# Observation of $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$

M. Ablikim1, M. N. Achasov10,c, P. Adlarson67, S. Ahmed15, M. Albrecht4, R.
Aliberti28, A. Amoroso66A,66C, M. R. An32, Q. An63,49, X. H. Bai57, Y. Bai48,
O. Bakina29, R. Baldini Ferroli23A, I. Balossino24A, Y. Ban38,k, K.
Begzsuren26, N. Berger28, M. Bertani23A, D. Bettoni24A, F. Bianchi66A,66C, J.
Bloms60, A. Bortone66A,66C, I. Boyko29, R. A. Briere5, H. Cai68, X. Cai1,49,
A. Calcaterra23A, G. F. Cao1,54, N. Cao1,54, S. A. Cetin53A, J. F. Chang1,49,
W. L. Chang1,54, G. Chelkov29,b, D. Y. Chen6, G. Chen1, H. S. Chen1,54, M. L.
Chen1,49, S. J. Chen35, X. R. Chen25, Y. B. Chen1,49, Z. J Chen20,l, W. S.
Cheng66C, G. Cibinetto24A, F. Cossio66C, X. F. Cui36, H. L. Dai1,49, X. C.
Dai1,54, A. Dbeyssi15, R. E. de Boer4, D. Dedovich29, Z. Y. Deng1, A. Denig28,
I. Denysenko29, M. Destefanis66A,66C, F. De Mori66A,66C, Y. Ding33, C. Dong36,
J. Dong1,49, L. Y. Dong1,54, M. Y. Dong1,49,54, X. Dong68, S. X. Du71, Y. L.
Fan68, J. Fang1,49, S. S. Fang1,54, Y. Fang1, R. Farinelli24A, L. Fava66B,66C,
F. Feldbauer4, G. Felici23A, C. Q. Feng63,49, J. H. Feng50, M. Fritsch4, C. D.
Fu1, Y. Gao64, Y. Gao38,k, Y. Gao63,49, Y. G. Gao6, I. Garzia24A,24B, P. T.
Ge68, C. Geng50, E. M. Gersabeck58, A Gilman61, K. Goetzen11, L. Gong33, W. X.
Gong1,49, W. Gradl28, M. Greco66A,66C, L. M. Gu35, M. H. Gu1,49, S. Gu2, Y. T.
Gu13, C. Y Guan1,54, A. Q. Guo22, L. B. Guo34, R. P. Guo40, Y. P. Guo9,h, A.
Guskov29, T. T. Han41, W. Y. Han32, X. Q. Hao16, F. A. Harris56, N
Hüsken22,28, K. L. He1,54, F. H. Heinsius4, C. H. Heinz28, T. Held4, Y. K.
Heng1,49,54, C. Herold51, M. Himmelreich11,f, T. Holtmann4, Y. R. Hou54, Z. L.
Hou1, H. M. Hu1,54, J. F. Hu47,m, T. Hu1,49,54, Y. Hu1, G. S. Huang63,49, L.
Q. Huang64, X. T. Huang41, Y. P. Huang1, Z. Huang38,k, T. Hussain65, W.
Ikegami Andersson67, W. Imoehl22, M. Irshad63,49, S. Jaeger4, S. Janchiv26,j,
Q. Ji1, Q. P. Ji16, X. B. Ji1,54, X. L. Ji1,49, Y. Y. Ji41, H. B. Jiang41, X.
S. Jiang1,49,54, J. B. Jiao41, Z. Jiao18, S. Jin35, Y. Jin57, T. Johansson67,
N. Kalantar-Nayestanaki55, X. S. Kang33, R. Kappert55, M. Kavatsyuk55, B. C.
Ke43,1, I. K. Keshk4, A. Khoukaz60, P. Kiese28, R. Kiuchi1, R. Kliemt11, L.
Koch30, O. B. Kolcu53A,e, B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc67, M.
G. Kurth1,54, W. Kühn30, J. J. Lane58, J. S. Lange30, P. Larin15, A.
Lavania21, L. Lavezzi66A,66C, Z. H. Lei63,49, H. Leithoff28, M. Lellmann28, T.
Lenz28, C. Li39, C. H. Li32, Cheng Li63,49, D. M. Li71, F. Li1,49, G. Li1, H.
Li43, H. Li63,49, H. B. Li1,54, H. J. Li16, J. L. Li41, J. Q. Li4, J. S. Li50,
Ke Li1, L. K. Li1, Lei Li3, P. R. Li31, S. Y. Li52, W. D. Li1,54, W. G. Li1,
X. H. Li63,49, X. L. Li41, Xiaoyu Li1,54, Z. Y. Li50, H. Liang63,49, H.
Liang1,54, H. Liang27, Y. F. Liang45, Y. T. Liang25, G. R. Liao12, L. Z.
Liao1,54, J. Libby21, C. X. Lin50, B. J. Liu1, C. X. Liu1, D. Liu63,49, F. H.
Liu44, Fang Liu1, Feng Liu6, H. B. Liu13, H. M. Liu1,54, Huanhuan Liu1, Huihui
Liu17, J. B. Liu63,49, J. L. Liu64, J. Y. Liu1,54, K. Liu1, K. Y. Liu33, Ke
Liu6, L. Liu63,49, M. H. Liu9,h, P. L. Liu1, Q. Liu54, Q. Liu68, S. B.
Liu63,49, Shuai Liu46, T. Liu1,54, W. M. Liu63,49, X. Liu31, Y. Liu31, Y. B.
Liu36, Z. A. Liu1,49,54, Z. Q. Liu41, X. C. Lou1,49,54, F. X. Lu16, F. X.
Lu50, H. J. Lu18, J. D. Lu1,54, J. G. Lu1,49, X. L. Lu1, Y. Lu1, Y. P. Lu1,49,
C. L. Luo34, M. X. Luo70, P. W. Luo50, T. Luo9,h, X. L. Luo1,49, S. Lusso66C,
X. R. Lyu54, F. C. Ma33, H. L. Ma1, L. L. Ma41, M. M. Ma1,54, Q. M. Ma1, R. Q.
Ma1,54, R. T. Ma54, X. X. Ma1,54, X. Y. Ma1,49, F. E. Maas15, M.
Maggiora66A,66C, S. Maldaner4, S. Malde61, Q. A. Malik65, A. Mangoni23B, Y. J.
Mao38,k, Z. P. Mao1, S. Marcello66A,66C, Z. X. Meng57, J. G. Messchendorp55,
G. Mezzadri24A, T. J. Min35, R. E. Mitchell22, X. H. Mo1,49,54, Y. J. Mo6, N.
Yu. Muchnoi10,c, H. Muramatsu59, S. Nakhoul11,f, Y. Nefedov29, F. Nerling11,f,
I. B. Nikolaev10,c, Z. Ning1,49, S. Nisar8,i, S. L. Olsen54, Q. Ouyang1,49,54,
S. Pacetti23B,23C, X. Pan9,h, Y. Pan58, A. Pathak1, P. Patteri23A, M.
Pelizaeus4, H. P. Peng63,49, K. Peters11,f, J. Pettersson67, J. L. Ping34, R.
G. Ping1,54, R. Poling59, V. Prasad63,49, H. Qi63,49, H. R. Qi52, K. H. Qi25,
M. Qi35, T. Y. Qi9, T. Y. Qi2, S. Qian1,49, W. B. Qian54, Z. Qian50, C. F.
Qiao54, L. Q. Qin12, X. P. Qin9, X. S. Qin41, Z. H. Qin1,49, J. F. Qiu1, S. Q.
Qu36, K. H. Rashid65, K. Ravindran21, C. F. Redmer28, A. Rivetti66C, V.
Rodin55, M. Rolo66C, G. Rong1,54, Ch. Rosner15, M. Rump60, H. S. Sang63, A.
Sarantsev29,d, Y. Schelhaas28, C. Schnier4, K. Schoenning67, M.
Scodeggio24A,24B, D. C. Shan46, W. Shan19, X. Y. Shan63,49, J. F. Shangguan46,
M. Shao63,49, C. P. Shen9, P. X. Shen36, X. Y. Shen1,54, H. C. Shi63,49, R. S.
Shi1,54, X. Shi1,49, X. D Shi63,49, J. J. Song41, W. M. Song27,1, Y. X.
Song38,k, S. Sosio66A,66C, S. Spataro66A,66C, K. X. Su68, P. P. Su46, F. F.
Sui41, G. X. Sun1, H. K. Sun1, J. F. Sun16, L. Sun68, S. S. Sun1,54, T.
Sun1,54, W. Y. Sun34, W. Y. Sun27, X Sun20,l, Y. J. Sun63,49, Y. K. Sun63,49,
Y. Z. Sun1, Z. T. Sun1, Y. H. Tan68, Y. X. Tan63,49, C. J. Tang45, G. Y.
Tang1, J. Tang50, J. X. Teng63,49, V. Thoren67, W. H. Tian43, Y. T. Tian25, I.
Uman53B, B. Wang1, C. W. Wang35, D. Y. Wang38,k, H. J. Wang31, H. P. Wang1,54,
K. Wang1,49, L. L. Wang1, M. Wang41, M. Z. Wang38,k, Meng Wang1,54, W. Wang50,
W. H. Wang68, W. P. Wang63,49, X. Wang38,k, X. F. Wang31, X. L. Wang9,h, Y.
Wang50, Y. Wang63,49, Y. D. Wang37, Y. F. Wang1,49,54, Y. Q. Wang1, Y. Y.
Wang31, Z. Wang1,49, Z. Y. Wang1, Ziyi Wang54, Zongyuan Wang1,54, D. H. Wei12,
