* Chaker et al. [2021] Jamil Chaker, Minhyun Kim, and Marvin Weidner.  Regularity for nonlocal problems with non-standard growth.  _arXiv:2111.09182 [math]_ , November 2021.  URL http://arxiv.org/abs/2111.09182.
  * Cozzi [2017] Matteo Cozzi.  Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes.  _Journal of Functional Analysis_ , 272(11):4762–4837, June 2017.  ISSN 0022-1236.  doi:10.1016/j.jfa.2017.02.016.  URL https://www.sciencedirect.com/science/article/pii/S0022123617300770.
  * Cozzi [2019] Matteo Cozzi.  Fractional De Giorgi Classes and Applications to Nonlocal Regularity Theory.  In Serena Dipierro, editor, _Contemporary Research in Elliptic PDEs and Related Topics_ , Springer INdAM Series, pages 277–299. Springer International Publishing, Cham, 2019.  ISBN 978-3-030-18921-1.  doi:10.1007/978-3-030-18921-1_7.  URL https://doi.org/10.1007/978-3-030-18921-1_7.
  * De Filippis and Mingione [2022] Cristiana De Filippis and Giuseppe Mingione.  Gradient regularity in mixed local and nonlocal problems.  _arXiv:2204.06590 [math], forthcoming in Mathematische Annalen_ , arXiv:2204.06590 [math], April 2022.  URL http://arxiv.org/abs/2204.06590.
  * De Filippis and Palatucci [2019] Cristiana De Filippis and Giampiero Palatucci.  Hölder regularity for nonlocal double phase equations.  _Journal of Differential Equations_ , 267(1):547–586, June 2019.  ISSN 0022-0396.  doi:10.1016/j.jde.2019.01.017.  URL https://www.sciencedirect.com/science/article/pii/S0022039619300397.
  * Di Castro et al. [2014] Agnese Di Castro, Tuomo Kuusi, and Giampiero Palatucci.  Nonlocal Harnack inequalities.  _Journal of Functional Analysis_ , 267(6):1807–1836, September 2014.  ISSN 0022-1236.  doi:10.1016/j.jfa.2014.05.023.  URL https://www.sciencedirect.com/science/article/pii/S0022123614002407.
  * Di Castro et al. [2016] Agnese Di Castro, Tuomo Kuusi, and Giampiero Palatucci.  Local behavior of fractional p-minimizers.  _Annales de l’Institut Henri Poincaré C, Analyse non linéaire_ , 33(5):1279–1299, September 2016.  ISSN 0294-1449.  doi:10.1016/j.anihpc.2015.04.003.  URL https://www.sciencedirect.com/science/article/pii/S0294144915000451.
  * DiBenedetto [1986] Emmanuele DiBenedetto.  On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients.  _Annali della Scuola Normale Superiore di Pisa - Classe di Scienze_ , 13(3):487–535, 1986.  URL http://www.numdam.org/item/?id=ASNSP_1986_4_13_3_487_0.
  * DiBenedetto [1993] Emmanuele DiBenedetto.  _Degenerate Parabolic Equations_.  Universitext. Springer-Verlag, New York, 1993.  ISBN 0-387-94020-0.  doi:10.1007/978-1-4612-0895-2.  URL https://doi.org/10.1007/978-1-4612-0895-2.
  * DiBenedetto et al. [2008] Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri.  Harnack estimates for quasi-linear degenerate parabolic differential equations.  _Acta Mathematica_ , 200(2):181–209, January 2008\.  ISSN 0001-5962, 1871-2509.  doi:10.1007/s11511-008-0026-3.  URL https://projecteuclid.org/journals/acta-mathematica/volume-200/issue-2/Harnack-estimates-for-quasi-linear-degenerate-parabolic-differential-equations/10.1007/s11511-008-0026-3.full.
  * DiBenedetto et al. [2012] Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri.  _Harnack’s Inequality for Degenerate and Singular Parabolic Equations_.  Springer Monographs in Mathematics. Springer, New York, 2012\.  ISBN 978-1-4614-1583-1.  doi:10.1007/978-1-4614-1584-8.  URL https://mathscinet.ams.org/mathscinet-getitem?mr=2865434.
  * Ding et al. [2021] Mengyao Ding, Chao Zhang, and Shulin Zhou.  Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations.  _Calculus of Variations and Partial Differential Equations_ , 60(1):38, January 2021.  ISSN 1432-0835.  doi:10.1007/s00526-020-01870-x.  URL https://doi.org/10.1007/s00526-020-01870-x.
  * Garain and Lindgren [2022] Prashanta Garain and Erik Lindgren.  Higher hölder regularity for mixed local and nonlocal degenerate elliptic equations.  _arXiv:2204.13196 [math]_ , April 2022.  URL http://arxiv.org/abs/2204.13196.
  * Gianazza et al. [2010] Ugo Gianazza, Mikhail Surnachev, and Vincenzo Vespri.  A new proof of the Hölder continuity of solutions to $\displaystyle p$-laplace type parabolic equations.  _Advances in Calculus of Variations_ , 3(3):263–278, 2010.  ISSN 1864-8258.  doi:10.1515/ACV.2010.009.  URL https://mathscinet.ams.org/mathscinet-getitem?mr=2660688.
  * Hwang and Lieberman [2015a] Sukjung Hwang and Gary M. Lieberman.  Hölder continuity of bounded weak solutions to generalized parabolic $\displaystyle p$-laplacian equations II: singular case.  _Electronic Journal of Differential Equations_ , pages No. 288, 24, 2015a.  URL https://mathscinet.ams.org/mathscinet-getitem?mr=3441670.
  * Hwang and Lieberman [2015b] Sukjung Hwang and Gary M. Lieberman.  Hölder continuity of bounded weak solutions to generalized parabolic $\displaystyle p$-laplacian equations I: degenerate case.  _Electronic Journal of Differential Equations_ , pages No. 287, 32, 2015b.  URL https://mathscinet.ams.org/mathscinet-getitem?mr=3441669.
  * Iannizzotto et al. [2016] Antonio Iannizzotto, Sunra J. N. Mosconi, and Marco Squassina.  Global Hölder regularity for the fractional p-Laplacian.  _Revista Matemática Iberoamericana_ , 32(4):1353–1392, December 2016.  ISSN 0213-2230.  doi:10.4171/rmi/921.  URL https://ems.press/journals/rmi/articles/14483.
  * Ishii and Nakamura [2010] Hitoshi Ishii and Gou Nakamura.  A class of integral equations and approximation of p-Laplace equations.  _Calculus of Variations and Partial Differential Equations_ , 37(3):485–522, March 2010.  ISSN 1432-0835.  doi:10.1007/s00526-009-0274-x.  URL https://doi.org/10.1007/s00526-009-0274-x.
  * Kassmann [2009] Moritz Kassmann.  A priori estimates for integro-differential operators with measurable kernels.  _Calculus of Variations and Partial Differential Equations_ , 34(1):1–21, January 2009.  ISSN 1432-0835.  doi:10.1007/s00526-008-0173-6.  URL https://doi.org/10.1007/s00526-008-0173-6.
  * Liao [2020] Naian Liao.  A unified approach to the Hölder regularity of solutions to degenerate and singular parabolic equations.  _Journal of Differential Equations_ , 268(10):5704–5750, May 2020.  ISSN 0022-0396.  doi:10.1016/j.jde.2019.11.023.  URL https://www.sciencedirect.com/science/article/pii/S0022039619305571.
  * Liao and Schätzler [2021] Naian Liao and Leah Schätzler.  On the Hölder Regularity of Signed Solutions to a Doubly Nonlinear Equation. Part III.  _International Mathematics Research Notices_ , page rnab339, December 2021.  ISSN 1073-7928.  doi:10.1093/imrn/rnab339.  URL https://doi.org/10.1093/imrn/rnab339.
  * Liao et al. [2020] Naian Liao, Igor I. Skrypnik, and Vincenzo Vespri.  Local regularity for an anisotropic elliptic equation.  _Calculus of Variations and Partial Differential Equations_ , 59(4):116, June 2020.  ISSN 1432-0835.  doi:10.1007/s00526-020-01781-x.  URL https://doi.org/10.1007/s00526-020-01781-x.
  * Mosconi [2018] Sunra J. N. Mosconi.  Optimal elliptic regularity: A comparison between local and nonlocal equations.  _Discrete & Continuous Dynamical Systems - S_, 11(3):547, 2018.  doi:10.3934/dcdss.2018030.  URL https://www.aimsciences.org/article/doi/10.3934/dcdss.2018030.
  * Prasad and Tewary [2021] Harsh Prasad and Vivek Tewary.  Local boundedness of variational solutions to nonlocal double phase parabolic equations.  _arXiv:2112.02345 [math], accepted for publication in Journal of Differential Equations_ , December 2021.  URL http://arxiv.org/abs/2112.02345.
  * Prasad and Tewary [2022] Harsh Prasad and Vivek Tewary.  Existence of variational solutions to nonlocal evolution equations via convex minimization.  _arXiv:2112.00402 [math], accepted for publication in ESAIM: Control, Optimisation, and Calculus of Variations_ , January 2022.  URL http://arxiv.org/abs/2112.00402.
  * Silvestre [2006] Luis Silvestre.  Hölder Estimates for Solutions of Integro-differential Equations Like The Fractional Laplace.  _Indiana University Mathematics Journal_ , 55(3):1155–1174, 2006.  ISSN 0022-2518.  URL https://www.jstor.org/stable/24902437.
  * Strömqvist [2019] Martin Strömqvist.  Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian.  _Journal of Differential Equations_ , 266(12):7948–7979, June 2019.  ISSN 0022-0396.  doi:10.1016/j.jde.2018.12.021.  URL https://www.sciencedirect.com/science/article/pii/S0022039618307095.
  * Ya-zhe and DiBenedetto [1988] Chen Ya-zhe and Emmanuele DiBenedetto.  On the local behavior of solutions of singular parabolic equations.  _Archive for Rational Mechanics and Analysis_ , 103(4):319–345, December 1988.  ISSN 1432-0673.  doi:10.1007/BF00251444.  URL https://doi.org/10.1007/BF00251444.


