Letter
Open Access

Search for the decay $D^{0} \to π^{0} ν \bar{ν}$

M. Ablikim¹, M. N. Achasov^10,b, P. Adlarson⁶⁶, S. Ahmed¹⁴, M. Albrecht⁴, R. Aliberti²⁷, A. Amoroso^65a,65c, M. R. An³¹, Q. An^62,48 et al. (BESIII Collaboration)

Q. An^62,48, X. H. Bai⁵⁶, Y. Bai⁴⁷, O. Bakina²⁸, R. Baldini Ferroli^22a, I. Balossino^23a, Y. Ban^37,g, K. Begzsuren²⁵, N. Berger²⁷, M. Bertani^22a, D. Bettoni^23a, F. Bianchi^65a,65c, J. Bloms⁵⁹, A. Bortone^65a,65c, I. Boyko²⁸, R. A. Briere⁵, A. Brueggemann⁵⁹, H. Cai⁶⁷, X. Cai^1,48, A. Calcaterra^22a, G. F. Cao^1,53, N. Cao^1,53, S. A. Cetin^52a, J. F. Chang^1,48, W. L. Chang^1,53, G. Chelkov^28,a, G. Chen¹, H. S. Chen^1,53, M. L. Chen^1,48, S. J. Chen³⁴, X. R. Chen^24,53, Y. B. Chen^1,48, Z. J. Chen^19,h, W. S. Cheng^65c, G. Cibinetto^23a, F. Cossio^65c, H. L. Dai^1,48, X. C. Dai^1,53, A. Dbeyssi¹⁴, R. E. de Boer⁴, D. Dedovich²⁸, Z. Y. Deng¹, A. Denig²⁷, I. Denysenko²⁸, M. Destefanis^65a,65c, F. De Mori^65a,65c, Y. Ding³², J. Dong^1,48, L. Y. Dong^1,53, M. Y. Dong^1,48,53, X. Dong⁶⁷, S. X. Du⁷⁰, Y. L. Fan⁶⁷, J. Fang^1,48, S. S. Fang^1,53, Y. Fang¹, R. Farinelli^23a, L. Fava^65b,65c, F. Feldbauer⁴, G. Felici^22a, C. Q. Feng^62,48, J. H. Feng⁴⁹, M. Fritsch⁴, C. D. Fu¹, Y. N. Gao^37,g, Ya Gao⁶³, Yang Gao^62,48, I. Garzia^23a,23b, P. T. Ge⁶⁷, C. Geng⁴⁹, E. M. Gersabeck⁵⁷, A. Gilman⁶⁰, K. Goetzen¹¹, L. Gong³², W. X. Gong^1,48, W. Gradl²⁷, M. Greco^65a,65c, L. M. Gu³⁴, M. H. Gu^1,48, C. Y. Guan^1,53, L. B. Guo³³, R. P. Guo³⁹, Y. P. Guo^9,f, A. Guskov^28,a, T. T. Han⁴⁰, W. Y. Han³¹, X. Q. Hao¹⁵, F. A. Harris⁵⁵, K. L. He^1,53, F. H. Heinsius⁴, C. H. Heinz²⁷, Y. K. Heng^1,48,53, C. Herold⁵⁰, M. Himmelreich^11,d, T. Holtmann⁴, G. Y. Hou^1,53, Y. R. Hou⁵³, Z. L. Hou¹, H. M. Hu^1,53, J. F. Hu^46,i, T. Hu^1,48,53, Y. Hu¹, G. S. Huang^62,48, L. Q. Huang⁶³, X. T. Huang⁴⁰, Y. P. Huang¹, Z. Huang^37,g, T. Hussain⁶⁴, N. Hüsken^21,27, W. Imoehl²¹, M. Irshad^62,48, J. Jackson²¹, S. Jaeger⁴, S. Janchiv²⁵, Q. Ji¹, Q. P. Ji¹⁵, X. B. Ji^1,53, X. L. Ji^1,48, Y. Y. Ji⁴⁰, H. B. Jiang⁴⁰, X. S. Jiang^1,48,53, Y. Jiang⁵³, J. B. Jiao⁴⁰, Z. Jiao¹⁷, S. Jin³⁴, Y. Jin⁵⁶, M. Q. Jing^1,53, T. Johansson⁶⁶, N. Kalantar-Nayestanaki⁵⁴, X. S. Kang³², R. Kappert⁵⁴, M. Kavatsyuk⁵⁴, B. C. Ke⁷⁰, I. K. Keshk⁴, A. Khoukaz⁵⁹, P. Kiese²⁷, R. Kiuchi¹, R. Kliemt¹¹, L. Koch²⁹, O. B. Kolcu^52a, B. Kopf⁴, M. Kuemmel⁴, M. Kuessner⁴, A. Kupsc⁶⁶, M. G. Kurth^1,53, W. Kühn²⁹, J. J. Lane⁵⁷, J. S. Lange²⁹, P. Larin¹⁴, A. Lavania²⁰, L. Lavezzi^65a,65c, Z. H. Lei^62,48, H. Leithoff²⁷, M. Lellmann²⁷, T. Lenz²⁷, C. Li³⁸, C. H. Li³¹, Cheng Li^62,48, D. M. Li⁷⁰, F. Li^1,48, G. Li¹, H. Li^62,48, H. Li⁴², H. B. Li^1,53, H. J. Li¹⁵, J. Q. Li⁴, J. S. Li⁴⁹, J. W. Li⁴⁰, Ke Li¹, L. K. Li¹, Lei Li³, P. R. Li^30,j,k, S. Y. Li⁵¹, W. D. Li^1,53, W. G. Li¹, X. H. Li^62,48, X. L. Li⁴⁰, Xiaoyu Li^1,53, Z. Y. Li⁴⁹, H. Liang^62,48, H. Liang^1,53, H. Liang²⁶, Y. F. Liang⁴⁴, Y. T. Liang^24,53, G. R. Liao¹², L. Z. Liao^1,53, L. Z. Liao⁴⁰, J. Libby²⁰, A. Limphirat⁵⁰, C. X. Lin⁴⁹, T. Lin¹, B. J. Liu¹, C. X. Liu¹, D. Liu^14,62, F. H. Liu⁴³, Fang Liu¹, Feng Liu⁶, H. M. Liu^1,53, Huanhuan Liu¹, Huihui Liu¹⁶, J. B. Liu^62,48, J. L. Liu⁶³, J. Y. Liu^1,53, K. Liu¹, K. Y. Liu³², L. Liu^62,48, M. H. Liu^9,f, P. L. Liu¹, Q. Liu⁶⁷, Q. Liu⁵³, S. B. Liu^62,48, Shuai Liu⁴⁵, T. Liu^1,53, W. M. Liu^62,48, X. Liu^30,j,k, Y. Liu^30,j,k, Y. B. Liu³⁵, Z. A. Liu^1,48,53, Z. Q. Liu⁴⁰, X. C. Lou^1,48,53, F. X. Lu⁴⁹, H. J. Lu¹⁷, J. D. Lu^1,53, J. G. Lu^1,48, X. L. Lu¹, Y. Lu¹, Y. P. Lu^1,48, C. L. Luo³³, M. X. Luo⁶⁹, T. Luo^9,f, X. L. Luo^1,48, X. R. Lyu⁵³, F. C. Ma³², H. L. Ma¹, L. L. Ma⁴⁰, M. M. Ma^1,53, Q. M. Ma¹, R. Q. Ma^1,53, R. T. Ma⁵³, X. X. Ma^1,53, X. Y. Ma^1,48, Y. Ma^37,g, F. E. Maas¹⁴, M. Maggiora^65a,65c, S. Maldaner⁴, S. Malde⁶⁰, Q. A. Malik⁶⁴, A. Mangoni^22b, Y. J. Mao^37,g, Z. P. Mao¹, S. Marcello^65a,65c, Z. X. Meng⁵⁶, J. G. Messchendorp^54,11, G. Mezzadri^23a, T. J. Min³⁴, R. E. Mitchell²¹, X. H. Mo^1,48,53, N. Yu. Muchnoi^10,b, H. Muramatsu⁵⁸, S. Nakhoul^11,d, Y. Nefedov²⁸, F. Nerling^11,d, I. B. Nikolaev^10,b, Z. Ning^1,48, S. Nisar^8,l, S. L. Olsen⁵³, Q. Ouyang^1,48,53, S. Pacetti^22b,22c, X. Pan^9,f, Y. Pan⁵⁷, A. Pathak¹, A. Pathak²⁶, P. Patteri^22a, M. Pelizaeus⁴, H. P. Peng^62,48, K. Peters^11,d, J. Pettersson⁶⁶, J. L. Ping³³, R. G. Ping^1,53, S. Pogodin²⁸, R. Poling⁵⁸, V. Prasad^62,48, H. Qi^62,48, H. R. Qi⁵¹, K. H. Qi²⁴, M. Qi³⁴, T. Y. Qi^9,f, S. Qian^1,48, W. B. Qian⁵³, Z. Qian⁴⁹, C. F. Qiao⁵³, L. Q. Qin¹², X. P. Qin^9,f, X. S. Qin⁴⁰, Z. H. Qin^1,48, J. F. Qiu¹, S. Q. Qu³⁵, S. Q. Qu⁵¹, K. H. Rashid⁶⁴, K. Ravindran²⁰, C. F. Redmer²⁷, A. Rivetti^65c, V. Rodin⁵⁴, M. Rolo^65c, G. Rong^1,53, Ch. Rosner¹⁴, M. Rump⁵⁹, H. S. Sang⁶², A. Sarantsev^28,c, Y. Schelhaas²⁷, C. Schnier⁴, K. Schoenning⁶⁶, M. Scodeggio^23a,23b, D. C. Shan⁴⁵, W. Shan¹⁸, X. Y. Shan^62,48, J. F. Shangguan⁴⁵, M. Shao^62,48, C. P. Shen^9,f, H. F. Shen^1,53, P. X. Shen³⁵, X. Y. Shen^1,53, H. C. Shi^62,48, R. S. Shi^1,53, X. Shi^1,48, X. D. Shi^62,48, W. M. Song^26,1, Y. X. Song^37,g, S. Sosio^65a,65c, S. Spataro^65a,65c, K. X. Su⁶⁷, P. P. Su⁴⁵, G. X. Sun¹, H. K. Sun¹, J. F. Sun¹⁵, L. Sun⁶⁷, S. S. Sun^1,53, T. Sun^1,53, W. Y. Sun²⁶, W. Y. Sun³³, X. Sun^19,h, Y. J. Sun^62,48, Y. Z. Sun¹, Z. T. Sun⁴⁰, Y. H. Tan⁶⁷, Y. X. Tan^62,48, C. J. Tang⁴⁴, G. Y. Tang¹, J. Tang⁴⁹, J. X. Teng^62,48, V. Thoren⁶⁶, W. H. Tian⁴², Y. Tian^24,53, I. Uman^52b, B. Wang¹, B. L. Wang⁵³, C. W. Wang³⁴, D. Y. Wang^37,g, H. J. Wang^30,j,k, H. P. Wang^1,53, K. Wang^1,48, L. L. Wang¹, M. Wang⁴⁰, M. Z. Wang^37,g, Meng Wang^1,53, S. Wang^9,f, W. Wang⁴⁹, W. H. Wang⁶⁷, W. P. Wang^62,48, X. Wang^37,g, X. F. Wang^30,j,k, X. L. Wang^9,f, Y. Wang^62,48, Y. D. Wang³⁶, Y. F. Wang^1,48,53, Y. Q. Wang¹, Ying