#  Measurement of the absolute branching fraction of the inclusive decay
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$

M. Ablikim1, M. N. Achasov12,b, P. Adlarson72, M. Albrecht4, R. Aliberti33, A.
Amoroso71A,71C, M. R. An37, Q. An68,55, Y. Bai54, O. Bakina34, R. Baldini
Ferroli27A, I. Balossino28A, Y. Ban44,g, V. Batozskaya1,42, D. Becker33, K.
Begzsuren30, N. Berger33, M. Bertani27A, D. Bettoni28A, F. Bianchi71A,71C, E.
Bianco71A,71C, J. Bloms65, A. Bortone71A,71C, I. Boyko34, R. A. Briere5, A.
Brueggemann65, H. Cai73, X. Cai1,55, A. Calcaterra27A, G. F. Cao1,60, N.
Cao1,60, S. A. Cetin59A, J. F. Chang1,55, W. L. Chang1,60, G. R. Che41, G.
Chelkov34,a, C. Chen41, Chao Chen52, G. Chen1, H. S. Chen1,60, M. L.
Chen1,55,60, S. J. Chen40, S. M. Chen58, T. Chen1,60, X. R. Chen29,60, X. T.
Chen1,60, Y. B. Chen1,55, Z. J. Chen24,h, W. S. Cheng71C, S. K. Choi 52, X.
Chu41, G. Cibinetto28A, F. Cossio71C, J. J. Cui47, H. L. Dai1,55, J. P. Dai76,
A. Dbeyssi18, R. E. de Boer4, D. Dedovich34, Z. Y. Deng1, A. Denig33, I.
Denysenko34, M. Destefanis71A,71C, F. De Mori71A,71C, Y. Ding32, Y. Ding38, J.
Dong1,55, L. Y. Dong1,60, M. Y. Dong1,55,60, X. Dong73, S. X. Du78, Z. H.
Duan40, P. Egorov34,a, Y. L. Fan73, J. Fang1,55, S. S. Fang1,60, W. X. Fang1,
Y. Fang1, R. Farinelli28A, L. Fava71B,71C, F. Feldbauer4, G. Felici27A, C. Q.
Feng68,55, J. H. Feng56, K Fischer66, M. Fritsch4, C. Fritzsch65, C. D. Fu1,
H. Gao60, Y. N. Gao44,g, Yang Gao68,55, S. Garbolino71C, I. Garzia28A,28B, P.
T. Ge73, Z. W. Ge40, C. Geng56, E. M. Gersabeck64, A Gilman66, K. Goetzen13,
L. Gong38, W. X. Gong1,55, W. Gradl33, M. Greco71A,71C, L. M. Gu40, M. H.
Gu1,55, Y. T. Gu15, C. Y Guan1,60, A. Q. Guo29,60, L. B. Guo39, R. P. Guo46,
Y. P. Guo11,f, A. Guskov34,a, W. Y. Han37, X. Q. Hao19, F. A. Harris62, K. K.
He52, K. L. He1,60, F. H. Heinsius4, C. H. Heinz33, Y. K. Heng1,55,60, C.
Herold57, G. Y. Hou1,60, Y. R. Hou60, Z. L. Hou1, H. M. Hu1,60, J. F. Hu53,i,
T. Hu1,55,60, Y. Hu1, G. S. Huang68,55, K. X. Huang56, L. Q. Huang29,60, X. T.
Huang47, Y. P. Huang1, Z. Huang44,g, T. Hussain70, N Hüsken26,33, W. Imoehl26,
M. Irshad68,55, J. Jackson26, S. Jaeger4, S. Janchiv30, E. Jang52, J. H.
Jeong52, Q. Ji1, Q. P. Ji19, X. B. Ji1,60, X. L. Ji1,55, Y. Y. Ji47, Z. K.
Jia68,55, P. C. Jiang44,g, S. S. Jiang37, X. S. Jiang1,55,60, Y. Jiang60, J.
B. Jiao47, Z. Jiao22, S. Jin40, Y. Jin63, M. Q. Jing1,60, T. Johansson72, S.
Kabana31, N. Kalantar-Nayestanaki61, X. L. Kang9, X. S. Kang38, R. Kappert61,
M. Kavatsyuk61, B. C. Ke78, I. K. Keshk4, A. Khoukaz65, R. Kiuchi1, R.
Kliemt13, L. Koch35, O. B. Kolcu59A, B. Kopf4, M. Kuemmel4, M. Kuessner4, A.
Kupsc42,72, W. Kühn35, J. J. Lane64, J. S. Lange35, P. Larin18, A. Lavania25,
L. Lavezzi71A,71C, T. T. Lei68,k, Z. H. Lei68,55, H. Leithoff33, M.
Lellmann33, T. Lenz33, C. Li41, C. Li45, C. H. Li37, Cheng Li68,55, D. M.
Li78, F. Li1,55, G. Li1, H. Li68,55, H. B. Li1,60, H. J. Li19, H. N. Li53,i,
Hui Li41, J. Q. Li4, J. S. Li56, J. W. Li47, Ke Li1, L. J Li1,60, L. K. Li1,
Lei Li3, M. H. Li41, P. R. Li36,j,k, S. X. Li11, S. Y. Li58, T. Li47, W. D.
Li1,60, W. G. Li1, X. H. Li68,55, X. L. Li47, Xiaoyu Li1,60, Y. G. Li44,g, Z.
X. Li15, Z. Y. Li56, C. Liang40, H. Liang32, H. Liang1,60, H. Liang68,55, Y.
F. Liang51, Y. T. Liang29,60, G. R. Liao14, L. Z. Liao47, J. Libby25, A.
Limphirat57, D. X. Lin29,60, T. Lin1, B. J. Liu1, C. Liu32, C. X. Liu1, D.
Liu18,68, F. H. Liu50, Fang Liu1, Feng Liu6, G. M. Liu53,i, H. Liu36,j,k, H.
B. Liu15, H. M. Liu1,60, Huanhuan Liu1, Huihui Liu20, J. B. Liu68,55, J. L.
Liu69, J. Y. Liu1,60, K. Liu1, K. Y. Liu38, Ke Liu21, L. Liu68,55, L. C.
Liu21, Lu Liu41, M. H. Liu11,f, P. L. Liu1, Q. Liu60, S. B. Liu68,55, T.
Liu11,f, W. K. Liu41, W. M. Liu68,55, X. Liu36,j,k, Y. Liu36,j,k, Y. B. Liu41,
Z. A. Liu1,55,60, Z. Q. Liu47, X. C. Lou1,55,60, F. X. Lu56, H. J. Lu22, J. G.
Lu1,55, X. L. Lu1, Y. Lu7, Y. P. Lu1,55, Z. H. Lu1,60, C. L. Luo39, M. X.
Luo77, T. Luo11,f, X. L. Luo1,55, X. R. Lyu60, Y. F. Lyu41, F. C. Ma38, H. L.
Ma1, L. L. Ma47, M. M. Ma1,60, Q. M. Ma1, R. Q. Ma1,60, R. T. Ma60, X. Y.
Ma1,55, Y. Ma44,g, F. E. Maas18, M. Maggiora71A,71C, S. Maldaner4, S. Malde66,
Q. A. Malik70, A. Mangoni27B, Y. J. Mao44,g, Z. P. Mao1, S. Marcello71A,71C,
Z. X. Meng63, J. G. Messchendorp13,61, G. Mezzadri28A, H. Miao1,60, T. J.
Min40, R. E. Mitchell26, X. H. Mo1,55,60, N. Yu. Muchnoi12,b, Y. Nefedov34, F.
Nerling18,d, I. B. Nikolaev12,b, Z. Ning1,55, S. Nisar10,l, Y. Niu 47, S. L.
Olsen60, Q. Ouyang1,55,60, S. Pacetti27B,27C, X. Pan52, Y. Pan54, A. Pathak32,
Y. P. Pei68,55, M. Pelizaeus4, H. P. Peng68,55, K. Peters13,d, J. L. Ping39,
R. G. Ping1,60, S. Plura33, S. Pogodin34, V. Prasad68,55, F. Z. Qi1, H.
Qi68,55, H. R. Qi58, M. Qi40, T. Y. Qi11,f, S. Qian1,55, W. B. Qian60, Z.
Qian56, C. F. Qiao60, J. J. Qin69, L. Q. Qin14, X. P. Qin11,f, X. S. Qin47, Z.
H. Qin1,55, J. F. Qiu1, S. Q. Qu58, K. H. Rashid70, C. F. Redmer33, K. J.
Ren37, A. Rivetti71C, V. Rodin61, M. Rolo71C, G. Rong1,60, Ch. Rosner18, S. N.
Ruan41, A. Sarantsev34,c, Y. Schelhaas33, C. Schnier4, K. Schoenning72, M.
Scodeggio28A,28B, K. Y. Shan11,f, W. Shan23, X. Y. Shan68,55, J. F.
Shangguan52, L. G. Shao1,60, M. Shao68,55, C. P. Shen11,f, H. F. Shen1,60, W.
H. Shen60, X. Y. Shen1,60, B. A. Shi60, H. C. Shi68,55, J. Y. Shi1, Q. Q.
Shi52, R. S. Shi1,60, X. Shi1,55, J. J. Song19, W. M. Song32,1, Y. X.
Song44,g, S. Sosio71A,71C, S. Spataro71A,71C, F. Stieler33, P. P. Su52, Y. J.
Su60, G. X. Sun1, H. Sun60, H. K. Sun1, J. F. Sun19, L. Sun73, S. S. Sun1,60,
T. Sun1,60, W. Y. Sun32, Y. J. Sun68,55, Y. Z. Sun1, Z. T. Sun47, Y. X.
Tan68,55, C. J. Tang51, G. Y. Tang1, J. Tang56, Y. A. Tang73, L. Y Tao69, Q.
T. Tao24,h, M. Tat66, J. X. Teng68,55, V. Thoren72, W. H. Tian49, Y.
Tian29,60, I. Uman59B, B. Wang1, B. Wang68,55, B. L. Wang60, C. W. Wang40, D.
Y. Wang44,g, F. Wang69, H. J. Wang36,j,k, H. P. Wang1,60, K. Wang1,55, L. L.
Wang1, M. Wang47, Meng Wang1,60, S. Wang14, S. Wang11,f, T. Wang11,f, T. J.
Wang41, W. Wang56, W. H. Wang73, W. P. Wang68,55, X. Wang44,g, X. F.
Wang36,j,k, X. L. Wang11,f, Y. Wang58, Y. D. Wang43, Y. F. Wang1,55,60, Y. H.
Wang45, Y. Q. Wang1, Yaqian Wang17,1, Z. Wang1,55, Z. Y. Wang1,60, Ziyi
Wang60, D. H. Wei14, F. Weidner65, S. P. Wen1, D. J. White64, U. Wiedner4, G.
Wilkinson66, M. Wolke72, L. Wollenberg4, J. F. Wu1,60, L. H. Wu1, L. J.
Wu1,60, X. Wu11,f, X. H. Wu32, Y. Wu68, Y. J Wu29, Z. Wu1,55, L. Xia68,55, T.
Xiang44,g, D. Xiao36,j,k, G. Y. Xiao40, H. Xiao11,f, S. Y. Xiao1, Y. L.
Xiao11,f, Z. J. Xiao39, C. Xie40, X. H. Xie44,g, Y. Xie47, Y. G. Xie1,55, Y.
H. Xie6, Z. P. Xie68,55, T. Y. Xing1,60, C. F. Xu1,60, C. J. Xu56, G. F. Xu1,
H. Y. Xu63, Q. J. Xu16, X. P. Xu52, Y. C. Xu75, Z. P. Xu40, F. Yan11,f, L.
Yan11,f, W. B. Yan68,55, W. C. Yan78, H. J. Yang48,e, H. L. Yang32, H. X.
Yang1, Tao Yang1, Y. F. Yang41, Y. X. Yang1,60, Yifan Yang1,60, M. Ye1,55, M.
H. Ye8, J. H. Yin1, Z. Y. You56, B. X. Yu1,55,60, C. X. Yu41, G. Yu1,60, T.
Yu69, X. D. Yu44,g, C. Z. Yuan1,60, L. Yuan2, S. C. Yuan1, X. Q. Yuan1, Y.
Yuan1,60, Z. Y. Yuan56, C. X. Yue37, A. A. Zafar70, F. R. Zeng47, X. Zeng6, Y.
Zeng24,h, X. Y. Zhai32, Y. H. Zhan56, A. Q. Zhang1,60, B. L. Zhang1,60, B. X.
Zhang1, D. H. Zhang41, G. Y. Zhang19, H. Zhang68, H. H. Zhang56, H. H.
Zhang32, H. Q. Zhang1,55,60, H. Y. Zhang1,55, J. J. Zhang49, J. L. Zhang74, J.
Q. Zhang39, J. W. Zhang1,55,60, J. X. Zhang36,j,k, J. Y. Zhang1, J. Z.
Zhang1,60, Jianyu Zhang1,60, Jiawei Zhang1,60, L. M. Zhang58, L. Q. Zhang56,
Lei Zhang40, P. Zhang1, Q. Y. Zhang37,78, Shuihan Zhang1,60, Shulei Zhang24,h,
X. D. Zhang43, X. M. Zhang1, X. Y. Zhang47, X. Y. Zhang52, Y. Zhang66, Y. T.
Zhang78, Y. H. Zhang1,55, Yan Zhang68,55, Yao Zhang1, Z. H. Zhang1, Z. L.
Zhang32, Z. Y. Zhang41, Z. Y. Zhang73, G. Zhao1, J. Zhao37, J. Y. Zhao1,60, J.
Z. Zhao1,55, Lei Zhao68,55, Ling Zhao1, M. G. Zhao41, S. J. Zhao78, Y. B.
Zhao1,55, Y. X. Zhao29,60, Z. G. Zhao68,55, A. Zhemchugov34,a, B. Zheng69, J.
P. Zheng1,55, Y. H. Zheng60, B. Zhong39, C. Zhong69, X. Zhong56, H. Zhou47, L.
P. Zhou1,60, X. Zhou73, X. K. Zhou60, X. R. Zhou68,55, X. Y. Zhou37, Y. Z.
Zhou11,f, J. Zhu41, K. Zhu1, K. J. Zhu1,55,60, L. X. Zhu60, S. H. Zhu67, S. Q.
Zhu40, W. J. Zhu11,f, Y. C. Zhu68,55, Z. A. Zhu1,60, J. H. Zou1, J. Zu68,55
(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 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
11 Fudan University, Shanghai 200433, People’s Republic of China
12 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
21 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
22 Huangshan College, Huangshan 245000, People’s Republic of China
23 Hunan Normal University, Changsha 410081, People’s Republic of China
24 Hunan University, Changsha 410082, People’s Republic of China
25 Indian Institute of Technology Madras, Chennai 600036, India
26 Indiana University, Bloomington, Indiana 47405, USA
27 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
28 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
29 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
30 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
31 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D,
Arica, Chile
32 Jilin University, Changchun 130012, People’s Republic of China
33 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
34 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
35 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
36 Lanzhou University, Lanzhou 730000, People’s Republic of China
37 Liaoning Normal University, Dalian 116029, People’s Republic of China
38 Liaoning University, Shenyang 110036, People’s Republic of China
39 Nanjing Normal University, Nanjing 210023, People’s Republic of China
40 Nanjing University, Nanjing 210093, People’s Republic of China
41 Nankai University, Tianjin 300071, People’s Republic of China
42 National Centre for Nuclear Research, Warsaw 02-093, Poland
43 North China Electric Power University, Beijing 102206, People’s Republic of
China
44 Peking University, Beijing 100871, People’s Republic of China
45 Qufu Normal University, Qufu 273165, People’s Republic of China
46 Shandong Normal University, Jinan 250014, People’s Republic of China
47 Shandong University, Jinan 250100, People’s Republic of China
48 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
49 Shanxi Normal University, Linfen 041004, People’s Republic of China
50 Shanxi University, Taiyuan 030006, People’s Republic of China
51 Sichuan University, Chengdu 610064, People’s Republic of China
52 Soochow University, Suzhou 215006, People’s Republic of China
53 South China Normal University, Guangzhou 510006, People’s Republic of China
54 Southeast University, Nanjing 211100, People’s Republic of China
55 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
56 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
57 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
58 Tsinghua University, Beijing 100084, People’s Republic of China
59 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
60 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
61 University of Groningen, NL-9747 AA Groningen, The Netherlands
62 University of Hawaii, Honolulu, Hawaii 96822, USA
63 University of Jinan, Jinan 250022, People’s Republic of China
64 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
65 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
66 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
67 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
68 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
69 University of South China, Hengyang 421001, People’s Republic of China
70 University of the Punjab, Lahore-54590, Pakistan
71 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
72 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
73 Wuhan University, Wuhan 430072, People’s Republic of China
74 Xinyang Normal University, Xinyang 464000, People’s Republic of China
75 Yantai University, Yantai 264005, People’s Republic of China
76 Yunnan University, Kunming 650500, People’s Republic of China
77 Zhejiang University, Hangzhou 310027, People’s Republic of China
78 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