# Study of $e^{+}e^{-}\to 2(p\bar{p})$ at center-of-mass energies between 4.0
and 4.6 GeV

M. Ablikim1, M. N. Achasov10,c, P. Adlarson67, S. Ahmed15, M. Albrecht4, R.
Aliberti28, A. Amoroso66A,66C, Q. An63,50, Anita21, X. H. Bai57, Y. Bai49, O.
Bakina29, R. Baldini Ferroli23A, I. Balossino24A, Y. Ban39,k, K. Begzsuren26,
N. Berger28, M. Bertani23A, D. Bettoni24A, F. Bianchi66A,66C, J Biernat67, J.
Bloms60, A. Bortone66A,66C, I. Boyko29, R. A. Briere5, H. Cai68, X. Cai1,50,
A. Calcaterra23A, G. F. Cao1,55, N. Cao1,55, S. A. Cetin54B, J. F. Chang1,50,
W. L. Chang1,55, G. Chelkov29,b, D. Y. Chen6, G. Chen1, H. S. Chen1,55, M. L.
Chen1,50, S. J. Chen36, X. R. Chen25, Y. B. Chen1,50, Z. J Chen20,l, W. S.
Cheng66C, G. Cibinetto24A, F. Cossio66C, X. F. Cui37, H. L. Dai1,50, X. C.
Dai1,55, A. Dbeyssi15, R. B. de Boer4, D. Dedovich29, Z. Y. Deng1, A. Denig28,
I. Denysenko29, M. Destefanis66A,66C, F. De Mori66A,66C, Y. Ding34, C. Dong37,
J. Dong1,50, L. Y. Dong1,55, M. Y. Dong1,50,55, X. Dong68, S. X. Du71, J.
Fang1,50, S. S. Fang1,55, Y. Fang1, R. Farinelli24A, L. Fava66B,66C, F.
Feldbauer4, G. Felici23A, C. Q. Feng63,50, M. Fritsch4, C. D. Fu1, Y. Fu1, Y.
Gao39,k, Y. Gao64, Y. Gao63,50, Y. G. Gao6, I. Garzia24A,24B, E. M.
Gersabeck58, A. Gilman59, K. Goetzen11, L. Gong34, W. X. Gong1,50, W. Gradl28,
M. Greco66A,66C, L. M. Gu36, M. H. Gu1,50, S. Gu2, Y. T. Gu13, C. Y Guan1,55,
A. Q. Guo22, L. B. Guo35, R. P. Guo41, Y. P. Guo9,h, Y. P. Guo28, A. Guskov29,
T. T. Han42, X. Q. Hao16, F. A. Harris56, K. L. He1,55, F. H. Heinsius4, C. H.
Heinz28, T. Held4, Y. K. Heng1,50,55, C. Herold52, M. Himmelreich11,f, T.
Holtmann4, Y. R. Hou55, Z. L. Hou1, H. M. Hu1,55, J. F. Hu48,m, T. Hu1,50,55,
Y. Hu1, G. S. Huang63,50, L. Q. Huang64, X. T. Huang42, Y. P. Huang1, Z.
Huang39,k, N. Huesken60, T. Hussain65, W. Ikegami Andersson67, W. Imoehl22, M.
Irshad63,50, S. Jaeger4, S. Janchiv26,j, Q. Ji1, Q. P. Ji16, X. B. Ji1,55, X.
L. Ji1,50, H. B. Jiang42, X. S. Jiang1,50,55, X. Y. Jiang37, J. B. Jiao42, Z.
Jiao18, S. Jin36, Y. Jin57, T. Johansson67, N. Kalantar-Nayestanaki31, X. S.
Kang34, R. Kappert31, M. Kavatsyuk31, B. C. Ke44,1, I. K. Keshk4, A.
Khoukaz60, P. Kiese28, R. Kiuchi1, R. Kliemt11, L. Koch30, O. B. Kolcu54B,e,
B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc67, M. G. Kurth1,55, W. Kühn30,
J. J. Lane58, J. S. Lange30, P. Larin15, L. Lavezzi66A,66C, Z. H. Lei63,50, H.
Leithoff28, M. Lellmann28, T. Lenz28, C. Li40, C. H. Li33, Cheng Li63,50, D.
M. Li71, F. Li1,50, G. Li1, H. Li44, H. Li63,50, H. B. Li1,55, H. J. Li9,h, J.
L. Li42, J. Q. Li4, Ke Li1, L. K. Li1, Lei Li3, P. L. Li63,50, P. R. Li32, S.
Y. Li53, W. D. Li1,55, W. G. Li1, X. H. Li63,50, X. L. Li42, Z. Y. Li51, H.
Liang63,50, H. Liang1,55, Y. F. Liang46, Y. T. Liang25, L. Z. Liao1,55, J.
Libby21, C. X. Lin51, B. J. Liu1, C. X. Liu1, D. Liu63,50, F. H. Liu45, Fang
Liu1, Feng Liu6, H. B. Liu13, H. M. Liu1,55, Huanhuan Liu1, Huihui Liu17, J.
B. Liu63,50, J. Y. Liu1,55, K. Liu1, K. Y. Liu34, Ke Liu6, L. Liu63,50, M. H.
Liu9,h, Q. Liu55, S. B. Liu63,50, Shuai Liu47, T. Liu1,55, W. M. Liu63,50, X.
Liu32, Y. B. Liu37, Z. A. Liu1,50,55, Z. Q. Liu42, X. C. Lou1,50,55, F. X.
Lu16, H. J. Lu18, J. D. Lu1,55, J. G. Lu1,50, X. L. Lu1, Y. Lu1, Y. P. Lu1,50,
C. L. Luo35, M. X. Luo70, P. W. Luo51, T. Luo9,h, X. L. Luo1,50, S. Lusso66C,
X. R. Lyu55, F. C. Ma34, H. L. Ma1, L. L. Ma42, M. M. Ma1,55, Q. M. Ma1, R. Q.
Ma1,55, R. T. Ma55, X. N. Ma37, X. X. Ma1,55, X. Y. Ma1,50, F. E. Maas15, M.
Maggiora66A,66C, S. Maldaner28, S. Malde61, Q. A. Malik65, A. Mangoni23B, Y.
J. Mao39,k, Z. P. Mao1, S. Marcello66A,66C, Z. X. Meng57, J. G.
Messchendorp31, G. Mezzadri24A, T. J. Min36, R. E. Mitchell22, X. H.
Mo1,50,55, Y. J. Mo6, N. Yu. Muchnoi10,c, H. Muramatsu59, S. Nakhoul11,f, Y.
Nefedov29, F. Nerling11,f, I. B. Nikolaev10,c, Z. Ning1,50, S. Nisar8,i, S. L.
Olsen55, Q. Ouyang1,50,55, S. Pacetti23B,23C, X. Pan9,h, Y. Pan58, A. Pathak1,
P. Patteri23A, M. Pelizaeus4, H. P. Peng63,50, K. Peters11,f, J. Pettersson67,
J. L. Ping35, R. G. Ping1,55, A. Pitka4, R. Poling59, V. Prasad63,50, H.
Qi63,50, H. R. Qi53, K. H. Qi25, M. Qi36, T. Y. Qi9, T. Y. Qi2, S. Qian1,50,
W.-B. Qian55, Z. Qian51, C. F. Qiao55, L. Q. Qin12, X. S. Qin4, Z. H. Qin1,50,
J. F. Qiu1, S. Q. Qu37, K. H. Rashid65, K. Ravindran21, C. F. Redmer28, A.
Rivetti66C, V. Rodin31, M. Rolo66C, G. Rong1,55, Ch. Rosner15, M. Rump60, H.
S. Sang63, A. Sarantsev29,d, Y. Schelhaas28, C. Schnier4, K. Schoenning67, M.
Scodeggio24A, D. C. Shan47, W. Shan19, X. Y. Shan63,50, M. Shao63,50, C. P.
Shen9, P. X. Shen37, X. Y. Shen1,55, H. C. Shi63,50, R. S. Shi1,55, X.
Shi1,50, X. D Shi63,50, W. M. Song27,1, Y. X. Song39,k, S. Sosio66A,66C, S.
