Searching for the massless dark photon in $c\to uγ'$
BESIII Collaboration M. Ablikim M. N. Achasov P. Adlarson O. Afedulidis X. C. Ai R. Aliberti A. Amoroso Y. Bai O. Bakina I. Balossino Y. Ban H. -R. Bao V. Batozskaya K. Begzsuren N. Berger M. Berlowski M. Bertani D. Bettoni F. Bianchi E. Bianco A. Bortone I. Boyko R. A. Briere A. Brueggemann H. Cai X. Cai A. Calcaterra G. F. Cao N. Cao S. A. Cetin X. Y. Chai J. F. Chang G. R. Che Y. Z. Che G. Chelkov C. Chen C. H. Chen Chao Chen G. Chen H. S. Chen H. Y. Chen M. L. Chen S. J. Chen S. L. Chen S. M. Chen T. Chen X. R. Chen X. T. Chen Y. B. Chen Y. Q. Chen Z. J. Chen Z. Y. Chen S. K. Choi G. Cibinetto F. Cossio J. J. Cui H. L. Dai J. P. Dai A. Dbeyssi R. E. de Boer D. Dedovich C. Q. Deng Z. Y. Deng A. Denig I. Denysenko M. Destefanis F. De Mori B. Ding X. X. Ding Y. Ding Y. Ding J. Dong L. Y. Dong M. Y. Dong X. Dong M. C. Du S. X. Du Y. Y. Duan Z. H. Duan P. Egorov Y. H. Fan J. Fang J. Fang S. S. Fang W. X. Fang Y. Fang Y. Q. Fang R. Farinelli L. Fava F. Feldbauer G. Felici C. Q. Feng J. H. Feng Y. T. Feng M. Fritsch C. D. Fu J. L. Fu Y. W. Fu H. Gao X. B. Gao Y. N. Gao Yang Gao S. Garbolino I. Garzia L. Ge P. T. Ge Z. W. Ge C. Geng E. M. Gersabeck A. Gilman K. Goetzen L. Gong W. X. Gong W. Gradl S. Gramigna M. Greco M. H. Gu Y. T. Gu C. Y. Guan A. Q. Guo L. B. Guo M. J. Guo R. P. Guo Y. P. Guo A. Guskov J. Gutierrez K. L. Han T. T. Han F. Hanisch X. Q. Hao F. A. Harris K. K. He K. L. He F. H. Heinsius C. H. Heinz Y. K. Heng C. Herold T. Holtmann P. C. Hong G. Y. Hou X. T. Hou Y. R. Hou Z. L. Hou B. Y. Hu H. M. Hu J. F. Hu Q. P. Hu S. L. Hu T. Hu Y. Hu G. S. Huang K. X. Huang L. Q. Huang X. T. Huang Y. P. Huang Y. S. Huang T. Hussain F. Hölzken N. Hüsken N. in der Wiesche J. Jackson S. Janchiv J. H. Jeong Q. Ji Q. P. Ji W. Ji X. B. Ji X. L. Ji Y. Y. Ji X. Q. Jia Z. K. Jia D. Jiang H. B. Jiang P. C. Jiang S. S. Jiang T. J. Jiang X. S. Jiang Y. Jiang J. B. Jiao J. K. Jiao Z. Jiao S. Jin Y. Jin M. Q. Jing X. M. Jing T. Johansson S. Kabana N. Kalantar-Nayestanaki X. L. Kang X. S. Kang M. Kavatsyuk B. C. Ke V. Khachatryan A. Khoukaz R. Kiuchi O. B. Kolcu B. Kopf M. Kuessner X. Kui N. Kumar A. Kupsc W. Kühn L. Lavezzi T. T. Lei Z. H. Lei M. Lellmann T. Lenz C. Li C. Li C. H. Li Cheng Li D. M. Li F. Li G. Li H. B. Li H. J. Li H. N. Li Hui Li J. R. Li J. S. Li K. Li K. L. Li L. J. Li L. K. Li Lei Li M. H. Li P. R. Li Q. M. Li Q. X. Li R. Li S. X. Li T. Li W. D. Li W. G. Li X. Li X. H. Li X. L. Li X. Y. Li X. Z. Li Y. G. Li Z. J. Li Z. Y. Li C. Liang H. Liang H. Liang Y. F. Liang Y. T. Liang G. R. Liao Y. P. Liao J. Libby A. Limphirat C. C. Lin C. X. Lin D. X. Lin T. Lin B. J. Liu B. X. Liu C. Liu C. X. Liu F. Liu F. H. Liu Feng Liu G. M. Liu H. Liu H. B. Liu H. H. Liu H. M. Liu Huihui Liu J. B. Liu J. Y. Liu K. Liu K. Y. Liu Ke Liu L. Liu L. C. Liu Lu Liu M. H. Liu P. L. Liu Q. Liu S. B. Liu T. Liu W. K. Liu W. M. Liu X. Liu X. Liu Y. Liu Y. Liu Y. B. Liu Z. A. Liu Z. D. Liu Z. Q. Liu X. C. Lou F. X. Lu H. J. Lu J. G. Lu X. L. Lu Y. Lu Y. P. Lu Z. H. Lu C. L. Luo J. R. Luo M. X. Luo T. Luo X. L. Luo X. R. Lyu Y. F. Lyu F. C. Ma H. Ma H. L. Ma J. L. Ma L. L. Ma L. R. Ma M. M. Ma Q. M. Ma R. Q. Ma T. Ma X. T. Ma X. Y. Ma Y. M. Ma F. E. Maas I. MacKay M. Maggiora S. Malde Y. J. Mao Z. P. Mao S. Marcello Z. X. Meng J. G. Messchendorp G. Mezzadri H. Miao T. J. Min R. E. Mitchell X. H. Mo B. Moses N. Yu. Muchnoi J. Muskalla Y. Nefedov F. Nerling L. S. Nie I. B. Nikolaev Z. Ning S. Nisar Q. L. Niu W. D. Niu Y. Niu S. L. Olsen S. L. Olsen Q. Ouyang S. Pacetti X. Pan Y. Pan A. Pathak Y. P. Pei M. Pelizaeus H. P. Peng Y. Y. Peng K. Peters J. L. Ping R. G. Ping S. Plura V. Prasad F. Z. Qi H. Qi H. R. Qi M. Qi T. Y. Qi S. Qian W. B. Qian C. F. Qiao X. K. Qiao J. J. Qin L. Q. Qin L. Y. Qin X. P. Qin X. S. Qin Z. H. Qin J. F. Qiu Z. H. Qu C. F. Redmer K. J. Ren A. Rivetti M. Rolo G. Rong Ch. Rosner M. Q. Ruan S. N. Ruan N. Salone A. Sarantsev Y. Schelhaas K. Schoenning M. Scodeggio K. Y. Shan W. Shan X. Y. Shan Z. J. Shang J. F. Shangguan L. G. Shao M. Shao C. P. Shen H. F. Shen W. H. Shen X. Y. Shen B. A. Shi H. Shi H. C. Shi J. L. Shi J. Y. Shi Q. Q. Shi S. Y. Shi X. Shi J. J. Song T. Z. Song W. M. Song Y. J. Song Y. X. Song S. Sosio S. Spataro F. Stieler S. S Su Y. J. Su G. B. Sun G. X. Sun H. Sun H. K. Sun J. F. Sun K. Sun L. Sun S. S. Sun T. Sun W. Y. Sun Y. Sun Y. J. Sun Y. Z. Sun Z. Q. Sun Z. T. Sun C. J. Tang G. Y. Tang J. Tang M. Tang Y. A. Tang L. Y. Tao Q. T. Tao M. Tat J. X. Teng V. Thoren W. H. Tian Y. Tian Z. F. Tian I. Uman Y. Wan S. J. Wang B. Wang B. L. Wang Bo Wang D. Y. Wang F. Wang H. J. Wang J. J. Wang J. P. Wang K. Wang L. L. Wang M. Wang N. Y. Wang S. Wang S. Wang T. Wang T. J. Wang W. Wang W. Wang W. P. Wang X. Wang X. F. Wang X. J. Wang X. L. Wang X. N. Wang Y. Wang Y. D. Wang Y. F. Wang Y. H. Wang Y. L. Wang Y. N. Wang Y. Q. Wang Yaqian Wang Yi Wang Z. Wang Z. L. Wang Z. Y. Wang Ziyi Wang D. H. Wei F. Weidner S. P. Wen Y. R. Wen U. Wiedner G. Wilkinson M. Wolke L. Wollenberg C. Wu J. F. Wu L. H. Wu L. J. Wu X. Wu X. H. Wu Y. Wu Y. H. Wu Y. J. Wu Z. Wu L. Xia X. M. Xian B. H. Xiang T. Xiang D. Xiao G. Y. Xiao S. Y. Xiao Y. L. Xiao Z. J. Xiao C. Xie X. H. Xie Y. Xie Y. G. Xie Y. H. Xie Z. P. Xie T. Y. Xing C. F. Xu C. J. Xu G. F. Xu H. Y. Xu M. Xu Q. J. Xu Q. N. Xu W. Xu W. L. Xu X. P. Xu Y. Xu Y. C. Xu Z. S. Xu F. Yan L. Yan W. B. Yan W. C. Yan X. Q. Yan H. J. Yang H. L. Yang H. X. Yang J. H. Yang T. Yang Y. Yang Y. F. Yang Y. F. Yang Y. X. Yang Z. W. Yang Z. P. Yao M. Ye M. H. Ye J. H. Yin Junhao Yin Z. Y. You B. X. Yu C. X. Yu G. Yu J. S. Yu M. C. Yu T. Yu X. D. Yu Y. C. Yu C. Z. Yuan J. Yuan J. Yuan L. Yuan S. C. Yuan Y. Yuan Z. Y. Yuan C. X. Yue A. A. Zafar F. R. Zeng S. H. Zeng X. Zeng Y. Zeng Y. J. Zeng Y. J. Zeng X. Y. Zhai Y. C. Zhai Y. H. Zhan A. Q. Zhang B. L. Zhang B. X. Zhang D. H. Zhang G. Y. Zhang H. Zhang H. Zhang H. C. Zhang H. H. Zhang H. H. Zhang H. Q. Zhang H. R. Zhang H. Y. Zhang J. Zhang J. Zhang J. J. Zhang J. L. Zhang J. Q. Zhang J. S. Zhang J. W. Zhang J. X. Zhang J. Y. Zhang J. Z. Zhang Jianyu Zhang L. M. Zhang Lei Zhang P. Zhang Q. Y. Zhang R. Y. Zhang S. H. Zhang Shulei Zhang X. M. Zhang X. Y Zhang X. Y. Zhang Y. Zhang Y. Zhang Y. T. Zhang Y. H. Zhang Y. M. Zhang Yan Zhang Z. D. Zhang Z. H. Zhang Z. L. Zhang Z. Y. Zhang Z. Y. Zhang Z. Z. Zhang G. Zhao J. Y. Zhao J. Z. Zhao L. Zhao Lei Zhao M. G. Zhao N. Zhao R. P. Zhao S. J. Zhao Y. B. Zhao Y. X. Zhao Z. G. Zhao A. Zhemchugov B. Zheng B. M. Zheng J. P. Zheng W. J. Zheng Y. H. Zheng B. Zhong X. Zhong H. Zhou J. Y. Zhou L. P. Zhou S. Zhou X. Zhou X. K. Zhou X. R. Zhou X. Y. Zhou Y. Z. Zhou Z. C. Zhou A. N. Zhu J. Zhu K. Zhu K. J. Zhu K. S. Zhu L. Zhu L. X. Zhu S. H. Zhu T. J. Zhu W. D. Zhu Y. C. Zhu Z. A. Zhu J. H. Zou J. Zu












