Formation of light nuclei at chemical freezeout: Description within a statistical thermal model
aa r X i v : . [ nu c l - t h ] J u l Light nuclei formation at chemical freezeout: A Statistical thermal model descritption
Deeptak Biswas ∗ Department of Physics, Center for Astroparticle Physics & Space ScienceBose Institute, EN-80, Sector-5, Bidhan Nagar, Kolkata-700091, India
We have reviewed the thermal description of light nuclei at the chemical freezeout. First, wehave verified the equilibration of the light nuclei, and then we have introduced a new method toinvestigate the light nuclei formation. We have studied the proximity between the phase spacedensity of light nuclei ratios and their hadronic constituents e.g ¯ d/d and (¯ p ¯ n/pn ). We have foundthat if we exclude the decay feed-down from the hadronic yields from the thermal model, then thehadronic representations have good agreement with the light nuclei ratios. We performed a similaranalysis with the ratio of Λ hypernuclei and He , which relates to the ratio Λ /p . In this context,we have also addressed the strangeness population factor S . These results indicate that the nucleiand hypernuclei formation may occur near the standard chemical freezeout and before the decay ofthe hadronic resonances. This method will serve as a guideline to discuss the light nuclei formationand the inclusion of decay into their hadronic constituents. PACS numbers: 12.38.Mh, 21.65.Mn, 24.10.Pa, 25.75.qKeywords: Heavy Ion collision, Light nuclei, Hypernuclei, Chemical freeze-out, Hadron Resonance Gasmodel
I. INTRODUCTION
The light nuclei and hypernuclei yields are available fora wide range of collision energies, from AGS [1, 2], SPS[3] to RHIC [4, 5] and LHC [6–8]. The existence of theselight nuclei at the chemical freezeout boundary is uncer-tain, as their binding energies (few MeV) are much lowerthan the typical freezeout temperature (150 MeV)[9].Despite these difficulties, a thermal model representa-tion of these bound states is important to understandthe degree of equilibration of the produced fireball. Theformation of light nuclei is also crucial in the cosmologicalcontext. As an example, the generated deuterons couldbe dissociated into their constituent nucleons if producedin an earlier epoch. Their production could be favorableonly when photon decoupled from baryons and the pro-cess n + p → d + γ became dominant in the detailedbalance [10].Statistical Hadronization Model (SHM) is a standardprescription to discuss the hadronic yields of heavy-ioncollision. This formalism is quite successful in explain-ing the final abundance of hadrons, with only a limitednumber of thermodynamic parameters (T, µ B , µ Q , µ S ,V) [9, 11–16]. The surface of these parameters is knownas the Chemical Freeze-out (CFO), as inelastic collisionterminates and the p T integrated hadron yields are frozenonward this boundary. The contradiction arises while de-scribing the light nuclei in this framework of this ther-mal model. These nuclei should not survive the chem-ical freezeout due to their smaller binding energy, andcollisions with pions will dissociate these nuclei into con-stituent nucleons [17].The Coalescence model also addresses the hadrons for-mation of heavy-ion collisions [18–21]. In this model, ∗ [email protected] depending on the momentum and spatial distribution,nearby partons confine to form a hadron. At the phe-nomenological level, this method relies on the momen-tum spectra of both the constituents and the final boundstate. A complete description of local correlation andenergy conservation is not possible due to the absence ofexperimental measurement of the parton spectra. On theother hand, the discussion of the light nuclei formation issimpler as the measured momentum spectra are availablefor both the light nuclei and their hadronic constituents[22]. Two or more hadrons coalesce to form the light nu-clei near the kinetic freezeout surface. The momentumspectra of a light nuclei with Z protons and A − Z numberof neutrons is proportional to, (cid:16) E p dN p d p (cid:17) Z (cid:16) E n dN n d p (cid:17) A − Z .This method has to implement several parameters to dis-cuss the experimental data. We can calculate the hadronyield and their ratio from the thermodynamic descrip-tion