# Measurement of the absolute branching fraction of the singly Cabibbo
suppressed decay $\Lambda^{+}_{c}\to p\eta^{\prime}$

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.
Bianchi69A,69C, J. Bloms63, A. Bortone69A,69C, I. Boyko32, R. A. Briere5, A.
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.
Chen23,h, W. S. Cheng69C, S. K. Choi 50, X. Chu39, G. Cibinetto27A, F.
Cossio69C, J. J. Cui45, H. L. Dai1,53, J. P. Dai73, A. Dbeyssi17, R. E. de
Boer4, D. Dedovich32, Z. Y. Deng1, A. Denig31, I. Denysenko32, M.
Destefanis69A,69C, F. De Mori69A,69C, Y. Ding36, J. Dong1,53, L. Y. Dong1,58,
M. Y. Dong1,53,58, X. Dong71, S. X. Du75, P. Egorov32,a, Y. L. Fan71, J.
Fang1,53, S. S. Fang1,58, W. X. Fang1, Y. Fang1, R. Farinelli27A, L.
Fava69B,69C, F. Feldbauer4, G. Felici26A, C. Q. Feng66,53, J. H. Feng54, K
Fischer64, M. Fritsch4, C. Fritzsch63, C. D. Fu1, H. Gao58, Y. N. Gao42,g,
Yang Gao66,53, S. Garbolino69C, I. Garzia27A,27B, P. T. Ge71, Z. W. Ge38, C.
Geng54, E. M. Gersabeck62, A Gilman64, K. Goetzen12, L. Gong36, W. X.
Gong1,53, W. Gradl31, M. Greco69A,69C, L. M. Gu38, M. H. Gu1,53, Y. T. Gu14,
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. Himmelreich31,d, G. Y. Hou1,58, Y. R. Hou58, Z. L. Hou1, H. M. Hu1,58, J.
F. Hu51,i, T. Hu1,53,58, Y. Hu1, G. S. Huang66,53, K. X. Huang54, L. Q.
Huang28,58, X. T. Huang45, Y. P. Huang1, Z. Huang42,g, T. Hussain68, N
Hüsken25,31, W. Imoehl25, M. Irshad66,53, J. Jackson25, S. Jaeger4, S.
Janchiv29, E. Jang50, J. H. Jeong50, Q. Ji1, Q. P. Ji18, X. B. Ji1,58, X. L.
Ji1,53, Y. Y. Ji45, Z. K. Jia66,53, H. B. Jiang45, S. S. Jiang35, X. S.
Jiang1,53,58, Y. Jiang58, J. B. Jiao45, Z. Jiao21, S. Jin38, Y. Jin61, M. Q.
Jing1,58, T. Johansson70, N. Kalantar-Nayestanaki59, X. S. Kang36, R.
Kappert59, M. Kavatsyuk59, B. C. Ke75, I. K. Keshk4, A. Khoukaz63, R. Kiuchi1,
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.
Lenz31, C. Li43, C. Li39, C. H. Li35, Cheng Li66,53, D. M. Li75, F. Li1,53, G.
Li1, H. Li66,53, H. Li47, H. B. Li1,58, H. J. Li18, H. N. Li51,i, J. Q. Li4,
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, Y. G. Li42,g, 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. Libby24, A. Limphirat55, C. X. Lin54, D. X. Lin28,58, T. Lin1, B.
J. Liu1, C. X. Liu1, D. Liu17,66, F. H. Liu48, Fang Liu1, Feng Liu6, G. M.
Liu51,i, H. Liu34,j,k, H. B. Liu14, H. M. Liu1,58, Huanhuan Liu1, Huihui
Liu19, J. B. 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, T. Luo10,f, X. L. Luo1,53, X. R. Lyu58, Y. F. Lyu39, F. C.
Ma36, H. L. Ma1, L. L. Ma45, M. M. Ma1,58, Q. M. Ma1, R. Q. Ma1,58, R. T.
Ma58, X. Y. Ma1,53, Y. Ma42,g, F. E. Maas17, M. Maggiora69A,69C, S. Maldaner4,
S. Malde64, Q. A. Malik68, A. Mangoni26B, Y. J. Mao42,g, Z. P. Mao1, S.
Marcello69A,69C, Z. X. Meng61, G. Mezzadri27A, H. Miao1, T. J. Min38, R. E.
Mitchell25, X. H. Mo1,53,58, N. Yu. Muchnoi11,b, Y. Nefedov32, F. Nerling17,d,
I. B. Nikolaev11,b, Z. Ning1,53, S. Nisar9,l, Y. Niu 45, S. L. Olsen58, Q.
Ouyang1,53,58, S. Pacetti26B,26C, X. Pan10,f, Y. Pan52, A. Pathak30, M.
Pelizaeus4, H. P. Peng66,53, K. Peters12,d, J. L. Ping37, R. G. Ping1,58, S.
Plura31, S. Pogodin32, V. Prasad66,53, F. Z. Qi1, H. Qi66,53, H. R. Qi56, M.
Qi38, T. Y. Qi10,f, S. Qian1,53, W. B. Qian58, Z. Qian54, C. F. Qiao58, J. J.
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.
Su50, Y. J. Su58, G. X. Sun1, H. Sun58, H. K. Sun1, J. F. Sun18, L. Sun71, S.
S. Sun1,58, T. Sun1,58, W. Y. Sun30, X. Sun23,h, Y. J. Sun66,53, Y. Z. Sun1,
Z. T. Sun45, Y. H. Tan71, Y. X. Tan66,53, C. J. Tang49, G. Y. Tang1, J.
Tang54, L. Y Tao67, Q. T. Tao23,h, M. Tat64, J. X. Teng66,53, V. Thoren70, W.
H. Tian47, Y. Tian28,58, I. Uman57B, B. Wang1, B. L. Wang58, C. W. Wang38, D.
Y. 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, Y. J Wu28, 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. Xiao37, C. Xie38, X. H. Xie42,g, Y. Xie45, Y. G. Xie1,53, Y.
H. Xie6, Z. P. 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. Ye1,53, M. H. Ye8, J. H. Yin1, Z. Y. You54, B. X. Yu1,53,58, C.
X. Yu39, G. 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. Zhang71, Z. Y. Zhang39, 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

The singly Cabibbo suppressed decay $\Lambda^{+}_{c}\to p\eta^{\prime}$ is
measured using 4.5 $\mathrm{fb}^{-1}$ of $e^{+}e^{-}$ collision data collected
at center-of-mass energies between 4.600 and 4.699 GeV with the BESIII
detector at BEPCII. Evidence for $\Lambda^{+}_{c}\to p\eta^{\prime}$ with a
statistical significance of $3.6\sigma$ is reported with a double-tag
approach. The $\Lambda^{+}_{c}\to p\eta^{\prime}$ absolute branching fraction
is determined to be $(5.62^{+2.46}_{-2.04}\pm 0.26)\times 10^{-4}$, where the
first and second uncertainties are statistical and systematic, respectively.
Our result is consistent with the branching fraction obtained by the Belle
collaboration within the uncertainty of $1\sigma$.

##  I Introduction

The weak decays of the ground state charmed baryon $\Lambda^{+}_{c}$ play an
essential role in understanding the interplay of weak and strong interactions
in the charm region Cheng:2015iom . In addition, information about the
lightest charmed baryon provides key input for investigations of heavier
charmed baryons Sheng:2018dc  and bottom baryons Rupak:2016bb ; Detmold:2015bb
. In contrast to charmed meson decays, which are usually dominated by
factorizable amplitudes, decays of charmed baryons receive sizable non-
factorizable contributions which arise from internal $W$-emission and
$W$-exchange Chau:1986bb ; Kohara:1991bb .