P. Weidenkaff28, F. Weidner60, S. P. Wen1, D. J. White58, U. Wiedner4, G.
Wilkinson61, M. Wolke67, L. Wollenberg4, J. F. Wu1,54, L. H. Wu1, L. J.
Wu1,54, X. Wu9,h, Z. Wu1,49, L. Xia63,49, H. Xiao9,h, S. Y. Xiao1, Z. J.
Xiao34, X. H. Xie38,k, Y. G. Xie1,49, Y. H. Xie6, T. Y. Xing1,54, G. F. Xu1,
Q. J. Xu14, W. Xu1,54, X. P. Xu46, Y. C. Xu54, F. Yan9,h, L. Yan9,h, W. B.
Yan63,49, W. C. Yan71, Xu Yan46, H. J. Yang42,g, H. X. Yang1, L. Yang43, S. L.
Yang54, Y. X. Yang12, Yifan Yang1,54, Zhi Yang25, M. Ye1,49, M. H. Ye7, J. H.
Yin1, Z. Y. You50, B. X. Yu1,49,54, C. X. Yu36, G. Yu1,54, J. S. Yu20,l, T.
Yu64, C. Z. Yuan1,54, L. Yuan2, X. Q. Yuan38,k, Y. Yuan1, Z. Y. Yuan50, C. X.
Yue32, A. Yuncu53A,a, A. A. Zafar65, Y. Zeng20,l, B. X. Zhang1, Guangyi
Zhang16, H. Zhang63, H. H. Zhang50, H. H. Zhang27, H. Y. Zhang1,49, J. J.
Zhang43, J. L. Zhang69, J. Q. Zhang34, J. W. Zhang1,49,54, J. Y. Zhang1, J. Z.
Zhang1,54, Jianyu Zhang1,54, Jiawei Zhang1,54, L. M. Zhang52, L. Q. Zhang50,
Lei Zhang35, S. Zhang50, S. F. Zhang35, Shulei Zhang20,l, X. D. Zhang37, X. Y.
Zhang41, Y. Zhang61, Y. H. Zhang1,49, Y. T. Zhang63,49, Yan Zhang63,49, Yao
Zhang1, Yi Zhang9,h, Z. H. Zhang6, Z. Y. Zhang68, G. Zhao1, J. Zhao32, J. Y.
Zhao1,54, J. Z. Zhao1,49, Lei Zhao63,49, Ling Zhao1, M. G. Zhao36, Q. Zhao1,
S. J. Zhao71, Y. B. Zhao1,49, Y. X. Zhao25, Z. G. Zhao63,49, A.
Zhemchugov29,b, B. Zheng64, J. P. Zheng1,49, Y. Zheng38,k, Y. H. Zheng54, B.
Zhong34, C. Zhong64, L. P. Zhou1,54, Q. Zhou1,54, X. Zhou68, X. K. Zhou54, X.
R. Zhou63,49, X. Y. Zhou32, A. N. Zhu1,54, J. Zhu36, K. Zhu1, K. J.
Zhu1,49,54, S. H. Zhu62, T. J. Zhu69, W. J. Zhu9,h, W. J. Zhu36, Y. C.
Zhu63,49, Z. A. Zhu1,54, 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 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9 Fudan University, Shanghai 200443, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 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
24 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
25 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27 Jilin University, Changchun 130012, People’s Republic of China
28 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31 Lanzhou University, Lanzhou 730000, People’s Republic of China
32 Liaoning Normal University, Dalian 116029, People’s Republic of China
33 Liaoning University, Shenyang 110036, People’s Republic of China
34 Nanjing Normal University, Nanjing 210023, People’s Republic of China
35 Nanjing University, Nanjing 210093, People’s Republic of China
36 Nankai University, Tianjin 300071, People’s Republic of China
37 North China Electric Power University, Beijing 102206, People’s Republic of
China
38 Peking University, Beijing 100871, People’s Republic of China
39 Qufu Normal University, Qufu 273165, People’s Republic of China
40 Shandong Normal University, Jinan 250014, People’s Republic of China
41 Shandong University, Jinan 250100, People’s Republic of China
42 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
43 Shanxi Normal University, Linfen 041004, People’s Republic of China
44 Shanxi University, Taiyuan 030006, People’s Republic of China
45 Sichuan University, Chengdu 610064, People’s Republic of China
46 Soochow University, Suzhou 215006, People’s Republic of China
47 South China Normal University, Guangzhou 510006, People’s Republic of China
48 Southeast University, Nanjing 211100, People’s Republic of China
49 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
50 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
51 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
52 Tsinghua University, Beijing 100084, People’s Republic of China
53 Turkish Accelerator Center Particle Factory Group, (A)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (B)Near East University, Nicosia,
North Cyprus, Mersin 10, Turkey
54 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
55 University of Groningen, NL-9747 AA Groningen, The Netherlands
56 University of Hawaii, Honolulu, Hawaii 96822, USA
57 University of Jinan, Jinan 250022, People’s Republic of China
58 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
59 University of Minnesota, Minneapolis, Minnesota 55455, USA
60 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
61 University of Oxford, Keble Rd, Oxford, UK OX13RH
62 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
63 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
64 University of South China, Hengyang 421001, People’s Republic of China
65 University of the Punjab, Lahore-54590, Pakistan
66 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
67 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
68 Wuhan University, Wuhan 430072, People’s Republic of China
69 Xinyang Normal University, Xinyang 464000, People’s Republic of China
70 Zhejiang University, Hangzhou 310027, People’s Republic of China
71 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Bogazici University, 34342 Istanbul, Turkey
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
d Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
e Also at Istanbul Arel University, 34295 Istanbul, Turkey
f Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
g 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
h 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
i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA
j Currently at: Institute of Physics and Technology, Peace Ave.54B,
Ulaanbaatar 13330, Mongolia
k Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
l School of Physics and Electronics, Hunan University, Changsha 410082, China
m Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China