§ ABSTRACT 
In the effective field theory, the massless dark photon γ' can only couple with the Standard Model particle through operators of dimension higher than four, thereby offering a high sensitivity to the new physics energy scale.
Using 7.9 fb^-1 of e^+e^- collision data collected at √(s)=3.773 GeV with the BESIII detector at the BEPCII collider, we measure the effective flavor-changing neutral current coupling of cuγ' in D^0→ωγ' and D^0→γγ' processes to search for the massless dark photon.
No significant signals are observed, and the upper limits at the 90% confidence level on the massless dark photon branching fraction are set to be 1.1×10^-5 and 2.0×10^-6 for D^0→ωγ' and D^0→γγ', respectively. These results provide the most stringent constraint on the new physics energy scale associated with cuγ' coupling in the world, with the new physics energy scale related parameter |ℂ|^2+|ℂ_5|^2<8.2×10^-17^-2 at the 90% confidence level, playing a unique role in the dark sector search with the charm sector.



  


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Searching for the massless dark photon in c→ uγ'
    

M. Ablikim^1, M. N. Achasov^4,c, P. Adlarson^76, O. Afedulidis^3, X. C. Ai^81, R. Aliberti^35, A. Amoroso^75A,75C, Y. Bai^57, O. Bakina^36, I. Balossino^29A, Y. Ban^46,h, H.-R. Bao^64, V. Batozskaya^1,44, K. Begzsuren^32, N. Berger^35, M. Berlowski^44, M. Bertani^28A, D. Bettoni^29A, F. Bianchi^75A,75C, E. Bianco^75A,75C, A. Bortone^75A,75C, I. Boyko^36, R. A. Briere^5, A. Brueggemann^69, H. Cai^77, X. Cai^1,58, A. Calcaterra^28A, G. F. Cao^1,64, N. Cao^1,64, S. A. Cetin^62A, X. Y. Chai^46,h, J. F. Chang^1,58, G. R. Che^43, Y. Z. Che^1,58,64, G. Chelkov^36,b, C. Chen^43, C. H. Chen^9, Chao Chen^55, G. Chen^1, H. S. Chen^1,64, H. Y. Chen^20, M. L. Chen^1,58,64, S. J. Chen^42, S. L. Chen^45, S. M. Chen^61, T. Chen^1,64, X. R. Chen^31,64, X. T. Chen^1,64, Y. B. Chen^1,58, Y. Q. Chen^34, Z. J. Chen^25,i, Z. Y. Chen^1,64, S. K. Choi^10, G. Cibinetto^29A, F. Cossio^75C, J. J. Cui^50, H. L. Dai^1,58, J. P. Dai^79, A. Dbeyssi^18, R.  E. de Boer^3, D. Dedovich^36, C. Q. Deng^73, Z. Y. Deng^1, A. Denig^35, I. Denysenko^36, M. Destefanis^75A,75C, F. De Mori^75A,75C, B. Ding^67,1, X. X. Ding^46,h, Y. Ding^40, Y. Ding^34, J. Dong^1,58, L. Y. Dong^1,64, M. Y. Dong^1,58,64, X. Dong^77, M. C. Du^1, S. X. Du^81, Y. Y. Duan^55, Z. H. Duan^42, P. Egorov^36,b, Y. H. Fan^45, J. Fang^59, J. Fang^1,58, S. S. Fang^1,64, W. X. Fang^1, Y. Fang^1, Y. Q. Fang^1,58, R. Farinelli^29A, L. Fava^75B,75C, F. Feldbauer^3, G. Felici^28A, C. Q. Feng^72,58, J. H. Feng^59, Y. T. Feng^72,58, M. Fritsch^3, C. D. Fu^1, J. L. Fu^64, Y. W. Fu^1,64, H. Gao^64, X. B. Gao^41, Y. N. Gao^46,h, Yang Gao^72,58, S. Garbolino^75C, I. Garzia^29A,29B, L. Ge^81, P. T. Ge^19, Z. W. Ge^42, C. Geng^59, E. M. Gersabeck^68, A. Gilman^70, K. Goetzen^13, L. Gong^40, W. X. Gong^1,58, W. Gradl^35, S. Gramigna^29A,29B, M. Greco^75A,75C, M. H. Gu^1,58, Y. T. Gu^15, C. Y. Guan^1,64, A. Q. Guo^31,64, L. B. Guo^41, M. J. Guo^50, R. P. Guo^49, Y. P. Guo^12,g, A. Guskov^36,b, J. Gutierrez^27, K. L. Han^64, T. T. Han^1, F. Hanisch^3, X. Q. Hao^19, F. A. Harris^66, K. K. He^55, K. L. He^1,64, F. H. Heinsius^3, C. H. Heinz^35, Y. K. Heng^1,58,64, C. Herold^60, T. Holtmann^3, P. C. Hong^34, G. Y. Hou^1,64, X. T. Hou^1,64, Y. R. Hou^64, Z. L. Hou^1, B. Y. Hu^59, H. M. Hu^1,64, J. F. Hu^56,j, Q. P. Hu^72,58, S. L. Hu^12,g, T. Hu^1,58,64, Y. Hu^1, G. S. Huang^72,58, K. X. Huang^59, L. Q. Huang^31,64, X. T. Huang^50, Y. P. Huang^1, Y. S. Huang^59, T. Hussain^74, F. Hölzken^3, N. Hüsken^35, N. in der Wiesche^69, J. Jackson^27, S. Janchiv^32, J. H. Jeong^10, Q. Ji^1, Q. P. Ji^19, W. Ji^1,64, X. B. Ji^1,64, X. L. Ji^1,58, Y. Y. Ji^50, X. Q. Jia^50, Z. K. Jia^72,58, D. Jiang^1,64, H. B. Jiang^77, P. C. Jiang^46,h, S. S. Jiang^39, T. J. Jiang^16, X. S. Jiang^1,58,64, Y. Jiang^64, J. B. Jiao^50, J. K. Jiao^34, Z. Jiao^23, S. Jin^42, Y. Jin^67, M. Q. Jing^1,64, X. M. Jing^64, T. Johansson^76, S. Kabana^33, N. Kalantar-Nayestanaki^65, X. L. Kang^9, X. S. Kang^40, M. Kavatsyuk^65, B. C. Ke^81, V. Khachatryan^27, A. Khoukaz^69, R. Kiuchi^1, O. B. Kolcu^62A, B. Kopf^3, M. Kuessner^3, X. Kui^1,64, N.  