The complicated physics in charmed baryon decays is described by
phenomenological model calculations, which strongly rely on experimental
results. Experimentally, some Cabibbo favored decays Zupanc:2013iki ;
Ablikim:2015flg ; Ablikim:2016mcr ; Belle:2017tfw ; Ablikim:2018czr ;
Ablikim:2020ffk  have been measured with relatively high precision, while the
singly Cabibbo suppressed (SCS) decays are limited by statistics. Recently,
measurements of SCS decay branching fractions have been carried out by the
BESIII and Belle Collaborations Ablikim:2016phi ; Ablikim:2017eta ;
Belle:2021pi0 . The SCS decay $\Lambda_{c}^{+}\to n\pi^{+}$ was observed for
the first time at BESIII Ablikim:2022SCS , and the measured branching fraction
is consistent with the SU(3) prediction Geng:2018rse  but twice as large as
that from current algebra Cheng:2018hwl .

The Belle Collaboration reported the first observation of the SCS decay
$\Lambda^{+}_{c}\to p\eta^{\prime}$ with a statistical significance of
5.4$\sigma$ and measured its decay branching fraction with respect to the
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ to be $\mathcal{B}(\Lambda^{+}_{c}\to
p\eta^{\prime})/\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})=(7.54\pm 1.32\pm
0.75)\times 10^{-3}$ Belle:2022etap . The two-body SCS decay
$\Lambda^{+}_{c}\to p\eta^{\prime}$ can proceed via the internal $W$-emission
and $W$-exchange mechanisms, with the lowest-order Feynman diagrams shown in
Fig. 1. The branching fraction of $\Lambda^{+}_{c}\to p\eta^{\prime}$ is
predicted to be in the range of $10^{-3}-10^{-4}$ Cheng:2018hwl . To improve
the knowledge of charmed baryons, more experimental studies of SCS decays are
highly desirable.

Fig. 1:  Lowest-order Feynman diagrams for $\Lambda^{+}_{c}\to p\eta^{\prime}$
mediated via (a) $W$-emission and (b) $W$-exchange mechanisms.

In this paper we report a measurement of the absolute branching fraction of
the SCS decay $\Lambda^{+}_{c}\to p\eta^{\prime}$ 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
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ data sets are above
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ threshold and provide a clean
environment with which to measure the absolute branching fractions of
$\Lambda^{+}_{c}$. The CM energy and the integrated luminosity for each energy
point are listed in Table 1 BESIII:lumi0 ; BESIII:lumi1 ; BESIII:lumi2 .
Throughout the text, the charge conjugate states are always implied.

Table 1: The CM energy and the integrated luminosity ($\mathcal{L}_{\rm int}$) for each energy point. The first and the second uncertainties are statistical and systematic, respectively. Data set | CM energy (MeV) |  $\mathcal{L}_{\rm int}$ (pb-1)
---|---|---
4600 | 4599.53 $\pm$ 0.07 $\pm$ 0.74 |  586.90 $\pm$ 0.10 $\pm$ 3.90
4612 | 4611.86 $\pm$ 0.12 $\pm$ 0.32 |  103.83 $\pm$ 0.05 $\pm$ 0.55
4628 | 4628.00 $\pm$ 0.06 $\pm$ 0.32 |  521.52 $\pm$ 0.11 $\pm$ 2.76
4641 | 4640.91 $\pm$ 0.06 $\pm$ 0.38 |  552.41 $\pm$ 0.12 $\pm$ 2.93
4661 | 4661.24 $\pm$ 0.06 $\pm$ 0.29 |  529.63 $\pm$ 0.12 $\pm$ 2.81
4682 | 4681.92 $\pm$ 0.08 $\pm$ 0.29 |  1669.31 $\pm$ 0.21 $\pm$ 8.85
4699 | 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 Yu:IPAC2016-TUYA01  in the CM energy range
from 2.0 to 4.95 GeV Ablikim:2019hff , with a peak luminosity of $1\times
10^{33}\;\text{cm}^{-2}\text{s}^{-1}$ achieved at CM energy of
$3.77\;\text{GeV}$. BESIII has collected large data samples in this energy
region Ablikim:2019hff . The cylindrical core of the BESIII detector covers
93% of the full solid angle and consists of a multilayer drift chamber (MDC)
operating with a helium-based gas mixture, 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 of
magnetic field. The solenoid is supported by an octagonal flux-return yoke
with resistive plate counter muon identification modules interleaved with
steel. The charged-particle momentum resolution at $1~{}{\rm GeV}/c$ is
$0.5\%$, and the 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. 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 .

Simulated data samples are produced with geant4-based Agostinelli:2002hh
Monte Carlo (MC) software, which describes the geometry of the BESIII detector
and simulates the performance of the 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 $\Lambda^{+}_{c}\to p\eta^{\prime}$, which
are used to determine the detection efficiencies, are generated for each
individual CM energy using the generator kkmc Jadach:2000ir  incorporating
initial-state radiation (ISR) effects and the beam energy spread. The
$\Lambda^{+}_{c}\to p\eta^{\prime}$ decay is modeled with a uniform phase-
space distribution. 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.
Particle decays are modeled with evtgen Lange:2001uf ; Ping:2008zz  using
branching fractions taken from the Particle Data Group (PDG) PDG:2020 , when
available, or otherwise estimated with lundcharm Chen:2000tv ; YANG:2014 .
Final state radiation from charged final state particles is incorporated using
photos Richter-Was:1992hxq .

##  III METHODOLOGY

A double-tag (DT) approach MarkIII:DT  is implemented to study the SCS decay
$\Lambda^{+}_{c}\to p\eta^{\prime}$. A data sample of $\bar{\Lambda}_{c}^{-}$
baryons, referred to as the single-tag (ST) sample, is reconstructed with ten
exclusive hadronic decay modes, as listed in Table 2. Events in which the
signal decay $\Lambda^{+}_{c}\to p\eta^{\prime}$ is reconstructed in the
system recoiling against the $\bar{\Lambda}_{c}^{-}$ candidates of the ST
sample are denoted as DT candidates. The branching fraction of
$\Lambda^{+}_{c}\to p\eta^{\prime}$ is determined as

$\mathcal{B}(\Lambda^{+}_{c}\to
p\eta^{\prime})=\frac{N_{\mathrm{sig}}}{\mathcal{B}_{\rm
inter}\cdot\sum_{ij}N_{ij}^{\mathrm{ST}}\cdot(\epsilon_{ij}^{\mathrm{DT}}/\epsilon_{ij}^{\mathrm{ST}})},$
(1)

where $N_{\mathrm{sig}}$ is the signal yield of the DT candidates and
$\mathcal{B}_{\rm inter}$ is the $\eta^{\prime}$ decay branching fraction
taken from the PDG PDG:2020 . 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 Single Tag Event Selections

The selection criteria of ST events are same as the Ref. Ablikim:2022SCS .
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 $z$-axis, which is the symmetry axis of the MDC. Except for
tracks from $K_{S}^{0}$ and $\bar{\Lambda}$ decays, their distances of closest
approach to the interaction point (IP) must be less than 10 cm along the
$z$-axis, and less than 1 cm in the transverse plane (referred to as a tight
track hereafter). Particle identification (PID) is implemented by combining
the measurements of $\mathrm{d}E/\mathrm{d}x$ in the MDC and the flight time
in the TOF, and each charged track is assigned a particle type of pion, kaon
or proton, according to which assignment has the highest probability.

Photon candidates are identified from showers 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. A $\pi^{0}$ candidate is
reconstructed with a photon pair within the invariant mass region (0.115,
0.150) GeV/$c^{2}$. To improve the resolution, a kinematic fit is performed by
constraining the invariant mass of the photon pair to the world average
$\pi^{0}$ mass PDG:2020 . The momentum updated by the kinematic fit is used in
further analysis.