Spataro66A,66C, K. X. Su68, F. F. Sui42, G. X. Sun1, H. K. Sun1, J. F. Sun16,
L. Sun68, S. S. Sun1,55, T. Sun1,55, W. Y. Sun35, X Sun20,l, Y. J. Sun63,50,
Y. K. Sun63,50, Y. Z. Sun1, Z. T. Sun1, Y. H. Tan68, Y. X. Tan63,50, C. J.
Tang46, G. Y. Tang1, J. Tang51, J. X. Teng63,50, V. Thoren67, I. Uman54D, C.
W. Wang36, D. Y. Wang39,k, H. P. Wang1,55, K. Wang1,50, L. L. Wang1, M.
Wang42, M. Z. Wang39,k, Meng Wang1,55, W. H. Wang68, W. P. Wang63,50, X.
Wang39,k, X. F. Wang32, X. L. Wang9,h, Y. Wang51, Y. Wang63,50, Y. D. Wang38,
Y. F. Wang1,50,55, Y. Q. Wang1, Z. Wang1,50, Z. Y. Wang1, Ziyi Wang55,
Zongyuan Wang1,55, D. H. Wei12, P. Weidenkaff28, F. Weidner60, S. P. Wen1, D.
J. White58, U. Wiedner4, G. Wilkinson61, M. Wolke67, L. Wollenberg4, J. F.
Wu1,55, L. H. Wu1, L. J. Wu1,55, X. Wu9,h, Z. Wu1,50, L. Xia63,50, H. Xiao9,h,
S. Y. Xiao1, Y. J. Xiao1,55, Z. J. Xiao35, X. H. Xie39,k, Y. G. Xie1,50, Y. H.
Xie6, T. Y. Xing1,55, G. F. Xu1, J. J. Xu36, Q. J. Xu14, W. Xu1,55, X. P.
Xu47, F. Yan9,h, L. Yan66A,66C, L. Yan9,h, W. B. Yan63,50, W. C. Yan71, Xu
Yan47, H. J. Yang43,g, H. X. Yang1, L. Yang44, R. X. Yang63,50, S. L. Yang55,
S. L. Yang1,55, Y. H. Yang36, Y. X. Yang12, Yifan Yang1,55, Zhi Yang25, M.
Ye1,50, M. H. Ye7, J. H. Yin1, Z. Y. You51, B. X. Yu1,50,55, C. X. Yu37, G.
Yu1,55, J. S. Yu20,l, T. Yu64, C. Z. Yuan1,55, L. Yuan2, W. Yuan66A,66C, X. Q.
Yuan39,k, Y. Yuan1, Z. Y. Yuan51, C. X. Yue33, A. Yuncu54B,a, A. A. Zafar65,
Y. Zeng20,l, B. X. Zhang1, Guangyi Zhang16, H. Zhang63, H. H. Zhang51, H. Y.
Zhang1,50, J. J. Zhang44, J. L. Zhang69, J. Q. Zhang4, J. W. Zhang1,50,55, J.
Y. Zhang1, J. Z. Zhang1,55, Jianyu Zhang1,55, Jiawei Zhang1,55, Lei Zhang36,
S. Zhang51, S. F. Zhang36, Shulei Zhang20,l, X. D. Zhang38, X. Y. Zhang42, Y.
Zhang61, Y. H. Zhang1,50, Y. T. Zhang63,50, Yan Zhang63,50, Yao Zhang1, Yi
Zhang9,h, Z. H. Zhang6, Z. Y. Zhang68, G. Zhao1, J. Zhao33, J. Y. Zhao1,55, J.
Z. Zhao1,50, Lei Zhao63,50, Ling Zhao1, M. G. Zhao37, Q. Zhao1, S. J. Zhao71,
Y. B. Zhao1,50, Y. X. Zhao25, Z. G. Zhao63,50, A. Zhemchugov29,b, B. Zheng64,
J. P. Zheng1,50, Y. Zheng39,k, Y. H. Zheng55, B. Zhong35, C. Zhong64, L. P.
Zhou1,55, Q. Zhou1,55, X. Zhou68, X. K. Zhou55, X. R. Zhou63,50, A. N.
Zhu1,55, J. Zhu37, K. Zhu1, K. J. Zhu1,50,55, S. H. Zhu62, T. J. Zhu69, W. J.
Zhu37, X. L. Zhu53, Y. C. Zhu63,50, Z. A. Zhu1,55, 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 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
32 Lanzhou University, Lanzhou 730000, People’s Republic of China
33 Liaoning Normal University, Dalian 116029, People’s Republic of China
34 Liaoning University, Shenyang 110036, People’s Republic of China
35 Nanjing Normal University, Nanjing 210023, People’s Republic of China
36 Nanjing University, Nanjing 210093, People’s Republic of China
37 Nankai University, Tianjin 300071, People’s Republic of China
38 North China Electric Power University, Beijing 102206, People’s Republic of
China
39 Peking University, Beijing 100871, People’s Republic of China
40 Qufu Normal University, Qufu 273165, People’s Republic of China
41 Shandong Normal University, Jinan 250014, People’s Republic of China
42 Shandong University, Jinan 250100, People’s Republic of China
43 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
44 Shanxi Normal University, Linfen 041004, People’s Republic of China
45 Shanxi University, Taiyuan 030006, People’s Republic of China
46 Sichuan University, Chengdu 610064, People’s Republic of China
47 Soochow University, Suzhou 215006, People’s Republic of China
48 South China Normal University, Guangzhou 510006, People’s Republic of China
49 Southeast University, Nanjing 211100, People’s Republic of China
50 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
51 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
52 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
53 Tsinghua University, Beijing 100084, People’s Republic of China
54 Turkish Accelerator Center Particle Factory Group, (A)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (B)Near East University, Nicosia,
North Cyprus, Mersin 10, Turkey
55 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
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 data taken at 23 center-of-mass energies between 4.0 and 4.6 GeV with
the BESIII detector at the BEPCII collider and with a total integrated
luminosity of approximately $15$ fb-1, the process $e^{+}e^{-}\to 2(p\bar{p})$
is studied for the first time. The Born cross sections for $e^{+}e^{-}\to
2(p\bar{p})$ are measured, and no significant structure is observed in the
lineshape. The baryon pair ($pp$ and $\bar{p}\bar{p}$) invariant mass spectra
are consistent with phase space, therefore no hexaquark or di-baryon state is
found.