###### Abstract

Using 2.93 fb-1 of $e^{+}e^{-}$ collision data taken with the BESIII detector
at a center-of-mass energy of 3.773 $\rm\,GeV$, the observation of the
$D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ semileptonic decay is presented. The
statistical significance of the decay $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$
is greater than $10\sigma$. The branching fraction of $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$ is measured to be $(1.09\pm
0.13^{+0.09}_{-0.16}\pm 0.12)\times 10^{-3}$. Here, the first uncertainty is
statistical, the second is systematic, and the third originates from the
assumed branching fraction of $K_{1}(1270)^{-}\rightarrow
K^{-}\pi^{+}\pi^{-}$. The fraction of longitudinal polarization in $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$ is determined for the first time to be $0.50\pm
0.19_{\rm stat}\pm 0.08_{\rm syst}$.

###### pacs:

13.20.Fc, 12.15.Hh

Semileptonic (SL) $D$ decays offer a good testbed to understand
nonperturbative strong-interaction dynamics in weak decays isgw ; isgw2 .
Studies of the SL $D^{0(+)}$ decays into the strange axial-vector mesons
$K_{1}(1270)$ or $K_{1}(1400)$ are especially appealing. Reference wangwei
points out that the combined measurements of
$D^{0(+)}\to\bar{K}_{1}(1270)\ell^{+}\nu_{\ell}$ and $B\to K_{1}(1270)\gamma$
provide a possible way to determine the photon polarization in $b\to s\gamma$
transitions without considerable theoretical ambiguity. Knowledge of the $b\to
s\gamma$ photon polarization plays a unique role in probing right-handed
couplings in new physics wangwei ; Atwood ; Becirevic . As LHCb reported a
large up-down asymmetry in $B^{-}\to K^{-}_{\rm res}(\to
K^{-}\pi^{+}\pi^{-})\gamma$ in the $K^{-}\pi^{+}\pi^{-}$ invariant mass bin of
[1.1, 1.3] GeV/$c^{2}$ which is dominated by $K_{1}(1270)^{-}$ contribution
lhcbk1g . Therefore, the decay $D^{0}\to K_{1}(1270)^{-}(\to
K^{-}\pi^{+}\pi^{-})\ell^{+}\nu_{\ell}$ is particularly desired to quantify
the hadronic effects of $K_{1}(1270)^{-}\to K^{-}\pi^{+}\pi^{-}$. Throughout
this Letter, charged conjugated modes are always implied.

