Effects of QCD critical point on light nuclei production
EEffects of QCD critical point on light nuclei production
Kai-Jia Sun ∗ and Che Ming Ko † Cyclotron Institute and Department of Physics and Astronomy,Texas A&M University, College Station, Texas 77843, USA
Feng Li ‡ School of Physical Science and Technology, Lanzhou University, Lanzhou, Gansu, 073000, China (Dated: August 7, 2020)Using the nucleon coalescence model, which can naturally take into account the correlations in thenucleon density distribution, we study the effects of QCD critical point on light nuclei productionin relativistic heavy-ion collisions. We find that the yield ratio N t N p /N of proton ( p ), deuteron( d ) and triton ( t ) increases monotonically with the nucleon density correlation length, which isexpected to increase significantly near the critical point in the QCD phase diagram. Our study thusdemonstrates that the yield ratio N t N p /N can be used as a sensitive probe of the QCD criticalphenomenon. We further discuss the relation between the QCD phase transitions in heavy-ioncollisions and the possible non-monotonic behavior of N t N p /N in its collision energy dependence. PACS numbers:
Introduction.
According to the quantum chromody-namics (QCD), for a hadronic matter at sufficiently hightemperatures and/or densities, the quarks and gluons in-side the hadrons can be liberated to form a new phase ofmatter called the quark-gluon plasma (QGP) [1]. Fromcalculations based on the lattice quantum chromodynam-ics (LQCD), the transition between the hadronic matterand the QGP is a smooth crossover if the matter has alow baryon chemical potential ( µ B ). This smooth phasetransition has been confirmed in experiments on ultrarel-ativistic heavy-ion collisions [2, 3] in which the initiallyproduced matter has a small µ B and high temperature( T ). Due to the fermion sign problem in LQCD [4], itremains, however, an open question whether this smoothcrossover changes to a first-order phase transition at high µ B and low T [5], with a possible critical endpoint (CEP),corresponding to a second-order phase transition, on thefirst-order phase transition line in the µ B − T plane ofthe QCD phase diagram. Besides its intrinsic interest,knowledge on the properties of QCD phase diagram atfinite µ B is also useful for understanding the structure ofthe inner core of a neutron star [6] and the gravitationalwave from neutron-star mergers [7].Locating the possible CEP and the phase boundaryin the QCD phase diagram is one of the main goals forthe heavy-ion collision experiments being carried out atthe Relativistic Heavy Ion Collider (RHIC), the Facilityfor Antiproton and Ion Research (FAIR), the Nuclotron-based Ion Collider Facility (NICA), the High-IntensityHeavy Ion Accelerator Facility (HIAF), and the JapanProton Accelerator Research Complex (J-PARC). For a ∗ Corresponding author: [email protected] † [email protected] ‡ Corresponding author: [email protected] recent review, see e.g. Ref. [8]. By changing the beamenergy ( √ s ) in heavy-ion collisions, different regions inthe µ B − T plane of the QCD phase diagram can beexplored. Although at very high √ s , the evolution tra-jectory of produced matter in the QCD phase diagramonly passes across the crossover line, it could move closeto the CEP or pass across the first-order phase transitionline as the √ s becomes lower.Since the hadronic matter, which is formed from thephase transition of the short-lived QGP created in heavy-ion collisions, undergoes a relatively long expansion, it isa great challenge to find observables that are sensitiveto the phase transitions occurred during the earlier stageof the collisions. Based on the generic feature of diver-gent correlation length ( ξ ) at the critical point of phasetransitions and the resulting properties of self similaritiesand universality classes [9], observables sensitive to thecorrelation length, such as the correlations and fluctua-tions of conserved charges [10–13], have been proposed.In particular, the fourth-order or kurtosis of