Searching for QCD critical point with light nuclei 1 corresp first-author ko@comp.tamu.edu ko@comp.tamu.edu ko@comp.tamu.edu https://orcid.org/0000-0002-1896-8787 Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-3366, USA Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-3366, USA Density fluctuations and correlations due to a first-order quark-gluon plasma to hadronic matter phase transition and its critical end point, if they remain present after the hadronic evolution in a heavy ion collisions, can lead to an enhanced production of light nuclei in these collisions. This would then result in a non-monotonic collision energy dependence of the yield ratio <math id="M1"><mrow><msub><mi>N</mi><mtext>t</mtext></msub><msub><mi>N</mi><mtext>p</mtext></msub><mo>/</mo><msubsup><mi>N</mi><mtext>d</mtext><mn>2</mn></msubsup></mrow></math> of proton number N p , deuteron number N d , and triton number N t . Measurements of this yield ratio as a function of collision energy thus provides the possibility to probe the equation of state of strong-interaction matter and its phase diagram. Studying the properties of baryon-rich quark-gluon plasma (QGP) is the main focus of the beam energy scan (BES) experiments [ 1 1 - 3 3 1-3 ] at the Relativistic Heavy Ion Collider (RHIC) as well as at the future Facility for Antiproton and Ion Research (FAIR) and the Nuclotron-based Ion Collider Facility (NICA). There have been many theoretical studies on the QCD phase diagram and its structure, based on, for example, the lattice QCD [ 4 4 ], holographic models [ 5 5 ], and effective field theories [ 6 6 ]. Also, the QCD phase diagram has been explored in the hydrodynamic framework by the BEST Collaboration [ 7 7 ] and used in machine learning methods to study its effects in heavy ion collisions [ 8 8 ]. In particular, using the hydrodynamic [ 9 9 - 12 12 9-12 ] or transport [ 13 13 ] model with an equation of state that has a first-order partonic to hadronic matter phase transition, it has been shown that the produced matter in heavy ion collisions could be mechanically unstable due to the spinodal instability associated with the first-order phase transition. This would then lead to an amplification of the density inhomogeniety, and the resulting density fluctuations could enhance the production of composite particles, such as hadrons and nuclei, which can then be used as a signal of the first-order phase transition. For example, using a partonic transport model based on a three-flavor Nambu-Jona-Lasinio (NJL) model [ 14 14 , 15 15 ] for Au+Au collisions with the initial quark temperature and density in the spinodal region of the corresponding strong-interaction QCD phase diagram, it was shown in Ref. [ 13 13 ] that the quark density fluctuation after the evolution would be larger for the case with a first-order chiral phase transition than for the case without a first-order chiral phase transition in the equation of state. These results are illustrated in panels (b) and (a) of Fig. 1 Fig. 1 , respectively. Also shown in Fig. 1 Fig. 1 are the light nuclei produced from the the hadronic matter resulting from the clumping quark matter after hadronization and further evolution as well as their yield ratio <math id="M2"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub><mo>=</mo><msub><mi>N</mi><mtext>t</mtext></msub><msub><mi>N</mi><mtext>p</mtext></msub><mo>/</mo><msubsup><mi>N</mi><mtext>d</mtext><mn>2</mn></msubsup></mrow></math> , where N p , N d , and N t denote, respectively, the proton, deuteron, and triton numbers. According to Ref. [ 16 16 ], where the coalescence model [ 17 17 - 20 20 17-20 ] is used for light nuclei production, this yield ratio is related to the neutron relative density fluctuation <math id="M3"><mrow><mtext>Δ</mtext><mi>n</mi><mo>=</mo><mo>〈</mo><msup><mrow><mo stretchy="false">(</mo><mi>δ</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>〉</mo><mo>/</mo><msup><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow><mn>2</mn></msup></mrow></math> , with <math id="M4"><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow></math> and <math id="M5"><mrow><mo>〈</mo><msup><mrow><mo stretchy="false">(</mo><mi>δ</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>〉</mo></mrow></math> being the average neutron density and its variance, respectively, by <math id="M6"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub><mo>≈</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext>Δ</mtext><mi>n</mi><mo stretchy="false">)</mo></mrow></math> . One thus expects an enhanced <math id="M7"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> if the neutron density fluctuation remains present at the kinetic freeze-out of produced hadronic matter. A similar density fluctuation effect on the yield ratio <math id="M8"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> also holds if deuteron and triton are produced thermally or statistically at the kinetic freeze out of the hadronic matter [ 21 21 , 22 22 ]. It was further argued in Ref. [ 23 23 ] that the critical fluctuations in the vicinity of the critical end point (CEP) of a first-order phase transition could also lead to an enhancement of the yield ratio <math id="M9"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> as shown in Ref. [ 24 24 ]. An enhanced <math id="M10"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> has also been interpreted as a signal for the CEP according to Refs. [ 25 25 , 26 26 ], which showed that light nuclei production would be enhanced as a result of their increased binding energies when the attractive part of nucleon-nucleon potential due to the σ meson exchange becomes deeper because of its decreasing mass as the system is close to the CEP. Reviews on this very interesting