Using an $e^{+}e^{-}$ collision data sample with a total integrated luminosity
of $3.19$ fb-1 collected with the BESIII detector at a center-of-mass energy
of 4.178 GeV, the branching fraction of the inclusive decay of the $D^{+}_{s}$
meson to final states including at least three charged pions is measured for
the first time to be ${\cal
B}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)=(32.81\pm 0.35_{\rm
stat}\pm{0.63_{\rm syst}})\%$. In this measurement the charged pions from
$K^{0}_{\scriptscriptstyle S}$ meson decays are excluded. The partial
branching fractions of $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ are also
measured as a function of the $\pi^{+}\pi^{+}\pi^{-}$ invariant mass.

##  I Introduction

Lepton flavor universality (LFU) is a very hot topic in the field of particle
physics. Recent reports from several high energy experiments studying $B$
meson semileptonic decays of the form $B\rightarrow
D^{(*)}{\ell}^{+}\nu_{\ell}$ ($\ell=e$, $\mu$, or $\tau$) suggest tensions
with Standard Model based predictions hflav , and have drawn attention from
both experimentalists and theorists. Among these results, the LHCb measurement
of the ratio of the branching fractions (BFs) of the $B^{0}$ meson
semileptonic decays, ${\cal R}(D^{*-})\equiv{\cal B}(B^{0}\rightarrow
D^{*-}\tau^{+}\nu_{\tau})/{\cal B}(B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu})$,
has attracted a lot of attention. The ${\cal R}(D^{*-})$ measurement with the
$\tau$ lepton reconstructed from three prongs, i.e. charged pions
($\tau^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$, where $X$ stands for all
possible particle combinations) rdstlhcbprl , based on a 3 $\mbox{\,fb}^{-1}$
data sample of $pp$ collisions, has one of the smallest statistical
uncertainties. However, it is systematically limited and one of the limiting
systematic uncertainty is associated with the background from
$B^{0}\rightarrow D^{*-}D_{s}^{+}(X)$, followed by
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decays. In the LHCb measurement,
the total systematic uncertainty is 9.1%, out of which 2.5% is attributed to
the “$D^{+}_{s}\rightarrow 3\pi X$ decay model” for these background events
rdstlhcbprl . Here, $3\pi\equiv\pi^{+}\pi^{+}\pi^{-}$. A better understanding
of the $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decay dynamics and of the
related $\pi^{+}\pi^{+}\pi^{-}$-system kinematics would directly enhance the
sensitivity of the ${\cal R}(D^{*-})$ measurement at LHCb, thus becoming more
and more important in view of the steady increase of the integrated luminosity
collected by the experiment. Similarly, the measurement of ${\cal R}(\mathchar
28931\relax_{c}^{+})\equiv{\cal B}(\mathchar
28931\relax_{b}^{0}\rightarrow\mathchar
28931\relax_{c}^{+}\tau^{-}\bar{\nu}_{\tau})/{\cal B}(\mathchar
28931\relax_{b}^{0}\rightarrow\mathchar
28931\relax_{c}^{+}\mu^{-}\bar{\nu}_{\mu})$ at LHCb would strongly benefit
from the knowledge on the $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decay
as input LHCb:2022piu .

Table 1: Major exclusive decay modes that contribute to the inclusive decay of $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$, along with the known BFs. The modes with BFs of large uncertainties ($K^{+}\eta^{\prime}$, $\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}\pi^{0}$ (without contributions from $\eta^{\prime}/\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$), etc.) are not shown. None of the charged pions in the $3\pi$ system comes from $K^{0}_{\scriptscriptstyle S}$ meson decays. The values of ${\cal B}({\cal M}\rightarrow\pi^{+}\pi^{-}X),\,{\cal M}=\eta,\omega,\eta^{\prime},\phi$ and ${\cal B}(\tau^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)$ are estimated by combining all known exclusive decays that contribute pdg .  Decay mode |  ${\cal B}$ (%) | Note | Ref.
---|---|---|---
$D^{+}_{s}\rightarrow\eta^{\prime}\rho^{+}$ | $4.6\pm 1.2$ | Scaled by ${\cal B}(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}X)\sim 80$% | pdg ; besiii_etaprrhop
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}\eta$ | $3.1\pm 0.2$ | — | ds2eta3pi
$D^{+}_{s}\rightarrow\eta^{\prime}\pi^{+}$ | $3.2\pm 0.2$ | Scaled by ${\cal B}(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}X)\sim 80$% | pdg
$D^{+}_{s}\rightarrow\eta\pi^{+}\pi^{0}$ | $2.6\pm 0.1$ | Scaled by ${\cal B}(\eta\rightarrow\pi^{+}\pi^{-}X)\sim 27$% | pdg
$D^{+}_{s}\rightarrow\omega\pi^{+}\pi^{0}$ | $2.5\pm 0.6$ | Scaled by ${\cal B}(\omega\rightarrow\pi^{+}\pi^{-}X)\sim 91$% | pdg
$D^{+}_{s}\rightarrow\phi a_{1}(1260)^{+}(\rightarrow\pi^{+}\pi^{+}\pi^{-})$ | $1.2\pm 0.1$ |  | BESIIIkkpipipi
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}\pi^{+}\pi^{-}$ nonresonant | $0.14\pm 0.02$ |  | BESIIIkkpipipi
$D^{+}_{s}\rightarrow\eta\pi^{+}$ | $0.46\pm 0.03$ | Scaled by ${\cal B}(\eta\rightarrow\pi^{+}\pi^{-}X)\sim 27$% | pdg
$D^{+}_{s}\rightarrow\omega\pi^{+}$ | $0.17\pm 0.03$ | Scaled by ${\cal B}(\omega\rightarrow\pi^{+}\pi^{-}X)\sim 91$% | pdg
$D^{+}_{s}\rightarrow\omega\pi^{+}\pi^{+}\pi^{-}$ | $1.6\pm 0.5$ | — | pdg
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ | $1.08\pm 0.04$ | — | pdg
$D^{+}_{s}\rightarrow\phi\pi^{+}$ | $0.72\pm 0.03$ | Scaled by ${\cal B}(\phi\rightarrow\pi^{+}\pi^{-}X)\sim 15.6$% | BESIIIkkpi
$D^{+}_{s}\rightarrow\phi\rho^{+}$ | $0.97\pm 0.05$ | Scaled by ${\cal B}(\phi\rightarrow\pi^{+}\pi^{-}X)\sim 15.6$% | BESIIIkkpipi0
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ | $0.79\pm 0.08$ | — | pdg
$D^{+}_{s}\rightarrow\tau^{+}\nu_{\tau}$ | $0.72\pm 0.01$ | Scaled by ${\cal B}(\tau^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)\sim 13.5$% | pdg ; BESIIItaunu
$D^{+}_{s}\rightarrow K^{0}\pi^{+}\pi^{+}\pi^{-}$ | $0.6\pm 0.2$ | $2\times{\cal B}(D^{+}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{+}\pi^{+}\pi^{-})$ | pdg
$D^{+}_{s}\rightarrow\eta^{\prime}e^{+}\nu_{e}$ | $0.10\pm 0.01$ | Scaled by ${\cal B}(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta(\rightarrow\pi^{+}\pi^{-}X))\sim 12$% | pdg
$D^{+}_{s}\rightarrow\eta^{\prime}\mu^{+}\nu_{\mu}$ | $0.10\pm 0.01$ | Same as above by assuming LFU | pdg
Sum | $24.7\pm 1.5\phantom{0}$ |  |

The inclusive decay $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ has never
been studied directly in experiments. Table 1 lists the BFs of major exclusive
$D^{+}_{s}$ channels with at least three charged pions of
$\pi^{+}\pi^{+}\pi^{-}$ in the final states that have been measured with
relatively good precision to date pdg ; ds2eta3pi ; BESIIIkkpipipi ;
BESIIIkkpi ; BESIIIkkpipi0 ; BESIIItaunu . The decays
$D^{+}_{s}\rightarrow\eta^{(\prime)}\pi^{+}(X)$ make up about half of the
contributions. Measurements on the inclusive decay rate of $D^{+}_{s}$ into
into final states with at least one combination of $\pi^{+}\pi^{+}\pi^{-}$,
$\cal B$$(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)$, will shed light on yet
unobserved decay modes with rich hadronic contents, such as
$D^{+}_{s}\rightarrow\eta^{\prime}\pi^{+}\pi^{+}\pi^{-}$. Charge-conjugate
states are implied throughout this paper.

