{h}_{t,b}^{Q}$. The point $(x,y)$ represents
attention weight. As seen in the figure, the model gives different weights to
the contextual frames. For illustration, the spectrum of a speech utterance is
shown in Figure 5. The original speech signal is superimposed with the café
and car noise to form noisy speech with an SNR of 0 dB.

Figure 4: The visualization of bidirectional attention weights, $\omega=15$,
$\xi=5$. Top: positive direction. Bottom: negative direction. Figure 5: A test
example before and after speech enhancement. Top: noisy speech. Middle: speech
processed by our model. Bottom: clean speech.

##  4 Conclusion

This paper demonstrates a new attention-based LSTM architecture for speech
enhancement. The primary innovation is the use of a two-way attention network
to take full advantage of latent information. Mel filter bank features are
used to reduce complexity. Results demonstrate that the resulting perceptual
quality of the enhanced speech is higher than comparative baseline methods in
most cases. The attention-based approach also shows better generalization
ability to irregular noise conditions. In addition, asymmetric forward and
backward attention weights are implemented and evaluated. Overall, the
proposed approach demonstrates the positive impact of increasing the extent of
speech and language context for noise suppression applications.

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