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Precision Measurement of Net-Proton-Number Fluctuations in $Au + Au$ Collisions at RHIC

B. E. Aboona⁵⁷, J. Adam¹⁷, L. Adamczyk³, I. Aggarwal⁴⁴, M. M. Aggarwal⁴⁴, Z. Ahammed⁶⁵, A. K. Alshammri³³, E. C. Aschenauer⁷, S. Aslam²² et al. (STAR Collaboration)

S. Aslam²², J. Atchison², V. Bairathi⁵⁵, X. Bao⁵¹, K. Barish¹², S. Behera²⁸, R. Bellwied²⁵, P. Bhagat³², A. Bhasin³², S. Bhatta⁵⁴, S. R. Bhosale³, J. Bielcik¹⁷, J. Bielcikova⁴², J. D. Brandenburg⁴³, C. Broodo²⁵, X. Z. Cai⁵², H. Caines⁶⁹, M. Calderón de la Barca Sánchez¹⁰, D. Cebra¹⁰, J. Ceska¹⁷, I. Chakaberia³⁶, P. Chaloupka¹⁷, B. K. Chan¹¹, Z. Chang³⁰, A. Chatterjee¹⁹, D. Chen¹², J. Chen⁵¹, J. H. Chen²², Q. Chen²³, Z. Chen⁵¹, J. Cheng⁶⁰, Y. Cheng¹¹, W. Christie⁷, X. Chu⁷, S. Corey⁴³, H. J. Crawford⁹, M. Csanád²⁰, G. Dale-Gau¹⁴, A. Das¹⁷, I. M. Deppner²⁴, A. Deshpande⁵⁴, A. Dhamija⁴⁴, A. Dimri⁵⁴, P. Dixit²⁷, X. Dong³⁶, J. L. Drachenberg², E. Duckworth³³, J. C. Dunlop⁷, J. Engelage⁹, G. Eppley⁴⁶, S. Esumi⁶¹, O. Evdokimov¹⁴, O. Eyser⁷, R. Fatemi³⁴, S. Fazio⁸, Y. Feng⁴⁵, E. Finch⁵³, Y. Fisyak⁷, F. A. Flor⁶⁹, C. Fu³¹, T. Fu⁵¹, C. A. Gagliardi⁵⁷, T. Galatyuk¹⁸, T. Gao⁵¹, F. Geurts⁴⁶, N. Ghimire⁵⁶, A. Gibson⁶⁴, K. Gopal²⁸, X. Gou⁵¹, D. Grosnick⁶⁴, A. Gu²⁶, A. Gupta³², W. Guryn⁷, A. Hamed⁵, R. J. Hamilton⁶⁹, X. Han⁴³, S. Harabasz¹⁸, M. D. Harasty¹⁰, J. W. Harris⁶⁹, H. Harrison-Smith³⁴, L. B. Havener⁶⁹, X. H. He³¹, Y. He⁵¹, N. Herrmann²⁴, L. Holub¹⁷, C. Hu⁶², Q. Hu³¹, Y. Hu³⁶, H. Huang^1,41, H. Z. Huang¹¹, S. L. Huang⁵⁴, T. Huang¹⁴, Y. Huang²⁰, Y. Huang¹³, T. J. Humanic⁴³, M. Isshiki⁶¹, W. W. Jacobs³⁰, A. Jalotra³², C. Jena²⁸, A. Jentsch⁷, Y. Ji³⁶, J. Jia^54,7, C. Jin⁴⁶, N. Jindal⁴³, X. Ju⁴⁸, E. G. Judd⁹, S. Kabana⁵⁵, D. Kalinkin³⁴, K. Kang⁶⁰, D. Kapukchyan¹², K. Kauder⁷, D. Keane³³, M. Kesler³³, A. Khanal⁶⁷, Y. V. Khyzhniak⁴³, D. P. Kikoła⁶⁶, J. Kim⁷, D. Kincses²⁰, I. Kisel²¹, A. Kiselev⁷, A. G. Knospe³⁷, J. Kołaś⁶⁶, B. Korodi⁴³, L. K. Kosarzewski⁴³, L. Kumar⁴⁴, M. C. Labonte¹⁰, R. Lacey⁵⁴, J. M. Landgraf⁷, C. Larson³⁴, J. Lauret⁷, A. Lebedev⁷, J. H. Lee⁷, Y. H. Leung²⁴, C. Li¹³, D. Li⁴⁸, H-S. Li⁴⁵, H. Li⁶⁸, H. Li²³, W. Li⁴⁶, X. Li⁴⁸, X. Li⁴⁸, Y. Li⁶⁰, Z. Li⁴⁹, Z. Li⁴⁸, X. Liang¹², Y. Liang³³, R. Licenik^42,17, T. Lin⁵¹, Y. Lin²³, M. A. Lisa⁴³, C. Liu³¹, G. Liu⁴⁹, H. Liu²⁶, L. Liu¹³, Z. Liu¹³, T. Ljubicic⁴⁶, O. Lomicky¹⁷, R. S. Longacre⁷, E. M. Loyd¹², T. Lu³¹, J. Luo⁴⁸, X. F. Luo¹³, L. Ma²², R. Ma⁷, Y. G. Ma²², N. Magdy⁵⁸, D. Mallick¹³, R. Manikandhan²⁵, S. Margetis³³, C. Markert⁵⁹, O. Matonoha¹⁷, O. Mezhanska¹⁷, K. Mi¹³, S. Mioduszewski⁵⁷, B. Mohanty⁴⁰, B. Mondal⁴⁰, M. M. Mondal⁴⁰, I. Mooney⁶⁹, J. Mrazkova^42,17, M. I. Nagy²⁰, C. J. Naim⁵⁴, A. S. Nain⁴⁴, J. D. Nam⁵⁶, M. Nasim²⁷, H. Nasrulloh⁴⁸, D. Neff¹¹, J. M. Nelson⁹, M. Nie⁵¹, G. Nigmatkulov¹⁴, T. Niida⁶¹, T. Nonaka⁶¹, G. Odyniec³⁶, A. Ogawa⁷, S. Oh⁵⁰, K. Okubo⁶¹, B. S. Page⁷, S. Pal¹⁷, A. Pandav³⁶, A. Panday²⁷, A. K. Pandey³¹, T. Pani⁴⁷, A. Paul¹², S. Paul⁵⁴, D. Pawlowska⁶⁶, C. Perkins⁹, J. Pluta⁶⁶, B. R. Pokhrel⁵⁶, I. D. Ponce Pinto⁶⁹, M. Posik⁵⁶, E. Pottebaum⁶⁹, S. Prodhan²⁸, T. L. Protzman³⁷, A. Prozorov¹⁷, V. Prozorova¹⁷, N. K. Pruthi⁴⁴, M. Przybycien³, J. Putschke⁶⁷, Z. Qin⁶⁰, H. Qiu³¹, C. Racz¹², S. K. Radhakrishnan³³, A. Rana⁴⁴, R. L. Ray⁵⁹, R. Reed³⁷, C. W. Robertson⁴⁵, M. Robotkova^42,17, M. A. Rosales Aguilar³⁴, D. Roy⁴⁷, P. Roy Chowdhury⁶⁶, L. Ruan⁷, A. K. Sahoo²⁷, N. R. Sahoo²⁸, H. Sako⁶¹, S. Salur⁴⁷, S. S. Sambyal³², J. K. Sandhu³⁷, S. Sato⁶¹, B. C. Schaefer³⁷, N. Schmitz³⁸, F-J. Seck¹⁸, J. Seger¹⁶, R. Seto¹², P. Seyboth³⁸, N. Shah²⁹, P. V. Shanmuganathan⁷, T. Shao²², M. Sharma³², N. Sharma²⁷, R. Sharma²⁸, S. R. Sharma²⁸, A. I. Sheikh³³, D. Shen⁵¹, D. Y. Shen³¹, K. Shen⁴⁸, S. Shi¹³, Y. Shi⁵¹, F. Si⁴⁸, J. Singh⁵⁵, S. Singha³¹, P. Sinha²⁸, M. J. Skoby^6,45, N. Smirnov⁶⁹, Y. Söhngen²⁴, Y. Song⁶⁹, T. D. S. Stanislaus⁶⁴, M. Stefaniak⁴³, Y. Su⁴⁸, M. Sumbera⁴², X. Sun³¹, Y. Sun⁴⁸, B. Surrow⁵⁶, M. Svoboda^42,17, Z. W. Sweger¹⁰, A. C. Tamis⁶⁹, A. H. Tang⁷, Z. Tang⁴⁸, T. Tarnowsky³⁹, J. H. Thomas³⁶, A. R. Timmins²⁵, D. Tlusty¹⁶, T. Todoroki⁶¹, D. Torres Valladares⁴⁶, S. Trentalange¹¹, P. Tribedy⁷, S. K. Tripathy⁶⁶, T. Truhlar¹⁷, B. A. Trzeciak¹⁷, O. D. Tsai^11,7, C. Y. Tsang^33,7, Z. Tu⁷, J. Tyler⁵⁷, T. Ullrich⁷, D. G. Underwood^4,64, G. Van Buren⁷, J. Vanek⁷, I. Vassiliev²¹, F. Videbæk⁷, S. A. Voloshin⁶⁷, F. Wang⁴⁵, G. Wang¹¹, J. S. Wang²⁶, J. Wang⁵¹, K. Wang⁴⁸, X. Wang⁵¹, Y. Wang⁴⁸, Y. Wang¹³, Y. Wang⁶⁰, Z. Wang⁵¹, A. J. Watroba³, J. C. Webb⁷, P. C. Weidenkaff²⁴, G. D. Westfall³⁹, D. Wielanek⁶⁶, H. Wieman³⁶, G. Wilks¹⁴, S. W. Wissink³⁰, R. Witt⁶³, C. P. Wong⁷, J. Wu¹³, J. Wu⁶², X. Wu¹¹, Wu X.⁴⁸, B. Xi²², Z. G. Xiao⁶⁰, G. Xie⁶², W. Xie⁴⁵, H. Xu²⁶, N. Xu¹³, Q. H. Xu⁵¹, Y. Xu⁵¹, Y. Xu¹³, Z. Xu³³, Z. Xu⁴, G. Yan⁵¹, Z. Yan⁵⁴, C. Yang⁵¹, Q. Yang⁵¹, S. Yang⁴⁹, Y. Yang^1,41, Z. Ye⁴⁹, Z. Ye³⁶, L. Yi⁵¹, Y. Yu⁵¹, H. Zbroszczyk⁶⁶, W. Zha⁴⁸, C. Zhang²², D. Zhang⁴⁹, J. Zhang⁵¹, S. Zhang¹⁵, W. Zhang⁴⁹, X. Zhang³¹, Y. Zhang³¹, Y. Zhang⁴⁸, Y. Zhang⁵¹, Y. Zhang²³, Z. Zhang⁷, Z. Zhang¹⁴, F. Zhao³⁵, J. Zhao²², M. Zhao⁷, S. Zhou¹³, Y. Zhou¹³, X. Zhu⁶⁰, M. Zurek^4,7, and M. Zyzak²¹ (STAR Collaboration)