To date, the $K_{1}(1270)$ and $K_{1}(1400)$ mesons have been extensively
investigated in $\tau$, $D$, $B$, and charmonium decays 4 ; 5 ; 6 ; 7 ; 8 ; 14
; jpsi ; chic0 ; 1 ; 2 . In theory, the physical mass eigenstates of
$K_{1}(1270)$ and $K_{1}(1400)$ are decomposed as mixtures of the ${}^{1}\rm
P_{1}$ and ${}^{3}\rm P_{1}$ states with a mixing angle $\theta_{K_{1}}$.
Various approaches were proposed to extract $\theta_{K_{1}}$, but with very
different results 18 ; 19 ; 20 ; 21 ; 22 ; 24 ; 26 ; 27 . Experimental
measurements of $D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$ offer deeper
insight into the mixing angle $\theta_{K_{1}}$, which is essential for
reliable calculations describing the $\tau$ 18 , $B$ 20 ; B-K1g , and $D$ D-AP
; D-AP-1  decays involving $K_{1}$, and for investigations in the field of
hadron spectroscopy Burakovsky .

The branching fractions (BFs) of $D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$
have been computed with different models: the Isgur-Scora-Grinstein-Wise
(ISGW) quark model isgw  and its update, ISGW2 isgw2 , three-point QCD sum
rules (3PSR) Khosravi , covariant light-front quark model (CLFQM) theory , and
the light-cone QCD sum rules (LCSR) formfactor ; Momeni . The predicted BFs,
which are sensitive to $\theta_{K_{1}}$ and its sign, vary from $10^{-3}$ to
$10^{-2}$ Khosravi ; theory ; formfactor . Measurements of these decay BFs and
related longitudinal polarization are key to testing different theoretical
calculations and understanding the weak-decay mechanisms of $D$ mesons. For
example, assuming isospin symmetry, the ratio of the partial decay widths for
the SL $D^{0(+)}$ decays, which are both mediated via $c\to se^{+}\nu_{e}$, is
expected to be unity soni2 . Measuring the BFs thus allows a test of isospin
invariance in $D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$. Large $D^{0}\to
K_{1}(1270)^{-}\ell^{+}\nu_{\ell}$ samples also supply a clean environment,
with no additional hadrons in the final state, to accurately determine the
mass and width of $K_{1}(1270)$, and to explore the relative strengths and
phases of $K_{1}(1270)^{-}$ decays into various final states that differ
considerably with its neutral counterpart $\bar{K}_{1}(1270)^{0}$, which
currently all suffer large uncertainties.

An observation of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ was previously
reported by BESIII bes3-k1ev . However, the only evidence for $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$ was reported by CLEO cleores . This Letter
presents an observation of $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ by using
2.93 fb-1 of $e^{+}e^{-}$ collision data lum  recorded at a center-of-mass
energy $\sqrt{s}=3.773$ GeV with the BESIII detector BESIII .

Details about the design and performance of the BESIII detector are given in
Ref. BESIII . Simulated samples produced with a geant4-based geant4  Monte
Carlo (MC) package, which includes the geometric description of the BESIII
detector and the detector response, are used to determine the detection
efficiency and to estimate the backgrounds. The simulation includes the beam-
energy spread and initial-state radiation (ISR) in the $e^{+}e^{-}$
annihilations modeled with the generator kkmc ref:kkmc . The inclusive MC
samples consist of the production of the $D\bar{D}$ pairs, the non-$D\bar{D}$
decays of the $\psi(3770)$, the ISR production of the $J/\psi$ and
$\psi(3686)$ states, and the continuum processes incorporated in kkmc ref:kkmc
. The known decay modes are modeled with evtgen ref:evtgen  using BFs taken
from the Particle Data Group pdg2020 , and the remaining unknown decays from
the charmonium states with lundcharm ref:lundcharm . Final-state radiation
(FSR) from charged final-state particles is incorporated with the photos
package photos . The $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ decay is simulated
with the ISGW2 model isgw2  and the $K_{1}(1270)^{-}$ is allowed to decay into
all intermediate processes with final state of $K^{-}\pi^{+}\pi^{-}$. The
$K_{1}(1270)^{-}$ resonance shape is parameterized by a relativistic Breit-
Wigner function with mass of $(1.253\pm 0.007)$ GeV/$c^{2}$ and width of
$(90\pm 20)$ MeV pdg2020 . The BFs of $K_{1}(1270)$ subdecays measured by
Belle non-resonance  are input, since they give better data/MC consistency
than those reported in Ref. pdg2020 .