##  I Introduction

Since 2003, a series of charmonium-like states, such as $X(3872)$ x3872 ,
$Y(4260)$ y42601 , and $Z_{c}(3900)$ zc39001 ; zc39002 , have been discovered.
The $Y(4260)$ was first observed by the BABAR experiment via the initial-state
radiative (ISR) process $e^{+}e^{-}\to\gamma_{\rm ISR}\pi^{+}\pi^{-}J/\psi$,
and it was confirmed by the CLEO isry42602  and Belle experiments isry42603 .
In 2017, BESIII reported precise measurements of the
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ cross sections in the energy region
between 3.77 and 4.60 GeV ppjpsibes3 . Two structures are observed with masses
of $4222.0\pm 3.1\pm 1.4$ MeV and $4320.0\pm 10.4\pm 7.0$ MeV, and the former,
regarded previously as the $Y(4260)$, is renamed to $Y(4220)$. The $Y(4220)$
mass is confirmed by cross-section measurements of
$e^{+}e^{-}\to\omega\chi_{c0}$ omegachic0 , $\pi^{+}\pi^{-}h_{c}$ pphcbes3 ,
$\pi^{+}\pi^{-}\psi(3686)$ pppsip , and $D^{0}D^{*-}\pi^{+}$ ddstpi .

Currently, the known decays of $Y(4220)$ occur only to open or hidden charm
final states. However, some theories argue that the charmonium-like states,
such as $Y(4220)$, are very likely to also decay to light hadrons
hadrocharmonium . BESIII has reported measurements of the cross sections for
the light hadron processes of $e^{+}e^{-}\to
K_{S}^{0}K^{\pm}\pi^{\mp}\pi^{0}(\eta)$ kskp , $K_{S}^{0}K^{\pm}\pi^{\mp}$
kskpi , $p\bar{n}K_{S}^{0}K^{-}+c.c.$ kskpn , and $p\bar{p}\pi^{0}$ pppi , but
no hint of charmless $Y(4220)$ decays has been found. Comprehensive
measurements of the cross sections for $e^{+}e^{-}\to light~{}hadrons$ are
important to search for charmless decays of $Y$ states and to deeply explore
the composition and properties of $Y$ states.