net-protonmultiplicity fluctuations has been suggested as an ob-servable for the critical point because of its dependenceon higher orders in ξ and its non-monotonic behavior asa function of √ s [11, 13]. However, the event-by-eventfluctuation in the number of net-protons in these the-oretical studies [11, 13] is for protons and antiprotonsin certain spatial volume, which is very different fromthose within certain momentum cut that are measuredin experiments [14, 15]. It is unclear how the measuredfluctuation in momentum space is related to the fluctu-ation in coordinate space due to the CEP, especially forheavy-ion collisions at low beam energies because of thelack of boost invariance [16] and the effects of thermalsmearing [17].On the other hand, light nuclei, such as the deuteron( d ), triton ( H or t ), helium-3 ( He), etc. that are a r X i v : . [ nu c l - t h ] A ug produced in relativistic heavy-ion collisions, provide apromising tool to probe the spatial density fluctuationand correlation in the produced matter as they areformed from nucleons that are located in a very re-stricted volume of ∆ x ∼ p ∼
100 MeV inphase space [18–21]. It has been shown that the fluc-tuation of nucleon density distributions can lead to anenhancement in the yield ratio N t N p /N through therelation N t N p /N ≈ √ (1 + ∆ ρ n ), where ∆ ρ n is theaverage neutron density fluctuation in the coordinatespace [20, 21]. Also, it has been proposed that this ra-tio could be enhanced because of the modification of thenucleon-nucleon potential near the CEP [22, 23]. Possi-ble non-monotonic behavior of this ratio has indeed beenseen in recent experimental data from heavy-ion collisionsat both SPS energies [20] and RHIC BES energies [24].However, no microscopic models have so far been able toexplain the data.In this Letter, we point out that the yield ratio N t N p /N in relativistic heavy-ion collisions is not onlyenhanced by a first-order phase transition in the pro-duced matter, it also receives an additional enhancementfrom the long-range spatial correlation if the producedmatter is near the CEP. To quantify the dependence ofthis ratio on the correlation length ξ in the producedmatter in heavy-ion collisions if its evolution trajectorypasses close to the CEP of the QCD phase diagram, wederive an expression to relate these two quantities. Be-cause of the intrinsic resolution scale of around 2 fm givenby the sizes of deuterons and tritons is comparable to theexpected correlation length generated near the CEP [25],the production of these nuclei is thus sensitive to theQCD critical fluctuations. Our finding also suggests thatthe information on the CEP can be obtained from exper-iments on heavy-ion collisions by studying the collisionenergy dependence of the yield ratio N t N p /N . Effects of critical point on light nuclei production inheavy-ion collisions.
Although rarely produced in highenergy heavy-ion collisions, light nuclei, such as d , t , He,helium-4 ( He), hypertriton ( H) and their antiparticles,have been observed in experiments at RHIC [26] and theLHC [27]. As shown in Refs. [20, 21, 28], the effectsof nucleon density correlations and fluctuations on lightnuclei production can be naturally studied by using thecoalescence model [29–34]. In this model, the number ofdeuterons produced from a hadronic matter can be cal-culated from the overlap of the proton and neutron jointdistribution function f np ( x , p ; x , p ) in phase spacewith the deuteron’s Wigner function W d ( x , p ), where x ≡ ( x − x ) / √ p ≡ ( p − p ) / √ N d = g d (cid:90) d x d p d x d p f np ( x , p ; x , p ) × W d ( x , p ) . (1) In the above, g d = 3 / W d by Gaussian functions inboth x and p , i.e., W d ( x , p ) = 8 exp (cid:16) − x σ d − σ d p (cid:17) , withthe normalization condition of (cid:82) d x (cid:82) d p W d ( x , p ) =(2 π ) [20, 21, 28]. For the width parameter σ d in thedeuteron Wigner function, it is related to the root-mean-square radius r d of deuteron by σ d = (cid:112) / r d ≈ . f np ( x , p ; x , p ) in phase space, we