¹Academia Sinica, Taipei 115201
²Abilene Christian University, Abilene, Texas 79699
³AGH University of Krakow, FPACS, Cracow 30-059, Poland
⁴Argonne National Laboratory, Argonne, Illinois 60439
⁵American University in Cairo, New Cairo 11835, Egypt
⁶Ball State University, Muncie, Indiana 47306
⁷Brookhaven National Laboratory, Upton, New York 11973
⁸University of Calabria and INFN-Cosenza, Rende 87036, Italy
⁹University of California, Berkeley, California 94720
¹⁰University of California, Davis, California 95616
¹¹University of California, Los Angeles, California 90095
¹²University of California, Riverside, California 92521
¹³Central China Normal University, Wuhan, Hubei 430079
¹⁴University of Illinois at Chicago, Chicago, Illinois 60607
¹⁵Chongqing University, Chongqing, 401331
¹⁶Creighton University, Omaha, Nebraska 68178
¹⁷Czech Technical University in Prague, FNSPE, Prague 115 19, Czech Republic
¹⁸Technische Universität Darmstadt, Darmstadt 64289, Germany
¹⁹National Institute of Technology Durgapur, Durgapur 713209, India
²⁰ELTE Eötvös Loránd University, Budapest, Hungary H-1117
²¹Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany
²²Fudan University, Shanghai, 200433
²³Guangxi Normal University, Guilin, 541004
²⁴University of Heidelberg, Heidelberg 69120, Germany
²⁵University of Houston, Houston, Texas 77204
²⁶Huzhou University, Huzhou, Zhejiang 313000
²⁷Indian Institute of Science Education and Research (IISER), Berhampur 760010, India
²⁸Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India
²⁹Indian Institute Technology, Patna, Bihar 801106, India
³⁰Indiana University, Bloomington, Indiana 47408
³¹Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000
³²University of Jammu, Jammu 180001, India
³³Kent State University, Kent, Ohio 44242
³⁴University of Kentucky, Lexington, Kentucky 40506-0055
³⁵Lanzhou University, Lanzhou 730000
³⁶Lawrence Berkeley National Laboratory, Berkeley, California 94720
³⁷Lehigh University, Bethlehem, Pennsylvania 18015
³⁸Max-Planck-Institut für Physik, Munich 80805, Germany
³⁹Michigan State University, East Lansing, Michigan 48824
⁴⁰National Institute of Science Education and Research, HBNI, Jatni 752050, India
⁴¹National Cheng Kung University, Tainan 70101
⁴²Nuclear Physics Institute of the CAS, Rez 250 68, Czech Republic
⁴³The Ohio State University, Columbus, Ohio 43210
⁴⁴Panjab University, Chandigarh 160014, India
⁴⁵Purdue University, West Lafayette, Indiana 47907
⁴⁶Rice University, Houston, Texas 77251
⁴⁷Rutgers University, Piscataway, New Jersey 08854
⁴⁸University of Science and Technology of China, Hefei, Anhui 230026
⁴⁹South China Normal University, Guangzhou, Guangdong 510631
⁵⁰Sejong University, Seoul, 05006, South Korea
⁵¹Shandong University, Qingdao, Shandong 266237
⁵²Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800
⁵³Southern Connecticut State University, New Haven, Connecticut 06515
⁵⁴State University of New York, Stony Brook, New York 11794
⁵⁵Instituto de Alta Investigación, Universidad de Tarapacá, Arica 1000000, Chile
⁵⁶Temple University, Philadelphia, Pennsylvania 19122
⁵⁷Texas A&M University, College Station, Texas 77843
⁵⁸Texas Southern University, Houston, Texas 77004
⁵⁹University of Texas, Austin, Texas 78712
⁶⁰Tsinghua University, Beijing 100084
⁶¹University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
⁶²University of Chinese Academy of Sciences, Beijing 101408
⁶³United States Naval Academy, Annapolis, Maryland 21402
⁶⁴Valparaiso University, Valparaiso, Indiana 46383
⁶⁵Variable Energy Cyclotron Centre, Kolkata 700064, India
⁶⁶Warsaw University of Technology, Warsaw 00-661, Poland
⁶⁷Wayne State University, Detroit, Michigan 48201
⁶⁸Wuhan University of Science and Technology, Wuhan, Hubei 430065
⁶⁹Yale University, New Haven, Connecticut 06520