of the chemical freezeout. As the nucleons furthercoalesce to form light nuclei, a one to one mapping inchemical composition between the light nuclei and theirconstituents is apparent.Despite these variations, both thermal and coales-cence models make similar predictions of light nucleiyields [11, 23]. These light nuclei and hypernuclei, espe-cially (anti-)deuterons are cleaner probes of the chemicalfreeze-out for having a negligible decay contribution fromthe higher mass clusters [24–26]. So from a parametriza-tion of the statistical thermal models, one can directlycalculate the yields of these nuclei and compare it withthe experimental data.Ref.[27] analyzed the ratio of light nuclei and their con-stituents, assuming the Boltzman approximation and ne-glecting the decay feed-downs into hadrons. Though thedeuteron to proton ratio was successfully reproduced inthis method, the hypernuclei to light nuclei ratio did notagree with the data. With hypernuclei data from RHIC-200 GeV, it remains a challenge for thermal models tosimultaneously describe all hadrons and hypernuclei ina single freezeout picture. Ref.[28] utilized two separatefreezeout surfaces for strange and non-strange particlesto address this issue. Recently, ref.[17] has shown iden-tical production and disintegration rates for deuteronsin a hydrodynamical approach, which holds even in thepresence of baryon-antibaryon annihilation.The Λ hypernuclei production is related to the primor-dial Λ-p phase space correlation. Referring to this, thestrangeness population factor S = H / (cid:16) He × Λ p (cid:17) wasproposed [29]. A multiphase transport model (AMPT)shows an enhancement of this ratio in case of a deconfinedinitial state, relative to a system with only a hadronicphase. This ratio is also important to investigate thestrangeness baryon correlation C BS .In the present work, we have reviewed the thermody-namics of the chemical freezeout and considered a uni-form thermal description for the hadrons and light nuclei.We have verified the equilibration of the light nuclei inthis prescription and also addressed the ratios concern-ing the hypernuclei and strangeness population factor S . Our parametrization has reasonably reproduced S at RHIC-200 GeV and LHC-2760 GeV. As these weaklybound states are composed of hadrons, so one can ask,whether these light nuclei formation happens near thehadronic chemical freeze-out or some later times, and dothese light nuclei experience a similar chemical freeze-out? In a thermal model, the inclusion of resonance de-cay may help to investigate these light nuclei formationand freezeout.We can represent the light nuclei ratios with theirhadronic constituents e.g the ratio ¯ d/d can be approx-imated with (¯ p/p ) . If the light nuclei are produced nearthe chemical freezeout boundary and immediately experi-ence the freezeout, then a hadronic description with onlythe primary yields of hadrons should be a reasonable rep-resentation for the phase space distribution of these nu-clei and hypernuclei ratios. Whereas, if the hadrons pro-duce these bound states long after the freezeout, then de-cay feed-downs from higher mass resonance will be addedto the final yields of the hadrons. On this occasion, thelight nuclei ratios will have a better resemblance to theratio of total yields (primary plus decay feed-downs) ofthe hadronic constituents. We have tried to address theseissues in our present manuscript. Though we have per-formed the parametrization with the proper decay con-tribution into final hadron states, we have found thatthe hadronic description provides a better estimation forthe light nuclei ratios while we exclude the feed-down ofhigher mass resonances. This study suggests that thelight nuclei yields attain an equilibrium value at freeze-out, and this formation of nuclei and hypernuclei occurslong before the decay feed-down to nucleons and hyper-ons take place.The manuscript is organized as follows. In section IIwe shall discuss our parametrization procedure and intro-duce essentials tools to discuss our findings. In section IIIwe shall discuss our results and summarize in section IV. II. FORMALISM
In this section, we shall briefly discuss our parametriza-tion method and available experimental data of the lightnuclei sector.