At $\sqrt{s}=3.773$ GeV, $\bar{D}^{0}$ and $D^{0}$ mesons are produced in
pairs. The momenta of $\bar{D}^{0}$ and $D^{0}$ are equal and in opposite
directions. This advantage allows to study the $D$ decays with the double-tag
(DT) technique first developed by Mark III dtref . The $\bar{D}^{0}$ mesons
are reconstructed by their hadronic decays to $K^{+}\pi^{-}$,
$K^{+}\pi^{-}\pi^{0}$, and $K^{+}\pi^{-}\pi^{-}\pi^{+}$. These inclusively
selected events are referred to as single-tag (ST) $\bar{D}^{0}$ mesons. In
the presence of the ST $\bar{D}^{0}$ mesons, candidates for $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$ are selected to form DT events. For a given tag
mode, the BF of $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$, ${\mathcal{B}}_{\rm
SL}$, is obtained by

${\mathcal{B}}_{\rm SL}=N_{\mathrm{DT}}/(N_{\mathrm{ST}}\cdot\varepsilon_{\rm
SL}\cdot{\mathcal{B}}_{\rm sub}),$ (1)

where $N_{\rm ST}$ and $N_{\rm DT}$ are the ST and DT yields in data,
$\varepsilon_{\rm SL}=\varepsilon_{\rm DT}/\varepsilon_{\rm ST}$ is the
efficiency of detecting the SL decay in the presence of the ST $\bar{D}^{0}$,
and ${\mathcal{B}}_{\rm sub}$ is the BF of $K_{1}(1270)^{-}\to
K^{-}\pi^{+}\pi^{-}$. $\varepsilon_{\rm ST}$ and $\varepsilon_{\rm DT}$ are
the efficiencies of selecting the ST and DT candidates, respectively.

This analysis uses the same selection criteria of $K^{\pm}$, $\pi^{\pm}$, and
$\pi^{0}$ as in Refs. BESIII:2018sjg ; BESIII:2018ccy ; BESIII:2018qmf ;
BES3_FSR . The ST $\bar{D}^{0}$ mesons are identified by the energy difference
$\Delta E\equiv E_{\bar{D}^{0}}-E_{\rm beam}$ and the beam-constrained mass
$M_{\rm BC}\equiv\sqrt{E_{\mathrm{beam}}^{2}-|\vec{p}_{\bar{D}^{0}}|^{2}}$,
where $E_{\mathrm{beam}}$ is the beam energy, $E_{\bar{D}^{0}}$ and
$\vec{p}_{\bar{D}^{0}}$ are the total energy and momentum of the ST
$\bar{D}^{0}$ in the $e^{+}e^{-}$ rest frame. If there are multiple
combinations in an event, the combination with the smallest $|\Delta E|$ is
chosen for each tag mode. Combinatorial backgrounds in the $M_{\rm BC}$
distributions are suppressed by requiring $\Delta E$ within $(-29,\,27)$,
$(-69,\,38)$, and $(-31,\,28)$ MeV for $\bar{D}^{0}\to K^{+}\pi^{-}$,
$K^{+}\pi^{-}\pi^{0}$, and $K^{+}\pi^{-}\pi^{-}\pi^{+}$, respectively.

The $M_{\rm BC}$ distributions of the accepted ST candidates in data for the
three tag modes are shown in Fig. 1. To extract the ST yield for each tag
mode, an unbinned maximum-likelihood fit is performed to the corresponding
$M_{\rm BC}$ distribution. The signal is described by the MC-simulated shape
convolved with a double-Gaussian function accounting for the resolution
difference between data and MC simulation, and the background is modeled by an
ARGUS function ARGUS . Fit results are shown in Fig. 1. Events within $M_{\rm
BC}\in(1.858,\,1.874)$ GeV/$c^{2}$ are kept for further analysis. The ST
yields for the $\bar{D}^{0}\to K^{+}\pi^{-}$, $K^{+}\pi^{-}\pi^{0}$, and
$K^{+}\pi^{-}\pi^{-}\pi^{+}$ tag modes are $542153\pm 774_{\rm stat}$,
$1080690\pm 1727_{\rm stat}$, and $737036\pm 1712_{\rm stat}$, respectively.

Fig. 1:  Fits to the $M_{\rm BC}$ distributions of the ST candidates in data.
Points with error bars are data. Blue solid curves are the fit results and red
dashed curves represent the background contributions of the fit. Pair of red
arrows in each subfigure indicate the $M_{\rm BC}$ window.

Particles recoiling against the ST $\bar{D}^{0}$ candidates are used to
reconstruct candidates for $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$. The
$K_{1}(1270)^{-}$ is reconstructed via $K_{1}(1270)^{-}\to
K^{-}\pi^{+}\pi^{-}$. The $K^{-}$ and $\pi^{\pm}$ candidates are selected with
the same criteria as the tag side. Positron particle identification (PID) uses
the combined information from the specific ionization energy loss ($dE/dx$),
time of flight, and electromagnetic calorimeter (EMC), with which we calculate
the combined confidence levels under positron, pion, and kaon hypotheses
$CL_{e}$, $CL_{\pi}$ and $CL_{K}$. Positron candidate is required to satisfy
$CL_{e}/(CL_{e}+CL_{\pi}+CL_{K})>0.8$. To reduce backgrounds from hadrons and
muons, the positron candidate is required to satisfy $E/p>0.8$, where $E$ is
the energy deposited in the EMC and $p$ is the momentum measured by the
multilayer drift chamber (MDC). No additional charged track is allowed in the
event.