topic of the relation of light nuclei and the related topic of net baryon number fluctuations to the QCD phase transition can be found, respectively, in Ref. [ 27 27 ] and Refs. [ 28 28 - 30 30 28-30 ]. For a recent review on experimental study of the QCD phase diagram in relativistic heavy-ion collisions, it can be found in Ref. [ 31 31 ]. Recently, the STAR Collaboration has published in Physical Review Letters [ 34 34 ] the yield ratio <math id="M13"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> from the data measured during the first phase of BES experiments. This result is based on the analysis led by Daniel Cebra, Matthew Harasty, Hui Liu, Xiaofeng Luo, Nu Xu, Ning Yu, and Dingwei Zhang. As shown in Fig. 2 Fig. 2 for Au+Au collisions in the center-of-mass energy range of <math id="M14"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>7.7</mn><mo>−</mo><mn>200</mn></mrow></math> GeV, this yield ratio is enhanced at 19.6 and 27 GeV in the most central collisions, although it shows a monotonic collision energy dependence. This result has been compared with theoretical predictions of the coalescence model for deuteron and triton production based on kinetically freeze-out protons and neutrons from microscopic models for relativistic heavy ion collisions. The latter include the hybrid model based on the (3+1)D viscous hydrodynamic model MUSIC for the quark-gluon plasma and the UrQMD transport model for the hadronic matter [ 32 32 , 35 35 ] as well as a multiphase transport (AMPT) model that includes both the partonic and hadronic phases [ 33 33 ]. With a smooth crossover transition between the quark-gluon plasma and hadronic matter, an essentially collision energy independent <math id="M15"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> is predicted by these models for both peripheral and central Au+Au collisions. There have been attempts to extend the study of Ref. [ 13 13 ] based on the NJL model for the quark matter to include its hadronization and the evolution of the resulting hadronic matter via the AMPT model [ 21 21 , 36 36 ]. These studies have shown that the quark density fluctuation can largely survive hadronization but the hadron density fluctuation is somewhat washed out by their scatterings. The resulting yield ratio <math id="M16"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> from the nucleon coalescence model based on kinetically freeze-out nucleons can, however, still be enhanced if the system goes through the spinodal region of the QCD phase diagram from the NJL model. Because of the low critical temperature and high baryon chemical potential in the NJL model, the enhanced <math id="M17"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> only happens in central Au+Au collisions at center-of-mass energies lower than where the peak is seen in the STAR data. More realistic models for the QCD equation of state are thus needed in this transport model study. These equations of state can also be used in the MUSIC+UrQMD hybrid model to study the collision energy dependence of <math id="M18"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> and to extract the critical temperature and baryon chemical potential from the STAR data. Besides the yield ratio <math id="M19"><mrow><msub><mi mathvariant="script">O</mi><mrow><mtext>p</mtext><mo>−</mo><mtext>d</mtext><mo>−</mo><mtext>t</mtext></mrow></msub></mrow></math> , there are other yield ratios of light nuclei that are also sensitive to nucleon density fluctuations and can be used to probe the QCD phase diagram. For example, the yield ratio <math id="M20"><mrow><mfrac><mrow><msub><mi>N</mi><mi>α</mi></msub><msub><mi>N</mi><mtext>p</mtext></msub></mrow><mrow><msub><mi>N</mi><mrow><msup><mrow/><mn>3</mn></msup><mtext>He</mtext></mrow></msub><msub><mi>N</mi><mtext>d</mtext></msub></mrow></mfrac><mo>≈</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>9</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext>Δ</mtext><mi>p</mi><mo stretchy="false">)</mo></mrow></math> is sensitive to the proton density fluctuation Δ p [ 24 24 ] and also to the closeness to the critical point [ 37 37 ]. To avoid the smearing effect of hadronic scatterings on the nucleon density fluctuations and to directly probe the large density fluctuation during the quark to hadronic matter first-order phase transition, one can consider the yield ratio of hadrons, such as <math id="M21"><mrow><mfrac><mrow><msub><mi>N</mi><mi>p</mi></msub><msub><mi>N</mi><mrow><msup><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup></mrow></msub></mrow><mrow><msub><mi>N</mi><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></msub><msub><mi>N</mi><mi>Λ</mi></msub></mrow></mfrac></mrow></math> and <math id="M22"><mrow><mfrac><mrow><msub><mi>N</mi><mrow><msup><mi>K</mi><mo>+</mo></msup></mrow></msub><msub><mi>N</mi><mrow><msup><mi>Ξ</mi><mo>−</mo></msup></mrow></msub></mrow><mrow><msub><mi>N</mi><mi>ϕ</mi></msub><msub><mi>N</mi><mi>Λ</mi></msub></mrow></mfrac></mrow></math> , which can be shown to be sensitive to the up quark and strange quark density fluctuations, respectively, if they are produced through quark coalescence [ 38 38 , 39 39 ] and their numbers do not change during the hadronic evolution as in the statistical hadronization model [ 40 40 , 41 41 ]. 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His work is supported by the U.S. Department of Energy under Award No. De-SC0015266. 10.1007/s41365-023-01231-1 His work is supported by the U.S. Department of Energy under Award No. De-SC0015266. 2023-05-03 2023-05-03 2023-05-11 2023-05-31 2023-07-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-023-01231-1 http://dx.doi.org/10.1007/s41365-023-01231-1 . 2023 39961205 Nuclear Science and Techniques 5 34