Kumar^26, A. Kupsc^44,76, W. Kühn^37, L. Lavezzi^75A,75C, T. T. Lei^72,58, Z. H. Lei^72,58, M. Lellmann^35, T. Lenz^35, C. Li^47, C. Li^43, C. H. Li^39, Cheng Li^72,58, D. M. Li^81, F. Li^1,58, G. Li^1, H. B. Li^1,64, H. J. Li^19, H. N. Li^56,j, Hui Li^43, J. R. Li^61, J. S. Li^59, K. Li^1, K. L. Li^19, L. J. Li^1,64, L. K. Li^1, Lei Li^48, M. H. Li^43, P. R. Li^38,k,l, Q. M. Li^1,64, Q. X. Li^50, R. Li^17,31, S. X. Li^12, T.  Li^50, W. D. Li^1,64, W. G. Li^1,a, X. Li^1,64, X. H. Li^72,58, X. L. Li^50, X. Y. Li^1,64, X. Z. Li^59, Y. G. Li^46,h, Z. J. Li^59, Z. Y. Li^79, C. Liang^42, H. Liang^1,64, H. Liang^72,58, Y. F. Liang^54, Y. T. Liang^31,64, G. R. Liao^14, Y. P. Liao^1,64, J. Libby^26, A.  Limphirat^60, C. C. Lin^55, C. X. Lin^64, D. X. Lin^31,64, T. Lin^1, B. J. Liu^1, B. X. Liu^77, C. Liu^34, C. X. Liu^1, F. Liu^1, F. H. Liu^53, Feng Liu^6, G. M. Liu^56,j, H. Liu^38,k,l, H. B. Liu^15, H. H. Liu^1, H. M. Liu^1,64, Huihui Liu^21, J. B. Liu^72,58, J. Y. Liu^1,64, K. Liu^38,k,l, K. Y. Liu^40, Ke Liu^22, L. Liu^72,58, L. C. Liu^43, Lu Liu^43, M. H. Liu^12,g, P. L. Liu^1, Q. Liu^64, S. B. Liu^72,58, T. Liu^12,g, W. K. Liu^43, W. M. Liu^72,58, X. Liu^38,k,l, X. Liu^39, Y. Liu^38,k,l, Y. Liu^81, Y. B. Liu^43, Z. A. Liu^1,58,64, Z. D. Liu^9, Z. Q. Liu^50, X. C. Lou^1,58,64, F. X. Lu^59, H. J. Lu^23, J. G. Lu^1,58, X. L. Lu^1, Y. Lu^7, Y. P. Lu^1,58, Z. H. Lu^1,64, C. L. Luo^41, J. R. Luo^59, M. X. Luo^80, T. Luo^12,g, X. L. Luo^1,58, X. R. Lyu^64, Y. F. Lyu^43, F. C. Ma^40, H. Ma^79, H. L. Ma^1, J. L. Ma^1,64, L. L. Ma^50, L. R. Ma^67, M. M. Ma^1,64, Q. M. Ma^1, R. Q. Ma^1,64, T. Ma^72,58, X. T. Ma^1,64, X. Y. Ma^1,58, Y. M. Ma^31, F. E. Maas^18, I. MacKay^70, M. Maggiora^75A,75C, S. Malde^70, Y. J. Mao^46,h, Z. P. Mao^1, S. Marcello^75A,75C, Z. X. Meng^67, J. G. Messchendorp^13,65, G. Mezzadri^29A, H. Miao^1,64, T. J. Min^42, R. E. Mitchell^27, X. H. Mo^1,58,64, B. Moses^27, N. Yu. Muchnoi^4,c, J. Muskalla^35, Y. Nefedov^36, F. Nerling^18,e, L. S. Nie^20, I. B. Nikolaev^4,c, Z. Ning^1,58, S. Nisar^11,m, Q. L. Niu^38,k,l, W. D. Niu^55, Y. Niu ^50, S. L. Olsen^10,64, S. L. Olsen^64, Q. Ouyang^1,58,64, S. Pacetti^28B,28C, X. Pan^55, Y. Pan^57, A.  Pathak^34, Y. P. Pei^72,58, M. Pelizaeus^3, H. P. Peng^72,58, Y. Y. Peng^38,k,l, K. Peters^13,e, J. L. Ping^41, R. G. Ping^1,64, S. Plura^35, V. Prasad^33, F. Z. Qi^1, H. Qi^72,58, H. R. Qi^61, M. Qi^42, T. Y. Qi^12,g, S. Qian^1,58, W. B. Qian^64, C. F. Qiao^64, X. K. Qiao^81, J. J. Qin^73, L. Q. Qin^14, L. Y. Qin^72,58, X. P. Qin^12,g, X. S. Qin^50, Z. H. Qin^1,58, J. F. Qiu^1, Z. H. Qu^73, C. F. Redmer^35, K. J. Ren^39, A. Rivetti^75C, M. Rolo^75C, G. Rong^1,64, Ch. Rosner^18, M. Q. Ruan^1,58, S. N. Ruan^43, N. Salone^44, A. Sarantsev^36,d, Y. Schelhaas^35, K. Schoenning^76, M. Scodeggio^29A, K. Y. Shan^12,g, W. Shan^24, X. Y. Shan^72,58, Z. J. Shang^38,k,l, J. F. Shangguan^16, L. G. Shao^1,64, M. Shao^72,58, C. P. Shen^12,g, H. F. Shen^1,8, W. H. Shen^64, X. Y. Shen^1,64, B. A. Shi^64, H. Shi^72,58, H. C. Shi^72,58, J. L. Shi^12,g, J. Y. Shi^1, Q. Q. Shi^55, S. Y. Shi^73, X. Shi^1,58, J. J. Song^19, T. Z. Song^59, W. M. Song^34,1, Y.  