Wang⁴⁹, Z. Wang^1,48, Z. Y. Wang^1,53, Ziyi Wang⁵³, Zongyuan Wang^1,53, D. H. Wei¹², F. Weidner⁵⁹, S. P. Wen¹, D. J. White⁵⁷, U. Wiedner⁴, G. Wilkinson⁶⁰, M. Wolke⁶⁶, L. Wollenberg⁴, J. F. Wu^1,53, L. H. Wu¹, L. J. Wu^1,53, X. Wu^9,f, Z. Wu^1,48, L. Xia^62,48, T. Xiang^37,g, H. Xiao^9,f, S. Y. Xiao¹, Z. J. Xiao³³, X. H. Xie^37,g, Y. Xie⁴⁰, Y. G. Xie^1,48, Y. H. Xie⁶, Z. P. Xie^62,48, T. Y. Xing^1,53, C. J. Xu⁴⁹, G. F. Xu¹, Q. J. Xu¹³, S. Y. Xu⁶¹, W. Xu^1,53, X. P. Xu⁴⁵, Y. C. Xu⁵³, F. Yan^9,f, L. Yan^9,f, W. B. Yan^62,48, W. C. Yan⁷⁰, Xu Yan⁴⁵, H. J. Yang^41,e, H. X. Yang¹, L. Yang⁴², S. L. Yang⁵³, Yifan Yang^1,53, Zhi Yang²⁴, M. Ye^1,48, M. H. Ye⁷, J. H. Yin¹, Z. Y. You⁴⁹, B. X. Yu^1,48,53, C. X. Yu³⁵, G. Yu^1,53, J. S. Yu^19,h, T. Yu⁶³, C. Z. Yuan^1,53, L. Yuan², X. Q. Yuan^37,g, Y. Yuan^1,53, Z. Y. Yuan⁴⁹, C. X. Yue³¹, A. A. Zafar⁶⁴, X. Zeng Zeng⁶, Y. Zeng^19,h, A. Q. Zhang¹, B. X. Zhang¹, G. Y. Zhang¹⁵, H. Zhang⁶², H. H. Zhang⁴⁹, H. H. Zhang²⁶, H. Y. Zhang^1,48, J. L. Zhang⁶⁸, J. Q. Zhang³³, J. W. Zhang^1,48,53, J. Y. Zhang¹, J. Z. Zhang^1,53, Jianyu Zhang^1,53, Jiawei Zhang^1,53, L. M. Zhang⁵¹, L. Q. Zhang⁴⁹, Lei Zhang³⁴, S. F. Zhang³⁴, Shulei Zhang^19,h, X. D. Zhang³⁶, X. Y. Zhang⁴⁰, Y. Zhang⁶⁰, Y. T. Zhang⁷⁰, Y. H. Zhang^1,48, Yan Zhang^62,48, Yao Zhang¹, Z. Y. Zhang⁶⁷, G. Zhao¹, J. Zhao³¹, J. Y. Zhao^1,53, J. Z. Zhao^1,48, Lei Zhao^62,48, Ling Zhao¹, M. G. Zhao³⁵, Q. Zhao¹, S. J. Zhao⁷⁰, Y. B. Zhao^1,48, Y. X. Zhao^24,53, Z. G. Zhao^62,48, A. Zhemchugov^28,a, B. Zheng⁶³, J. P. Zheng^1,48, Y. H. Zheng⁵³, B. Zhong³³, C. Zhong⁶³, H. Zhou⁴⁰, L. P. Zhou^1,53, Q. Zhou^1,53, X. Zhou⁶⁷, X. K. Zhou⁵³, X. R. Zhou^62,48, X. Y. Zhou³¹, A. N. Zhu^1,53, J. Zhu³⁵, K. Zhu¹, K. J. Zhu^1,48,53, S. H. Zhu⁶¹, T. J. Zhu⁶⁸, W. J. Zhu^9,f, W. J. Zhu³⁵, Y. C. Zhu^62,48, Z. A. Zhu^1,53, B. S. Zou¹, and J. H. Zou¹ (BESIII Collaboration)