Searches for di-baryon or hexaquark states via $pp$ and $pn$ scattering
processes have been carried out in fixed target experiments. A resonance
$d^{*}(2380)$ in the isoscalar two-pion fusion process $pn\to d\pi^{0}\pi^{0}$
was observed by WASA dp0p0 . This state was later confirmed by the other two-
pion fusion processes $pn\to d\pi^{+}\pi^{-}$ dppp0  and $pp\to
d\pi^{+}\pi^{0}$ dpppm , and the two-pion non-fusion processes $pn\to
pp\pi^{0}\pi^{-}$ ppp0pm  and $pn\to pn\pi^{0}\pi^{0}$ pnp0p0 . However, no
experimental information is available in any $e^{+}e^{-}$ collision
experiment.

In the analysis presented in this paper, we study for the first time the
$e^{+}e^{-}\to 2(p\bar{p})$ process in the center-of-mass (c.m.) energy
($\sqrt{s}$) region between 4.0 and 4.6 GeV. We search for the $Y(4220)$
structure by fitting the lineshape of the Born cross sections measured at
these c.m. energies. In addition, we search for a potential structure similar
to the $d^{*}(2380)$ in the $pp$ and $\bar{p}\bar{p}$ mass spectra.

##  II The BESIII detector and data samples

The BESIII detector is a magnetic spectrometer BESIII  located at the Beijing
Electron Positron Collider (BEPCII). The cylindrical core of the BESIII
detector consists of a helium-based multilayer drift chamber (MDC), a plastic
scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic
calorimeter (EMC), which are all enclosed in a superconducting solenoidal
magnet providing a $1.0$ T magnetic field. The solenoid is supported by an
octagonal flux-return yoke with resistive plate counter muon identifier
modules interleaved with steel. The acceptance of charged particles and
photons is $93$% over $4\pi$ solid angle. The charged-particle momentum
resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the specific ionization energy
loss (${\rm d}E/{\rm d}x$) resolution is $6\%$ for the electrons from Bhabha
scattering. The EMC measures photon energies with a resolution of $2.5\%$
($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution of the
TOF barrel part is 68 ps, while that of the end cap part is 110 ps.

The 23 data sets taken at $\sqrt{s}=4.0\sim 4.6$ GeV are used for this
analysis. The nominal energy of each data set is calibrated by the di-muon
process $e^{+}e^{-}\to(\gamma_{\rm ISR/FSR})\mu^{+}\mu^{-}$ dimu , where the
subscript ${\rm ISR/FSR}$ stands for the initial-state or final-state
radiative process, respectively. The integrated luminosity $\mathcal{L}$ is
determined using large angle Bhabha events rlum , and the total integrated
luminosity is approximately 15 fb-1.

The response of the BESIII detector is modeled with Monte Carlo (MC)
simulations using the software framework BOOST boost  based on GEANT4 geant4 ,
which includes the geometry and material description of the BESIII detectors,
the detector response and digitization models, as well as a database that
keeps track of the running conditions and the detector performance. MC samples
are used to optimize the selection criteria, evaluate the signal efficiency,
and estimate backgrounds.

Inclusive MC samples are generated at different c.m. energies to study the
potential backgrounds. The inclusive MC samples consist of the production of
open-charm processes, the ISR production of vector charmonium and charmonium-
like states, and the continuum processes incorporated in kkmc kkmc . The known
decay modes are modeled with evtgen besevtgen  using branching fractions taken
from the Particle Data Group (PDG) pdg , and the remaining unknown decays from
the charmonium states with lundcharm lundcharm . The FSR from charged final-
state particles are incorporated with photos photos . The signal MC samples
are generated with a phase-space (PHSP) distribution for the same 23 energy
points as data.

##  III data analysis

For each candidate event, it is required that there are four good charged
tracks. Two of them must be identified as protons and two as anti-protons. The
charged particles are required to be within the acceptance range of
$|\cos\theta|<0.93$, where $\theta$ is the polar angle with respect to the MDC
axis. All the charged tracks are required to originate from the interaction
region $R_{xy}<1$ cm and $|V_{z}|<10$ cm, where $R_{xy}$ and $|V_{z}|$ are the
distances of closest approach of the charged track to the interaction point in
the $xy$-plane and $z$ direction, respectively. For particle identification
(PID), the ${\rm d}E/{\rm d}x$ measured by the MDC and the TOF information are
used to calculate the confidence levels for the particle hypotheses of pion,
kaon and proton. If the confidence level for the proton (anti-proton)
hypothesis is larger than that for the other two hypotheses, it is identified
as a proton (anti-proton). The efficiency of PID is $80\%\sim 100\%$ as a
function of transverse momentum of proton (anti-proton).

A three-constraint ($3$C) kinematic fit imposing three-momentum conservation
under the hypothesis of $e^{+}e^{-}\to 2(p\bar{p})$ is performed for the four
candidate charged tracks to suppress background events. Since the energy will
be used in determining the signal yield, it is not constrained in the
kinematic fit. The candidate events with $\chi^{2}_{\rm 3C}<60$ are kept for
further analysis.

The signal yield is determined by a kinematic variable $R_{E}=E_{\rm
measure}/E_{\rm cm}$, where $E_{\rm measure}$ is the total energy of all final
particles and $E_{\rm cm}$ is the c.m. energy. Figure 1 shows the $R_{E}$
distribution of the accepted candidate events. The signal events concentrate
around $1.0$ in the $R_{E}$ distribution. The signal region is defined as the
region with $R_{E}\in(0.99,1.01)$, while the sideband region is defined as the
region with $R_{E}\in(0.966,0.986)\cup(1.014,1.034)$. Studies based on the
inclusive MC samples show that only a few background events survive at $4.180$
GeV, and they do not form a peak. The background in the $R_{E}$ signal region
is estimated by the events in the $R_{E}$ sideband region multiplied by a
scale factor of $0.5$ assuming that the background is flat. The numbers of
events in the $R_{E}$ signal region in data and the scaled background yields,
which are obtained by counting, are summarized in the third and fourth columns
of Table 1.

\begin{overpic}[width=173.44534pt]{figure1_RE.eps} \end{overpic} Figure 1:
Distributions of $R_{E}$ of the accepted candidate events in data (dots with
error bars) and signal MC simulation (histogram) from all c.m. energies. The
region between the two solid red arrows is the signal region, while the region
between the two dashed pink arrows is the sideband region.