Candidates for $K_{S}^{0}$ and $\bar{\Lambda}$ are reconstructed in their
decays to $\pi^{+}\pi^{-}$ and $\bar{p}\pi^{+}$, respectively, where the
charged tracks must have the distances of closest approach to the IP within 20
cm along the $z$-axis (referred to as a loose track hereafter). To improve the
signal purity, PID is required for the anti-proton candidate, but not for the
charged pion. The two daughter tracks are constrained to originate from a
common decay vertex, and the $\chi^{2}$ of the vertex fit is required to be
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. The
momenta of the $K_{S}^{0}$ or $\bar{\Lambda}$ candidates updated by the fit
are used in further analysis, and the invariant masses are required to be
within (0.487, 0.511) GeV/$c^{2}$ for $\pi^{+}\pi^{-}$ and (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 with invariant masses being within (1.179,
1.203) and (1.176, 1.200) GeV/$c^{2}$, respectively.

The ST $\bar{\Lambda}_{c}^{-}$ candidates are identified using the beam
constrained 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 and $E_{\bar{\Lambda}^{-}_{c}}$
and $\vec{p}_{\bar{\Lambda}^{-}_{c}}$ are the energy and momentum of the
$\bar{\Lambda}_{c}^{-}$ candidate, respectively. The $\bar{\Lambda}_{c}^{-}$
candidates are required to satisfy tag-mode dependent asymmetric $\Delta E$
requirements, listed in Table 2, which take into account the effects of ISR
and correspond to three times the resolution around the peak. If there is more
than one candidate satisfying the above requirements for a specific tag mode,
the one with the minimum $|\Delta E|$ is kept, and those with
$M_{\mathrm{BC}}\in(2.275,2.310)$ GeV/$c^{2}$ are retained for further
analysis.

Table 2: Branching fractions ($\mathcal{B}_{\mathrm{ST}}$), $\Delta E$ requirement, ST yield, and ST detection efficiency of $\Lambda^{+}_{c}\to p\eta^{\prime}$ of each tag mode for the data set 4682. The branching fractions have taken into account the intermediate particle decays, i.e. $\bar{\Lambda}\to\bar{p}\pi^{+}$, $\bar{\Sigma}^{0}\to\gamma\bar{\Lambda}$, $\bar{\Sigma}^{-}\to\bar{p}\pi^{0}$, $K_{S}^{0}\to\pi^{+}\pi^{-}$ and $\pi^{0}\to\gamma\gamma$. The uncertainty in the ST yield is statistical only.  Tag mode |  $\mathcal{B}_{\mathrm{ST}}$(%) |  $\Delta E$ (MeV) | $N_{i}^{\mathrm{ST}}$ |  $\epsilon_{i}^{\mathrm{ST}}$(%)
---|---|---|---|---
$\bar{p}K^{+}\pi^{-}$ | 6.28 | $(-34,~{}20)$ |  $17,415$ $\pm$ |  145 | 47.3
$\bar{p}K_{S}^{0}$ | 1.10 | $(-20,~{}20)$ |  $3,353$ $\pm$ |  61 | 48.1
$\bar{\Lambda}\pi^{-}$ | 0.83 | $(-20,~{}20)$ |  $2,012$ $\pm$ |  47 | 37.8
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | 4.41 | $(-30,~{}20)$ |  $4,005$ $\pm$ |  95 | 14.5
$\bar{p}K_{S}^{0}\pi^{0}$ | 1.35 | $(-30,~{}20)$ |  $1,454$ $\pm$ |  52 | 16.5
$\bar{\Lambda}\pi^{-}\pi^{0}$ | 4.48 | $(-30,~{}20)$ |  $3,576$ $\pm$ |  71 | 14.6
$\bar{p}K_{S}^{0}\pi^{+}\pi^{-}$ | 1.11 | $(-20,~{}20)$ |  $1,261$ $\pm$ |  49 | 17.7
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | 2.33 | $(-20,~{}20)$ |  $1,818$ $\pm$ |  52 | 12.3
$\bar{\Sigma}^{0}\pi^{-}$ | 0.82 | $(-20,~{}20)$ |  $1,047$ $\pm$ |  34 | 19.3
$\bar{\Sigma}^{-}\pi^{+}\pi^{-}$ | 2.29 | $(-30,~{}20)$ |  $2,275$ $\pm$ |  63 | 16.2

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 set 4682 are illustrated in Fig. 2, where clear
$\bar{\Lambda}_{c}^{-}$ signals are observed in each ST mode. No peaking
background is found with the investigation of 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 the resolution difference between data and MC simulation. The
parameters of the Gaussian function are float and determined in the fit, and
the resultant standard deviations are varied between 0.1 and 0.7 MeV/$c^{2}$
for different ST modes. The background shape is described by an ARGUS function
ARGUS:1990hfq , fixing the endpoint parameter to the corresponding
$E_{\mathrm{beam}}$. 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, and 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 $(1.0524\pm 0.0038)\times 10^{5}$,
where the uncertainty is statistical.

Fig. 2:  The $M_{\mathrm{BC}}$ distributions of the ST modes for the data set
4680. 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 Signal Reconstruction

Candidates for the signal decay $\Lambda^{+}_{c}\to p\eta^{\prime}$ are
selected with the remaining tracks recoiling against the
$\bar{\Lambda}_{c}^{-}$ candidates. The $\eta^{\prime}$ meson is reconstructed
in its two most prominent decay modes, $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$
and $\pi^{+}\pi^{-}\gamma$, where the $\eta$ meson is reconstructed with its
neutral decay modes, $i.e.$ $\eta\to 2\gamma$ and $\eta\to 3\pi^{0}$,
corresponding to $(72.12\pm 0.34)$ % PDG:2020  of $\eta$ decays.

For both $\eta^{\prime}$ decay modes, the signal side is required to include
exactly three tight tracks, which are then identified with PID to be
$p,\pi^{+},\pi^{-}$. To suppress contamination from long-lifetime particles in
the final state, the candidate events are further required to not contain any
loose tracks. Possible background from $\Lambda\to p\pi^{-}$ is rejected
completely by requiring $M_{p\pi^{-}}>1.125$ GeV/$c^{2}$, and candidate events
with $M_{\pi^{+}\pi^{-}}\in(0.490,0.510)$ GeV/$c^{2}$ are vetoed to reduce the
$K_{S}^{0}\to\pi^{+}\pi^{-}$ background by 86%.

For the reconstruction mode of $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$, to
increase the detection efficiency, the neutral decays of $\eta$ are not
reconstructed with individual final states, but considered as missing. Thus,
the $\eta$ signal is selected in the system recoiling against the ST
$\bar{\Lambda}_{c}^{-}$, $p$, $\pi^{+}$ and $\pi^{-}$. To improve the mass
resolution, we use $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}+M_{\bar{\Lambda}_{c}^{-}}-m_{\Lambda^{+}_{c}}$
instead of $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ to
reconstruct the $\eta$ signal, where $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ is the recoiling mass against the
ST $\bar{\Lambda}_{c}^{-}$, $p$, $\pi^{+}$ and $\pi^{-}$,
$M_{\bar{\Lambda}_{c}^{-}}$ is the reconstructed mass of the
$\bar{\Lambda}_{c}^{-}$ candidate, and $m_{\Lambda^{+}_{c}}$ is the world
average $\Lambda^{+}_{c}$ mass PDG:2020 . The $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ variable is required to be within
the $\eta$ mass interval $(0.533,0.561)$ GeV/$c^{2}$, corresponding to
$1\sigma$ resolution of the recoiling mass, to reject the $\Lambda^{+}_{c}\to
pK_{L}^{0}\pi^{+}\pi^{-}$ background. The recoiling mass against the ST
$\bar{\Lambda}_{c}^{-}$ is required to be within $(2.275,2.310)$ GeV$/c^{2}$.
Finally, the signal yield of $\Lambda^{+}_{c}\to p\eta^{\prime}$ is determined
by fitting the distribution of $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p)}$, which
is defined as the recoiling mass against the ST $\bar{\Lambda}_{c}^{-}$ and
$p$, as presented in Fig. 3(a).