take it to have theform f np ( x , p ; x , p ) = ρ np ( x , x )(2 πmT ) − e − p p mT , (2)by assuming that protons and neutrons are emitted froma thermalized source of temperature T and neutron andproton densities ρ n ( x ) and ρ p ( x ), respectively. In theabove equation, m is the nucleon mass and ρ np ( x , x )is the joint density distribution function of protons andneutrons in the coordinate space, which can be writtenas ρ np ( x , x ) = ρ n ( x ) ρ p ( x ) + C ( x , x ) , (3)in terms of the neutron and proton density correlationfunction C ( x , x ). Substituting f np from Eq. (2) andEq. (3) in Eq. (1) and integrating over the nucleon mo-menta, we obtain the deuteron number as N d ≈ N (0)d (1 + C np ) + 32 / (cid:18) πmT (cid:19) / × (cid:90) d x d x C ( x , x ) e − ( x − x σ d (2 πσ d ) . (4)In the above, N (0)d = / (cid:0) πmT (cid:1) / N p (cid:104) ρ n (cid:105) , with N p beingthe proton number in the emission source, denotes thedeuteron number in the usual coalescence model stud-ies without density fluctuations and correlations in theemission source, and C np = (cid:104) δρ n ( x ) δρ p ( x ) (cid:105) / ( (cid:104) ρ n (cid:105)(cid:104) ρ p (cid:105) ) isthe correlation between the neutron and proton densityfluctuations, with (cid:104)· · · (cid:105) denoting the average over the co-ordinate space [20, 21]. In obtaining Eq. (4), we haveused the fact that the width parameter σ d in the deuteronWigner function is much larger than the thermal wave-length of the nucleons in the emission source. We notethat the C np term in Eq. (4) for deuteron production hasbeen carefully studied in Refs. [20, 21, 28].The nucleon density correlation becomes importantnear the CEP, where it is dominated by its singular partgiven by [9, 36] C ( x , x ) ≈ λ (cid:104) ρ n (cid:105)(cid:104) ρ p (cid:105) e −| x − x | /ξ | x − x | η . (5)In the above, ξ is the correlation length, η is the criticalexponent of anomalous dimension, and λ is a parame-ter that varies smoothly with the temperature and thebaryon chemical potential of the emission source. Thenucleon correlation length is similar to the correlationlength of the (net-)baryon density because nucleons carrymost of the baryon charges in heavy-ion collisions. In thenon-linear sigma model [10, 37], the correlation length isgiven by ξ = 1 /m σ , where m σ denotes the in-mediummass of the sigma meson and decreases as the systemapproaches the CEP. Since the value of the anomalousexponent η ≈ .
04 is small [38], it is neglected in thepresent study. Using the fact that the nucleon numberfluctuation (cid:104) δN (cid:105) ∝ (cid:82) d x C ( x ) ∝ λξ near the criticalpoint is always positive and enhanced, the λ parameteris positive as well.With Eq. (5), the deuteron number in Eq. (4) thenbecomes N d ≈ N (0)d (cid:20) C np + λσ d G (cid:18) ξσ d (cid:19)(cid:21) , (6)where the function G denotes the contribution from thelong-range correlation between neutrons and protons,and it is given by G ( z ) = (cid:114) π − z e z erfc (cid:18) √ z (cid:19) , (7)with erfc( z ) being the complementary error function.The behavior of G ( z ) is depicted in Fig. 1. As the cor-relation length ξ increases, the function G ( z ) is seen toincrease monotonically and saturate to the value (cid:112) /π for z (cid:29) ξ (cid:29) σ d . This means that the divergenceof ξ does not lead to a divergence of the deuteron yield,which is different from observables like the kurtosis ofnet-proton number distribution function. For small ξ ,the function G increases as ξ . At ξ ∼ σ d , it increaseslinearly with ξ , and the increase becomes much slowerfor ξ > σ d .Similarly, the number of tritons from the coalescenceof two neutrons and one proton is given by N t ≈ / (cid:18) πmT (cid:19) (cid:90) d x d x d x ρ nnp ( x , x , x ) × / ( πσ t ) e − ( x1 − x2 )22 σ t − ( x1 + x2 − x3 )26 σ t , (8)if the triton Wigner function is also approximated byGaussian functions in the relative coordinates [28, 31, 32].In the above, σ t is related to the root-mean-square radius r t of triton by σ t = r t = 1 .