A. Paramterization with hadron resonance gas
The ideal hadron resonance gas is an effective toolto describe the matter at freezeout. For the last twodecades, several studies have successfully explained thebulk properties of heavy ion collision at freezeout by ap-plying this model [12, 13, 30–35]. At the chemical freeze-out, one can associate particle density with experimen-tally measured yield by, [36], dN i dy | Det = dVdy n T oti | Det (1)where the subscript
Det denotes the detected hadrons.The total number density of any hadron is, n totali = n primaryi ( T, µ B , µ Q , µ S ) + P j n j ( T, µ B , µ Q , µ S ) × Branching Ratio( j → i ) (2)where the summation runs over the heavier resonances( j ), which decay to the i th hadron and primary denotesthe thermal density of hadrons without decay contribu-tion.The number density n i is calculated using Eq.3. n i = TV (cid:18) ∂ ln Z i ∂µ i (cid:19) V,T = g i (2 π ) Z d p exp[( E i − µ i ) /T ] ± . (3)For the i th species of hadron, g i , E i and m i are respec-tively the degeneracy factor, energy, and mass, whereas µ i = B i µ B + S i µ S + Q i µ Q is the chemical potential, with B i , S i and Q i denoting the baryon number, strangenessand the electric charge respectively. Though this modelis commonly applied for hadrons and their resonances, wecan incorporate the light nuclei states with their respec-tive quantum numbers, mass, and degeneracy [9, 14, 15].Here we have followed our earlier introduced formal-ism of chemical freeze-out parameter extraction [37, 38].This approach relies on ratios of conserved current likenet baryon charge and entropy and suitably parameter-izes the freeze-out surface with good precision. We con-struct net charges and total charges, from the detectedparticle’s rapidity spectra. We equate the model estima-tion of the net baryon number normalized to the totalbaryon number with that of the experimental data, asin Eq.4. The other equation is constructed for detectednet baryon number normalized to total particle yield asEq.5. P Deti B i dN i dY P Deti | B i | dN i dY = P Deti B i n T oti P Deti | B i | n T oti (4) P Deti B i dN i dY P Deti dN i dY = P Deti B i n T oti P Deti n T oti (5)The last equation relies on the fact that the detectedtotal particle multiplicity is a good measure of total en-tropy [39]. We solve these two equations alongside twoconstraints of the colliding nuclei, i.e net electric chargeto net baryon and strangeness neutrality. The system-atics regarding the freezeout volume is nullified as wedeal with ratios only. It is important to note, thoughthis formalism is different from the standard χ analysis,the extracted parameter set is consistent with the resultsfrom [12, 35, 37, 38, 40].Here we want to mention that yields of these light nu-clei are considerably smaller than that of the hadrons.So, the addition of light nuclei in the parametrizationprocess should not significantly affect the extracted pa-rameters. B. Data analysis
We have used SPS data of deuterons, He, and H fol-lowing Ref.[3] and for Au-Au in BES from Ref.[5]. Onlyin LHC we have data of all four light nucleus d, He[6], H[7] and He[8]. We have included AGS data of11.6 AGeV/c beam energy for proton and deuteron fromRef.[1]. Ratios regarding hypertriton( H), H are givenin Ref.[4] for RHIC 200 GeV. As individual yields are notavailable, we could not utilize most of these yields in ouranalysis except at LHC energy. We have predicted ratiosfrom our parametrization and compared with availabledata.We have included midrapidity yields (dN / dy) ofhadrons for most central collision following AGS [41–49],SPS [50–59], RHIC [60–75] and LHC [76–79]. We haveused STAR BES data following [40, 80]. In our HRGspectrum, we have included all confirmed hadrons up to2 GeV, with masses and branching ratios following theParticle Data Group [81] and THERMUS [82]. Finally,we solve Eq.(4 −
5) and the constraints numerically, usingBroyden’s method with a minimum convergence crite-rion of 10 − . The variances of thermal parameters havebeen estimated by repeating the analysis at the givenextremum value of hadrons yields. The errors in experi-mental data points are the quadrature sum of statisticaland systematic uncertainties. III. RESULT AND DISCUSSION
The extracted parameter set (
T, µ B , µ Q , µ S ) has goodagreement with our previous analysis, which we obtainedwith only hadron yields [37, 38, 83, 84]. At the LHC en-ergy, the temperature decreases 1 MeV if we incorporateall available light nuclei yields. This variation is withinthe estimated variances of Ref.[37, 38, 83].It is a general exercise to reproduce particle ratios withthe extracted parameter set to verify the accuracy ofthe fitting procedure. We used all the available light . . . π − π + k − π − k + π + φπ + Ξ − Λ Ω − φ Ω − p d¯p ¯dd Hed He He ¯H H H He Data, LHC 2760 GeVModel, χ / N df = 8 . / FIG. 1. (color online) Ratio of various particles and light nu-clei yields for LHC 2.76 TeV. Data (Red line) are from Ref.