In this paper, the first measurement of the BF of the inclusive decay of
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ based on 3.19 $\mbox{\,fb}^{-1}$
of $e^{+}e^{-}$ collision data collected at a center-of-mass energy ($E_{\rm
cm}$) of 4.178 GeV with the BESIII detector in 2016 is presented. The paper is
organized as follows. Section II introduces the BESIII detector as well as the
data and Monte Carlo (MC) simulated samples used in this analysis. Section III
gives an overview of the analysis technique. Event selection requirements to
fully reconstruct $D_{s}^{-}$ decays are described in Sec. IV, while the
selection requirements for the 3$\pi$ system at the signal $D^{+}_{s}$ side
and the further analysis based on the selected signal candidates are presented
in Sec. V. The systematic uncertainties on our measurements are evaluated in
Sec. VI, and the final results are summarized in Sec. VII.

##  II Detector and Data Sets

The BESIII detector ABLIKIM2010345  records the final state particles of
symmetric $e^{+}e^{-}$ collisions provided by the BEPCII storage ring bepcii
in the $E_{\rm cm}$ range from 2.00 to 4.95 GeV. BESIII has collected large
data samples in this energy region dataset , while this analysis uses the
entirety of the 3.19 $\mbox{\,fb}^{-1}$ data sample collected at $E_{\rm
cm}=4.178$ GeV in 2016. 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 besiii_mag
. The solenoid is supported by an octagonal flux-return yoke instrumented with
resistive-plate-counter muon identification modules interleaved with steel.
The MDC reconstructs charged-particle momenta with a resolution of 0.5% at
$1~{}{\rm GeV}/c$, measuring the characteristic energy loss
($\mathrm{d}E/\mathrm{d}x$) with 6% precision. 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 chambers,
providing a time resolution of 60 ps detector2015 .

Simulated data samples are produced with a geant4 sim  based MC simulation
toolkit, which includes the geometric description of the BESIII detector and
the detector response. The simulation also exploits the kkmc KKMC  generator
to take into account the beam energy spread and the initial-state radiation in
the $e^{+}e^{-}$ annihilations. The MC simulation samples, referred to as
“inclusive MC simulation”, include the production of $D^{+}_{s}$ particles via
the process $e^{+}e^{-}\rightarrow D_{s}^{(*)\pm}D_{s}^{\mp}$ (“$D_{s}$
inclusive MC simulation”), and other open charm processes, as well as the
production of vector charmonium(-like) states, and the continuum processes
incorporated in kkmc. The known decay modes are modeled with evtgen EvtGen
using the BFs taken from the Particle Data Group pdg , and the remaining
unknown decays from the charmonium states with lundcharm LundCharm . Final
state radiation from charged final state particles is incorporated using
photos FSR . The effective luminosities of the inclusive MC simulation samples
correspond to 40 times the data luminosity. The inclusive MC simulation
samples are used to estimate background contributions. A subset of the $D_{s}$
inclusive MC simulation sample, with one $D^{+}_{s}$ meson decaying into final
state modes including at least three charged pions, is also used as the signal
MC simulation sample of $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ to
determine detection efficiencies.

##  III Analysis Technique

At $E_{\rm cm}=4.178$ GeV, $D_{s}^{\pm}$ mesons are produced in pairs
predominantly through the process of $e^{+}e^{-}\rightarrow
D_{s}^{*\pm}D_{s}^{\mp}$. The $D_{s}^{*\pm}$ mesons then primarily decay into
$D_{s}^{\pm}\gamma$ (93.5%) and $D_{s}^{\pm}\pi^{0}$ (5.8%) pdg . The double-
tag (DT) technique, first employed by the Mark III collaboration dtref , is
used to determine the BF for the inclusive decay mode of the $D^{+}_{s}$
meson. We first fully reconstruct a $D^{-}_{s}$ meson, named “single tag
(ST)”, in one of the hadron tag modes listed in Sec. IV ds2pn . Then in the ST
$D^{-}_{s}$ sample, we reconstruct the
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ signal in the side recoiling
against the $D^{-}_{s}$ candidate, referred to as the DT. The number of ST
candidates for a specific tag mode $\alpha$ ($N^{\alpha}_{\rm ST}$) determined
from the fit shown in Fig. 1 is used to obtain the total number of
$e^{+}e^{-}\rightarrow D_{s}^{(*)\pm}D_{s}^{\mp}$ events in the data ($N_{\rm
tot}$) by

$N^{\alpha}_{\rm ST}=2N_{\rm tot}\cdot{\cal
B}_{\alpha}\cdot\epsilon^{\alpha}_{\rm ST}\,,$ (1)

where ${\cal B}_{\alpha}$ is the BF for the $D^{-}_{s}$ tag mode $\alpha$, and
$\epsilon^{\alpha}_{\rm ST}$ is the ST detection efficiency.

Then the signal decay which contains at least three charged pions is selected
at the recoil (signal) side. The tag side and signal side selection criteria
are described in detail in Sec. IV and Sec. V, respectively. To ensure one
3$\pi$ system per signal decay, in the case of more than one $\pi^{-}$ (two
$\pi^{+}$s) at the signal side, we select the $\pi^{-}$ (two $\pi^{+}$s) with
the largest momentum (largest and second largest momenta), in the laboratory
frame. The true (reconstructed) $3\pi$ invariant mass
$M(\pi^{+}\pi^{+}\pi^{-})$ is calculated based on true (reconstructed) pion
momenta.

Since the reconstruction efficiency is dependent on the kinematics of the
$3\pi$ system at the signal side, which is not well known, the partial BFs of
the inclusive $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decay are measured
in intervals of $M(\pi^{+}\pi^{+}\pi^{-})$. The efficiency is parametrized as
a function of $M(\pi^{+}\pi^{+}\pi^{-})$, and its dependence on other
kinematic variables is later examined in Sec. VI.7. For tag mode $\alpha$, the
number of produced DT events and the numbers of observed DT events are related
in $M(\pi^{+}\pi^{+}\pi^{-})$ intervals through a detector response matrix
that accounts for detector efficiency and detector resolution for
$M(\pi^{+}\pi^{+}\pi^{-})$,

$N^{\alpha}_{{\rm obs},i}=\sum_{j}\epsilon^{\alpha}_{ij}N^{\alpha}_{{\rm
prod},j}\,,$ (2)

where $N^{\alpha}_{{\rm prod},j}$ ($N^{\alpha}_{{\rm obs},i}$) is the number
of signal events produced (observed) in the $j^{{th}}$ ($i^{th}$)
$M(\pi^{+}\pi^{+}\pi^{-})$ interval. The $ij^{th}$ element of the efficiency
matrix, $\epsilon^{\alpha}_{ij}$, describes the efficiency and migration
across the $M(\pi^{+}\pi^{+}\pi^{-})$ intervals. It is calculated as
$\epsilon^{\alpha}_{ij}\equiv N^{\alpha}_{{\rm obs},ij}/N^{\alpha}_{{\rm
prod},j}$ using simulated signal $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$
decays with $D_{s}^{-}$ decaying into the tag mode $\alpha$. Here,
$N^{\alpha}_{{\rm obs},ij}$ is the number of signal MC simulated events
produced in the $j^{th}$ interval and reconstructed in the $i^{th}$ interval
of the $M(\pi^{+}\pi^{+}\pi^{-})$ distribution.

The number of observed $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ signal
candidates in each $M(\pi^{+}\pi^{+}\pi^{-})$ interval is determined by
fitting the invariant mass distribution of ST $D_{s}^{-}$ candidates to
extract the raw signal yield $N^{\alpha}_{{\rm raw},i}$. The yields of
expected $K^{0}_{\scriptscriptstyle S}$ background contributions,
$N^{\alpha}_{{K^{0}_{\scriptscriptstyle S}},i}$, and contributions from
particle misidentification (misID), $N^{\alpha}_{{\rm misID},i}$, estimated
from MC simulation, are subtracted from the raw yield $N^{\alpha}_{{\rm
raw},i}$

$N^{\alpha}_{{\rm obs},i}=N^{\alpha}_{{\rm
raw},i}-N^{\alpha}_{{K^{0}_{\scriptscriptstyle S}},i}-N^{\alpha}_{{\rm
misID},i}.$ (3)

Here, $N^{\alpha}_{{\rm
misID},i}=N^{\alpha}_{e\rightarrow\pi,i}+N^{\alpha}_{\mu\rightarrow\pi,i}+N^{\alpha}_{K\rightarrow\pi,i}$
includes contributions from the three misID scenarios described in Sec. V.2.

For tag mode $\alpha$, the number of
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ signal candidates produced in the
$i^{th}$ interval of the $M(\pi^{+}\pi^{+}\pi^{-})$ distribution is then
obtained by solving Eq. (2), which gives

$N^{\alpha}_{{\rm
prod},i}=\sum_{j}(\epsilon^{-1})^{\alpha}_{ij}N^{\alpha}_{{\rm obs},j}\,,$ (4)

and its statistical uncertainty is given by

$\left(\sigma_{\rm stat}\left(N^{\alpha}_{{\rm
prod},i}\right)\right)^{2}=\sum_{j}\left(\left(\epsilon^{-1}\right)^{\alpha}_{ij}\right)^{2}\left(\sigma_{\rm
stat}\left(N^{\alpha}_{{\rm obs},j}\right)\right)^{2}\,,$ (5)

where $\sigma_{\rm stat}\left(N^{\alpha}_{{\rm obs},j}\right)$ is the
statistical uncertainty of $N^{\alpha}_{{\rm obs},j}$. The statistical
uncertainties of $\epsilon^{\alpha}_{ij}$ due to the limited size of the
signal MC simulation sample are later considered as a source of systematic
uncertainties, as discussed in Sec. VI.8.

Taking $N_{\rm tot}$ from Eq. (1), the corresponding partial BF of
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ is given by

$\displaystyle{\Delta{\cal B}}^{\alpha}_{3\pi X,i}=\frac{N^{\alpha}_{{\rm
prod},i}}{2N_{\rm tot}\cdot{\cal B}_{\alpha}}=\frac{N^{\alpha}_{{\rm
prod},i}}{N^{\alpha}_{\rm ST}}\cdot\epsilon^{\alpha}_{\rm ST}\,.$ (6)

In each tag mode $\alpha$, the partial BFs of the
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decays are summed to obtain the
total BF ${\cal
B}^{\alpha}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)=\sum_{i}\Delta{\cal
B}^{\alpha}_{3\pi X,i}$. The total BF and its associated statistical
uncertainty are

$\displaystyle{\cal B}\left(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X\right)$
$\displaystyle=\sum_{\alpha}\frac{{\cal B}^{\alpha}}{\sigma^{2}_{{\cal
B}^{\alpha}}}/\sum_{\alpha}\frac{1}{\sigma^{2}_{{\cal B}^{\alpha}}},$ (7)

and

$\displaystyle\sigma^{2}_{\rm stat}\left({\cal
B}\left(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X\right)\right)$
$\displaystyle=1/\sum_{\alpha}\frac{1}{\sigma^{2}_{{\cal B}^{\alpha}}},$ (8)

respectively, where $\sigma_{{\cal B}^{\alpha}}$ is the statistical
uncertainty of ${\cal
B}^{\alpha}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)$.