For the reconstruction mode of $\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$, the
$\gamma$ candidates are selected from photons not assigned to ST $\pi^{0}$s,
and the one with the minimum value of $|\Delta E_{p\pi^{+}\pi^{-}\gamma}|$ is
kept for further analysis, where $\Delta
E_{p\pi^{+}\pi^{-}\gamma}=E_{p\pi^{+}\pi^{-}\gamma}-E_{\mathrm{beam}}$ and
$E_{p\pi^{+}\pi^{-}\gamma}$ is the reconstructed energy of the candidate
events. The candidate events are further required to be within $\Delta
E_{p\pi^{+}\pi^{-}\gamma}\in(-0.017,0.008)$ GeV. The background containing
extra $\pi^{0}$s is vetoed with the requirement of $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}<0.1$ GeV/$c^{2}$. Additionally,
the $\gamma$ is selected with $\alpha_{\gamma}<20^{\degree}$, where
$\alpha_{\gamma}$ is the angle between $\gamma$ and the direction of recoiling
system of ST $\bar{\Lambda}_{c}^{-}$, $p$, $\pi^{+}$ and $\pi^{-}$.
Furthermore, the invariant mass of $p\pi^{+}\pi^{-}\gamma$ is required to be
within $M_{p\pi^{+}\pi^{-}\gamma}\in(2.275,2.310)$ GeV/$c^{2}$. Finally, the
signal yield of $\Lambda^{+}_{c}\to p\eta^{\prime}$ is determined by fitting
the distribution of $M_{\pi^{+}\pi^{-}\gamma}$, as presented in Fig. 3(b).

##  VI Background analysis

The potential background events are classified into two categories: those
directly originating from continuum hadron production in the $e^{+}e^{-}$
annihilation (denoted as $q\bar{q}$ background hereafter) and those from the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ events (denoted as
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ background hereafter). The
distributions and magnitudes of $q\bar{q}$ and
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ backgrounds are estimated with the
inclusive MC samples. The main residual background sources are
$\Lambda^{+}_{c}\to pK_{L}^{0}\pi^{+}\pi^{-}$,
$\Lambda^{+}_{c}\to\Sigma^{+}\eta$ and $\Lambda^{+}_{c}\to\Sigma^{+}\omega$.
The resultant $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p)}$ and
$M_{\pi^{+}\pi^{-}\gamma}$ distributions of the accepted candidates in data
are depicted in Fig. 3. There are small peaks in the $\eta^{\prime}$ signal
regions for both decay modes. The simulated shapes, which display no peaking
background, describe the backgrounds well.

Fig. 3:  Simultaneous fit to (a) the $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p)}$
distribution in the $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ mode and (b) the
$M_{\pi^{+}\pi^{-}\gamma}$ distribution in the
$\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ mode with the combined seven data
samples. The black points with error bars are data. The red and blue dashed
lines indicate the curves for the signal and background, respectively. The
black line is the sum over all the components in the fit. The brown and green
hatched histograms for the two background components are from the inclusive MC
samples.

##  VII Branching Fraction Measurement

The branching fraction of $\Lambda^{+}_{c}\to p\eta^{\prime}$ is determined
with Eq. (1) by performing a simultaneous unbinned maximum likelihood fit on
the distributions of $M_{\rm rec(\bar{\Lambda}_{c}^{-}p)}$ and
$M_{\pi^{+}\pi^{-}\gamma}$ in the two $\eta^{\prime}$ decay modes, constrained
to the same $\mathcal{B}(\Lambda^{+}_{c}\to p\eta^{\prime})$ and taking into
account different detection efficiencies and branching fractions of
$\eta^{\prime}$. The signal shapes are modeled by the MC-simulated shapes. The
background shapes are described by second-order polynomial functions with
fixed parameters, which are obtained by fitting the corresponding
distributions of inclusive MC samples, and the background yields are floating.
The fit curves are depicted in Fig. 3. In Eq. (1), the $\mathcal{B}_{\rm
inter}$ values are (30.7$\pm$0.4)% and (29.5$\pm$0.4)% for the modes of
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and
$\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$, respectively, the ST detection
efficiency $\epsilon_{ij}^{\mathrm{ST}}$ is obtained with the same procedure
as in Ref. Ablikim:2022SCS , and the DT detection efficiency
$\epsilon_{ij}^{\mathrm{DT}}$ is derived with the signal MC samples. The
efficiencies are summarized in Table 3 and Table 4 for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and $\pi^{+}\pi^{-}\gamma$ decay modes,
respectively. The statistical significance of $\Lambda^{+}_{c}\to
p\eta^{\prime}$ is $3.6\sigma$, which is calculated from the change of the
likelihood values between the fits with and without the signal component
included, and accounting for the change in the number of degrees-of-freedom.
The branching fraction of $\Lambda^{+}_{c}\to p\eta^{\prime}$ is determined to
be $\mathcal{B}(\Lambda^{+}_{c}\to
p\eta^{\prime})=(5.62^{+2.46}_{-2.04})\times 10^{-4}$, corresponding to signal
yields of $4.9^{+3.2}_{-2.6}$ and $4.3^{+2.6}_{-2.2}$ for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and $\pi^{+}\pi^{-}\gamma$ modes, where
the uncertainties are statistical.

Table 3:  The DT detection efficiencies in percentage in the $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ mode for the ten tag modes and seven data sets at different CM energies. The statistical uncertainties are lower than 0.3%. Data set | $4600$ | $4612$ | $4628$ | $4641$ | $4661$ | $4682$ | $4699$
---|---|---|---|---|---|---|---
$\bar{p}K^{+}\pi^{-}$ | 16.1 | 14.8 | 14.1 | 14.2 | 13.5 | 13.2 | 12.9
$\bar{p}K_{S}^{0}$ | 18.6 | 15.6 | 15.7 | 14.8 | 14.5 | 14.1 | 14.3
$\bar{\Lambda}\pi^{-}$ | 15.2 | 12.8 | 12.3 | 11.7 | 11.7 | 10.9 | 10.4
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | 3.9 | 3.7 | 3.4 | 3.3 | 3.3 | 3.2 | 3.2
$\bar{p}K_{S}^{0}\pi^{0}$ | 5.3 | 4.5 | 4.6 | 4.2 | 4.5 | 3.8 | 4.0
$\bar{\Lambda}\pi^{-}\pi^{0}$ | 4.7 | 3.8 | 3.6 | 3.6 | 3.4 | 3.2 | 3.2
$\bar{p}K_{S}^{0}\pi^{+}\pi^{-}$ | 6.3 | 5.3 | 5.1 | 4.9 | 4.5 | 4.8 | 4.6
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | 3.8 | 3.2 | 3.4 | 3.1 | 2.9 | 3.2 | 2.8
$\bar{\Sigma}^{0}\pi^{-}$ | 6.3 | 5.8 | 5.7 | 5.0 | 4.4 | 4.2 | 4.5
$\bar{\Sigma}^{-}\pi^{+}\pi^{-}$ | 5.3 | 4.7 | 4.7 | 4.3 | 4.1 | 3.9 | 4.0
Table 4:  The DT detection efficiencies in percentage in the $\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ mode for ten tag modes and seven data sets at different CM energies. The statistical uncertainties are lower than 0.3%. Data set | $4600$ | $4612$ | $4628$ | $4641$ | $4661$ | $4682$ | $4699$
---|---|---|---|---|---|---|---
$\bar{p}K^{+}\pi^{-}$ | 13.7 | 13.0 | 12.4 | 12.4 | 12.1 | 11.7 | 11.5
$\bar{p}K_{S}^{0}$ | 15.4 | 14.3 | 14.0 | 14.6 | 13.4 | 12.7 | 12.5
$\bar{\Lambda}\pi^{-}$ | 13.2 | 11.0 | 10.4 | 10.6 | 10.8 | 9.9 | 9.8
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | 3.7 | 3.5 | 3.4 | 3.3 | 3.3 | 3.2 | 3.2
$\bar{p}K_{S}^{0}\pi^{0}$ | 4.2 | 4.3 | 4.5 | 4.3 | 3.8 | 4.1 | 4.0
$\bar{\Lambda}\pi^{-}\pi^{0}$ | 4.1 | 3.8 | 3.6 | 3.4 | 3.4 | 3.1 | 3.2
$\bar{p}K_{S}^{0}\pi^{+}\pi^{-}$ | 5.4 | 4.8 | 4.4 | 4.6 | 4.8 | 4.8 | 4.6
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | 3.7 | 3.4 | 3.1 | 3.0 | 3.0 | 3.0 | 3.0
$\bar{\Sigma}^{0}\pi^{-}$ | 6.9 | 5.0 | 5.4 | 5.0 | 4.5 | 5.1 | 4.5
$\bar{\Sigma}^{-}\pi^{+}\pi^{-}$ | 4.6 | 4.4 | 4.3 | 3.8 | 3.8 | 3.8 | 3.9