[6–8, 76–79]. Blue lines are thermal prediction. Estimated χ bydegrees of freedom for the representative ratios is 8.45/11. nuclei yields in our fitting procedure for the LHC en-ergy. We have displayed the predicted ratios regardingmeson, baryon, and light nuclei, alongside their experi-mental data in Fig.1. We have successfully reproducedparticle ratios with excellent precision. We reiterate thatour method does not depend on individual yield ratio,so these ratios are independent predictions. The par-ticle and anti-particle yields become identical at LHC,which demands the chemical potentials to be zero. Theresemblance between k + /π + and k − /π − is also an indi-cation of the vanishing µ S . The agreement between dataand thermal model prediction establishes the fact thatthe light nuclei and hadrons experience the same chem-ical freezeout. This fact raises contradictions due to thesmaller binding energy of light nuclei. The light nucleishould melt immediately at a freezeout temperature of152 MeV. Despite this discrepancy, the beautiful agree-ment at LHC makes it interesting to investigate ratiosregarding light nuclei at the other collision energies. A. Light nuclei to proton ratio
Light nuclei yields are significant to review the baryonequilibrium for their high baryon content. We have nor-malized the light nuclei (d, He, He ) yields with theproton and have examined their variation with collisionenergy in Fig.2. Measured yields of the deuteron areavailable from RHIC-BES, LHC, whereas estimations for He are available at SPS and LHC. There is reasonableagreement between our model predictions and experi-mental data, which indicates the chemical equilibrium -8 -7 -6 -5 -4 -3 -2 -1
10 100 1000 10000 d / p , H e / p , H e / p √ s NN in GeV d/p He/p He/p
FIG. 2. (Color online) Variations of light nuclei to protonratio with √ s . The red points are the data from AGS[1]SPS[3], RHIC [5] and LHC [6, 8] . The blue points are themodel predictions. of these light nuclei states at freezeout.These three ratios show a similar variation with thecollision energy ( √ s NN ). They remain flat at the higherRHIC, LHC, and increase towards lower BES and AGSenergies. The relative difference between LHC and AGSvalues increases with the mass number of light nuclei .At the lower collision energies, a finite µ B favors the pro-duction of baryon clusters with a higher baryon number.Whereas, at the higher RHIC and LHC, the light nucleiyields are just mass suppressed. This explains the varia-tion shown. From the parametrization, we have observeda horn in the He / p and He / p at lower AGS energy.This peak arises as an interplay among the thermal pa-rameters and nucleon mass. Future data from CBM andNICA collaborations will help to investigate these claims. B. Anti-particle to particle ratio of d and p
In Fig.3a, we have presented antiproton to proton andanti-deuteron to deuteron ratio. Our model estimationssuitably match with the experimental data. Both of theseratios increase with the collision energy and become 1at LHC, as the particle and antiparticle yields becomeequal. On the other hand, due to a large baryon stoppingamong the colliding nuclei (which results in a finite µ B ),the baryons are more abundant than the anti-baryon atlower √ s NN . This demands ¯ d/d to be smaller than ¯ p/p , Two orders of magnitude for d/p , whereas He /p rises to 10 − in AGS, from 10 − of LHC energy as deuteron has a larger baryon content. The agreementof the thermal model with data elucidates the existenceof (anti-)deuterons at the hadronic chemical freezeout.The decay contribution from the higher mass clustersinto (anti-)deuteron is negligible [24–26], so these yieldscan be determined directly from the primary thermaldensity. The (anti-)deuteron is a weakly bound state ofneutron and proton. In a general coalescence picture, thelight nuclei density is proportional to their constituents’thermal abundances [18, 27, 28]. Neglecting the isospinasymmetry, we can assume proton and neutron density tobe equal. We can then approximate ¯d / d with the squaredanti-proton to proton ratio [27].¯ dd = C (cid:18) ¯ p ¯ npn (cid:19) ≃ C (cid:18) ¯ pp (cid:19) (6)We propose that, this C helps to investigate the lightnuclei formation by quantifying the similarity in chem-ical composition between ¯ d/d and (¯p / p) . To do thatwe have considered the hadronic ratios from our ther-mal parametrization in two scenarios. First, we estimate(¯p / p) with only primary yields of (anti-)proton. A bet-ter resemblance of ¯ d/d with this primary (¯p / p) will im-ply that the (anti-)deuteron formation happens from theprimordial (anti-)protons. In the second case, we con-struct (¯p / p) including the decay feed-down in the (anti-)proton yields. If the (anti-)deuterons are formed longafter the chemical freezeout, then the square of this totalantiproton-proton ratio will be a good representation for¯ d/d .In Fig.3b we have plotted collision energy variation of C , for both the cases. C increases with √ s NN andsaturates near 1 at RHIC and LHC. This variation iscomparatively smaller (0 . p/p ) entirely from primary density of the (anti-)proton. Onthe contrary, C decreases significantly in lower √ s NN with the inclusion of resonance decay. In lower AGS andBES energies, the feed-down contributions from the