##  IV Detection efficiency

Figure 2 compares momenta, transverse momenta, and polar angle distributions
between accepted candidate events of data and signal MC samples. Throughout
the paper, the data and MC distributions sum over all c.m. energies unless
stated otherwise, while the MC distributions have been weighted by the data
signal yields. There is good agreement between data and MC simulation.
Therefore, the MC events generated according to PHSP are used to determine the
detection efficiency. The $pp$ and $\bar{p}\bar{p}$ invariant-mass spectra are
shown in Fig. 3, and no obvious structure is found.

\begin{overpic}[width=433.62pt]{figure2_c_pt_p_com.eps} \end{overpic} Figure
2: Transverse-momentum distribution (left) and the momentum distribution
(middle), the polar-angle distribution (right) for the proton (top) and anti-
proton (bottom) for all data (filled circles with error bars) and signal MC
simulation (histogram). Please note that there are two entries for each event.
Figure 3:  Invariant-mass distributions of $pp$ (top) and $\bar{p}\bar{p}$
(bottom) from all data (filled circles with error bars) and signal MC
simulation (histogram). Figure 4: Born cross sections of the process
$e^{+}e^{-}\to 2(p\bar{p})$ as a function of c.m. energy. The data are
presented as filled triangles with error bars corresponding to the combination
of statistical and systematic uncertainties. The lines are fit results to
various model assumptions which are described in the text.

The Born cross section of $e^{+}e^{-}\to 2(p\bar{p})$ at each c.m. energy is
calculated as

$\sigma^{\rm Born}=\frac{N^{\rm
net}}{\mathcal{L}\times\epsilon\times(1+\delta^{\gamma})\times\frac{1}{|1-\Pi|^{2}}},$
(1)

where $N^{\rm net}$ is the net number of signal events after background
subtraction, $\mathcal{L}$ is the integrated luminosity of the data set,
$\epsilon$ is the detection efficiency, $(1+\delta^{\gamma})$ and
$\frac{1}{|1-\Pi|^{2}}$ are the ISR and vacuum polarization correction
factors, respectively.

To obtain $(1+\delta^{\gamma})$ and $\frac{1}{|1-\Pi|^{2}}$, we take the cross
section according to the energy-dependent lineshape of $1/s$ as the initial
input, and obtain the Born cross section by iteration until the efficiencies
become stable at all energies. The difference of
$\epsilon\cdot(1+\delta^{\gamma})$ between the last two iterations is required
to be less than 1%. The relevant numbers related to Born cross section
measurement are summarized in Table 1.

Table 1: The integrated luminosities ($\mathcal{L}$), detection efficiencies ($\epsilon$), radiative correction factors $(1+\delta^{\gamma})$, vacuum polarization factors $(1+\delta^{\nu})$, and the Born cross section ($\sigma^{\rm B}$) at different c.m. energies ($\sqrt{s}$). The first uncertainties for cross sections are statistical and the second are systematic, respectively, while those for $N^{\rm obs}$, $N^{\rm bkg}$, $N^{\rm net}$ and $\epsilon$ are statistical only.  $\sqrt{s}$ (GeV) | $N^{\rm obs}_{\rm data}$ | $N^{\rm bkg}_{\rm data}$ | $N^{\rm net}_{\rm data}$ |  $\mathcal{L}$ (pb-1) | $\epsilon(\%)$ | $(1+\delta^{\gamma})$ | $\frac{1}{|1-\Pi|^{2}}$ |  $\sigma^{\rm Born}$ (fb)
---|---|---|---|---|---|---|---|---
4.009 | $1.0_{-0.8}^{+2.3}$ | $0.0_{-0.0}^{+0.5}$ | $1.0_{-0.8}^{+2.4}$ | $482.0$ | $13.6\pm 0.1$ | 0.8211 | 1.0441 | $18_{-14}^{+43}\pm 2$
4.160 | $2.0_{-1.3}^{+2.6}$ | $0.0_{-0.0}^{+0.5}$ | $2.0_{-1.3}^{+2.7}$ | $406.9$ | $26.0\pm 0.1$ | 0.8492 | 1.0533 | $21_{-14}^{+29}\pm 2$
4.180 | $16.0_{-4.0}^{+5.1}$ | $0.0_{-0.0}^{+0.5}$ | $16.0_{-4.0}^{+5.1}$ | $3194.5$ | $28.6\pm 0.1$ | 0.8502 | 1.0541 | $20_{-5}^{+6}\pm 2$
4.190 | $10.0_{-3.1}^{+4.3}$ | $0.0_{-0.0}^{+0.5}$ | $10.0_{-3.1}^{+4.3}$ | $523.9$ | $29.3\pm 0.1$ | 0.8503 | 1.0558 | $73_{-22}^{+31}\pm 6$
4.200 | $5.0_{-2.2}^{+3.4}$ | $0.0_{-0.0}^{+0.5}$ | $5.0_{-2.2}^{+3.4}$ | $525.2$ | $30.0\pm 0.1$ | 0.8515 | 1.0565 | $35_{-16}^{+24}\pm 3$
4.210 | $1.0_{-0.8}^{+2.3}$ | $0.5_{-0.4}^{+1.2}$ | $0.5_{-0.9}^{+2.6}$ | $517.2$ | $30.7\pm 0.1$ | 0.8522 | 1.0568 | $3_{-6}^{+18}\pm 1$
4.220 | $8.0_{-2.8}^{+4.0}$ | $0.0_{-0.0}^{+0.5}$ | $8.0_{-2.8}^{+4.0}$ | $513.4$ | $31.4\pm 0.1$ | 0.8515 | 1.0563 | $55_{-19}^{+28}\pm 4$
4.230 | $12.0_{-3.4}^{+4.6}$ | $0.0_{-0.0}^{+0.5}$ | $12.0_{-3.4}^{+4.6}$ | $1056.4$ | $32.1\pm 0.1$ | 0.8529 | 1.0564 | $39_{-11}^{+15}\pm 3$
4.237 | $8.0_{-2.8}^{+4.0}$ | $0.0_{-0.0}^{+0.5}$ | $8.0_{-2.8}^{+4.0}$ | $529.1$ | $32.5\pm 0.1$ | 0.8527 | 1.0555 | $52_{-18}^{+26}\pm 4$
4.246 | $3.0_{-1.6}^{+2.9}$ | $0.0_{-0.0}^{+0.5}$ | $3.0_{-1.6}^{+3.0}$ | $536.3$ | $33.1\pm 0.1$ | 0.8535 | 1.0555 | $19_{-10}^{+19}\pm 2$
4.260 | $5.0_{-2.2}^{+3.4}$ | $0.0_{-0.0}^{+0.5}$ | $5.0_{-2.2}^{+3.4}$ | $828.4$ | $33.9\pm 0.1$ | 0.8543 | 1.0536 | $20_{-9}^{+13}\pm 2$
4.270 | $2.0_{-1.3}^{+2.6}$ | $0.0_{-0.0}^{+0.5}$ | $2.0_{-1.3}^{+2.7}$ | $529.7$ | $34.5\pm 0.2$ | 0.8545 | 1.0530 | $12_{-8}^{+16}\pm 1$
4.280 | $7.0_{-2.6}^{+3.8}$ | $0.0_{-0.0}^{+0.5}$ | $7.0_{-2.6}^{+3.8}$ | $175.2$ | $35.1\pm 0.2$ | 0.8545 | 1.0530 | $126_{-47}^{+69}\pm 9$
4.290 | $5.0_{-2.2}^{+3.4}$ | $0.0_{-0.0}^{+0.5}$ | $5.0_{-2.2}^{+3.4}$ | $491.5$ | $33.7\pm 0.1$ | 0.8541 | 1.0527 | $34_{-15}^{+23}\pm 3$
4.315 | $4.0_{-1.9}^{+3.2}$ | $0.0_{-0.0}^{+0.5}$ | $4.0_{-1.9}^{+3.2}$ | $492.1$ | $35.0\pm 0.2$ | 0.8554 | 1.0522 | $26_{-12}^{+21}\pm 2$
4.340 | $6.0_{-2.4}^{+3.6}$ | $0.0_{-0.0}^{+0.5}$ | $6.0_{-2.4}^{+3.6}$ | $501.1$ | $36.2\pm 0.2$ | 0.8557 | 1.0508 | $37_{-15}^{+22}\pm 3$
4.360 | $4.0_{-1.9}^{+3.2}$ | $0.0_{-0.0}^{+0.5}$ | $4.0_{-1.9}^{+3.2}$ | $543.9$ | $39.2\pm 0.2$ | 0.8556 | 1.0511 | $21_{-10}^{+17}\pm 2$
4.380 | $8.0_{-2.8}^{+4.0}$ | $0.0_{-0.0}^{+0.5}$ | $8.0_{-2.8}^{+4.0}$ | $522.8$ | $38.0\pm 0.2$ | 0.8560 | 1.0513 | $45_{-16}^{+22}\pm 4$
4.400 | $5.0_{-2.2}^{+3.4}$ | $0.0_{-0.0}^{+0.5}$ | $5.0_{-2.2}^{+3.4}$ | $505.0$ | $38.9\pm 0.2$ | 0.8558 | 1.0510 | $28_{-12}^{+19}\pm 3$
4.420 | $16.0_{-4.0}^{+5.1}$ | $0.0_{-0.0}^{+0.5}$ | $16.0_{-4.0}^{+5.1}$ | $1043.9$ | $39.7\pm 0.2$ | 0.8552 | 1.0524 | $43_{-11}^{+14}\pm 4$
4.440 | $17.0_{-4.1}^{+5.2}$ | $0.0_{-0.0}^{+0.5}$ | $17.0_{-4.1}^{+5.2}$ | $568.1$ | $40.4\pm 0.2$ | 0.8548 | 1.0537 | $82_{-20}^{+25}\pm 6$
4.470 | $4.0_{-1.9}^{+3.2}$ | $0.0_{-0.0}^{+0.5}$ | $4.0_{-1.9}^{+3.2}$ | $111.1$ | $41.5\pm 0.2$ | 0.8547 | 1.0548 | $96_{-46}^{+77}\pm 7$
4.600 | $24.0_{-4.9}^{+6.0}$ | $0.0_{-0.0}^{+0.5}$ | $24.0_{-4.9}^{+6.0}$ | $586.9$ | $45.0\pm 0.2$ | 0.8551 | 1.0546 | $101_{-21}^{+25}\pm 8$