##  VIII Systematic Uncertainty

The systematic uncertainties for the branching fraction measurement comprise
those associated with the ST yields ($N_{ij}^{\mathrm{ST}}$), detection
efficiencies of the ST $\bar{\Lambda}_{c}^{-}$
($\epsilon_{ij}^{\mathrm{ST}}$), detection efficiencies of the DT events
($\epsilon_{ij}^{\mathrm{DT}}$) and signal yield ($N_{\mathrm{sig}}$). As the
DT technique is adopted, the systematic uncertainties originating from
reconstructing the ST side largely cancel. The systematic uncertainties are
evaluated relative to the measured branching fraction. The details are
described in the following.

The uncertainties associated with the tracking and PID efficiencies for the
proton and $\pi$ are determined with the control samples $J/\psi\to
p\bar{p}\pi^{+}\pi^{-}$ Uncer:pidTracking  and
$J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ Ablikim:2017systpi , respectively. The
systematic uncertainties for tracking and PID are both assigned to be 1.0% for
each proton and $\pi$, respectively.

The uncertainty in the reconstruction efficiency for the $\gamma$ in the
$\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decay mode is assigned to be 1.0% based
on a study with the control sample $J/\psi\to\rho^{0}\pi^{0}$ events
Ablikim:gamerr .

The uncertainty in the ST yields is 0.5%, which arises from the statistical
uncertainty and the fits to the $M_{\mathrm{BC}}$ distributions. The
uncertainty in the fitting procedure is evaluated by floating the truncation
parameter of the ARGUS function and changing the single Gaussian function to a
double Gaussian function.

The systematic uncertainty of the $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ requirement for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ decay mode is estimated by correcting the
variable $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ in the MC
samples according to the observed resolution difference between data and MC
simulation. The resolution difference is studied with the control sample
$\Lambda_{c}^{+}\to pK_{L}^{0}\pi^{+}\pi^{-}$. The change of the obtained
efficiency of the corrected MC samples from the nominal efficiency, 1.3%, is
taken as the corresponding systematic uncertainty.

The systematic uncertainty due to the $\Delta E_{p\pi^{+}\pi^{-}\gamma}$
requirement for the $\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decay mode is
studied with the control sample $\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ with
$\Sigma^{0}\to\Lambda\gamma$, which has a similar final state as the signal
process. The difference between the efficiency in MC simulation and that in
the control sample, 1.1%, is assigned to be the systematic uncertainty.

The systematic uncertainties arising from the $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ and $\alpha_{\gamma}$
requirements for the $\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decay mode are
studied with the control sample $\Lambda_{c}^{+}\to pK_{S}^{0}\pi^{0}$. The
differences of the efficiencies between data and MC simulation, 1.3% and 0.8%,
are taken as the systematic uncertainties due to the $M_{{\rm
rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ and $\alpha_{\gamma}$
requirements, respectively.

The uncertainty in the fit strategy of $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p)}$
and $M_{\pi^{+}\pi^{-}\gamma}$ is assigned to be 2.9%, which is estimated by
shifting the fit range higher by 50 MeV and varying the background shape to a
third-order polynomial function.

The systematic uncertainties due to the signal shape for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and
$\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decay modes are studied with the
control samples $\Lambda_{c}^{+}\to pK_{S}^{0}$ and
$\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ with $\Sigma^{0}\to\Lambda\gamma$,
respectively. The distributions of $M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p)}$ and
$M_{\Sigma^{0}}$ are fitted by MC shapes, with or without a convolution with a
Gaussian function. The differences of yields between the two cases, 0.6% and
0.1%, are taken as the corresponding systematic uncertainties for the two
modes, respectively.

The systematic uncertainty arising from the signal modeling is investigated by
generating a new set of signal MC $\Lambda^{+}_{c}\to p\eta^{\prime}$ events,
with proton polar angle distribution parameterized by $1+\alpha{\rm
cos}^{2}\theta$ Ablikim:alpha  with decay parameter $\alpha=\pm 1$. Here
$\theta$ is the polar angle of the proton with respect to the
$\Lambda^{+}_{c}$ in the $e^{+}e^{-}$ CM system. Comparing the differences in
detection efficiencies between the nominal and alternative samples, the
resultant uncertainties are obtained to be 1.0% for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decay mode, and 0.1% for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ mode.

The uncertainties in the branching fractions of the intermediate state decays
from the PDG PDG:2020  are 0.5% and 0.4% for the
$\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and
$\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decay modes, respectively.

According to Eq. (1), the uncertainty related to the ST efficiency mostly
cancels. However, due to different multiplicities, the ST efficiencies
estimated with the generic and signal MC samples are expected to differ from
each other slightly and result in a so called “tag bias” uncertainty. The
difference between the ST efficiencies given by the generic and signal MC
samples, 0.9%, is assigned as the corresponding uncertainty.

Table 5 summarizes the individual relative systematic uncertainties, where the
correlated systematic uncertainties are listed in the top and the uncorrelated
systematic uncertainties are listed in the bottom.

Table 5: The relative systematic uncertainties in percent for $\Lambda^{+}_{c}\to p\eta^{\prime}$ in the decay modes of $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and $\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$. Source | $\pi^{+}\pi^{-}\eta$ | $\pi^{+}\pi^{-}\gamma$
---|---|---
$p$ tracking | 1.0
$p$ PID | 1.0
$\pi$ tracking | 2.0
$\pi$ PID | 2.0
ST yield | 0.5
$M_{{\rm rec}(\bar{\Lambda}_{c}^{-}p\pi^{+}\pi^{-})}$ requirement | 1.3
Fit strategy | 2.9
Tag bias | 0.9
$\gamma$ detection | - | 1.0
$\Delta E_{p\pi^{+}\pi^{-}\gamma}$ requirement | - | 1.1
$\alpha_{\gamma}$ requirement | - | 0.8
$\mathcal{B}_{\rm inter}$ | 0.5 | 0.4
Signal shape | 0.6 | 0.1
Signal modeling | 0.1 | 1.0
Total | 4.7 | 5.0

##  IX Summary

In summary, the SCS decay $\Lambda^{+}_{c}\to p\eta^{\prime}$ is observed with
a statistical significance of $3.6\sigma$ by using $e^{+}e^{-}$ collision data
samples corresponding to a total integrated luminosity of 4.5
$\mathrm{fb}^{-1}$ collected at seven CM energies between 4.600 and 4.699 GeV
with the BESIII detector. The absolute branching fraction of
$\Lambda^{+}_{c}\to p\eta^{\prime}$ is measured to be
$(5.62^{+2.46}_{-2.04}\pm 0.26)\times 10^{-4}$, where the first and second
uncertainties are statistical and systematic, respectively. The branching
fraction measured in this work is consistent with the Belle result
Belle:2022etap  and the predictions in Refs. Sharma:1996sc ; Geng:2018rse ,
but significantly higher than that in Ref. Khanna:1994sc , as shown in Table
6. The result from this analysis provides an input to understand the dynamics
of charmed baryon decays, and helps to improve different theoretical models.