¹Institute of High Energy Physics, Beijing 100049, People’s Republic of China
²Beihang University, Beijing 100191, People’s Republic of China
³Beijing Institute of Petrochemical Technology, Beijing 102617, People’s Republic of China
⁴Bochum Ruhr-University, D-44780 Bochum, Germany
⁵Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
⁶Central China Normal University, Wuhan 430079, People’s Republic of China
⁷China Center of Advanced Science and Technology, Beijing 100190, People’s Republic of China
⁸COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan
⁹Fudan University, Shanghai 200433, People’s Republic of China
¹⁰G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia
¹¹GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
¹²Guangxi Normal University, Guilin 541004, People’s Republic of China
¹³Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
¹⁴Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
¹⁵Henan Normal University, Xinxiang 453007, People’s Republic of China
¹⁶Henan University of Science and Technology, Luoyang 471003, People’s Republic of China
¹⁷Huangshan College, Huangshan 245000, People’s Republic of China
¹⁸Hunan Normal University, Changsha 410081, People’s Republic of China
¹⁹Hunan University, Changsha 410082, People’s Republic of China
²⁰Indian Institute of Technology Madras, Chennai 600036, India
²¹Indiana University, Bloomington, Indiana 47405, USA
^22aINFN Laboratori Nazionali di Frascati, INFN Laboratori Nazionali di Frascati I-00044, Frascati, Italy
^22bINFN Laboratori Nazionali di Frascati, INFN Sezione di Perugia, I-06100, Perugia, Italy
^22cINFN Laboratori Nazionali di Frascati, University of Perugia, I-06100, Perugia, Italy
^23aINFN Sezione di Ferrara, INFN Sezione di Ferrara, I-44122, Ferrara, Italy
^23bINFN Sezione di Ferrara, University of Ferrara, I-44122, Ferrara, Italy
²⁴Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
²⁵Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330, Mongolia
²⁶Jilin University, Changchun 130012, People’s Republic of China
²⁷Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany
²⁸Joint Institute for Nuclear Research, 141980 Dubna, Moscow, Russia
²⁹Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany
³⁰Lanzhou University, Lanzhou 730000, People’s Republic of China
³¹Liaoning Normal University, Dalian 116029, People’s Republic of China
³²Liaoning University, Shenyang 110036, People’s Republic of China
³³Nanjing Normal University, Nanjing 210023, People’s Republic of China
³⁴Nanjing University, Nanjing 210093, People’s Republic of China
³⁵Nankai University, Tianjin 300071, People’s Republic of China
³⁶North China Electric Power University, Beijing 102206, People’s Republic of China
³⁷Peking University, Beijing 100871, People’s Republic of China
³⁸Qufu Normal University, Qufu 273165, People’s Republic of China
³⁹Shandong Normal University, Jinan 250014, People’s Republic of China
⁴⁰Shandong University, Jinan 250100, People’s Republic of China
⁴¹Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
⁴²Shanxi Normal University, Linfen 041004, People’s Republic of China
⁴³Shanxi University, Taiyuan 030006, People’s Republic of China
⁴⁴Sichuan University, Chengdu 610064, People’s Republic of China
⁴⁵Soochow University, Suzhou 215006, People’s Republic of China
⁴⁶South China Normal University, Guangzhou 510006, People’s Republic of China
⁴⁷Southeast University, Nanjing 211100, People’s Republic of China
⁴⁸State Key Laboratory of Particle Detection and Electronics, Beijing 100049, Hefei 230026, People’s Republic of China
⁴⁹Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
⁵⁰Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand
⁵¹Tsinghua University, Beijing 100084, People’s Republic of China
^52aTurkish Accelerator Center Particle Factory Group, Istinye University, 34010, Istanbul, Turkey
^52bTurkish Accelerator Center Particle Factory Group, Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
⁵³University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
⁵⁴University of Groningen, NL-9747 AA Groningen, The Netherlands
⁵⁵University of Hawaii, Honolulu, Hawaii 96822, USA
⁵⁶University of Jinan, Jinan 250022, People’s Republic of China
⁵⁷University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
⁵⁸University of Minnesota, Minneapolis, Minnesota 55455, USA
⁵⁹University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
⁶⁰University of Oxford, Keble Rd, Oxford OX1 3RH, United Kingdom
⁶¹University of Science and Technology Liaoning, Anshan 114051, People’s Republic of China
⁶²University of Science and Technology of China, Hefei 230026, People’s Republic of China
⁶³University of South China, Hengyang 421001, People’s Republic of China
⁶⁴University of the Punjab, Lahore-54590, Pakistan
^65aUniversity of Turin and INFN, University of Turin, I-10125 Turin, Italy
^65bUniversity of Turin and INFN, University of Eastern Piedmont, I-15121, Alessandria, Italy
^65cUniversity of Turin and INFN, INFN, I-10125 Turin, Italy
⁶⁶Uppsala University, Box 516, SE-75120 Uppsala, Sweden
⁶⁷Wuhan University, Wuhan 430072, People’s Republic of China
⁶⁸Xinyang Normal University, Xinyang 464000, People’s Republic of China
⁶⁹Zhejiang University, Hangzhou 310027, People’s Republic of China
⁷⁰Zhengzhou University, Zhengzhou 450001, People’s Republic of China