##  IV Single Tag Analysis

ST $D^{-}_{s}$ candidates are selected in two hadronic tag modes,
$D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ and
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$. Other tag modes are not considered
due to their relatively high background level.

All charged track candidates detected in the MDC must 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. For charged
tracks not originating from $K^{0}_{\scriptscriptstyle S}$ decays, the
distance of closest approach to the interaction point (IP) is required to be
less than 10 cm along the $z$ axis, and less than 1 cm in the transverse
plane.

Charged tracks are identified as pions or kaons with the particle
identification algorithm (PID), which combines measurements of
$\mathrm{d}E/\mathrm{d}x$ in the MDC and the time of flight in the TOF to form
likelihoods ${\cal L}(h)(h=K,\pi)$ for different hadron $h$ hypotheses.
Charged kaons and pions are identified by comparing the likelihoods for the
kaon and pion hypotheses, ${\cal L}(K)>{\cal L}(\pi)$ and ${\cal L}(\pi)>{\cal
L}(K)$, respectively. Furthermore, charged pion candidates with momenta below
100 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ are rejected to suppress soft pions
from $D^{*+}\rightarrow D^{0}\pi^{+}$.

The $K^{0}_{\scriptscriptstyle S}$ candidates are reconstructed from two
oppositely charged tracks each with the distance of closest approach to the IP
less than 20 cm along the $z$ axis. The two charged tracks are assigned as
$\pi^{+}\pi^{-}$ without imposing further PID criteria. They are constrained
to originate from a common vertex and are required to have an invariant mass
satisfying $|M(\pi^{+}\pi^{-})-m_{K^{0}_{\scriptscriptstyle S}}|<12$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where $M(\pi^{+}\pi^{-})$ and
$m_{K^{0}_{\scriptscriptstyle S}}$ are the invariant mass of the pion pair and
the known $K^{0}_{\scriptscriptstyle S}$ mass pdg , respectively.

For each $D_{s}^{-}$ candidate, the reconstructed invariant mass
$M(D_{s}^{-})$ is required to be in the range 1.90 and 2.05
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The recoil mass of the $D^{-}_{s}$
candidate is defined as

$\displaystyle M_{\rm rec}c^{2}\equiv\sqrt{\left(E_{\rm
cm}-\sqrt{\left|\vec{p}_{D_{s}}c\right|^{2}+\left(m_{D_{s}}c^{2}\right)^{2}}\right)^{2}-\left|\vec{p}_{D_{s}}c\right|^{2}}\,,$
(9)

where $\vec{p}_{D_{s}}$ is the reconstructed momentum of the $D^{-}_{s}$
candidate in the $e^{+}e^{-}$ center-of-mass frame, and $m_{D_{s}}$ is the
known $D^{-}_{s}$ mass pdg .

Furthermore, for the tag mode $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle
S}K^{-}$, background contributions from $D^{0}\rightarrow K^{-}\pi^{+}$ decays
are suppressed by rejecting $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle
S}K^{-}$ candidates with $|M(K^{-}\pi^{+}_{K_{S}})-m_{D^{-}}|<10$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where $\pi^{+}_{K_{S}}$ is from
$K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$, and $m_{D^{-}}$ is
the known $D^{-}$ mass pdg .

Finally, for either tag mode, in about 6% of the cases, more than one
$D_{s}^{-}$ candidate per event passes the tag side event selection
requirements. In these cases, only the $D_{s}^{-}$ candidate with $M_{\rm
rec}$ closest to the known $D^{*+}_{s}$ mass per event per tag mode is
retained for further study. The systematic uncertainties related to this
single-candidate requirement are discussed in Sec. VI.11.

For each tag mode, the number of correctly reconstructed ST $D^{-}_{s}$
candidates is determined by performing an unbinned maximum likelihood fit to
the distribution of $M(D^{-}_{s})$, as shown in Fig. 1. The combinatorial
background is modeled using a linear probability density function (PDF). The
shapes of the $D^{-}$ background contributions from $D^{-}\rightarrow
K^{0}_{\scriptscriptstyle S}\pi^{-}$ ($D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$)
decays in the tag mode $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle
S}K^{-}$ ($D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$) when $K^{-}$ is
misidentified as $\pi^{-}$ are modeled with MC simulation. The yields of the
$D^{-}$ meson background contributions are estimated with the simulation and
fixed in the fits. The signal lineshapes are modeled by the sum of a Gaussian
PDF and a double-sided Crystal-ball (DSCB) PDF dscb  with a common mean value
$m_{0}$. The Gaussian PDF has a width of $\sigma_{0}$ that is allowed to float
along with $m_{0}$. The DSCB PDF tail parameters and the width ratio between
the DSCB and Gaussian PDFs are determined from signal MC simulated events and
fixed in the fits to data, as in Ref. ds23pi . The fitted yields are
summarized in Table 2, along with the ST efficiencies.

$D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$

$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$

Figure 1: The invariant mass $M(D^{-}_{s})$ distributions for the ST modes $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ (top) and $D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$ (bottom). Data (black points) are shown overlaid with the fit results including the total (solid blue), signal PDF (dashed magenta), $D^{-}$ background PDF (dotted red), and combinatorial background PDF (long-dashed blue) components.  Table 2: The ST yields and efficiencies for the two tag modes. Tag mode | $N_{\rm ST}^{\alpha}$ | $\epsilon_{\rm ST}^{\alpha}$
---|---|---
$D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ | $\phantom{0}30673\pm 230$ | 46.3%
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$ | $134370\pm 565$ | 38.8%

##  V Double Tag Analysis

In each event, once an ST $D^{-}_{s}$ candidate is found, three charged pions
with the sum of charges opposite to the charge of the $D^{-}_{s}$ tag in the
rest of the event (recoil side) are searched for. The tracking and PID
requirements of charged pions are the same as those in the tag side as
described in Sec. IV.

There are two major background sources that contribute to the $3\pi$ inclusive
final states. The first source of background contribution is due to events
with three correctly identified charged pions where at least one of them comes
from $K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays. The other
type of background comes from events for which at least one charged pion is
misidentified. These two types of background sources are studied separately
using dedicated MC simulations of inclusive $D^{+}_{s}$ decays to determine
additional selection criteria, and to estimate the remaining contribution
after imposing all the requirements. The background subtracted pion momentum
spectra and mass spectra from the $3\pi$ system for the tag mode
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$ are shown in Figs. 2 and 3, while the
spectra for the tag mode $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle
S}K^{-}$ follow very similar patterns with lower yields. There is reasonable
agreement between data and the simulated sample, and any differences in the
pion momentum spectra are later considered as a source of systematic
uncertainties.

Figure 2: Normalized momentum spectra after background subtraction from the
$3\pi$ system of the inclusive decay
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ for the tag mode
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$. The points are obtained from data
and the solid line histograms are from inclusive MC simulation samples. The
subscripts for $\pi^{+}$s indicate the order $p(\pi_{1}^{+})>p(\pi_{2}^{+})$.

Figure 3: Normalized invariant mass spectra after background subtraction from
the $3\pi$ system of the inclusive decay
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ for the tag mode
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$. The points are obtained from data
and the solid line histograms are from inclusive MC simulation samples. The
subscripts for the two $\pi^{+}\pi^{-}$ combinations indicate the order
$M_{\rm high}(\pi^{+}\pi^{-})>M_{\rm low}(\pi^{+}\pi^{-})$. The dips in the
$M(\pi^{+}\pi^{-})$ spectra are due to the 10
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ $K^{0}_{\scriptscriptstyle S}$ mass
veto defined in Sec. V.1.

###  V.1 Background from $K^{0}_{\scriptscriptstyle S}$ decays

To suppress the background contribution due to $K^{0}_{\scriptscriptstyle S}$
decays, each of the three charged pions of the $3\pi$ system is checked
against a list of $K^{0}_{\scriptscriptstyle S}$ candidates in an event. Each
of the $K^{0}_{\scriptscriptstyle S}$ candidates is reconstructed with the
criteria defined in Sec. IV . If at least one pion is part of one
reconstructed $K^{0}_{\scriptscriptstyle S}$ candidate that has
$L/\sigma_{L}>2$ ($L$ is defined as the distance between the
$K^{0}_{\scriptscriptstyle S}$ vertex and the IP, and $\sigma_{L}$ is the
uncertainty of $L$), or within the mass window
$|M(\pi^{+}\pi^{-})-m_{K^{0}_{\scriptscriptstyle
S}}|<10$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, this signal candidate is
rejected. With this requirement, about 88% of the $K^{0}_{\scriptscriptstyle
S}$ meson related background contributions are rejected, while losing about
11% of the signal events.

###  V.2 Particle misidentification background

For one charged pion, there are three possible misID scenarios:

  * •

$\mu^{\pm}\rightarrow\pi^{\pm}$,

  * •

$e^{\pm}\rightarrow\pi^{\pm}$,

  * •

$K^{\pm}\rightarrow\pi^{\pm}$.

As the charged pions are identified against kaons, the main misID background
contributions are from the semileptonic $D^{+}_{s}$ decays, and the
misidentified pion in the $3\pi$ system has the same charge as the signal
$D^{+}_{s}$ side. Electron background can be largely removed by rejecting
signal candidates for which at least one $e^{+}$ has been identified at the
signal $D^{+}_{s}$ side. The electron PID is based on the likelihoods for the
electron, pion, and kaon hypotheses ${\cal L}(h)(h=e,\pi,K)$ using information
from the MDC, TOF, and EMC, as in Ref. BESIIIDs2eX . With this, about 90% of
the $e^{+}\rightarrow\pi^{+}$ misID background contributions are removed at
the cost of mere $\sim$5% of the signal events. The residual misID backgrounds
are then dominated by the semimuonic decays $D^{+}_{s}\rightarrow
X\mu^{+}\nu_{\mu}$, and are estimated from the $D_{s}$ inclusive MC simulation
sample ($\sim$70%), , and the related systematic uncertainties are discussed
in Sec. V.2.

###  V.3 Partial branching fractions

The $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ partial BFs are measured in
eleven $M(\pi^{+}\pi^{+}\pi^{-})$ intervals. The intervals are populated with
approximately equal numbers of DT candidates, except for the last interval,
where the signal candidates are predominantly from the exclusive decay of
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}$. The interval boundaries in the
$M(\pi^{+}\pi^{+}\pi^{-})$ distribution are chosen as [0.40, 0.74, 0.90, 0.99,
1.08, 1.15, 1.23, 1.30, 1.41, 1.57, 1.90, 2.05]
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The number of $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ signal candidates
in each $M(\pi^{+}\pi^{+}\pi^{-})$ interval is determined by fitting the
corresponding $M(D^{-}_{s})$ distribution of the tag side. From this fit, the
raw signal yield $N^{\alpha}_{{\rm raw},i}$ is measured for each interval $i$
of the $M(\pi^{+}\pi^{+}\pi^{-})$ distribution, for tag mode $\alpha$. The
signal shape for each tag mode is determined in the corresponding ST fit
(results shown in Fig. 1). The same shape is used in all
$M(\pi^{+}\pi^{+}\pi^{-})$ intervals, and the linear background shape
parameter is left unconstrained in the fit. The fitted raw signal yields
($N^{\alpha}_{{\rm raw},i}$) are presented in Tables 3 and 4 for the tag modes
of $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ and
$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$, respectively. Two examples of the 22
fits are shown in Fig. 4, and the full set of fit results is made available as
Supplemental Material supp . Tables 3 and 4 summarize the observed signal
yields, after subtracting the $K^{0}_{\scriptscriptstyle S}$-decay related and
misID background contributions from the raw signal yields as described earlier
in this section.