Table 6: Comparison of the measured branching fraction (in $10^{-4}$) of $\Lambda^{+}_{c}\to p\eta^{\prime}$ to theoretical predictions and the Belle result. | $\Lambda^{+}_{c}\to p\eta^{\prime}$
---|---
BESIII | $5.62^{+2.46}_{-2.04}\pm 0.26$
Belle Belle:2022etap  | $4.73\pm 0.97$
Sharma et al. Sharma:1996sc  | $4-6$
Uppal et al. Khanna:1994sc  | $0.4-2$
Geng et al. Geng:2018rse  | $12.2^{+14.3}_{-8.7}$

## ACKNOWLEDGMENT

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center and the supercomputing center of the University of Science and
Technology of China (USTC) for their strong support. 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 Nos. 11635010, 11735014, 11835012, 11935015, 11935016, 11935018,
11961141012, 12022510, 12025502, 12035009, 12035013, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265, 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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# Manifestation of the Normal Intensity Distribution Law (NIDL) in the
rovibrational emission spectrum of hydroxyl radical

Emile S. Medvedeva, Aleksander Yu. Ermilovb, and Vladimir G. Ushakova CONTACT:
E. S. Medvedev. Email<EMAIL_ADDRESS>aFederal Research Center of
Problems of Chemical Physics and Medicinal Chemistry (former Institute of
Problems of Chemical Physics), Russian Academy of Sciences, 142432
Chernogolovka, 1 Prospekt Akademika N. N. Semenova, Russian Federation; bM. V.
Lomonosov Moscow State University, 119991 Moscow, Leninskie Gory 1/3, Russian
Federation

###### Abstract

The latest experimental [Noll _et al._ Atmos. Chem. Phys. 20(2020)5269] and
theoretical [Brooke _et al._ J. Quant. Spectr. Rad. Transfer 168(2016)142]
data on the OH emission intensities are analyzed with use of the NIDL. It is
found that the calculated intensities of the $\Delta v>6$ transitions should
not be trusted. The analysis of the OH data revealed that the NIDL theory is
not applicable to the satellite bands. The effect of small reduced mass
previously discovered in H2 [Ushakov _et al._ J. Mol. Spectrosc.
399(2024)111863], causing the NIDL straight-line slope to be larger than the
one associated with the repulsive branch of the potential, is demonstrated in
OH, and the same should be true of all the diatomic hydrides. We performed _ab
initio_ calculations of the OH repulsive branch and compared it with the one
of Brooke _et al._ and the other due to Varandas and Voronin [Chem. Phys.
194(1995)91]. We found that the _ab initio_ PEF closely follows the Varandas-
Voronin potential in the repulsive region important for calculating the
overtone intensities [Medvedev, J. Chem. Phys. 137(2012)174307]. Assumption is
made that different potentials should be used to calculate the transition
frequencies and the intensities of the overtone bands for spectroscopic
databases.

###### keywords:

overtone transitions, repulsive branch, _ab initio_ calculations

##  1 Introduction

More than thirty years ago, two of us in collaboration with our beautiful
friend and colleague Aleksander Nemukhin published paper [1] in support of the
NIDL theory (see review [2] and references therein). The theory has been
verified by the available experimental and theoretical data for a number of
diatomic molecules and quasi-diatomic local vibrations in polyatomic
molecules, and it has proven to be a powerful tool to control the precision of
the calculated intensities of the overtone transitions [3, 4]. In particular,
the OH observational data of Krassovsky _et al._ [5] and Cosby and Slanger [6]
were used to demonstrate the NIDL behavior of the relative emission
intensities [7, 2].

In this paper, we use the contemporary data on hydroxyl radical [8, 9, 10] to
verify the NIDL for emission and to demonstrate the utility of the NIDL for
the analysis of the calculated overtone intensities.

We perform the _ab initio_ calculations of the PEF repulsive branch important
for calculations of the overtone intensities [2]. The calculated repulsive
branch is compared with the ones of two literature PEFs to determine which one
is more suited for calculations of the overtone intensities.

Section 2 gives a brief review of the predictions of the NIDL theory for
emission. Sections 3 and 4 provide for verifications of these predictions in
the emission spectra. Section 5 describes the _ab initio_ calculations and
their significance for the calculations of the overtone intensities. Section 6
summarize our findings.

##  2 Predictions of the NIDL theory

The NIDL theory is based on the quasi-classical approximation, which states
that, at high-enough energies, the vibrational wave function can be
represented in the form $p^{-1/2}\exp{\left(iS/\hbar\right)}$, where $p$ and
$S$ are classical momentum and action in a given vibrational state, $E_{v}$,
which is formally equivalent to $v\gg 1$. In practice, however, $v\geq 2$ is
sufficient.111Actually, even $v=0$ and 1 can be treated quasi-classically
because the NIDL formalism considers the wave functions in the complex plane
far enough from the turning points where the quasi-classical behavior is
assured, but we leave aside this issue since it is beyond the scope of the
present paper.

Another important feature of the theory is application of the Franck-Condon
principle, which states that, due to a large difference between the electron
and nuclear masses, any optical transition at frequency $\nu$ between states 1
and 2 occurs at a fixed nuclear configuration, $r^{\star}$, where the nuclear
momenta coincide, $p_{1}(r^{\star})=p_{2}(r^{\star})$ and the potentials
differ by the photon energy, $U_{1}(r^{\star})=U_{2}(r^{\star})+h\nu$.

In application to the rovibrational transitions within the ground electronic
state, where $U_{1}(r)=U_{2}(r)\equiv U(r)$, the Franck-Condon principle means
[11, Secs. 5.4-5.6] that the main contribution to the transition-dipole-moment
(TDM) integral is provided by a vicinity of point $r^{\star}$ in the complex
plane where the potential-energy function (PEF) has singularity,
$U(r)\rightarrow\infty$, see §51 in Ref. [12].

There is one and only one _physical_ singularity of $U(r)$, namely that at
$r=0$, due to the Coulomb nuclear repulsion,222This important notion means
that any model theoretical PEF must not have any other singularities affecting
the TDMs. therefore the repulsive branch of $U(r)$ plays a crucial role in
determining the overtone intensities. If we approximate the repulsive branch
with a simple exponential function, $U(r)\propto\exp{\left(-2\beta r\right)}$,
then the TDM squared333For brevity, we will call the product,
$A\lambda^{3}\propto\textrm{TDM}^{2}$, the intensity. for the overtone
transition $\left(\Delta v\equiv v^{\prime}-v^{\prime\prime}\geq 2\right)$
from the upper level $v^{\prime}$ to the lower level $v^{\prime\prime}$ obeys
the NIDL,

$\log\textrm{TDM}^{2}_{v^{\prime}v^{\prime\prime}}=\textrm{const}-a\sqrt{E_{v^{\prime}}/\omega},$
(1)

where the energy and harmonic frequency of vibration, $\omega$, are in cm-1,
and the upper level is assumed to be high, $v^{\prime}\geq 2$, which occurs in
both absorption and emission. The const is assumed to be a slow function of
$v^{\prime}$ at a given $v^{\prime\prime}$ 444But see below, Sec. 4. except
for the anomalies [2, Sec. II], which do not obey the NIDL and must be
excluded from the data fitting.

If the lower level is also high, $v^{\prime\prime}\geq 2$, which is often met
in emission, then the NIDL for the ratio of the intensities, TDM2, for two
overtone transitions starting at a common upper level takes the form (we omit
primes for brevity)

$\log\left(\frac{\textrm{TDM}_{vv_{1}}}{\textrm{TDM}_{vv_{2}}}\right)^{2}=a\left(\sqrt{\frac{E_{v_{1}}}{\omega}}-\sqrt{\frac{E_{v_{2}}}{\omega}}\right),\hskip
10.0ptv_{1}>v_{2}\geq 2,\hskip 10.0ptv-v_{1}\geq 2,$ (2)

where $a$ is the same as in Eq. (1). Here, the const disappears because the
left-hand side vanishes at $v_{1}=v_{2}$. In fact, due to approximate nature
of the NIDL, there is a small const, on the order of the statistical error,
which, however, becomes large at the anomalies; yet, the anomalies are ignored
in the NIDL plots.