59 fm [28, 31, 32, 35]. Thethree-nucleon joint density distribution function ρ nnp canbe expressed as ρ nnp ( x , x , x ) ≈ ρ n ( x ) ρ n ( x ) ρ p ( x )+ C ( x , x ) ρ p ( x ) + C ( x , x ) ρ n ( x )+ C ( x , x ) ρ n ( x ) + C ( x , x , x ) (9) FIG. 1: The dependence of the function G ( ξ/σ ) on the cor-relation length ξ with σ being the width parameter in thedeuteron or triton Wigner function. in terms of three two-nucleon correlation functions C ( x i , x j ) with i (cid:54) = j , if one neglects the isospin depen-dence of two-nucleon correlation functions [10], and thethree-nucleon correlation function C ( x , x , x ). Thecontribution from the two-nucleon correlation functionsto triton production can be similarly evaluated as inEq. (6) for deuteron production. Keeping only theleading-order term in the function G , the triton numberis then given by N t ≈ N (0)t (cid:20) ρ n + 2 C np + 3 λσ t G (cid:18) ξσ t (cid:19) + O ( G ) (cid:21) , (10)where N (0)t = / (cid:0) πmT (cid:1) N p (cid:104) ρ n (cid:105) is the triton num-ber in the absence of nucleon density fluctuationsand correlations in the emission source, and ∆ ρ n = (cid:104) δρ n ( x ) (cid:105) / (cid:104) ρ n (cid:105) ≥ O ( G ) in Eq. (10)comes from the contribution of the three-nucleon cor-relation. We note that the ∆ ρ n is closely related tothe second-order scaled density moment y by y ≡ [ (cid:82) d x ρ n ( x )][ (cid:82) d x ρ n ( x )] / [ (cid:82) d x ρ n ( x )] ≈ ρ n [28],which has been frequently used to describe the densityfluctuation or inhomogeneity in coordinate space [18, 28,39].By considering the yield ratio N t N p /N , which canbe considered as the double ratio of N t /N d and N d /N p ,one can eliminate the pre-factors of temperature and theproton number in Eqs. (6) and (10), which depend onthe beam energy and the collision system. Neglecting thedifference between σ d and σ t for simplicity by denoting σ ≈ σ d ≈ σ t and keeping only the leading-order term in G , the yield ratio N t N p /N can be simplified to N t N p N ≈ √ (cid:20) ρ n + λσ G (cid:18) ξσ (cid:19)(cid:21) . (11)Eq. (11) shows that besides its enhancement by theneutron density fluctuation ∆ ρ n [20, 21], the yield ra-tio N t N p /N is also enhanced by the nucleon den-sity correlations characterized by the correlation length ξ . Although the density fluctuations ∆ ρ n in a homo-geneous system vanishes, the correlation length ξ inthe system becomes divergent if the system is closeto the CEP. On the other hand, in the presence of afirst-order phase transition with the coexistence of twophases, the system could have large density inhomogene-ity [18, 28, 39, 40] and thus non-vanishing density fluctu-ations, even though the correlation length in each phaseis significantly smaller than in the case that the systemis near the critical point. As a result, the existence ofa first-order phase transition and the CEP in the sys-tem can both lead to enhancements of the yield ratio N t N p /N if their effects can survive the hadronic evolu-tion in a heavy-ion collision.The above result can be generalized to the yield ratiosinvolving the heavier He ( α ) [20, 22, 23], e.g. the ratios N α N p N He N d ≈ √ √ (cid:20) C np + ∆ ρ p + 2 λσ G (cid:18) ξσ (cid:19)(cid:21) , (12) N α N t N p N He N ≈ √ (cid:20) C np + 2∆ ρ n + 3 λσ G (cid:18) ξσ (cid:19)(cid:21) . (13)Compared to the yield ratio N t N p /N in Eq. (11), thesetwo yield ratios show a larger sensitivity to the correla-tion length ξ . However, the yield of α particle in heavy-ion collisions is much more difficult to measure preciselyat high collision energies because of its small value dueto the large penalty factor e − A ( m − µ B ) /T for the yield ofa nucleus with A nucleons [41].As can be seen from Eq. (11), the yield ratio N t N p /N encodes directly the spatial density fluctua-tions and correlations of nucleons due to, respectively,the first-order QGP to hadronic matter phase transitionand the critical point in the produced matter from rel-ativistic heavy-ion collisions. This is in contrast to thehigher-order net-proton multiplicity fluctuations [11, 13]based on the measurement of the event-by-event fluctua-tion of the net-proton number distribution in momentumspace with its relation to the spatial correlations due