Table 3: Raw signal yields and estimated background contributions for $D_{s}^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ versus $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ events. The associated uncertainties are statistical only. The uncertainties from different background contributions are due to the limited size of our inclusive MC simulation samples, whose integrated luminosity is 40 times of data.  $M(\pi^{+}\pi^{+}\pi^{-})$ interval | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
Raw yield | 540.3 $\pm$ 28.7 | 625.9 $\pm$ 31.8 | 514.4 $\pm$ 28.4 | 627.7 $\pm$ 31.3 | 418.8 $\pm$ 26.1 | 486.8 $\pm$ 27.2
$K^{0}_{\scriptscriptstyle S}$ contribution | 16.6 $\pm$ 0.7 | 37.9 $\pm$ 1.0 | 28.3 $\pm$ 0.9 | 22.4 $\pm$ 0.8 | 12.3 $\pm$ 0.6 | 11.3 $\pm$ 0.6
MisID contribution | 40.7 $\pm$ 1.1 | 64.1 $\pm$ 1.4 | 44.8 $\pm$ 1.1 | 46.8 $\pm$ 1.2 | 35.5 $\pm$ 1.0 | 36.3 $\pm$ 1.0
Total background | 57.4 $\pm$ 1.3 | 102.0 $\pm$ 1.70 | 73.1 $\pm$ 1.5 | 69.2 $\pm$ 1.4 | 47.7 $\pm$ 1.2 | 47.6 $\pm$ 1.2
Background subtracted yield | 482.9 $\pm$ 28.8 | 523.9 $\pm$ 31.8 | 441.3 $\pm$ 28.4 | 558.5 $\pm$ 31.3 | 371.0 $\pm$ 26.1 | 439.2 $\pm$ 27.2
$M(\pi^{+}\pi^{+}\pi^{-})$ interval | 7 | 8 | 9 | 10 | 11 |
Raw yield | 403.3 $\pm$ 24.4 | 515.3 $\pm$ 27.8 | 400.5 $\pm$ 25.4 | 494.8 $\pm$ 27.4 | 153.3 $\pm$ 14.4 |
$K^{0}_{\scriptscriptstyle S}$ contribution |  06.8 $\pm$ 0.4 |  09.7 $\pm$ 0.5 |  08.1 $\pm$ 0.5 |  02.7 $\pm$ 0.3 |  00.4 $\pm$ 0.1 |
MisID contribution | 28.5 $\pm$ 0.9 | 35.9 $\pm$ 1.0 | 36.5 $\pm$ 1.0 | 27.6 $\pm$ 0.9 |  00.8 $\pm$ 0.2 |
Total background | 35.3 $\pm$ 1.0 | 45.6 $\pm$ 1.1 | 44.6 $\pm$ 1.1 | 30.3 $\pm$ 0.9 |  01.3 $\pm$ 0.2 |
Background subtracted yield | 368.1 $\pm$ 24.4 | 469.6 $\pm$ 27.8 | 356.0 $\pm$ 25.4 | 464.5 $\pm$ 27.4 | 152.0 $\pm$ 14.4 |
Table 4: Raw signal yields and estimated background contributions for $D_{s}^{-}\rightarrow K^{-}K^{+}\pi^{-}$ versus $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ events. The associated uncertainties are statistical only. The uncertainties from different background contributions are due to the limited size of our inclusive MC simulation samples, whose integrated luminosity is 40 times of data.  $M(\pi^{+}\pi^{+}\pi^{-})$ interval | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
Raw yield | 2300.8 $\pm$ 71.1 | 2883.4 $\pm$ 85.8 | 2405.7 $\pm$ 77.6 | 2630.0 $\pm$ 81.3 | 1944.3 $\pm$ 70.6 | 2276.2 $\pm$ 74.3
$K^{0}_{\scriptscriptstyle S}$ contribution |  059.1 $\pm$ 1.3 | 148.2 $\pm$ 2.1 | 115.0 $\pm$ 1.8 |  093.6 $\pm$ 1.7 |  047.7 $\pm$ 1.2 |  048.4 $\pm$ 1.2
MisID contribution | 190.1 $\pm$ 2.4 | 297.3 $\pm$ 3.0 | 214.6 $\pm$ 2.5 | 226.6 $\pm$ 2.6 | 164.3 $\pm$ 2.2 | 171.8 $\pm$ 2.2
Total background | 249.2 $\pm$ 2.7 | 445.5 $\pm$ 3.6 | 329.6 $\pm$ 3.1 | 320.2 $\pm$ 3.1 | 212.0 $\pm$ 2.5 | 220.2 $\pm$ 2.5
Background subtracted yield | 2051.6 $\pm$ 71.1 | 2437.9 $\pm$ 85.9 | 2076.1 $\pm$ 77.7 | 2309.8 $\pm$ 81.4 | 1732.2 $\pm$ 70.7 | 2056.0 $\pm$ 74.4
$M(\pi^{+}\pi^{+}\pi^{-})$ interval | 7 | 8 | 9 | 10 | 11 |
Raw yield | 1924.3 $\pm$ 65.2 | 2182.5 $\pm$ 68.8 | 1926.0 $\pm$ 65.5 | 1993.0 $\pm$ 63.7 | 767.7 $\pm$ 34.0 |
$K^{0}_{\scriptscriptstyle S}$ contribution |  031.1 $\pm$ 1.0 |  036.9 $\pm$ 1.0 |  031.1 $\pm$ 1.0 |  010.7 $\pm$ 0.6 |  01.4 $\pm$ 0.2 |
MisID contribution | 127.5 $\pm$ 1.9 | 169.0 $\pm$ 2.2 | 168.3 $\pm$ 2.2 | 126.3 $\pm$ 1.9 |  03.9 $\pm$ 0.3 |
Total background | 158.7 $\pm$ 2.2 | 205.9 $\pm$ 2.5 | 199.4 $\pm$ 2.4 | 137.0 $\pm$ 2.0 |  05.3 $\pm$ 0.4 |
Background subtracted yield | 1765.6 $\pm$ 65.2 | 1976.6 $\pm$ 68.8 | 1726.6 $\pm$ 65.5 | 1856.0 $\pm$ 63.7 | 762.4 $\pm$ 34.0 |

Using the observed signal yields and the efficiency matrix elements
$\epsilon^{\alpha}_{ij}$ presented in Tables 5 and 6, the partial BFs in
various $M(\pi^{+}\pi^{+}\pi^{-})$ intervals are calculated using Eqs. 4 and 6
and shown in Table 7 after combining the measurements from both tag modes.

Table 5: Efficiency matrix $\epsilon_{ij}$ in percentage for $D_{s}^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ versus $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ determined from signal MC simulated events. Each column gives the true $M(\pi^{+}\pi^{+}\pi^{-})$ interval $j$, while each row gives the reconstructed $M(\pi^{+}\pi^{+}\pi^{-})$ interval $i$.  $\epsilon_{ij}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---|---
1 | 15.11 |  00.25 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00
2 |  00.19 | 15.86 |  00.67 |  00.08 |  00.01 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00
3 |  00.01 |  00.41 | 16.76 |  00.85 |  00.12 |  00.03 |  00.01 |  00.00 |  00.00 |  00.01 |  00.00
4 |  00.01 |  00.02 |  00.62 | 20.99 |  01.00 |  00.19 |  00.05 |  00.01 |  00.00 |  00.00 |  00.00
5 |  00.01 |  00.01 |  00.02 |  00.73 | 21.76 |  00.98 |  00.13 |  00.05 |  00.01 |  00.00 |  00.00
6 |  00.00 |  00.00 |  00.01 |  00.03 |  00.88 | 21.93 |  01.18 |  00.21 |  00.05 |  00.00 |  00.00
7 |  00.00 |  00.00 |  00.00 |  00.01 |  00.04 |  00.90 | 22.46 |  00.91 |  00.11 |  00.00 |  00.00
8 |  00.00 |  00.00 |  00.00 |  00.00 |  00.01 |  00.06 |  01.11 | 23.46 |  00.88 |  00.05 |  00.00
9 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.01 |  00.03 |  00.66 | 24.84 |  00.77 |  00.02
10 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.01 |  00.48 | 27.10 |  00.75
11 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.10 | 24.53
Table 6: Efficiency matrix $\epsilon_{ij}$ in percentage for $D_{s}^{-}\rightarrow K^{-}K^{+}\pi^{-}$ versus $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ determined from signal MC simulated events. Each column gives the true $M(\pi^{+}\pi^{+}\pi^{-})$ interval $j$, while each row gives the reconstructed $M(\pi^{+}\pi^{+}\pi^{-})$ interval $i$.  $\epsilon_{ij}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---|---
1 | 12.80 |  00.20 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00
2 |  00.17 | 13.65 |  00.54 |  00.06 |  00.01 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00
3 |  00.01 |  00.33 | 14.64 |  00.67 |  00.09 |  00.02 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00
4 |  00.00 |  00.01 |  00.47 | 17.96 |  00.96 |  00.14 |  00.04 |  00.01 |  00.00 |  00.00 |  00.00
5 |  00.00 |  00.00 |  00.01 |  00.62 | 18.36 |  00.83 |  00.13 |  00.05 |  00.00 |  00.00 |  00.00
6 |  00.00 |  00.00 |  00.01 |  00.03 |  00.83 | 18.89 |  01.01 |  00.16 |  00.04 |  00.00 |  00.00
7 |  00.00 |  00.00 |  00.00 |  00.01 |  00.03 |  00.75 | 19.19 |  00.77 |  00.09 |  00.01 |  00.00
8 |  00.00 |  00.00 |  00.00 |  00.00 |  00.01 |  00.06 |  00.88 | 20.16 |  00.76 |  00.06 |  00.00
9 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.01 |  00.02 |  00.56 | 21.15 |  00.55 |  00.01
10 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.01 |  00.40 | 23.13 |  00.67
11 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.00 |  00.07 | 21.29

$D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$

$D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$

Figure 4: The $M(D^{-}_{s})$ distributions in the first $M(\pi^{+}\pi^{+}\pi^{-})$ interval for the tag modes of $D^{-}_{s}\rightarrow K^{0}_{\scriptscriptstyle S}K^{-}$ (top) and $D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$ (bottom). Data (black points) are shown overlaid with the fit results including the total (solid blue), signal PDF (dashed magenta), $D^{-}$ background PDF (dotted red), and combinatorial background PDF (long-dashed blue) components. Table 7: Partial BFs (in percentage) of the decay $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ combined from the two tag modes. The first and second uncertainties are statistical and systematic, respectively.  $M(\pi^{+}\pi^{+}\pi^{-})$ interval |  ${\Delta{\cal B}}_{3\pi X,i}$ (%)
---|---
1 | $4.63\pm 0.14\pm 0.14$
2 | $4.92\pm 0.16\pm 0.19$
3 | $3.79\pm 0.13\pm 0.10$
4 | $3.55\pm 0.12\pm 0.09$
5 | $2.42\pm 0.10\pm 0.07$
6 | $2.87\pm 0.10\pm 0.09$
7 | $2.39\pm 0.09\pm 0.07$
8 | $2.69\pm 0.09\pm 0.07$
9 | $2.19\pm 0.08\pm 0.05$
10 | $2.32\pm 0.07\pm 0.05$
11 | $1.01\pm 0.04\pm 0.04$

##  VI Systematic uncertainties

The methods to determine the systematic uncertainties in the measured total
and partial BFs of inclusive $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$
decays are described below.

###  VI.1 Signal shape in $M(D^{-}_{s})$

The signal $M(D^{-}_{s})$ PDF is changed from the DSCB shape to a shape
modeled with signal MC simulated events convolved with a Gaussian function.
The width and mean parameters of the Gaussian function are floating in the
fits to account for the resolution difference between data and simulation.
Alternatively, the fits are performed by allowing the two signal shape
parameters $m_{0}$ and $\sigma_{0}$ in different $M(\pi^{+}\pi^{+}\pi^{-})$
intervals to differ. The largest variations are taken as the relative
systematic uncertainties.

###  VI.2 Background shape in $M(D^{-}_{s})$

The background shape is changed from the linear PDF to an exponential or a
histogram PDF based on MC simulation. Also the fits are performed with the
yields of $D^{-}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{-}$
($D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$) in the tag mode $D^{-}_{s}\rightarrow
K^{0}_{\scriptscriptstyle S}K^{-}$ ($D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}$)
floating, rather than fixed to the numbers from simulation. The largest
variations are taken as the relative systematic uncertainties.

###  VI.3  Pion tracking and PID efficiency

The $\pi^{\pm}$ tracking efficiency is studied with the control sample of
$e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ decay. The data-MC tracking
efficiency ratios of $\pi^{+}$ and $\pi^{-}$ are measured in bins of pion
transverse momentum. The signal MC samples are then weighted by these ratios
to recalculate signal efficiency matrices. This leads to 0.98% variation in
total BF, which is taken as the systematic uncertainty associated with
tracking efficiency.