Finally, if the lowest level is 0 or 1, equation (2) is modified as

$\log\left(\frac{\textrm{TDM}_{vv_{1}}}{\textrm{TDM}_{vv_{2}}}\right)^{2}=\textrm{const}+a\sqrt{\frac{E_{v_{1}}}{\omega}},\hskip
10.0ptv_{2}=0,1$ (3)

(for more details, see review [2] and references therein).

Graphically, Eqs. (1)-(3) are represented, in the respective coordinates, by
straight lines with slope $a$, which is connected to the steepness, $\beta$,
of the repulsive branch of the PEF by relation [2]

$a=\frac{\pi}{\beta
r_{\textrm{e}}\ln{10}}\sqrt{\frac{\omega}{B_{\textrm{e}}}},$ (4)

where $r_{\textrm{e}}$ is equilibrium bond length, harmonic frequency $\omega$
and rotational constant $B_{\textrm{e}}$ are in cm-1.

There are several consequences of the above equations that are easy to verify.
Equation (1) predicts the exponential decrease of the intensities with the
overtone number, of which the pace, $a$, given by Eq. (4) is inversely
proportional to the steepness of the repulsive branch of the PEF: the steeper
the PEF, the slowlier the decay. The decay rate in Eq. (4) depends solely on
the PEF, being independent of the dipole-moment function (DMF), which affects
only the const.555See, however, Sec. 4; the DMF is also responsible for the
anomalies, which drop off the NIDL line.

Equation (2) predicts that the ratio of the intensities of two lines emitted
from a common upper level, $v$, is the same for various $v$. This important
feature of the overtone transitions was first noted by Ferguson and Parkinson
[13], who even considered a possibility to use the linear DMF (“the relative
intensities in the high overtone sequences are likely to be similar to those
for a linear dipole moment”).

Finally, Eq. (4) permits direct verification of the NIDL theory by calculating
$\beta$ with the _ab initio_ methods and comparing it with the one derived
from the NIDL slope.

Here, we will use the theoretical [8] and experimental [9] data to verify the
NIDL and to derive the $\beta$ values from the NIDL slopes; then we perform
the _ab initio_ calculations to compare the above $\beta$ with the steepness
of the repulsive branches of both the present _ab initio_ PEF and the
literature PEFs.

##  3 Verification of Eqs. (1)-(3)

Figure 1: Einstein-_A_ coefficients for the main-band low-$J$ $v$-0
transitions, $v=1$-13, from Ref. [8] normalized to the frequency factor to
obtain TDM2. The NIDL line in the upper panel is plotted by least-squares
fitting of Eq. (1) to the $v=2$-8 intensities. Data at $v>8$ are unreliable.
The lower panel shows a wavy line, which testifies that the NIDL fails for the
intercombination lines.

When plotting the data in the NIDL coordinates according to Eq. (1), we
discovered that only the main-band (intra-multiplet, $\Delta F=0$) transitions
obey the NIDL. This is illustrated in Fig. 1, where the upper and lower panels
show examples of the intra- and intercombination ($\Delta F\neq 0$, satellite)
lines. The reason for this different behavior is explained by the fact that
the very existence of the satellite transitions is due to the rotational
mixing of two $X$ multiplet sub-states with different potentials. Moreover,
severe cancellation of various contributions to the TDM for the satellite
transitions takes place [14, 15, 16], hence this is a special case not covered
by the NIDL theory. We remind that the anomalies also do not obey the NIDL,
hence we have here a second reason for the NIDL to fail.

Table 1 shows the NIDL slopes for a few main-band low-$J$ transitions.666The
NIDL theory was developed for purely vibrational transitions, therefore only
low $J$ are in order to consider to minimize the effect of rotation.
Previously [7], it was found that $\beta=3.57\pm 0.22$ Å-1 from the NIDL
slope, $a=5.54\pm 0.33$, derived from the OH data by Krassovsky _et al._ [5];
a similar result was obtained in Ref. [2] from the HITRAN 2008 data. It is
seen from the table that the NIDL slopes are very close to the one cited
above.

Table 1: The NIDL slopes derived from fitting the data of Brooke16 [8] to Eq. (1). The NIDL line is drawn across the $v=2$-8 points. Line | $a$
---|---
pP1e(2.5) | $5.60\pm 0.10$
pP2e(2.5) | $5.59\pm 0.12$
qQ1e(1.5) | $5.61\pm 0.13$
qQ2e(1.5) | $5.84\pm 0.19$
rR1e(2.5) | $5.69\pm 0.17$
rR2e(2.5) | $5.74\pm 0.17$
average | $5.68\pm 0.15$
Ref. [2] | $5.22\pm 0.03^{a,b}$
| $5.77\pm 0.02^{a,c}$
aBased on the absorption data from HITRAN 2008.
bFrom the plot of oscillator strength.
cFrom the plot of oscillator strength divided by frequency.

In Fig. 1, the $v=9$ point looks like an anomaly, but in fact it manifests the
beginning of chaotic behavior of the intensities, which is also confirmed for
the other lines shown in Table 1. Therefore, the intensities at $v>8$ are
highly unreliable.

Figure 2 shows the relative intensities of the lines emitted from a common
upper level. In the upper panel (the main-band transitions), the NIDL is drawn
by the least-squares fitting of Eq. (2) to all data excluding some
anomalies.777It is difficult to exclude all the anomalies.  The lower panel
demonstrates the failure of the NIDL theory for the satellite transitions.
Indeed, the intensity ratio must be relatively insensitive to the upper level,
see Eq. (2). In the figure, vertical sets of points are seen that correspond
to transitions from various upper levels, $v$, to the same pair of the lower
ones. Within each set, in contrast to the above prediction, the ratio varies
dramatically, up to 5 orders of magnitude, which is comparable to the change
of this ratio over the full range of abscissa (about 6 orders of magnitude);
compare this with the upper panel where the variations within the vertical
sets are much less, about 1.5 order of magnitude.

Table 2 presents the NIDL slopes of Eq. (2) derived from the relative
intensities of transitions with $v_{2}\geq 2$ and $v$ up to the maximum value
of $v_{\textrm{max}}$. The average value of the NIDL slope agrees with the one
derived previously in Ref. [2] from the observational data of Krassovsky _et
al._ [5] and of Cosby and Slanger [6].

Figure 3 shows an example of the NIDL plot according to Eq. (3) for
transitions involving the lowest level $v_{2}=1$. The NIDL slopes derived for
some other low-$J$ lines are collected in Table 3.

The data for transitions involving the lowest state $v_{2}=0$ are not shown as
they contain too many anomalies that is not easy to exclude. The NIDL slope
for this kind of transitions found in Ref. [7] is $5.62\pm 0.48$.