J. Song^12,g, Y. X. Song^46,h,n, S. Sosio^75A,75C, S. Spataro^75A,75C, F. Stieler^35, S. S Su^40, Y. J. Su^64, G. B. Sun^77, G. X. Sun^1, H. Sun^64, H. K. Sun^1, J. F. Sun^19, K. Sun^61, L. Sun^77, S. S. Sun^1,64, T. Sun^51,f, W. Y. Sun^34, Y. Sun^9, Y. J. Sun^72,58, Y. Z. Sun^1, Z. Q. Sun^1,64, Z. T. Sun^50, C. J. Tang^54, G. Y. Tang^1, J. Tang^59, M. Tang^72,58, Y. A. Tang^77, L. Y. Tao^73, Q. T. Tao^25,i, M. Tat^70, J. X. Teng^72,58, V. Thoren^76, W. H. Tian^59, Y. Tian^31,64, Z. F. Tian^77, I. Uman^62B, Y. Wan^55,  S. J. Wang ^50, B. Wang^1, B. L. Wang^64, Bo Wang^72,58, D. Y. Wang^46,h, F. Wang^73, H. J. Wang^38,k,l, J. J. Wang^77, J. P. Wang ^50, K. Wang^1,58, L. L. Wang^1, M. Wang^50, N. Y. Wang^64, S. Wang^38,k,l, S. Wang^12,g, T.  Wang^12,g, T. J. Wang^43, W.  Wang^73, W. Wang^59, W. P. Wang^35,58,72,o, X. Wang^46,h, X. F. Wang^38,k,l, X. J. Wang^39, X. L. Wang^12,g, X. N. Wang^1, Y. Wang^61, Y. D. Wang^45, Y. F. Wang^1,58,64, Y. H. Wang^38,k,l, Y. L. Wang^19, Y. N. Wang^45, Y. Q. Wang^1, Yaqian Wang^17, Yi Wang^61, Z. Wang^1,58, Z. L.  Wang^73, Z. Y. Wang^1,64, Ziyi Wang^64, D. H. Wei^14, F. Weidner^69, S. P. Wen^1, Y. R. Wen^39, U. Wiedner^3, G. Wilkinson^70, M. Wolke^76, L. Wollenberg^3, C. Wu^39, J. F. Wu^1,8, L. H. Wu^1, L. J. Wu^1,64, X. Wu^12,g, X. H. Wu^34, Y. Wu^72,58, Y. H. Wu^55, Y. J. Wu^31, Z. Wu^1,58, L. Xia^72,58, X. M. Xian^39, B. H. Xiang^1,64, T. Xiang^46,h, D. Xiao^38,k,l, G. Y. Xiao^42, S. Y. Xiao^1, Y.  L. Xiao^12,g, Z. J. Xiao^41, C. Xie^42, X. H. Xie^46,h, Y. Xie^50, Y. G. Xie^1,58, Y. H. Xie^6, Z. P. Xie^72,58, T. Y. Xing^1,64, C. F. Xu^1,64, C. J. Xu^59, G. F. Xu^1, H. Y. Xu^67,2, M. Xu^72,58, Q. J. Xu^16, Q. N. Xu^30, W. Xu^1, W. L. Xu^67, X. P. Xu^55, Y. Xu^40, Y. C. Xu^78, Z. S. Xu^64, F. Yan^12,g, L. Yan^12,g, W. B. Yan^72,58, W. C. Yan^81, X. Q. Yan^1,64, H. J. Yang^51,f, H. L. Yang^34, H. X. Yang^1, J. H. Yang^42, T. Yang^1, Y. Yang^12,g, Y. F. Yang^1,64, Y. F. Yang^43, Y. X. Yang^1,64, Z. W. Yang^38,k,l, Z. P. Yao^50, M. Ye^1,58, M. H. Ye^8, J. H. Yin^1, Junhao Yin^43, Z. Y. You^59, B. X. Yu^1,58,64, C. X. Yu^43, G. Yu^1,64, J. S. Yu^25,i, M. C. Yu^40, T. Yu^73, X. D. Yu^46,h, Y. C. Yu^81, C. Z. Yuan^1,64, J. Yuan^34, J. Yuan^45, L. Yuan^2, S. C. Yuan^1,64, Y. Yuan^1,64, Z. Y. Yuan^59, C. X. Yue^39, A. A. Zafar^74, F. R. Zeng^50, S. H. Zeng^63A,63B,63C,63D, X. Zeng^12,g, Y. Zeng^25,i, Y. J. Zeng^59, Y. J. Zeng^1,64, X. Y. Zhai^34, Y. C. Zhai^50, Y. H. Zhan^59, A. Q. Zhang^1,64, B. L. Zhang^1,64, B. X. Zhang^1, D. H. Zhang^43, G. Y. Zhang^19, H. Zhang^81, H. Zhang^72,58, H. C. Zhang^1,58,64, H. H. Zhang^59, H. H. Zhang^34, H. Q. Zhang^1,58,64, H. R. Zhang^72,58, H. Y. Zhang^1,58, J. Zhang^59, J. Zhang^81, J. J. Zhang^52, J. L. Zhang^20, J. Q. Zhang^41, J. S. Zhang^12,g, J. W. Zhang^1,58,64, J. X. Zhang^38,k,l, J. Y. Zhang^1, J. Z. Zhang^1,64, Jianyu Zhang^64, L. M. Zhang^61, Lei Zhang^42, P. Zhang^1,64, Q. Y. Zhang^34, R. Y. Zhang^38,k,l, S. H. Zhang^1,64, Shulei Zhang^25,i, X. M. Zhang^1, X. Y Zhang^40, X. Y. Zhang^50, Y.  Zhang^73, Y. Zhang^1, Y.  T. Zhang^81, Y. H. Zhang^1,58, Y. M. Zhang^39, Yan Zhang^72,58, Z. D. Zhang^1, Z. H. Zhang^1, Z. L. Zhang^34, Z. Y. Zhang^77, Z. Y. Zhang^43, Z. Z.  Zhang^45, G. Zhao^1, J. Y. Zhao^1,64, J. Z. Zhao^1,58, L. Zhao^1, Lei Zhao^72,58, M. G. Zhao^43, N. Zhao^79, R. P. Zhao^64, S. J. Zhao^81, Y. B. Zhao^1,58, Y. X. Zhao^31,64, Z. G. Zhao^72,58, A. Zhemchugov^36,b, B. Zheng^73, B. M. Zheng^34, J. P. Zheng^1,58, W. J. Zheng^1,64, Y. H. Zheng^64, B. Zhong^41, X. Zhong^59, H.  Zhou^50, J. Y. Zhou^34, L. P. Zhou^1,64, S.  Zhou^6, X. Zhou^77, X. K. Zhou^6, X. R. Zhou^72,58, X. Y. Zhou^39, Y. Z. Zhou^12,g, Z. C. Zhou^20, A. N. Zhu^64, J. Zhu^43, K. Zhu^1, K. J. Zhu^1,58,64, K. S. Zhu^12,g, L. Zhu^34, L. X. Zhu^64, S. H. Zhu^71, T. J. Zhu^12,g, W. D. Zhu^41, Y. C. Zhu^72,58, Z. A. Zhu^1,64, J. H. Zou^1, J. Zu^72,58