The $\pi^{\pm}$ PID efficiency is studied with the control samples of
$e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}(\pi^{0})$ and
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}(\pi^{0})$ decays. The data-MC tracking
efficiency ratios of $\pi^{\pm}$ are measured in bins of pion momentum. The
signal MC samples are then weighted by these ratios to recalculate signal
efficiency matrices. This leads to 0.99% variation in the total BF, which is
taken as the systematic uncertainty associated with PID efficiency.

###  VI.4 Background from $K^{0}_{\scriptscriptstyle S}$ decays

As $N^{\alpha}_{{K^{0}_{\scriptscriptstyle S}},i}$ defined in Eq. (3) is
determined by the $D_{s}$ inclusive MC simulation sample, the data and
simulation differences on the $D^{+}_{s}\rightarrow K^{0}_{\scriptscriptstyle
S}\pi^{+}X$ contributions are studied by using control samples with dedicated
stringent requirements on the $3\pi$ system: one $\pi^{+}\pi^{-}$ pair forms
the reconstructed $K^{0}_{\scriptscriptstyle S}$ that has $L/\sigma_{L}>2$ and
$|M(\pi^{+}\pi^{-})-m_{K^{0}_{\scriptscriptstyle
S}}|<10$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. These requirements lead to
a data sample with more than 99% of the $D^{+}_{s}$ candidates having a
$\pi^{+}\pi^{-}$ pair from $K^{0}_{\scriptscriptstyle S}$. The agreement on
the observed number of $D^{+}_{s}$ candidates in data and the number from
simulation is at $\sim$10% level across different $M(\pi^{+}\pi^{+}\pi^{-})$
intervals. Therefore, the $N^{\alpha}_{{K^{0}_{\scriptscriptstyle S}},i}$
values are all scaled by $\pm 10$% in turn, and the BFs are recalculated
according to altered $N^{\alpha}_{{\rm obs},i}$ values based on Eq. (3). The
larger variations from the two measurements are taken as the relative
systematic uncertainties on the $K^{0}_{\scriptscriptstyle S}$ background.

Furthermore, the systematic uncertainty related to the
$K^{0}_{\scriptscriptstyle S}$ veto of
$|M(\pi^{+}\pi^{-})-m_{K^{0}_{\scriptscriptstyle S}}|<10$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is examined by varying the mass
window by $\pm 5$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The resulting
changes in the BFs are smaller than the statistical uncertainties on the
differences, so no uncertainty is assigned for this source according to Ref.
barlow .

###  VI.5 Particle misidentification background

The $D_{s}$ inclusive MC simulation sample is also used to estimate
$N^{\alpha}_{{\rm misID},i}$ defined in Eq. (3). The uncertainties on
$N^{\alpha}_{{\rm misID},i}$ come from limited knowledge on the BFs of
different decays that contribute to the misID backgrounds, and differences in
misID rates between data and simulation, that are considered separately in the
three misID scenarios as listed in Sec. V.2.

The $D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\ell^{+}X$ decays form the background
where the leptons $\ell=e,\mu$ are misidentified as pions, and the decays
$D^{+}_{s}\rightarrow K^{-}\pi^{+}\pi^{+}X$ and $D^{+}_{s}\rightarrow
K^{+}\pi^{+}\pi^{-}X$ are relevant for the background when a kaon is
misidentified instead. The $D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\ell^{+}X$
decays are dominated by the semileptonic decays of
$D^{+}_{s}\rightarrow\phi\ell^{+}\nu_{\ell}$ and
$D^{+}_{s}\rightarrow\eta^{(\prime)}\ell^{+}\nu_{\ell}$. By assuming LFU in
semileptonic charm decays, the data and simulation differences on modeling the
$D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\ell^{+}X$ decays are studied using samples
of $D^{+}_{s}\rightarrow\pi^{+}\pi^{-}e^{+}X$. This LFU assumption has been
found to hold experimentally, e.g. within a precision of 1.4% in
$D^{0}\rightarrow K^{-}\ell^{+}\nu_{\ell}$ decays bes3d2klnu . The electron
PID requirement used on one of the positively charged track in the 3$\pi$
system is the same as that in Ref. BESIIIDs2eX . As the relative difference
between data and MC simulation is below 5%, the sums of
$N^{\alpha}_{e\rightarrow\pi,\ i}+N^{\alpha}_{\mu\rightarrow\pi,\ i}$ are all
scaled by $\pm$5% in turn. The larger variations from the two measurements are
taken as the systematic uncertainties related to the lepton misID background.
Similarly, the dedicated simulation samples of $D^{+}_{s}\rightarrow
K^{-}\pi^{+}\pi^{+}X$ and $D^{+}_{s}\rightarrow K^{+}\pi^{+}\pi^{-}X$ decays
can be compared to data by imposing kaon identification requirements on one
track from the 3$\pi$ system. This study indicates a disagreement between data
and simulation at a level of about 15%. This is significantly larger than in
the lepton case due to the poor knowledge of hadronic $D^{+}_{s}$ decays with
at least one charged kaon. The $N^{\alpha}_{K\rightarrow\pi,\ i}$ values are
all scaled by $\pm$15% in turn, and the larger variations from the two
measurements are taken as the systematic uncertainties related to the kaon
misID. Separate studies on the probabilities of tracks being misidentified as
pions show the effects due to data and simulation differences are negligible.
Therefore, this is not included as part of the systematic uncertainties.

###  VI.6 Pion multiplicity

The number of charged pions in $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$
decays may be inaccurately modeled in the simulation mainly due to the limited
knowledge of hadronic $D^{+}_{s}$ decays with five or more charged pions in
the final states such as
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}(\pi^{0})$ pdg
. The signal MC simulated events are weighted so that the simulated
distribution of the number of charged pion candidates at the signal side
matches with that in the data. This causes changes in signal efficiencies, and
the resulting BF variations using the weighted MC simulated events are taken
as the relative systematic uncertainties.

###  VI.7 Pion momenta

The signal MC simulated events are weighted to improve the agreement of the
pion momentum distributions between data and simulation. The multivariate
algorithm gbrw  to determine the weights is trained on the distributions from
unweighted signal MC simulation and background subtracted data samples as
shown in Fig. 2. Using the weighted MC candidates based on the three-
dimensional weighting scheme to recalculate signal efficiency matrices for
both tag modes, the variations in the BF central values are taken as the
relative systematic uncertainties.

As the data-MC differences in the angular distributions of pion momenta may
result in differences in the mass spectra of the 3$\pi$ system, we also adopt
a five-dimensional weighting scheme with the addition of $M_{\rm
high}(\pi^{+}\pi^{-})$ and $M_{\rm low}(\pi^{+}\pi^{-})$ to account for the
data-MC differences shown in Fig. 3. The resulting variations in BFs are
smaller than those from the three-dimensional weighting scheme, and thus no
additional uncertainty is assigned.

###  VI.8 Monte Carlo simulation sample size

The BFs are recalculated 1000 times by randomly perturbing the efficiency
matrix elements shown in Tables 5 and 6 according to their uncertainties. The
distributions of the recalculated BF central values are fitted with a Gaussian
function and the fitted Gaussian widths are taken as the relative systematic
uncertainties.

###  VI.9 Measurement bias

The uncertainties due to the entire analysis procedure to extract the BFs are
estimated by using 40 inclusive MC simulation samples, each with a total
candidate number statistically matched to that in the data. While good
agreement between the input and measured BF values is found, the mean biases
are taken as the relative systematic uncertainties.

###  VI.10 Signal MC simulation model

The purpose of measuring ${\cal
B}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)$ in intervals of
$M(\pi^{+}\pi^{+}\pi^{-})$ is to be independent of the $D^{+}_{s}$ decay model
used in the signal simulation. A number of exclusive $D^{+}_{s}$ decay modes
such as $D^{+}_{s}\rightarrow\eta^{\prime}\rho^{+}$,
$D^{+}_{s}\rightarrow\phi\pi^{+}\pi^{+}\pi^{-}$, and
$D^{+}_{s}\rightarrow\phi\rho^{+}$ are used to perform the tests. The yield of
the respective signal MC simulation events is weighted by about 40%. This in
effect changes the related input BF by about 40% (up or down). The signal
efficiency matrices are then updated based on the weighted MC simulation
samples. The obtained BF results show good agreement with the baseline results
well within one standard deviation. Therefore, a systematic uncertainty
related to the signal MC simulation model is not assigned.

###  VI.11 Single-candidate requirement

The single-candidate requirement described in Sec. IV may bias the measurement
due to potentially different candidate multiplicities and $M_{\rm rec}$
distributions in data and MC simulation. The systematic uncertainties from the
single-candidate requirement are examined by rerunning the measurement with
this requirement removed. The resulting BF variations are smaller than the
statistical uncertainties on the differences, so no uncertainty is assigned
for this source.

###  VI.12 Summary of systematic uncertainties

Table 8 summarizes contributions from the different systematic sources in the
measurement of ${\cal B}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)$. The
largest sources of systematic uncertainties come from tracking and PID of the
$3\pi$ system. These contributions are combined in quadrature to determine the
total systematic uncertainty in the BF measurement. The summary table for the
systematic uncertainties of the partial BFs of
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ is given in the Supplemental
Material supp .

Table 8: Sources of systematic uncertainties in the measurement of ${\cal B}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)$.  Source | Relative uncertainty (%)
---|---
Signal shape | 0.49
Background shape | 0.40
$\pi^{+}\pi^{+}\pi^{-}$ tracking efficiency | 0.98
$\pi^{+}\pi^{+}\pi^{-}$ PID efficiency | 0.99
$K^{0}_{\scriptscriptstyle S}$ background | 0.34
Kaon misID background | 0.32
Lepton misID background | 0.35
Pion multiplicity | 0.51
Pion momenta | 0.86
MC simulation sample size | 0.01
Fit bias | 0.03
Total | 1.92

##  VII Summary

Figure 5: Partial BFs normalized by the total BF and interval widths as a
function of $M(\pi^{+}\pi^{+}\pi^{-})$ for the
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decay. The vertical gray lines
represent the uncertainties as the quadratic sums of the statistical and
systematic uncertainties.

Based on 3.19 $\mbox{\,fb}^{-1}$ of $e^{+}e^{-}$ collision data recorded with
the BESIII detector at $E_{\rm cm}=4.178$ $\mathrm{\,Ge\kern-1.00006ptV}$, the
BF of the inclusive $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decay is
measured for the first time. It is found to be

${\cal B}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X)=\left(32.81\pm 0.35_{\rm
stat}\pm{0.63}_{\rm syst}\right)\%.$

This is larger than the sum of all observed exclusive BFs of $\sim$25% based
on the PDG and recent measurements as summarized in Table 1, hinting at
potentially unobserved decay modes with at least three charged pions in the
final states.

Furthermore, the partial BFs of the
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ decay as a function of
$M(\pi^{+}\pi^{+}\pi^{-})$ are also measured, as presented in Table 7 and Fig.
5. As the feed-up contribution from lower intervals is expected to be
negligible ($\sim$1% of exclusive signals), the partial BF in the last
$M(\pi^{+}\pi^{+}\pi^{-})$ interval is consistent with the exclusive BF of
${\cal B}(D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-})=(1.08\pm 0.04)$% from the
PDG pdg , with somewhat lower precision. The measured total and partial BFs of
$D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}X$ offer important inputs to
constrain the systematic uncertainties in future LHCb measurements on
$R(D^{*-})$ and $R(\mathchar 28931\relax_{c}^{+})$ with much larger data
samples lhcb-white-paper .

## ACKNOWLEDGMENTS

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center 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; 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, U1932108; the CAS Center for
Excellence in Particle Physics (CCEPP); 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, 455635585, Collaborative Research Center
CRC 1044, FOR5327, 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; Olle
Engkvist Foundation under Contract No. 200-0605; 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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Unification of $\kappa$-Minkowski and extended Snyder spaces

S. Meljanac††† e-mail<EMAIL_ADDRESS>
Rudjer Bošković Institute, Theoretical Physics Division

Bljenička c. 54, 10002 Zagreb, Croatia

and

S. Mignemi††† e-mail<EMAIL_ADDRESS>
Dipartimento di Matematica, Università di Cagliari

via Ospedale 72, 09124 Cagliari, Italy

and INFN, Sezione di Cagliari

Abstract

In a recent paper, we have studied associative realizations of the
noncommutative extended Snyder model, obtained by including the Lorentz
generators (tensorial coordinates) and their conjugated momenta. In this
paper, we extend this result to also incorporate a covariant realization of
the $\kappa$-Poincaré spacetime. We obtain the coproduct, the associative star
product and the twist in a Weyl-ordered realization, to first order in the
noncommutativity parameters. This could help the construction of a quantum
field theory based on this geometry.