To distinguish positrons from backgrounds related to hadrons, the positron
candidates are required to satisfy $E/p-0.38>0.14\times\chi^{e}_{dE/dx}$,
where $\chi^{e}_{dE/dx}$ is the $dE/dx$ $\chi^{2}$ with the positron
hypothesis, respectively. To suppress the background from $D^{0}\to
K^{-}\pi^{+}\pi^{-}\pi^{+}$, we require
$M_{K^{-}\pi^{+}\pi^{-}\pi^{+}_{e\to\pi}}<1.8~{}{\rm GeV}/c^{2}$, where
$\pi^{+}_{e\to\pi}$ is the positron candidate reconstructed with the pion mass
hypothesis. To suppress the background from $D^{0}\to
K^{-}\pi^{+}\pi^{0}\,(\pi^{0})$, with $\pi^{0}\to e^{+}e^{-}\gamma$ (and
missing another $\pi^{0}$), the opening angle between $e^{+}$ and $\pi^{-}$
($\theta_{a}$) is required to satisfy $\cos\theta_{a}<0.94$. To suppress the
background from $D^{0}\to K^{-}\pi^{+}\pi^{-}\pi^{+}\pi^{0}$, we require
$M_{K^{-}\pi^{+}\pi^{-}\pi^{+}_{e\to\pi}\pi^{0}}<1.4~{}{\rm GeV}/c^{2}$ when
there is at least one reconstructed $\pi^{0}$ among the photons recoiling
against the ST $\bar{D}^{0}$ meson in an event. Furthermore, the opening angle
between the missing momentum (defined below) and the most energetic unused
shower ($\theta_{b}$) is required to satisfy $\cos\theta_{b}<0.81$. To
suppress the background from $D^{0}\to K^{-}\pi^{0}e^{+}\nu_{e}$ with
$\pi^{0}\to e^{+}e^{-}\gamma$, we require $M_{\rm\pi^{+}\pi^{-}}>0.31~{}{\rm
GeV}/c^{2}$. Background involving $K^{0}_{S}$ decay is suppressed by requiring
$M_{\rm\pi^{+}\pi^{-}}$ outside the interval $(0.488,0.508)$ GeV/$c^{2}$. For
the $\bar{D}^{0}\to K^{+}\pi^{-}\pi^{0}$ tag mode, combinatorial background
from $D^{-}\to K^{+}\pi^{-}\pi^{-}$ vs. $D^{+}\to K^{-}\pi^{+}X$ is suppressed
by requiring the difference between the beam-energy and the energy of the
$(K^{+}\pi^{-})_{\rm tag}\pi_{\rm sig}^{-}$ combination to be greater than 8
MeV.

Information concerning the undetectable neutrino is inferred by the kinematic
quantity $M^{2}_{\mathrm{miss}}\equiv
E^{2}_{\mathrm{miss}}-|\vec{p}_{\mathrm{miss}}|^{2}$, where
$E_{\mathrm{miss}}$ and $\vec{p}_{\mathrm{miss}}$ are the missing energy and
momentum of the SL candidate, respectively, calculated by
$E_{\mathrm{miss}}\equiv E_{\mathrm{beam}}-\Sigma_{j}E_{j}$ and
$\vec{p}_{\mathrm{miss}}\equiv-\vec{p}_{\bar{D}^{0}}-\Sigma_{j}\vec{p}_{j}$ in
the $e^{+}e^{-}$ center-of-mass frame. The index $j$ sums over the $K^{-}$,
$\pi^{+}$, $\pi^{-}$ and $e^{+}$ of the signal candidate, and $E_{j}$ and
$\vec{p}_{j}$ are the energy and momentum of the $j$-th particle,
respectively. To partially recover the energy lost to FSR and bremsstrahlung,
the four-momenta of photon(s) within $5^{\circ}$ of the initial positron
direction are added to the positron four-momentum measured by the MDC. To
improve the $M^{2}_{\mathrm{miss}}$ resolution, all the candidate tracks plus
the missing neutrino are subjected to a kinematic fit requiring energy and
momentum conservation, as well as the invariant masses of the $\bar{D}^{0}$
and $D^{0}$ candidate particles being constrained to the nominal $D^{0}$ mass.
The momenta from the kinematic fit are used to calculate
$M^{2}_{\mathrm{miss}}$.

Fig. 2:  (a) Distribution of $M_{K^{-}\pi^{+}\pi^{-}}$ vs. $M^{2}_{\rm miss}$
of the DT candidate events. Projections of the 2D fit to (b) $M^{2}_{\rm
miss}$ and (c) $M_{K^{-}\pi^{+}\pi^{-}}$. The distributions are summed over
all three tags. In (b) and (c), points with error bars are data; blue solid,
red dotted, green dashed, and black dashed curves are total fit, signal,
peaking background of $D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}$, and other
background, respectively. In (b), the peaking background concentrating around
0.033 GeV${}^{2}/c^{4}$ is from $D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$.

Figure 2(a) shows the distribution of $M_{K^{-}\pi^{+}\pi^{-}}$ vs.
$M^{2}_{\rm miss}$ of the accepted $D^{0}\to K^{-}\pi^{+}\pi^{-}e^{+}\nu_{e}$
candidate events in data after combining all tag modes. A clear signal, which
concentrates around the $K_{1}(1270)^{-}$ nominal mass in the
$M_{K^{-}\pi^{+}\pi^{-}}$ distribution and around zero in the $M^{2}_{\rm
miss}$ distribution, can be seen. The DT yield is obtained from a two-
dimensional (2D) unbinned extended maximum-likelihood simultaneous fit to the
data for the three tags. In the fit, the 2D signal shape is described by the
MC-simulated shape extracted from the signal MC events of $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$. The 2D shapes of the peaking background of
$D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}$ and the other backgrounds are modeled by
those derived from the inclusive MC sample. The number of peaking background
events from $D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}$ is fixed at the simulated
value, and the number of the other backgrounds is a free parameter. The smooth
2D probability density functions of signal and background are modeled by using
RooNDKeysPdf roofit ; class . The signal efficiencies with the ST modes
$\bar{D}^{0}\to K^{+}\pi^{-}$, $K^{+}\pi^{-}\pi^{0}$, and
$K^{+}\pi^{-}\pi^{-}\pi^{+}$ are $(14.08\pm 0.14_{\rm stat})\%$, $(13.38\pm
0.10_{\rm stat})\%$, and $(11.22\pm 0.10_{\rm stat})\%$, respectively. The BFs
given by the three tags are constrained to have the same value in the fit. The
2D fit projections to the $M^{2}_{\rm miss}$ and $M_{K^{-}\pi^{+}\pi^{-}}$
distributions are shown in Figs. 2(b) and 2(c), respectively. From the fit, we
obtain the DT yield of $N_{\rm DT}=109.0\pm 12.5_{\rm stat}$. The statistical
significance of the signal is estimated to be greater than 10$\sigma$, by
comparing the likelihoods with and without the signal component, and taking
the change in the number of degrees of freedom into account. The fitted
product of the BFs for $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ and
$K_{1}(1270)^{-}\to K^{-}\pi^{+}\pi^{-}$ is

${\mathcal{B}}_{\rm SL}\cdot{\mathcal{B}}_{\rm sub}=(3.59\pm 0.41_{\rm
stat}{{}^{+0.31}_{-0.52}}_{\rm syst})\times 10^{-4}.$

The reliability of the MC simulation is verified since the data distributions
of momenta and $\cos\theta$ of $K^{-}$, $\pi^{+}$, $\pi^{-}$ and $e^{+}$ as
well as invariant masses of $K^{-}\pi^{+}$ and $\pi^{+}\pi^{-}$ are consistent
with those of MC simulations.