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Phys. Rev. Lett. 135, 142301 – Published 29 September, 2025

DOI: https://doi.org/10.1103/9l69-2d7p

Abstract

We report precision measurements on cumulants ( $C_{n}$ ) and factorial cumulants ( $κ_{n}$ ) of (net) proton number distributions up to fourth order in $Au + Au$ collisions over center-of-mass energies $\sqrt{s_{N N}} = 7.7 - 27 GeV$ from phase II of the Beam Energy Scan program at RHIC. (Anti)protons are selected at midrapidity ( $| y | < 0.5$ ) within a transverse momentum range of $0.4 < p_{T} < 2.0 GeV / c$ . Relative to various noncritical-point model calculations and peripheral collision 70%–80% data, the net proton $C_{4} / C_{2}$ measurement in 0%–5% collisions shows a minimum around 19.6 GeV for significance of deviation at $\sim 2 - 5 σ$ . A minimum in $C_{4} / C_{2}$ with respect to a noncritical baseline is expected to be a characteristic feature of the signature associated with a critical point in the QCD phase diagram. In addition, deviations from noncritical baselines around the same collision energy region are also seen in proton factorial cumulant ratios, especially in $κ_{2} / κ_{1}$ and $κ_{3} / κ_{1}$ . Dynamical model calculations including a critical point are called for in order to understand these precision measurements.

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Criticality in Nature’s Strongest Force

Published 29 September, 2025

Experiments at the Relativistic Heavy Ion Collider give the first hints of a critical point in the hot quark–gluon “soup” that is thought to have pervaded the infant Universe.

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Precision Measurement of Net-Proton-Number Fluctuations in $Au + Au$ Collisions at RHIC