1\. Introduction

In the last decades, noncommutative geometry has become an important topic
both in mathematics and in theoretical physics, where it is considered a
promising candidate for describing the structure of spacetime at the Planck
scale. The mathematical framework most suitable for physical applications is
based on the formalism of Hopf algebra. This structure is simultaneously an
associative algebra and a coassociative coalgebra, satisfying some
compatibility conditions [1], which is apt to describe the quantization of
spacetime and the deformation of the momentum composition laws. Its physical
applications are based on the observation that most theories of quantum
gravity predict a granular structure of spacetime at small scales, with the
appearance of a minimal length, that renders the classical geometric
description inadequate [2].

Among the most well-known models of this kind are the Snyder space [3,4] and
the $\kappa$-Poincaré model [5] with its associated $\kappa$-Minkowski
spacetime [6]. Both of them are based on a deformation of the commutation
relations of the position operators $\hat{x}_{i}$: for Snyder space one has††∗
In this paper, Latin indices run from 0 to $N-1$, Greek indices from 0 to $N$.

$[\hat{x}_{i},\hat{x}_{j}]=i\beta\hat{M}_{ij},$ $None$

where $\beta$ is a constant with dimension length square, while the operators
$\hat{M}_{ij}$ are the generators of Lorentz transformations. For
$\kappa$-Minkowski geometry, in a covariant setting,

$[\hat{x}_{i},\hat{x}_{j}]=i(a_{i}\hat{x}_{j}-a_{j}\hat{x}_{i}),$ $None$

with $a_{i}$ a fixed vector with dimension of length. The length scale is
usually identified with the Planck length.

While in $\kappa$-Minkowski the algebra of the position operators is closed,
in the Snyder case a closed algebra is obtained only if one includes the
Lorentz generators with standard commutation relations as elements of the
defining coordinate algebra. As a consequence, one cannot construct a proper
Hopf algebra using a representation of the $\hat{M}_{ij}$ in terms of the
position and momentum operators, and in fact such attempt gives rise to a
noncoassociative bialgebra [4].

A way to obtain a coassociative Hopf algebra for the Snyder model was proposed
in [7] and discussed in detail in [8], and is based on the introduction of new
degrees of freedom given by antisymmetric tensors, identified with the Lorentz
generators $\hat{M}_{ij}$. In this way, the Snyder algebra in $N$ dimensions
becomes isomorphic to $so(1,N)$, and has been called extended Snyder algebra,
to distinguish it from the standard representation based solely on the $N$
position variables [4,7].

In this paper, using a result from [9], we extend the formalism developed in
ref. [8] to include also the $\kappa$-Minkowski deformation and then achieve
an unification of the extended Snyder model with the $\kappa$-Poincaré model.
This will be obtained by deforming $so(1,N)$ to an algebra which leaves
invariant a suitable metric $g$ and will be denoted $so(1,N;g)$. We then
consider a Weyl realization of this algebra in terms of an extended Heisenberg
algebra [8] and compute its coproduct, star product and twist.

We shall only consider the Weyl realization of the $\kappa$-Poincaré Snyder
algebra. Of course more general realizations can be obtained both for Snyder
[4,10] and $\kappa$-Poincaré [11], that can easily be implemented in our
formalism. We should also remark that a unification of the Snyder and
$\kappa$-Poincaré models has been previously obtained in the standard
formalism [12]. In that case, the resulting algebra is however not
coassociative and the star products are not associative.

2\. $\kappa$-Minkowski Snyder spacetime and $so(1,N;g)$ algebra In ref. [9] it
was proposed a formalism for constructing $\kappa$-deformations of orthogonal
groups by considering transformations that leave invariant generic metric
tensors. We apply this formalism to the extended Snyder model of ref. [8] to
get an unification of it with $\kappa$-Minkowski spacetime.

Let us consider the Lorentz algebra $so(1,N)$ with generators $M_{\mu\nu}$,

$[M_{\mu\nu},M_{\rho\sigma}]=i\lambda(\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\nu\sigma}M_{\mu\rho}),$
$None$

where $\eta={\rm diag}(-1,1,\dots,1)$, and perform a change of basis, defining

$\tilde{M}_{\mu\nu}=(OMO^{T})_{\mu\nu},\qquad g_{\mu\nu}=(O\eta
O^{T})_{\mu\nu},$ $None$

where $O$ is an $(N+1)\times(N+1)$ matrix with transpose $O^{T}$. Then,

$\tilde{M}^{T}=-\tilde{M},\qquad g^{T}=g,\qquad\det g\neq 0,$ $None$

and

$[\tilde{M}_{\mu\nu},\tilde{M}_{\rho\sigma}]=i\lambda(g_{\mu\rho}\tilde{M}_{\nu\sigma}-g_{\mu\sigma}\tilde{M}_{\nu\rho}-g_{\nu\rho}\tilde{M}_{\mu\sigma}+g_{\nu\sigma}\tilde{M}_{\mu\rho}).$
$None$

We shall denote this algebra with general metric $g$ as $so(1,N;g)$. In
particular, we are interested in matrices $g_{\mu\nu}$ of the form

$g_{\mu\nu}=\left(\matrix{-1&0&\dots&0&g_{0}\cr 0&1&\dots&0&g_{1}\cr&&\dots\cr
0&0&\dots&1&g_{N-1}&\cr g_{0}&g_{1}&\dots&g_{N-1}&g_{N}\cr}\right),$ $None$

with det $g=-g_{0}^{2}+\sum_{i=1}^{N-1}g_{i}^{2}-g_{N}$. If $g_{N}=0$, the
algebra (6) reduces to the $\kappa$-Poincaré algebra, if
$g_{0}=\dots,g_{N-1}=0$, to the Snyder one.

In fact, we can introduce extended coordinates $\hat{x}_{\mu\nu}=M_{\mu\nu}$
and $\hat{X}_{\mu\nu}=\tilde{M}_{\mu\nu}$, so that
$\hat{X}_{\mu\nu}=(O\hat{x}O^{T})_{\mu\nu}$. The $\hat{x}_{\mu\nu}$ satisfy
the standard $so(1,N)$ algebra, while we can split the relations (6) as

$\eqalignno{&[\hat{X}_{iN},\hat{X}_{jN}]=i\lambda(g_{iN}\hat{X}_{jN}-g_{jN}\hat{X}_{iN}+g_{NN}\hat{X}_{ij}),&\cr&[\hat{X}_{ij},\hat{X}_{kN}]=i\lambda(g_{ik}\hat{X}_{jN}-g_{jk}\hat{X}_{iN}-g_{iN}\hat{X}_{jk}+g_{jN}\hat{X}_{ik}),&\cr&[\hat{X}_{ij},\hat{X}_{kl}]=i\lambda(g_{ik}\hat{X}_{jl}-g_{il}\hat{X}_{jk}-g_{jk}\hat{X}_{il}+g_{jl}\hat{X}_{ik}).&(8)}$

If we now define $\hat{X}_{iN}=\kappa\,\hat{X}_{i}$ and

$g_{ij}=\eta_{ij},\quad g_{iN}=\kappa\,a_{i},\quad g_{NN}=\kappa^{2}\beta,$
$None$

we get the extended unified $\kappa$-Minkowski Snyder spacetime, with
commutation relations

$\eqalignno{&[\hat{X}_{i},\hat{X}_{j}]=i\lambda(a_{i}\hat{X}_{j}-a_{j}\hat{X}_{i}+\beta\hat{X}_{ij}),&\cr&[\hat{X}_{ij},\hat{X}_{k}]=i\lambda(\eta_{ik}\hat{X}_{j}-\eta_{jk}\hat{X}_{i}-a_{i}\hat{X}_{jk}+a_{j}\hat{X}_{ik}),&\cr&[\hat{X}_{ij},\hat{X}_{kl}]=i\lambda(\eta_{ik}\hat{X}_{jl}-\eta_{il}\hat{X}_{jk}-\eta_{jk}\hat{X}_{il}+\eta_{jl}\hat{X}_{ik}).&(10)}$

In this way we obtain an unification of the Snyder and $\kappa$-Poincaré
spacetimes using the formalism of extended coordinates $\hat{X}_{\mu\nu}$.

Example As an example, let us consider an $so(1,2;g)$ algebra. Generalization
to higher dimensions is straightforward. In this case,

$g_{\mu\nu}=\left(\matrix{-1&0&g_{0}\cr 0&1&g_{1}\cr
g_{0}&g_{1}&g_{2}\cr}\right).$ $None$

The matrix $O$ is defined up to an $SO(1,2)$ transformation. In fact, given an
$SO(1,2)$ matrix $R$, one has $OR\eta(OR)^{T}=O\eta O^{T}=g$. Assuming
$g_{2}+g_{0}^{2}-g_{1}^{2}>0$ and defining
$\rho=\sqrt{g_{2}+g_{0}^{2}-g_{1}^{2}}$, one can write down a particularly
simple expression for $O$, namely

$O_{\mu\nu}=\left(\matrix{1&0&0\cr 0&1&0\cr-g_{0}&g_{1}&\rho\cr}\right).$
$None$

Using (12), $\hat{X}_{\mu\nu}$ can then be written in terms of
$\hat{x}_{\mu\nu}$ as

$\hat{X}_{01}=\hat{x}_{01},\qquad\hat{X}_{02}=\rho\,\hat{x}_{02}+g_{1}\,\hat{x}_{01},\qquad\hat{X}_{12}=\rho\,\hat{x}_{12}+g_{0}\,\hat{x}_{01}.$
$None$

3\. Weyl representation of $so(1,N;g)$ with general metric Consider the
generalized Heisenberg algebra

$[x_{\mu\nu},x_{\rho\sigma}]=[p^{\mu\nu},p^{\rho\sigma}]=0,\qquad[x_{\mu\nu},p^{\rho\sigma}]=i(\delta_{\mu}^{\
\rho}\delta_{\nu}^{\ \sigma}-\delta_{\mu}^{\ \sigma}\delta_{\nu}^{\ \rho}),$
$None$

and define $X_{\mu\nu}=(O\,x\,O^{T})_{\mu\nu}$ and
$P^{\mu\nu}=(O^{\ddagger}\,p\,O^{-1})^{\mu\nu}$, where
$O^{\ddagger}=(O^{-1})^{T}$. These variables still satisfy the commutation
relations (14), but their indices are raised and lowered by means of the
metric $g_{\mu\nu}$.

We want now to find a representation of the $\hat{X}_{\mu\nu}$ defined in
sect. 1 in terms of the Heisenberg algebra generated by $X_{\mu\nu}$ and
$P_{\mu\nu}$. The $\hat{X}_{\mu\nu}$ satisfy the $so(1,N;g)$ algebra (6), that
we write as

$[\hat{X}_{\mu\nu},\hat{X}_{\rho\sigma}]=i\lambda
C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}\,\hat{X}_{\alpha\beta},$ $None$

where the structure constants are given by

$C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}={1\over
2}\Big{[}-g_{\nu\rho}(\delta_{\mu}^{\ \alpha}\delta_{\sigma}^{\
\beta}-\delta_{\mu}^{\ \beta}\delta_{\sigma}^{\
\alpha})+g_{\mu\sigma}(\delta_{\rho}^{\ \alpha}\delta_{\nu}^{\
\beta}-\delta_{\rho}^{\ \beta}\delta_{\nu}^{\
\alpha})-(\mu\leftrightarrow\nu)\Big{]}.$ $None$

The symmetry properties
$C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}=-C_{\nu\mu,\rho\sigma}^{\qquad{\alpha\beta}}=-C_{{\mu\nu},\sigma\rho}^{\qquad{\alpha\beta}}=-C_{{\mu\nu},\rho\sigma}^{\qquad\beta\alpha}$
$=-C_{\rho\sigma,{\mu\nu}}^{\qquad{\alpha\beta}}$ hold.