In the BF measurement, the DT method ensures that most uncertainties arising
from the ST selection cancel. The uncertainty from the ST yield is assigned to
be 0.5%, by examining the relative change in the yield between data and MC
simulation after varying the signal shape and the endpoint of the ARGUS
function in the yield fits.

The systematic uncertainties originating from $e^{+}$ tracking and PID
efficiencies are studied by using the control samples of $e^{+}e^{-}\to\gamma
e^{+}e^{-}$ events and those for $K^{-}$ and $\pi^{\pm}$ are investigated with
the DT $D\bar{D}$ hadronic events. All samples provide good coverage on track
kinematics. The $e^{+}$ efficiencies for tracking and PID are also re-weighted
in 2D (momentum and $\cos\theta$) to match those of the $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$ data. For $K^{-}$ and $\pi^{+}$, similar
weighting is performed on momentum only since the data and MC angular
distributions already agree well. Small differences between the data and MC
efficiencies for $K^{-}$ tracking, $e^{+}$ tracking, and $e^{+}$ PID are
found, which are $+(2.6\pm 0.4)\%$, $+(1.0\pm 0.2)\%$, and $-(1.4\pm 0.2)\%$,
respectively. The MC efficiencies, corrected by the aforementioned
differences, are used for the BF determination. After corrections, the
residual uncertainties related to the tracking (PID) efficiencies of $e^{+}$,
$K^{-}$, $\pi^{+}$, and $\pi^{-}$ are assigned as 0.2% (0.2%), 0.4% (0.3%),
0.2% (0.2%), and 0.2% (0.2%), respectively.

Any systematic effects related to the requirements on
$M_{K^{-}\pi^{+}\pi^{-}\pi^{+}_{e\to\pi}}$,
$M_{K^{-}\pi^{+}\pi^{-}\pi^{+}_{e\to\pi}\pi^{0}}$, $M_{\pi^{+}\pi^{-}}$,
$\Delta E[(K^{-}\pi^{+})_{\rm tag}\pi_{\rm sig}^{+}]$, $\cos\theta_{a}$, and
$\cos\theta_{b}$, are examined by varying individual requirements by $\pm
0.05$ GeV/$c^{2}$, $\pm 0.05$ GeV/$c^{2}$, $\pm 0.01$ GeV/$c^{2}$, $\pm 0.004$
GeV, $\pm 0.02$, and $\pm 0.02$, respectively. Accounting for correlations in
the samples, the changes in the BFs are smaller than the statistical
uncertainty on the difference, so neither a systematic correction nor
uncertainty is applied from this source according to Ref. err . The systematic
uncertainty from the input BFs of $K_{1}(1270)^{-}$ subdecays is assigned to
be 3.0% by varying each of the quoted subdecay BFs of Belle non-resonance  by
$\pm 1\sigma$ and by comparing our nominal signal efficiency to the one based
on the world average BFs of $K_{1}(1270)^{-}$ decays.

The systematic uncertainty of the 2D fit is estimated to be
${}^{+6.9\%}_{-13.5\%}$ via two aspects. The uncertainty from signal shape is
mainly caused by varying the $K_{1}(1270)$ width by $\pm 1\sigma$ ($\pm
6.0$%). The uncertainty of background shape is mainly due to
non-$K_{1}(1270)^{-}$ sources of $K^{-}\pi^{+}\pi^{-}$
(${}^{+0.0\%}_{-8.7\%}$), which is the change of the fitted DT yield after
fixing a non-resonant component by referring to the non-resonant fraction in
$B\to J/\psi\bar{K}\pi\pi$ non-resonance . The uncertainty due to ignoring
$D^{+}\to K_{1}(1400)^{-}e^{+}\nu_{e}$ is assigned as ${}^{+0.0\%}_{-7.6\%}$,
by performing pseudoexperiments to evaluate fit biases and assuming its
contribution is one order of magnitude lower than our signal decay pdg2020 ;
theory ; formfactor , while the effects from $D^{0}\to
K^{*}(1410)^{-}e^{+}\nu_{e}$ and $D^{0}\to K_{2}^{*}(1430)^{-}e^{+}\nu_{e}$
are negligible.

The uncertainty due to the MC samples’ limited size, 1.0%, is considered as a
source of systematic uncertainty.

The uncertainty from FSR recovery is assigned as 0.3% by referring to Ref.
BES3_FSR . The uncertainty due to the kinematic fit is ignored since it is
only used to improve the $M^{2}_{\rm miss}$ resolution. The total systematic
uncertainty is estimated to be ${}^{+8.7\%}_{-14.5\%}$ by adding all the
individual contributions in quadrature.

Using the world average of ${\mathcal{B}}_{\rm sub}=(32.9\pm 3.6)\%$ pdg2020 ;
K10 , we obtain

${\mathcal{B}}_{\rm SL}={\mathcal{B}}_{D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}}=(1.09\pm 0.13^{+0.09}_{-0.16}\pm 0.12)\times
10^{-3},$

where the third uncertainty is from the external uncertainty of the assumed BF
${\mathcal{B}}_{\rm sub}$.