(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 Bochum  Ruhr-University, D-44780 Bochum, Germany

^4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia

^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 Chung-Ang University, Seoul, 06974, Republic of Korea

^11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan

^12 Fudan University, Shanghai 200433, People's Republic of China

^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, Kaifeng 475004, People's Republic of China

^21 Henan University of Science and Technology, Luoyang 471003, People's Republic of China

^22 Henan University of Technology, Zhengzhou 450001, People's Republic of China

^23 Huangshan College, Huangshan  245000, People's Republic of China

^24 Hunan Normal University, Changsha 410081, People's Republic of China

^25 Hunan University, Changsha 410082, People's Republic of China

^26 Indian Institute of Technology Madras, Chennai 600036, India

^27 Indiana University, Bloomington, Indiana 47405, USA

^28 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

^29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of Ferrara,  I-44122, Ferrara, Italy

^30 Inner Mongolia University, Hohhot 010021, People's Republic of China

^31 Institute of Modern Physics, Lanzhou 730000, People's Republic of China

^32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330, Mongolia

^33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile

^34 Jilin University, Changchun 130012, People's Republic of China

^35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany

^36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia

^37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany

^38 Lanzhou University, Lanzhou 730000, People's Republic of China

^39 Liaoning Normal University, Dalian 116029, People's Republic of China

^40 Liaoning University, Shenyang 110036, People's Republic of China

^41 Nanjing Normal University, Nanjing 210023, People's Republic of China

^42 Nanjing University, Nanjing 210093, People's Republic of China

^43 Nankai University, Tianjin 300071, People's Republic of China

^44 National Centre for Nuclear Research, Warsaw 02-093, Poland

^45 North China Electric Power University, Beijing 102206, People's Republic of China

^46 Peking University, Beijing 100871, People's Republic of China

^47 Qufu Normal University, Qufu 273165, People's Republic of China

^48 Renmin University of China, Beijing 100872, People's Republic of China

^49 Shandong Normal University, Jinan 250014, People's Republic of China

^50 Shandong University, Jinan 250100, People's Republic of China

^51 Shanghai Jiao Tong University, Shanghai 200240,  People's Republic of China

^52 Shanxi Normal University, Linfen 041004, People's Republic of China

^53 Shanxi University, Taiyuan 030006, People's Republic of China

^54 Sichuan University, Chengdu 610064, People's Republic of China

^55 Soochow University, Suzhou 215006, People's Republic of China

^56 South China Normal University, Guangzhou 510006, People's Republic of China

^57 Southeast University, Nanjing 211100, People's Republic of China

^58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049, Hefei 230026, People's Republic of China