##  V systematic uncertainty

The uncertainty in the measurement of the integrated luminosity of the data
set is $1.0$% rlum . The uncertainties of the tracking and PID efficiencies
have been studied with high purity control samples of $J/\psi\to
p\bar{p}\pi^{+}\pi^{-}$ and
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi\to\pi^{+}\pi^{-}p\bar{p}$ prd91112004 ;
prd99031101 . The differences of the tracking and PID efficiencies between
data and MC simulation in different transverse momentum and momentum ranges
are obtained separately. The averaged differences for tracking (PID)
efficiencies that are re-weighted by the proton and anti-proton momenta of the
signal MC events, $0.5$% (1.3%) per proton and $1.0$% (1.8%) per anti-proton,
are assigned as the tracking (PID) systematic uncertainties. Adding them
linearly gives our estimate of the total systematic uncertainty of the
tracking (PID) efficiency for all charged tracks resulting in $3.0$% (6.2%).

To determine the systematic error related to uncertainties in the signal
window, we define a ratio which is the number of net signal events in the
signal window obtained by counting to the number of signal events obtained by
fitting to the $R_{E}$ spectra. For data, the combined $R_{E}$ spectra are
fitted with a MC-derived shape convolved with a Gaussian to describe the
signal shape, while the background shape is described by a first-order
Chebychev polynomial. For MC-simulated data at each energy point, the ratio is
similarly obtained. The relative differences of the ratio between data and MC
simulations range from $0.5$% to $1.4$% depending on the energy point and are
taken as the uncertainties.

To obtain reliable detection efficiencies, the Born cross sections input in
the generator have been iterated until the
$(1+\delta^{\mathit{r}})\cdot\epsilon$ values converge. The differences of
$(1+\delta^{\mathit{r}})\cdot\epsilon$ between the last two iterations in the
cross section measurements, which range from 0.0% to 1.0%, are taken as the
systematic uncertainties due to the ISR correction factor.

The systematic uncertainty from the kinematic fit is estimated by changing the
$\chi^{2}_{\rm 3C}$ requirement by $\pm 15$. The largest changes of the cross
sections compared to the nominal $\chi^{2}_{\rm 3C}$ requirement range from
$0.3$% to $0.7$% and are taken as the corresponding uncertainties. The total
systematic uncertainty is determined, to be $7.0$% to $7.1$%, by summing the
individual values in quadrature under the assumption that all the sources are
independent.