A 2D fit is also performed in each of the five equal-sized $\cos\theta_{K}$
bins to determine the background subtracted angular distribution, where
$\theta_{K}$ is the angle between the opposite of $D^{0}$ flight direction and
the normal $\vec{p}_{\pi,{\rm slow}}\times\vec{p}_{\pi,{\rm fast}}$ to the
$K^{-}\pi^{+}\pi^{-}$ plane in the $K^{-}\pi^{+}\pi^{-}$ rest frame, where
$\vec{p}_{\pi,{\rm slow}}$ ($\vec{p}_{\pi,{\rm fast}}$) is the momentum of the
lower (higher) momentum pion lhcbk1g ; wangwei . Figure 3 shows the fit to the
$\theta_{K}$ distribution with a second-order polynomial function wangwei ,

$\frac{d\Gamma\left(D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}\right)}{d\cos\theta_{K}}\propto
1+k_{1}\cos\theta_{K}+k_{2}\cos^{2}\theta_{K},$ (2)

where $k_{1}$ is a free parameter, $k_{2}=(1-3F_{L})/(1+F_{L})$,
$F_{L}=\frac{|c_{0}|^{2}}{|c_{0}|^{2}+|c_{+}|^{2}+|c_{-}|^{2}}$ is the
fraction of $K_{1}$ longitudinal polarization, with $c_{0,\pm}$ representing
the nonperturbative amplitudes for $D\to K_{1}$ with different polarizations.
As $\theta_{K}$ is parity odd, the sign for $\cos\theta_{K}$ in $\bar{D}^{0}$
decays is flipped. We obtain $F_{L}=0.50\pm 0.17_{\rm stat}\pm 0.08_{\rm
syst}$, where the systematic uncertainty mainly comes from signal shape
modeling. Our $F_{L}$ result is compatible within 1$\sigma$ with the LCSR
predictions in Ref. formfactor .

Fig. 3:  Fit to the efficiency corrected signal yields in bins of
$\cos\theta_{K}$. Solid (dashed) lines with error bars are signal yields after
(before) efficiency correction that accounts for efficiency differences
between the first bin and other bins. Blue solid curve is the fit result.

In summary, using 2.93 $\rm fb^{-1}$ of $e^{+}e^{-}$ collision data taken at
$\sqrt{s}=3.773$ GeV, we report the first observation of $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$. The obtained product of the BFs for $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}$ and $K_{1}(1270)^{-}\to K^{-}\pi^{+}\pi^{-}$ is
consistent with the CLEO’s result but with precision improved by about
threefold cleores . Our BF of $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$
contributes $(1.68\pm 0.35)\%$ of the total SL decay width of $D^{0}$ pdg2020
, which lies between the ISGW prediction (1%) and the ISGW2 prediction (2%),
consistent with the BESIII result for the $D^{+}$ counterpart bes3-k1ev . Our
BF of $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ agrees with the CLFQM and LCSR
predictions when $\theta_{K_{1}}\approx 33^{\circ}$ or $57^{\circ}$ theory ;
Momeni  and clearly disfavors the prediction reported in Ref. formfactor .
Using the BF of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ measured by BESIII
bes3-k1ev  and the world-average lifetimes of $D^{0}$ and $D^{+}$ pdg2020 , we
determine the ratio of the partial decay widths of the two decays to be
$\Gamma_{D^{0}\to
K_{1}(1270)^{-}e^{+}\nu_{e}}/\Gamma_{D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}}=1.20\pm
0.20\pm 0.14\pm 0.04$, where the systematic uncertainties from the background
shape, the tracking and PID efficiencies of $K^{-}$, $\pi^{+}$, and $e^{+}$ as
well as FSR recovery are canceled, the uncertainties of the lifetimes of
$D^{0}$ and $D^{+}$ are included; the uncertainties of the quoted BFs for
$K_{1}(1270)$ decays are largely canceled. This result agrees with unity as
predicted by isospin symmetry. Our $F_{L}$ measurement is compatible with
theoretical predictions.

Further studies of the $K\pi\pi$ system with larger
$D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$ samples at BESIII in the near
future bes3-white-paper  and large $D^{*+}\to D^{0}\pi^{+},\ D^{0}\to
K_{1}(1270)^{-}(\to K^{-}\pi^{+}\pi^{-})\ell^{+}\nu_{\ell}$ samples at LHCb
adavis  allow to extract the hadronic-transition form factors, the mass,
width, and the subdecay BFs of $K_{1}(1270)$, and to quantify the hadronic
effects in $K_{1}(1270)\to K\pi\pi$. These will benefit the precise
determinations of photon polarization in $b\to s\gamma$ transitions with high
statistics sample of $B\to K_{1}(1270)\gamma$ at Belle II belle2-white-paper
and LHCb lhcb-white-paper , thereby effectively over-constraining the right-
handed couplings in new physics models.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. Authors thank Wei Wang and Fu-Sheng Yu for
helpful discussions. This work is supported in part by National Key Research
and Development Program of China under Contracts Nos. 2020YFA0406300,
2020YFA0406400; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11775230, 11605124, 11625523, 11635010, 11735014, 11822506,
11835012, 11935015, 11935016, 11935018, 11961141012; the Chinese Academy of
Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1732263,
U1832207, U1932108; CAS Key Research Program of Frontier Sciences under
Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS;
INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC
under Contract No. 758462; European Union Horizon 2020 research and innovation
programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790;
German Research Foundation DFG under Contracts Nos. 443159800, Collaborative
Research Center CRC 1044, FOR 2359, FOR 2359, GRK 214; Istituto Nazionale di
Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; National Science and Technology fund; Olle Engkvist
Foundation under Contract No. 200-0605; STFC (United Kingdom); The Knut and
Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal
Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research
Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374,
DE-SC-0012069.

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