^59 Sun Yat-Sen University, Guangzhou 510275, People's Republic of China

^60 Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand

^61 Tsinghua University, Beijing 100084, People's Republic of China

^62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University, 34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus, 99138, Mersin 10, Turkey

^63 University of Bristol, (A)H H Wills Physics Laboratory; (B)Tyndall Avenue; (C)Bristol; (D)BS8 1TL

^64 University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China

^65 University of Groningen, NL-9747 AA Groningen, The Netherlands

^66 University of Hawaii, Honolulu, Hawaii 96822, USA

^67 University of Jinan, Jinan 250022, People's Republic of China

^68 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

^69 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany

^70 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom

^71 University of Science and Technology Liaoning, Anshan 114051, People's Republic of China

^72 University of Science and Technology of China, Hefei 230026, People's Republic of China

^73 University of South China, Hengyang 421001, People's Republic of China

^74 University of the Punjab, Lahore-54590, Pakistan

^75 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

^76 Uppsala University, Box 516, SE-75120 Uppsala, Sweden

^77 Wuhan University, Wuhan 430072, People's Republic of China

^78 Yantai University, Yantai 264005, People's Republic of China

^79 Yunnan University, Kunming 650500, People's Republic of China

^80 Zhejiang University, Hangzhou 310027, People's Republic of China

^81 Zhengzhou University, Zhengzhou 450001, People's Republic of China



^a Deceased

^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 Goethe University Frankfurt, 60323 Frankfurt am Main, Germany

^f 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

^g 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

^h Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China

^i Also at School of Physics and Electronics, Hunan University, Changsha 410082, China

^j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China

^k Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China

^l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China

^m Also at the Department of Mathematical Sciences, IBA, Karachi 75270, Pakistan

^n Also at Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

^o Also at Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany





    September 9, 2024
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Although the standard model (SM) has achieved great success in high-energy physics, some questions like, e.g., dark matter, matter and anti-matter asymmetry, fermion mass hierarchy remain unresolved. So-called “dark" sectors have been theorized, named as such due to their assumed extremely faint interactions with the visible sector. Searching for these dark sectors offers a unique opportunity to uncover new physics (NP) beyond the SM, which would require extension to accommodate for these new particles and their interaction with SM matter. In the SM, the Abelian gauge group U(1)_Y (with hypercharge Y) describes the electromagnetic interaction and produces the associated gauge boson, the SM photon γ. 
A minimum extension of the SM with an extra Abelian gauge group, U(1)_D (D denoting dark) produces another gauge boson, the dark photon <cit.>. 
These two Abelian gauge fields could have a kinetic mixing because the field strengths can be multiplied to form a dimension-four operator <cit.>.
The dark photon may be a portal between the SM matter and the dark sector matter, and it could be massive or massless. If the symmetry of U(1)_D is broken spontaneously, the dark photon will acquire mass, which will be called A'. If the symmetry of U(1)_D remains unbroken, it will produce the massless dark photon, γ' <cit.>. 
This massless dark photon could provide a natural explanation for the fermion mass hierarchy puzzle <cit.>.

While the massive dark photon couples to SM matter via the coupling strength eε through the kinetic mixing, where ε is the mixing parameter <cit.>, the massless dark photon has no direct interaction with the SM particle in the dimension-four operator <cit.>.
In the massless dark photons case, the SM photon can couple with the dark sector particle, which is usually called milli-charge particle <cit.>.

The massive dark photon can be produced in many processes by replacing the SM photon, such as e^+e^-→γ A', a process that has not been confirmed yet despite extensive search. A stringent constraint is set on the mixing parameter ε with different m_A' <cit.>, indicating that the value of ε must be small. 

Since the massless dark photon has no interaction with SM particles within the dimension-four operator, the restrictions obtained from the massive dark photon do not apply to the massless dark photon. 


















                                                  



                                                   
    








A dimension-six operator has been proposed to provide a connection between SM matter and the massless dark photon <cit.>:


    ℒ_NP=   1/Λ^2_NP ( C^U_jkq̅_j σ^μν u_k H̃ + C^D_jkq̅_j σ^μν d_k H 
       + C^L_jkl̅_j σ^μν e_k H + h.c. ) F'_μν,


where Λ_NP is the effective mass, indicating the NP energy scale, C^U_jk, C^D_jk, and C^L_jk are the up-type, down-type, and charged-lepton-type dimensionless coefficients, respectively, depending on the NP and not necessarily related to one another, j(k)=1,2,3 is the generation tag of the SM particle. More details can be found in Ref. <cit.>.

The first three terms in this equations are the couplings between the massless dark photon and the up-type quarks, down-type quarks, and charged leptons, where the flavors of the two quarks or leptons could be identical or different, differing to flavor diagonal of the tree-level couplings of the massive dark photon.
This effective operator may cover some new dark-sector particles with very heavy mass in the NP energy scale Λ_NP <cit.>. Up to now, no new particles have been found up to the mass of ∼ 1 TeV, but some anomalies require an energy scale above the electroweak energy scale Λ_EW (∼ 100 GeV).
Similar to the β decay observed in low energy experiments <cit.>, where a missing neutrino within the four-fermion effective coupling <cit.> can predict the electroweak energy scale at about 100 GeV <cit.>, the massless dark photon could provide a portal for exploring the NP energy scale Λ_NP beyond the TeV magnitude.

In this Letter, we focus on the first item of the dimension-six operator in Eq. (<ref>), which causes the cuγ' coupling in the flavor-changing neutral current (FCNC) process of a charm quark with j=1,k=2.
In the SM, the FCNC processes are strongly suppressed by the Glashow-Iliopoulos-Maiani mechanism <cit.>, stating that these processes are forbidden at the tree level and can only happen through a loop diagram. In the SM, the branching fraction (BF) of the FCNC charm process is expected to be smaller than 10^-9 <cit.>. But for the cuγ' coupling, its FCNC process originates from the NP coupling in the NP energy scale, which is different from the SM. 