tothe CEP still lacking [16, 17]. Also, the N t N p /N ra-tio has a natural resolution scale of around 2 fm, whichis comparable to the correlation length ξ (cid:39) Collision energy dependence of the yield ratio N t N p /N d . Since both the density fluctuations and long-range correlations of nucleons in the emission source canlead to an enhanced yield ratio N t N p /N as shown in theabove, the effects of QCD phase transitions can thus bestudied in experiments from the collision energy depen-dence of this ratio. It has been demonstrated in Ref. [43] that the correlation length ξ along the chemical freeze-out line peaks near the CEP. This indicates that the col-lision energy dependence of ξ is likely to exhibit a peakstructure with its maximum value at certain collision en-ergy √ s H . Due to the critical slowing down [25] in thegrowth of the correlation length, the value of ξ is, how-ever, limited to be around 2-3 fm at the time the matterproduced in realistic heavy-ion collisions is near the CEP.The neutron density fluctuation ∆ ρ n is mostly relatedto the first-order phase transition during which largedensity inhomogeneity could be developed due to thespinodal instability [18, 28, 39, 40]. It was estimatedin Ref. [44] that the largest effect of a first-order phasetransition could be developed at around half the criticaltemperature [44] when the phase trajectory spends thelongest time in the spinodal unstable region of the QCDphase diagram. Therefore, the collision energy depen-dence of ∆ ρ n is also expected to have a peak structurewith its maximum value at a lower collision energy √ s L .As a result, the yield ratio N t N p /N as a func-tion of the collision energy √ s shows two possible non-monotonic behaviors. The first one has a double-peakstructure, with one peak at √ s H due to the critical pointand the other peak at a lower collision energy √ s L due tothe first-order phase transition. This double-peak struc-ture was conjectured in Ref. [21] without the explicit re-lation between N t N p /N and ξ given in Eq. (11). Dueto the flattening of the function G given by Eq. (7) forlarge ξ , the signal from the critical point is broadened. Itis thus also possible that √ s L and √ s H are so close thatthe signals from the CEP and the first-order phase tran-sition overlap, resulting in only one broad peak in thecollision energy dependence of the yield ratio N t N p /N .Possible non-monotonic behavior of the yield ratio N t N p /N has been seen in recent experimental data fromheavy-ion collisions at both SPS energies [20, 45] andRHIC BES energies [24]. In particular, the data for N t N p /N at collision energies from √ s NN = 6 . √ s NN = 200 GeV shows a possible double-peak struc-ture. However, due to the large error bars in the data,one cannot exclude the possibility that the double peaksare actually a single broad peak. To extract the collisionenergy dependence of ∆ ρ n and ξ from the experimentaldata by using Eq. (11), one needs to know the value of the λ parameter in the critical region and its evolution dur-ing the hadronic evolution of heavy ion collisions, whichrequires studies that are beyond the scope of present pa-per.In the statistical hadronization model, which assumesthat both yields of deuterons and trions remain constantduring the hadronic evolution of heavy ion collisions, thisratio would increase with increasing collision energy afterincluding the strong decay contribution to protons [46].Calculations based on transport models [47, 48] withoutthe CEP in the QCD phase diagram all give an essentiallyenergy-independent constant value for N t N p /N andthus fail to describe the data. However, a recent multi-phase transport model study [28], which includes a first-order QCD phase transition, shows that the density fluc-tuation or inhomogeneity induced during the first-orderphase transition can largely survive the hadronic evo-lution, because of the fast expansion of the producedmatter, and eventually leads to an enhanced yield ra-tio N t N p /N at the kinetic freeze out of nucleons whenthey undergo their last scatterings. One thus expectsthe long-range correlation to similarly persist until ki-netic freeze out and also lead to an enhancement of theyield ratio N t N p /N . Summary and outlook.