##  VI Fit to the Born cross sections

We fit to the Born cross sections under four assumptions with the least-square
method leastchi2 . In order to describe purely continuum production, the first
cases is based on a simple four-body energy-dependent PHSP lineshape phsp .
And the second case is based on an exponential function barbarexpf ; besexpf .
The exponential function is constructed as

$\sigma^{\rm Born}(s)=\frac{1}{s}\times e^{-p_{\rm
0}(\sqrt{s}-M_{\text{th}})}\times p_{\rm 1},$ (2)

where $p_{0}$ and $p_{1}$ are free parameters, and $M_{\text{th}}$ is
$(2m_{p}+2m_{\bar{p}})$. The third (fourth) case is based on an exponential
function for continuum production plus the well-established charmonium state
$\psi(4160)$ (charmonium-like state $Y(4220)$) for resonance production. For
the latter two cases, the light-hadron production is described as

$\sigma^{\rm Born}(s)=\left|{\rm BW}(\sqrt{{\it
s}})e^{i\phi}+\sqrt{f(\sqrt{{\it s}})}\right|^{2},$ (3)

where $\sqrt{f(\sqrt{s})}$ denotes the chosen continuum production amplitude,
the resonance amplitude is described by a relativistic Breit-Wigner amplitude
${\rm BW}(\rm\sqrt{{\it s}})=\frac{\sqrt{12\pi\Gamma^{ee}\Gamma^{tot}}}{{\it
s}-M^{2}+{\it i}M\Gamma^{tot}}$, and $\phi$ is the phase angle between the two
amplitudes. Moreover, $\rm M$, $\Gamma^{ee}$ and $\Gamma^{\rm tot}$ are the
mass, partial width to $e^{+}e^{-}$ and total width of the assumed resonance,
and the values are taken from the PDG pdg , which are $\rm M=4.191$ $(4.23)$
GeV/$c^{2}$ and $\Gamma^{\rm tot}=70$ $(55)$ MeV for $\psi(4160)$ ($Y(4220)$).
The fit results are shown in Fig. 4.

The goodness-of-fit is $\chi^{2}/{\rm NDF}=1.9$, 1.2, 1.3, and 1.2 for the
four fit scenarios, respectively. Here, ${\rm NDF}$ is the number of degrees
of freedom. The large goodness-of-fit for the first case implies that it is
less preferable to describe the $e^{+}e^{-}\rightarrow 2(p\bar{p})$ process by
a simple four-body energy-dependent PHSP assumption, while it can be by the
exponential function. The statistical significances of the resonances are
estimated by comparing the change of $\chi^{2}$ with and without the resonance
and taking the change of degrees of freedom into account. The statistical
significance is $0.83\sigma$ ($1.69\sigma$) for $\psi(4160)$ ($Y(4220)$),
which indicates it is unnecessary to include the $\psi(4160)$ or $Y(4220)$
resonance. This could also imply that the $\psi(4160)$ and $Y(4220)$ disfavor
a decay to $2(p\bar{p})$.

##  VII summary

In conclusion, the process of $e^{+}e^{-}\to 2(p\bar{p})$ is studied at 23
c.m. energies in the region from 4.0 to 4.6 GeV. The Born cross sections at
the different c.m. energies are measured, and the lineshape can be generally
described by an empirical exponential function. The significances for possible
contributions by a $\psi(4160)$ or $Y(4220)$ resonance are small, namely
$0.83\sigma$ and $1.69\sigma$, respectively. With the present statistics it is
impossible to draw any conclusion whether there are actual resonances or
structures in this lineshape.

The baryon-pair invariant-mass spectra are in good agreement with phase space,
and no hexaquark or di-baryon state is found with the currently available
statistics.

##  VIII ACKNOWLEDGMENT

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 Basic Research Program of China under Contract No. 2015CB856700; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 11975118,
11625523, 11635010, 11735014, 11822506, 11835012, 11935015, 11935016,
11935018, 11961141012; the Natural Science Foundation of Hunan Province under
Contract No. 2019JJ30019; 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; 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; German Research
Foundation DFG under Contracts Nos. Collaborative Research Center CRC 1044,
FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of
Development of Turkey under Contract No. DPT2006K-120470; National Science and
Technology fund; 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.

## References

  * (1) S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, 262001 (2003).
  * (2) B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 95, 142001 (2005).
  * (3) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 110, 252001 (2013).
  * (4) Z.Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110, 252002 (2013).
  * (5) T. E. Coan et al. (CLEO Collaboration), Phys. Rev. Lett. 96, 162003 (2006).
  * (6) C. Z. Yuan et al. (Belle Collaboration), Phys. Rev. Lett. 99, 182004 (2007).
  * (7) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092001 (2017).
  * (8) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 114, 092003 (2015).
  * (9) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092002 (2017).
  * (10) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 96, 032004 (2017).
  * (11) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 122, 102002 (2019).
  * (12) S. Dubynskiy, M. B. Voloshin. Phys. Lett. B 666, 344 (2008).
  * (13) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 99, 012003 (2019).
  * (14) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 99, 072005 (2019).
  * (15) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 98, 032014 (2018).
  * (16) M. Ablikim et al. (BESIII Collaboration), Phys. Lett. B 771, 45 (2017).
  * (17) M. Bashkanov et al. (CELSIUS/WASA Collaboration), Phys. Rev. Lett. 102, 052301 (2009).
P. Adlarson et al. (WASA-at-COSY Collaboration), Phys. Rev. Lett. 106, 242302
(2011).

  * (18) F. Kren et al. (CELSIUS/WASA Collaboration), Phys. Lett. B 684, 110 (2010), Erratum: Phys. Lett. B 702, 312 (2011).
  * (19) P. Adlarson et al. (WASA-at-COSY Collaboration), Phys. Rev. Lett. 106, 242302 (2011).
  * (20) P. Adlarson et al. (WASA-at-COSY Collaboration), Phys. Rev. C 88, 055208 (2013).
  * (21) P. Adlarson et al. (WASA-at-COSY Collaboration), Phys. Lett. B 743, 325 (2015).
  * (22) M. Ablikim et al. (BESIII Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 614, 345 (2010).
  * (23) M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 40, 063001 (2016).
  * (24) M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 39, 093001 (2015).
  * (25) Z. Y. Deng et al., High Energy Physics & Nuclear Physics 30, 371 (2006).
  * (26) S. Agostinelli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003).
  * (27) S. Jadach, B. F. L. Ward, and Z. Was, Phys. Rev. D 63, 113009 (2001); Comput. Phys. Commun. 130, 260 (2000).
  * (28) R. G. Ping, Chin. Phys. C 32, 599 (2008); D. J. Lange, Nucl. Instr. Meth. A 462, 152 (2001).
  * (29) M. Tanabashi et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (30) J. C. Chen, G. S. Huang, X. R. Qi et al., Phys. Rev. D 62, 034003 (2000); R. L. Yang, R. G. Ping, D. Chen, Chin. Phys. Lett. 31, 061301 (2014).
  * (31) E. Richter-Was, Phys. Lett. B 303, 163 (1993).
  * (32) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 91, 112004 (2015).
  * (33) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 99, 031101 (2019).
  * (34) B. P. Roe, Probability and Statistics in Experimental Physics, 2nd edn., New York (2001).
  * (35) J. P. Lees et al. (BABAR Collaboration), Phys. Rev. D. 86, 051102(R) (2012).
  * (36) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092001 (2017).
  * (37) E. Byckling and K. Kajantie, Particle kinematics, Wiley, 1973.