In the charm sector, the massless dark photon can be searched for in D meson or Λ^+_c baryon decays, such as D→ Vγ', D→γγ', or Λ^+_c→ p γ', where V is a vector particle like ρ or ω. The BFs of these processes are directly related to |ℂ|^2+|ℂ_5|^2 <cit.>. Here ℂ=Λ_NP^-2(C^U_12+C^U*_12)ν/√(8) and ℂ_5=Λ_NP^-2(C^U_12-C^U*_12)ν/√(8) with the Higgs vacuum expectation value ν=246.2 GeV <cit.>, which are determined by the NP energy scale Λ_NP and the up-type dimensionless coefficient C^U_12. From the constraint of the dark matter (DM) and the vacuum stability (VS) in the universe <cit.>, the allowed BF of the massless dark photon in charm FCNC processes can be enhanced to 10^-7∼10^-5 <cit.>. Previously, BESIII searched for Λ^+_c→ p γ' and set the upper limit (UL) of its decay BF to be 8×10^-5 at the 90% confidence level (CL) <cit.>. This, however, is still above the allowed region obtained from DM and VS <cit.>. In this Letter, we search for the massless dark photon and probe the NP energy scale through the FCNC processes D^0→ωγ' and D^0→γγ' for the first time, which can be mediated via Feynman diagrams shown in FIG <ref>, by analyzing e^+e^- collision data of 7.9 fb^-1 at a center-of-mass energy of √(s)=3.773 GeV with the BESIII detector.












































































        






Details about the BESIII detector design and performance are provided elsewhere <cit.>. The simulated Monte Carlo (MC) samples, also described in Ref. <cit.>, are used to determine detection efficiencies and to estimate backgrounds.
The generator of the signal MC samples is parameterized by the helicity amplitudes same as the radiative decay of the D meson <cit.>.
At √(s)=3.773 GeV, the D^0D̅^0 meson pairs are produced from ψ(3770) decays without accompanying hadrons, which provide an ideal opportunity to study invisible massless dark photon decays of D mesons using the double tag (DT) method <cit.>. The D̅^0 mesons are first tagged with the main hadronic-decay modes D̅^0→ K^+π^-, D̅^0→ K^+π^-π^0 and D̅^0→ K^+π^-π^+π^-, and the selected candidates are referred to as the single tag (ST) sample. Here and throughout this letter, charge conjugations are always implied. Then, the signal processes D^0→ωγ' and D^0→γγ' are searched for in the system recoiling against the ST D̅^0 meson, and the selected candidates are denoted as the DT sample. Here, ω is reconstructed through its decay ω→π^+π^-π^0, π^0→γγ, and γ' is missing in the detector. The BFs of D^0→ωγ' and D^0→γγ' are calculated by


    ℬ(D^0→ω(γ)  γ')= N_sig/ϵ̂/ℬ_int∑_i N_i^ST,


with ∑_i N_i^ST=(6306.7±2.8)×10^3 <cit.> and the effective efficiency


    ϵ̂=∑_i ϵ_i^DT/ϵ_i^ST×N_i^ST/∑_i N_i^ST,


where i indicates each mode of D̅^0→hadrons, N_sig is the signal yield of the massless dark photon in data, N_i^ST is the ST yield of D̅^0 meson samples in data, ϵ_i^ST is the ST efficiency of D̅^0→hadrons, ϵ_i^DT is the DT efficiency of D̅^0→hadrons, D^0→ω(γ)γ', ℬ_int=ℬ(ω→π^+π^-π^0)×ℬ(π^0→γγ) is obtained from Particle Data Group <cit.> for D^0→ωγ' and ℬ_int=1 for D^0→γγ'.

















    








The selection criteria of ST samples, the ST yield N_i^ST, and the ST efficiency ϵ_i^ST can be found in Ref. <cit.>. The selection criteria of D^0→ωγ' and D^0→γγ', based on the tagged D̅^0 meson samples, are described below. To select D^0→γγ', no additional charged track is allowed. 




The good charged track, particle identification (PID), and photon candidates are selected with the same strategy as outlined in Ref. <cit.>.
To select D^0→ωγ', only events with exactly two selected charged tracks, both identified as pions with zero net charge, are retained for further analysis.
There should be at least one photon with energy larger than 0.5 GeV for D^0→γγ' and at least two photons for D^0→ωγ', where the two photons with minimum χ^2 value of the kinematic fit <cit.> constraining M_γγ to the nominal π^0 mass are regarded as the correct photons from the π^0 meson. To select the ω meson in the data samples, the invariant mass of the two photons M_γγ before the kinematic fit must be in the region of [0.115, 0.150], and the invariant mass M_π^+π^-π^0 of the ω candidate is required to be in the region of [0.700, 0.850]. To further reduce the non-ω background, a kinematic fit <cit.> constraining M_γγ to the nominal π^0 mass and M_π^+π^-π^0 to the nominal ω mass is performed to obtain the χ^2_2C value which is required to be less than 44, optimized with the Punzi-optimization method <cit.>. To suppress the background with additional photons or π^0, the total energy of photon candidates other than those from the π^0 (γ) and the D̅^0 meson (E^tot_oth.γ) is required to be less than 0.1 GeV for D^0→ωγ' (D^0→γγ'). After these selections, there may still be some background particles flying to the endcap of the detector that cannot be effectively detected <cit.>, so the recoiling angle of D̅^0ω (D̅^0γ) is applied to veto these associated background events. The cosine of the recoiling angle is defined as cosθ^reccoil_D̅ω(γ)=|p⃗_cms - p⃗_D̅^0 - p⃗_ω(γ)|_z/|p⃗_cms - p⃗_D̅^0 - p⃗_ω(γ)|, where p⃗_cms is the momentum of the center-of-mass in e^+e^- collision, p⃗_D̅^0 is the reconstructed momentum of the D̅^0 meson, p⃗_ω(γ) is the reconstructed momentum of ω (γ), and the subscript z refers to the z-component. To suppress these background events, a requirement of |cosθ_D̅ω(γ)^recoil|<0.7 is applied. 
With the above selection criteria, 

the effective efficiency is estimated from the MC samples, which is ϵ̂=(15.98±0.02)% for D^0→ωγ' and ϵ̂=(52.18±0.05)% for D^0→γγ'.
The main background after the selections comes from the K^0_L associated background events, such as D^0→ω K^0_L for D^0→ωγ' and D^0→π^0 K^0_L for D^0→γγ'.









The signals of the massless dark photon are extracted from an unbinned extended maximum likelihood fit on the distribution of the  square of the missing mass, M^2_miss, defined as


    M^2_miss=|p_cms - p_D̅^0 - p_ω(γ)|^2/c^4,


where p_cms is the four-momentum of the e^+e^- center-of-mass system in the laboratory frame, p_ω(γ) is the kinematic fitted (reconstructed) four-momentum of ω(γ), p_D̅^0 is the four-momentum of the D̅^0 meson, achieved by the kinematic fit <cit.> constraining M_γγ to the nominal π^0 mass and M_K^+π^-, M_K^+π^-π^0, M_K^+π^-π^+π^- to the nominal D̅^0 meson mass. In the fit, the background is separated into the K^0_L-related and non-K^0_L backgrounds <cit.>. 
The background shape is derived from the inclusive MC sample <cit.>, with the number of non-K^0_L background events assumed to follow a Gaussian distribution and constrained by the MC simulation (referred to as the Gaussian constraint), while the number of K^0_L-related background events is left as a floating parameter.
The signal is derived by the simulated shape convolved with a Gaussian function G(μ,σ), where μ and σ are restrained to the values obtained from the control samples D^0→ω K^0_S (D^0→π^0 K^0_S) for D^0→ωγ' (D^0→γγ'). The fit results are shown in FIG <ref>, with the massless dark photon signal yield N_sig=-15±8 for D^0→ωγ' and N_sig=-6±4 for D^0→γγ'.


