Figure 2: The relative intensities of the low-$J$ $v$-$v_{1}$ and $v$-$v_{2}$ transitions from a common upper level $v$ to lower levels $v_{1}>v_{2}\geq 2$, $v\geq v_{1}+2$. Circles, data from Brooke16 [8]; crosses, the anomalies partly excluded${}^{\textrm{\ref{ft_an}}}$ from the NIDL plot of Eq. (2); squares, the observational data from Noll20 [9]. The upper panel shows the main-band transitions, the lower the satellite transitions not observed in Noll20; for the latter, the NIDL fails because the variations along the ordinate at a given abscissa are huge. Figure 3: The same as in Fig. 2 with $v_{2}=1$. Table 2: The NIDL slopes derived from fitting the data of Brooke16 [8] to Eq. (2). Line | $a$ | $v_{\textrm{max}}$ | No.⋆
---|---|---|---
pP1e(2.5) | $5.72\pm 0.29$ | 9 | 7
pP2e(2.5) | $5.76\pm 0.28$ | 9 | 5
qQ1e(1.5) | $5.73\pm 0.29$ | 9 | 7
qQ2e(1.5) | $5.76\pm 0.29$ | 9 | 7
rR1e(1.5) | $5.76\pm 0.32$ | 9 | 0
rR2e(1.5) | $5.82\pm 0.33$ | 9 | 0
rR1e(2.5) | $5.81\pm 0.35$ | 9 | 1
rR2e(2.5) | $5.88\pm 0.38$ | 9 | 7
average | $5.78\pm 0.32$ |  |
Ref. [7] | $5.54\pm 0.33$ |  |
Ref. [2] | $5.36\pm 0.22$ |  |
⋆The number of the Noll20 [9]
experimental points in the plot.
Table 3: The NIDL slopes derived from fitting the data of Brooke16 [8] to Eq. (3), $v_{2}=1$. Line | $a$ | $v_{\textrm{max}}$ | No.⋆
---|---|---|---
pP1e(2.5) | $5.18\pm 0.29$ | 10 | 1
pP2e(2.5) | $5.22\pm 0.28$ | 10 | 1
qQ1e(1.5) | $5.22\pm 0.29$ | 10 | 1
qQ2e(1.5) | $5.17\pm 0.29$ | 10 | 1
rR1e(1.5) | $5.23\pm 0.32$ | 10 | 0
rR2e(1.5) | $5.27\pm 0.33$ | 10 | 0
rR1e(2.5) | $5.25\pm 0.35$ | 11 | 1
rR2e(2.5) | $5.30\pm 0.38$ | 10 | 1
average | $5.22\pm 0.32$ |  |
Ref. [7] | $5.88\pm 0.31$ |  |
Ref. [2] | $5.34\pm 0.27$ |  |
⋆The number of the Noll20 [9]
experimental points in the plot.

##  4 Verification of Eq. (4)

Figure 4: The repulsive branches of two PEFs with linear fits, which give
$\beta=4.72\pm 0.01$ Å-1 for the Brooke16 [8] PEF and $4.05\pm 0.02$ Å-1 for
the Varandas-Voronin95 [17] PEF.

Thus, we have got the NIDL slopes in the range of 5.22-5.88, which results in
the $\beta$ values in the range of (3.37-3.79) Å-1. Taking the largest
statistical error of $\delta a=\pm 0.48$, we obtain
$\delta\beta=(\beta/a)\delta a=\pm 0.35$ Å-1. This results will be compared
with the available data on the repulsive branch of PEF.

The PEF of Brooke16 is based on the RKR points and is presented in the tabular
form in Supplementary material to Ref. [8]. One more PEF was developed by
Varandas and Voronin in an analytic form [17]. The Varandas-Voronin95 [17]
potential was constructed so as to reproduce the asymptotic united- and
separated-atoms limits. Figure 4 shows the repulsive branches of both these
potentials. The boudaries of the repulsive branch are determined as follows:
it must extend from the left turning point of the $v=2$ level (0.67 Å) down to
a point $r^{\star}$ where the PEF reaches the value of
$D_{\textrm{e}}\left(v_{\textrm{max}}/2\right)^{2}$ where $v_{\textrm{max}}$
is the desirable upper level [2]. Taking $v_{\textrm{max}}=10$, we obtain
$r^{\star}=0.34$ Å for the Brooke16 PEF and 0.3 Å for the Varandas-Voronin95
PEF.

The $\beta$ values obtained from the linear fits to both the PEFs are
indicated in the figure caption, they are significantly larger that the above
estimates. Inserting them into Eq. (4), we obtain that the NIDLs associated
with these two PEF’s repulsive branches must have the slopes of $a=4.19\pm
0.01$ and $4.89\pm 0.02$, respectively, which are appreciably lower than those
obtained from the NIDL plots shown in Tables 1-3. Thus we conclude that _Eq. (
4) fails_ in the present case.

We have already encountered such a situation in molecular hydrogen [18], where
it was found that the prefactor in the quasi-classical expression for the TDM
is NOT a slow function of the vibrational quantum numbers, as was the case in
CO [19] or PN [4]. It was shown that this is due to the small reduced mass of
H2, which entails relatively small actions (on the order of 20 in the $\hbar$
units) as compared to heavier molecules ($\sim 1500$). OH also has a small
mass, hence, the prefactor also contributes to the NIDL slope.

In summary, _in the H-containing diatomics, the NIDL slope derived from the
calculated and/or observed intensities will always be larger than the one
predicted from the steepness of the repulsive branch by Eq. ( 4)_.

##  5 The _ab initio_ calculations of the repulsive branch

In Fig. 4, it is seen that the repulsive branches of the two PEFs are
essentially different. Since the steepness of the repulsive branch affects the
overtone intensities [2], it is interesting to learn which one of these PEFs
has correct $\beta$.

To this end, we performed _ab initio_ calculations of the PEFs of the
$X^{2}\Pi$ and $A^{2}\Sigma^{+}$ electronic states by the MRDCI approach. The
active space of the MOs is prepared by the CASSCF method with the state
averaging technique888Averaging over two $\Pi$ and one $\Sigma$ states. for
one double occupied core MO and 7 electrons distributed over 9 active MOs. The
total amount of configurations in MRDCI are about 6 millions for each of the
$X$ and $A$ states. The aug-cc-pV5Z basis set is used at the O and H atoms.
All calculations are carried out by GAMESS-US program package [20].

The results are shown in Fig. 4.999The _ab initio_ data for the repulsive
branch of the $X^{2}\Pi$ state are given in Supplementary material. It is seen
that the _ab initio_ points perfectly follow the Varandas-Voronin95 PEF. This
result is very significant because the analytical results of Varandas and
Voronin are confirmed by the first-principles calculation. From the point of
view of the NIDL theory, this means that the analytical Varandas-Voronin95 PEF
is more suited for calculations of the overtone intensities than the point-
wise Brooke16 one since it has correct repulsive branch. Therefore, the
Varandas-Voronin95 potential is a good candidate for calculations of the
overtone intensities. It should be noted that the Brooke16 PEF is perfectly
adjusted to describe the line positions. However, it cannot be excluded that
the best description of the overtone intensities will be reached with a
different PEF.

##  6 Conclusions

With use of the NIDL, we performed the analysis of the intensities calculated
by Brooke _et al._ [8] and found that the intensities of the $\Delta v>6$
transitions should not be trusted, in accord with our preliminary estimates
[21].

The analysis of the OH data revealed one more limitation to the NIDL theory,
in addition to the anomalies. Namely, the theory is not applicable to the
satellite bands.

The effect of the small reduced mass, previously discovered in H2 [18],
causing the NIDL straight-line slope to be larger than the one associated with
the repulsive branch of the potential, is demonstrated for OH as well. The
same should be true of all the diatomic hydrides, X-H.

Finally, we performed _ab initio_ calculations of the OH repulsive branch and
compared it with two OH potentials, one (point-wise, spline-interpolated) PEF
of Brooke _et al._ [8] and the other (analytical) PEF due to Varandas and
Voronin [17]. We found that the _ab initio_ PEF closely follows the Varandas-
Voronin95 potential in the repulsive region, which is not surprising since the
latter was specially constructed to correctly describe both asymptotic limits,
in particular the united-atom limit so important in determination of the
overtone intensities [2]. On the other hand, the Brooke16 potential is
perfectly suited to describe the line positions. Therefore, we dare to assume
that, when selecting data for the spectroscopic databases like HITRAN, HITEMP,
_etc._ , different potentials can be used to calculate transition frequencies
and transition intensities.

## Acknowledgement

This work was performed under a “FRC of PCP and MC RAS” state task, state
registration number 124013000760-0.

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