Based on the nucleon coales-cence model, we have obtained an explicit expression thatrelates the yield ratio N t N p /N to the nucleon densitycorrelation length ξ in the hadronic matter produced inheavy-ion collisions, which could be appreciable if theproduced matter is initially close to the CEP in the QCDphase diagram. This ratio is found to increase monotoni-cally with the dimensionless quantity ξ/σ where σ ≈ ρ n that could be developed during afirst-order QGP to hadronic matter phase transition pre-viously studied in Refs. [20, 21, 28]. Consequently, thecollision energy dependence of this ratio is expected tohave a double-peak or a broad one-peak structure de-pending on the closeness in √ s between the signal of theCEP and that of the first-order phase transition. Sucha non-monotonic behavior in the collision energy depen-dence of the yield ratio N t N p /N has indeed been seenin the preliminary data from the STAR Collaboration.Our study has thus led to the possibility of extractingthe information of the CEP and the phase boundary ofQCD phase diagram from comparing the precisely mea-sured data on the yields of light nuclei in heavy-ion col-lisions with those from theoretical models based on thetransport approach [28] and the various hydrodynamicapproaches [49–53].This work was supported in part by the US Depart-ment of Energy under Contract No.de-sc0015266 and theWelch Foundation under Grant No. A-1358. [1] E. V. Shuryak, Phys. Rept. , 71 (1980).[2] M. Gyulassy and L. McLerran, Nucl. Phys. A750 , 30(2005), nucl-th/0405013.[3] A. Andronic, P. Braun-Munzinger, K. Redlich, andJ. Stachel, Nature , 321 (2018), 1710.09425.[4] R. V. Gavai, Pramana , 757 (2015), 1404.6615.[5] M. A. Stephanov, Prog. Theor. Phys. Suppl. ,139 (2004), [Int. J. Mod. Phys.A20,4387(2005)], hep-ph/0402115.[6] E. Annala, T. Gorda, A. Kurkela, J. N¨attil¨a, andA. Vuorinen (2019), 1903.09121. [7] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.Lett. , 161101 (2017), 1710.05832.[8] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov,and N. Xu, Phys. Rept. , 1 (2020), 1906.00936.[9] K. G. Wilson and J. B. Kogut, Phys. Rept. , 75 (1974).[10] Y. Hatta and M. A. Stephanov, Phys. Rev.Lett. , 102003 (2003), [Erratum: Phys. Rev.Lett.91,129901(2003)], hep-ph/0302002.[11] M. A. Stephanov, Phys. Rev. Lett. , 032301 (2009),0809.3450.[12] M. Asakawa, S. Ejiri, and M. Kitazawa, Phys. Rev. Lett. , 262301 (2009), 0904.2089.[13] M. A. Stephanov, Phys. Rev. Lett. , 052301 (2011),1104.1627.[14] J. Adamczewski-Musch et al. (HADES) (2020),2002.08701.[15] J. Adam et al. (STAR) (2020), 2001.02852.[16] M. Asakawa, M. Kitazawa, and B. M¨uller, Phys. Rev. C , 034913 (2020), 1912.05840.[17] Y. Ohnishi, M. Kitazawa, and M. Asakawa, Phys. Rev. C94 , 044905 (2016), 1606.03827.[18] J. Steinheimer and J. Randrup, Phys. Rev. Lett. ,212301 (2012), 1209.2462.[19] J. Steinheimer, J. Randrup, and V. Koch, Phys. Rev.
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