The systematic uncertainty sources for the BF measurement include ST yield, intermediate BF, signal generator, DT signal efficiency, and signal extraction. With the DT method, several systematic uncertainties associated with the ST selection can be canceled without impacting the BF measurement. The uncertainty of ST yield is assigned as 0.1% <cit.>. 
The uncertainty of the generator is estimated from the efficiency difference compared with a flat angular generation of γ' (phase space model), which is 1.3% (0.6%) for D^0→ωγ' (D^0→γγ').
The uncertainty of the BF of ω→π^+π^-π^0 is 0.8% and that of π^0→γγ is negligible <cit.>. 
The uncertainty of photon detection is assigned as 1.0% per photon <cit.>. The uncertainties of pion tracking and PID are studied from the control sample J/ψ→π^+π^-π^0, which is assigned as 0.9% for tracking and 1.1% for PID of the two pions.
The uncertainties of other selections are estimated from the control sample D^0→ω K^0_S (D^0→π^0 K^0_S) for D^0→ωγ' (D^0→γγ'), where the K^0_S meson is regarded as a missing particle. For D^0→ωγ', the uncertainty is 0.2% for the M_γγ selection, negligible for the M_π^+π^-π^0 selection, 3.1% for the χ^2_2C selection, 5.9% for the E^tot_oth.γ selection and 1.1% for the cosθ^recoil_D̅ω selection, respectively. For D^0→γγ',  the uncertainty is 2.4% for the E^tot_oth.γ selection and is 0.6% for the cosθ^recoil_D̅γ selection, respectively. The total systematic uncertainty is calculated by summing up all sources in quadrature, yielding 7.4% for D^0→ωγ' and 2.7% for D^0→γγ'.
For the uncertainty from the signal extraction, the signal shape is convolved with a Gaussian function to describe the difference where the parameter of the Gaussian function is in a Gaussian constraint within its uncertainty, the K^0_L background yield is floating in the fit, and the non-K^0_L background yield is also floating in a Gaussian constraint within its uncertainty. The uncertainty of signal extraction is negligible.

Since no significant excess of signal above the background is observed, a UL on the BF is set using a Bayesian approach following Ref. <cit.>, where the BF is calculated by Eq. (<ref>) and the systematic uncertainty is estimated with the method in Refs. <cit.>. The UL on the BF at the 90% CL is calculated by integrating the likelihood distribution with different signal assumptions to the 90% region, which is ℬ(D^0→ωγ')<1.1×10^-5 and ℬ(D^0→γγ')<2.0×10^-6. 
Note that in the D^0→ωγ' measurement, the non-ω contribution can not be fully removed, and the current UL of ℬ(D^0→ωγ') is a conservative estimation.
Since the BF of massless dark photon production is related to |ℂ|^2+|ℂ_5|^2 and directly includes the NP energy scale <cit.>, the constraint on |ℂ|^2+|ℂ_5|^2 can be performed as well.















The UL of |ℂ|^2+|ℂ_5|^2 is shown in FIG <ref>. For D^0→ωγ', one sees that |ℂ|^2+|ℂ_5|^2<8.2×10^-17^-2, reaching the DM and VS allowed region <cit.> for the first time. 
The channel D^0→γγ' has a better UL of the BF but a worse constraint on cuγ' coupling due to an additional electromagnetic vertex in FIG <ref>(b).
The two-dimensional constraint on the NP energy scale Λ_NP and the up-type dimensionless coefficient C^U_12 is given in FIG <ref> (b). Our result currently resemble the best constraint on the NP parameter space. Assuming |C^U_12|=1, our results can exclude NP energy scales below 138 TeV in the dark sector, which is approximately ten times the energy reached at the Large Hadron Collider <cit.>, suggesting a challenge of directly detecting superheavy particles at the NP energy scale associated with the cuγ' coupling in the present collider settings. Note that the value of |C^U_12| is model-dependent.


















            
		








In the dimension-six operator of the massless dark photon (Eq. (<ref>)), there are three terms with the same NP energy scale but different types of dimensionless coefficients C^U_jk, C^D_jk, C^L_jk. For each type, the massless dark photon can couple with a pair of SM particles involving identical or different flavors. In principle, these dimensionless coefficients are not necessarily related to one another. 

To compare the difference between different couplings, we follow the method from Ref. <cit.>, constructing a uniform variable F_NP=Λ_NP/√(|C^i_jk|ν/√(2m^2)) 
to perform the comparison on the first two generations, where i=U,D,L, |C^i_jk|ν/√(2m^2) indicates the strength of the massless dark photon coupled with the SM particle. Here, m is the mass of the heavier SM particle in the coupling, and F_NP represents the NP energy scale when |C^i_jk| equals to the Higgs-fermions coupling √(2)m/v <cit.>. The summary of the constraints in different couplings is shown in FIG <ref>, depicting only the best constraint on the coupling.
The constraints on the massless dark photon coupled with the same-flavor SM particles are mainly from astrophysics and cosmology <cit.>, while the constraints on the different flavors are mainly from laboratory physics <cit.>. This Letter provides the best constraint for the cuγ' coupling, which plays a unique role in the dark sector of the charm sector.

In summary, 

we search for the massless dark photon and constraint the NP scale through the cuγ' coupling in D^0→ωγ' and D^0→γγ' for the first time. Based on 7.9 fb^-1 of e^+e^- collision data at √(s)=3.773 GeV, no significant signals are observed. The constraints on the BF and the NP energy scale of the massless dark photon production are given. The result of D^0→ωγ' gives the most stringent constraint on cuγ' coupling to date, exploring the DM and VS allowed space for the first time. The result of D^0→γγ' has a 5.5 times better UL of the BF than D^0→ωγ' but a worse constraint on cuγ' coupling.



Acknowledgement

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. 2023YFA1606000, 2020YFA0406400, 2020YFA0406300; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11635010, 11735014, 11935015, 11935016, 11935018, 12025502, 12035009, 12035013, 12061131003, 12175321, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017, 12361141819; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207, U1932101; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. FOR5327, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Knut and Alice Wallenberg Foundation under Contracts Nos. 2021.0174, 2021.0299; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology fund of Mongolia; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation of Thailand under Contracts Nos. B16F640076, B50G670107; Polish National Science Centre under Contract No. 2019/35/O/ST2/02907; Swedish Research Council under Contract No. 2019.04595; The Swedish Foundation for International Cooperation in Research and Higher Education under Contract No. CH2018-7756; U. S. Department of Energy under Contract No. DE-FG02-05ER41374.

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