^aAlso at the Moscow Institute of Physics and Technology, Moscow 141700, Russia.
^bAlso at the Novosibirsk State University, Novosibirsk 630090, Russia.
^cAlso at the NRC “Kurchatov Institute”, PNPI, 188300 Gatchina, Russia.
^dAlso at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany.
^eAlso 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.
^fAlso 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.
^gAlso at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People’s Republic of China.
^hAlso at School of Physics and Electronics, Hunan University, Changsha 410082, China.
ⁱAlso at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China.
^jAlso at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People’s Republic of China.
^kAlso at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People’s Republic of China.
^lAlso at the Department of Mathematical Sciences, IBA, Karachi, Pakistan.

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Phys. Rev. D 105, L071102 – Published 21 April, 2022

DOI: https://doi.org/10.1103/PhysRevD.105.L071102

Abstract

We present the first experimental search for the rare charm decay $D^{0} \to π^{0} ν \bar{ν}$ . It is based on an $e^{+} e^{-}$ collision sample consisting of $10.6 \times 10^{6}$ pairs of $D^{0} {\bar{D}}^{0}$ mesons collected by the BESIII detector at $\sqrt{s} = 3.773 GeV$ , corresponding to an integrated luminosity of $2.93 {fb}^{- 1}$ . A data-driven method is used to ensure the reliability of the background modeling. No significant $D^{0} \to π^{0} ν \bar{ν}$ signal is observed in data and an upper limit of the branching fraction is set to be $2.1 \times 10^{- 4}$ at the 90% confidence level. This is the first experimental constraint on charmed-hadron decays into dineutrino final states.

Physics Subject Headings (PhySH)

Article Text

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Phys. Rev. D 105, L071102– Published 21 April, 2022

Vol. 105, Iss. 7 — 1 April 2022

Received 31 December 2021
Accepted 4 April 2022

DOI: https://doi.org/10.1103/PhysRevD.105.L071102

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Search for the decay $D^{0} \to π^{0} ν \bar{ν}$

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Search for the decay D0→π0νν¯

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Search for the decay $D^{0} \to π^{0} ν \bar{ν}$