In general, if the operators $\hat{X}_{\mu\nu}$ generate a Lie algebra with
structure constants $C_{{\mu\nu},\rho\sigma}^{\qquad{\alpha\beta}}$, the
universal realization of $\hat{X}_{\mu\nu}$ in terms of the Heisenberg algebra
(14), corresponding to Weyl-symmetric ordering, is given by [8,13,14]

$\hat{X}_{\mu\nu}=X_{\alpha\beta}\left[{\lambda\,{\cal C}\over
1-e^{-\lambda\cal C}}\right]_{{\mu\nu}}^{\quad{\alpha\beta}},$ $None$

where ${\cal C}_{{\mu\nu}}^{\quad{\alpha\beta}}=-{1\over
2}\,C_{{\mu\nu}{\rho\sigma}}^{\qquad{\alpha\beta}}P^{\rho\sigma}$. This
realization enjoys the property

$e^{ik^{\mu\nu}\hat{X}_{\mu\nu}}\triangleright
1=e^{ik^{\mu\nu}X_{\mu\nu}},\qquad k^{\mu\nu}\in{\bf R},$ $None$

where the action $\triangleright$ is defined as

$X_{\mu\nu}\triangleright
f(X_{\alpha\beta})=X_{\mu\nu}f(X_{\alpha\beta}),\qquad
P^{\mu\nu}\triangleright f(X_{\alpha\beta})=-i{\partial
f(X_{\alpha\beta})\over\partial X_{\mu\nu}}=[P^{\mu\nu},f(X_{\alpha\beta})].$
$None$

From (17) it follows that the Weyl realization of $\hat{X}_{\mu\nu}$ in terms
of the generalized Heisenberg algebra generated by $X_{\mu\nu}$ and
$P^{\mu\nu}$ reads

$\hat{X}_{\mu\nu}=X_{\mu\nu}+{\lambda\over 2}X_{\alpha\beta}\,{\cal
C}_{{\mu\nu}}^{\quad{\alpha\beta}}+{\lambda^{2}\over
12}X_{\alpha\beta}\left({\cal
C}^{2}\right)_{{\mu\nu}}^{\quad{\alpha\beta}}+{\cal O}(\lambda^{4}).$ $None$

where

$\eqalignno{&{\cal C}_{{\mu\nu}}^{\quad{\alpha\beta}}={1\over
2}\Big{(}\delta_{\mu}^{\ \alpha}P_{\nu}^{\ \beta}+\delta_{\nu}^{\
\beta}P_{\mu}^{\ \alpha}-(\alpha\leftrightarrow\beta)\Big{)},&\cr&\left({\cal
C}^{2}\right)_{\mu\nu}^{\quad{\alpha\beta}}={1\over 2}\Big{(}2P_{\mu}^{\
\alpha}P_{\nu}^{\ \beta}+\delta_{\nu}^{\
\beta}P_{\mu\rho}P^{\rho\alpha}+\delta_{\mu}^{\
\alpha}P_{\nu\rho}P^{\rho\beta}-(\alpha\leftrightarrow\beta)\Big{)},&(21)}$

and the indices are lowered by means of the metric $g_{\mu\nu}$.

Inserting $\cal C$ in (20), we find up to order $\lambda^{2}$,

$\hat{X}_{\mu\nu}=\ X_{\mu\nu}+{\lambda\over 2}\left(X_{\mu\alpha}P_{\nu}^{\
\alpha}-X_{\nu\alpha}P_{\mu}^{\ \alpha}\right)-{\lambda^{2}\over
12}(X_{\mu\alpha}P_{\nu\beta}P^{\alpha\beta}-X_{\nu\alpha}P_{\mu\beta}P^{\alpha\beta}-2X_{\alpha\beta}P_{\mu}^{\
\alpha}P_{\nu}^{\ \beta}),$ $None$

and

$\eqalignno{&[\hat{X}_{\mu\nu},P^{\rho\sigma}]=i(\delta_{\mu}^{\
\rho}\delta_{\nu}^{\ \sigma}-\delta_{\mu}^{\ \sigma}\delta_{\nu}^{\
\rho})+{i\lambda\over 2}(\delta_{\mu}^{\ \rho}P_{\nu}^{\
\sigma}-\delta_{\nu}^{\ \rho}P_{\mu}^{\ \sigma}+\delta_{\nu}^{\
\sigma}P_{\mu}^{\ \rho}-\delta_{\mu}^{\ \sigma}P_{\nu}^{\
\rho})\cr&-{i\lambda^{2}\over 12}(\delta_{\mu}^{\
\rho}P_{\nu\alpha}P^{\sigma\alpha}-\delta_{\mu}^{\
\sigma}P_{\nu\alpha}P^{\rho\alpha}-\delta_{\nu}^{\
\rho}P_{\mu\alpha}P^{\sigma\alpha}+\delta_{\nu}^{\
\sigma}P_{\mu\alpha}P^{\rho\alpha}+2P_{\mu}^{\ \rho}P_{\nu}^{\
\sigma}-2P_{\nu}^{\ \rho}P_{\mu}^{\ \sigma}).\cr&&(23)}$

Using the decomposition of $\hat{X}_{\mu\nu}$ introduced in sect. 1 and
defining $X_{i}={1\over\kappa}X_{iN}$ and $P_{i}=\kappa P_{iN}$, we can then
write

$\eqalignno{\hat{X}_{i}&=X_{i}+{\lambda\over 2}\Big{(}X_{k}P_{i}^{\ k}-\beta
X_{ik}P^{k}-a_{i}X_{k}P^{k}+a_{k}X_{i}P^{k}+a_{j}X_{ik}P^{jk}\Big{)}+{\cal
O}(\lambda^{2}),&\cr\hat{X}_{ij}&=X_{ij}+{\lambda\over
2}\Big{(}X_{i}P_{j}+X_{ik}P_{j}^{\ k}+a_{i}X_{jk}P^{k}-(i\leftrightarrow
j)\Big{)}+{\cal O}(\lambda^{2}).&(24)}$

4\. Coproduct and star product in Weyl realization Formulae for coproduct and
deformed addition of momenta can be deduced from those of ref. [8], with the
difference that now the sums are performed with the curved metric (7) instead
of the flat metric. In particular, defining

$e^{{i\over 2}k^{{\mu\nu}}\hat{X}_{{\mu\nu}}}e^{{i\over
2}q^{\rho\sigma}\hat{X}_{{\rho\sigma}}}=e^{{i\over 2}(k^{{\mu\nu}}\oplus
q^{\mu\nu})\hat{X}_{{\mu\nu}}}\equiv e^{{i\over 2}{\cal
D}^{\mu\nu}(k,q)\hat{X}_{\mu\nu}},$ $None$

one has

${\cal D}^{\mu\nu}(k^{\alpha\beta},q^{\alpha\beta})=\
k^{\mu\nu}+q^{\mu\nu}-{\lambda\over 2}\Big{(}k^{\mu\alpha}q^{\nu}_{\
\alpha}-k^{\nu\alpha}q^{\mu}_{\ \alpha}\Big{)}+{\cal O}(\lambda^{2}).$ $None$

The coproduct $\Delta P^{\mu\nu}$ is then

$\Delta P^{\mu\nu}={\cal D}^{\mu\nu}(P^{\mu\nu}\otimes 1,1\otimes
P^{\mu\nu})=\ \Delta_{0}P^{\mu\nu}-{\lambda\over 2}\Big{(}P^{\mu\alpha}\otimes
P^{\nu}_{\ \alpha}-P^{\nu\alpha}\otimes P^{\mu}_{\ \alpha}\Big{)}+{\cal
O}(\lambda^{2}),\quad$ $None$

where $\Delta_{0}P^{\mu\nu}=P^{\mu\nu}\otimes 1+1\otimes P^{\mu\nu}$. The
coproduct (27) is coassociative. Generally, Lie-deformed quantum Minkowski
spaces admit both Hopf algebra and Hopf algebroid structures [15] and
coproduct of momenta are coassociative [14].

The functions ${\cal D}^{\mu\nu}(q,k)$ satisfy the symmetry properties

${\cal D}^{\mu\nu}(q,k)\big{|}_{\lambda}={\cal
D}^{\mu\nu}(k,q)\big{|}_{-\lambda}.$ $None$

Moreover,

$e^{{i\over 2}k^{\mu\nu}X_{\mu\nu}}\star e^{{i\over
2}q^{\rho\sigma}X_{\rho\sigma}}=e^{{i\over 2}{\cal
D}^{\mu\nu}(k,q)X_{\mu\nu}}.$ $None$

This star product is associative. ${\cal D}^{\mu\nu}$ can be decomposed as

$\eqalignno{{\cal D}^{i}(k,q)=&\ k^{i}+q^{i}-{\lambda\over
2}\Big{(}k_{k}q^{ik}-k^{ik}q_{k}+a_{k}(k^{k}q^{i}-k^{i}q^{k})\Big{)}+{\cal
O}(\lambda^{2})&\cr{\cal D}^{ij}(k,q)=&\ k^{ij}+q^{ij}-{\lambda\over
2}\Big{(}k^{ik}q^{j}_{\
k}+\beta\,k^{i}q^{j}+a_{k}(k^{ik}q^{j}+k^{i}q^{jk})-(i\leftrightarrow
j)\Big{)}+{\cal O}(\lambda^{2}),&(30)}$

with symmetry properties

${\cal D}^{i}(q,k)\big{|}_{\lambda}={\cal
D}^{i}(k,q)\big{|}_{-\lambda},\qquad{\cal D}^{ij}(q,k)\big{|}_{\lambda}={\cal
D}^{ij}(k,q)\big{|}_{-\lambda}.$ $None$

5\. The twist for the Weyl realization In this section, we construct the twist
operator at first order in $\lambda$, using the results of [8]. The twist is
defined as a bilinear operator such that $\Delta h={\cal F}\Delta_{0}h{\cal
F}^{-1}$ for each $h\in so(1,N)$.

The twist in a Hopf algebroid sense can be computed by means of the formula
[16]

${\cal F}^{-1}\equiv e^{F}=e^{-{i\over 2}P^{\mu\nu}\otimes
X_{\mu\nu}}e^{{i\over 2}P^{\rho\sigma}\otimes\hat{X}_{\rho\sigma}}.$ $None$

Using the CBH formula one gets

$F={i\over 2}\,P^{\mu\nu}\otimes(\hat{X}_{\mu\nu}-X_{\mu\nu})+\dots.$ $None$

Substituting (20) in (33), one obtains

$F={i\lambda\over 2}P^{\alpha\gamma}\otimes X_{\alpha\beta}P_{\gamma}^{\
\beta}+{\cal O}(\lambda^{2}).$ $None$

It is easy to check that

${\cal F}\Delta_{0}P^{\mu\nu}{\cal F}^{-1}=\Delta P^{\mu\nu},$ $None$

with $\Delta P^{\mu\nu}$ given in (27). In particular,

$\eqalignno{\Delta P^{i}=&\ \Delta_{0}P^{i}-{\lambda\over
2}\Big{(}P^{k}\otimes P^{i}_{\ k}-P^{ik}\otimes P_{k}+a_{k}(P^{k}\otimes
P^{i}-P^{i}\otimes P^{k})\Big{)}+{\cal O}(\lambda^{2}),\cr\Delta P^{ij}=&\
\Delta_{0}P^{ij}-{\lambda\over 2}\Big{(}P^{ik}\otimes P^{j}_{\
k}+\beta\,P^{i}\otimes P^{j}+a_{k}(P^{ik}\otimes P^{j}+P^{i}\otimes
P^{jk})\cr&-(i\leftrightarrow j)\Big{)}+{\cal O}(\lambda^{2}),&(36)}$

where

$\Delta_{0}P^{i}=P^{i}\otimes 1+1\otimes
P^{i},\qquad\Delta_{0}P^{ij}=P^{ij}\otimes 1+1\otimes P^{ij}.$ $None$

6\. Final remarks Using the formalism introduced in ref. [9], we have been
able to unify the extended Snyder model with the $\kappa$-Minkowski spacetime.
In contrast with the unified model proposed in ref. [12], the resulting
coalgebra is coassociative, but this is obtained at the price of introducing
new tensorial variables, whose physical interpretation is not evident,
although they should be considered as internal degrees of freedom.

Our result could be useful for the construction of dispersion relations [17]
and an associative field theory on a general $\kappa$-deformed Snyder
spacetime.

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