bary-onic resonances are larger than anti-baryons due to thefinite µ B , which increases the asymmetry between thetotal yields of proton and antiproton. The higher valueof C for the primary case denotes that (¯ p/p ) with theprimordial yields of (anti-)proton is a better represen-tation of ¯ d/d . In this case, the little deviation from 1at lower collision energy can be reduced by consideringthe isospin asymmetry and neutron yields properly. Thisfinding means that the (anti-)deuterons are formed fromthe primary (anti-)nucleons, near the chemical freezeoutboundary. As we have already presented a good agree-ment between the thermal model and experimental datafor both the ratios, this finding will act as a benchmarkto study the light nuclei formation.This conclusion is in agreement with the findings ofref.[17]. They have observed that the deuteron yieldsbecome fixed near the chemical freezeout, though the in-elastic interactions may continue further. Here we want -9 -8 -7 -6 -5 -4 -3 -2 -1
1 10 100 1000 10000 d / d , p / p √ s NN in GeV Data p/pModelData d/dModel (a) C √ s NN in GeV PrimaryTotal (b)
FIG. 3. (Upper panel) Red and black points are data [5, 6]for ¯ p/p and ¯ d/d respectively. Blue and violet points denotemodel estimations. (Lower panel) Variation of C with √ s .The red and blue points denote estimations with and withoutdecay feed-down into (anti-)proton yield respectively. to mention that at LHC, the yields of baryon and an-tibaryon are equal, so the antiproton to proton ratio doesnot vary with the inclusion of feed-down. The hypernu-clei to light nuclei ratios will be relevant in this context C. Hypertriton to He ratio Hypernuclei are produced in high-energy interactionsvia hyperon capture by nuclei [85]. The lowest mass hy- pernuclei are Λ hypertriton ( H). In a thermal model,yields and ratios regarding this hypernuclei support tounderstand the phase space occupancy for strangenessat the freeze-out. For example, a hypertriton is a boundstate of n, p, and Λ. On the other hand, He has twoprotons and one neutron. This resemblance of thesetwo states makes their ratio important for investigatingstrangeness equilibration. In a coalescence picture, theratio H / He should follow the Λ /p ratio. A ratio S ,namely the strangeness population factor has been pro-posed [29], where (cid:18) H He (cid:19) = (cid:18) Λpnppn (cid:19) = S (cid:18) Λp (cid:19) (7)and S = (cid:18) H He (cid:19) / (cid:18) Λp (cid:19) (8)In fig.4a, we have displayed ratios Λ /p and H / He.Data are only available at LHC [7] and RHIC 200 Gev[4] for the hypernuclei to nuclei ratio. Our predicted Λ /p has good agreement with experimental data. In RHICenergies, the difference between data and model predic-tion is an influence of the uncertainties in weak decayinclusion into the proton yield. Though we have repro-duced the H / He ratio in LHC energy, our predictionhas slight down-shift at RHIC 200 GeV.Alike the C , this S is important to relate the lightnuclei and hypernuclei states to their composing nucle-ons and hyperons. We have estimated S with and with-out decay contribution in lambda and proton and haveshown the variation in fig.4b. First, we shall discuss thecase with the decay feed-downs and check whether it canexplain the available data or not, then we shall follow upwithout the decay and check the similarity between theratios Λ / p and H / He.With the decay feed-down, the phase space occupancyfactor increases from 0.6 (AGS value) to 1 at RHIC 200GeV, and it drops to 0.6 at LHC. S remains flat near 0.6SPS energies, which was previously shown by [14]. Avail-able data from experimental collaborations also supportthis non-monotonic behavior. Our prediction for AGSenergy is within the uncertainty band of data. The vari-ation with collision energies arises due to the difference indecay contribution from hyperons and non-strange bary-onic resonances. Contrarily, when we consider only theprimary yields of Λ and p, the thermal model predictionfor S stays near 0 . √ s NN .As we have suitably reproduced the experimental dataof S with the total yields, the result regarding the pri-mary density will be a guideline to investigate the Λ-hypernuclei formation. If the nuclei and hypernucleiformation occur near the hadronic chemical freeze-outand before the feed-down into Λ and proton takes place,then there will be no significant differences between the primary Λ /p and H / He. In that case, the S will stay .
11 1 10 100 1000 10000 Λ / p , Λ H / H e √ s NN in GeVData Λ /p ModelData H / HeModel (a) S √ s NN in GeV DataTotalPrimary (b)
FIG. 4. (Upper panel) Collision energy variation of Λ /p and H / He. The ratio regarding hypernuclei are only availablein LHC [7] and RHIC 200 Gev [4]. Red and black are thedata points. Blue and violet points are the model predictionsfor Λ /p and H / He respectively. (Lower panel) Variationof S with √ s . Here black points are estimations with totalΛ, proton yield and blue denotes S without the decay feed-down. Red points are the experimental data regarding S .AGS Data are from Ref.[2]. near 1 at all √ s NN . We have observed this flatness of S in our thermal model predictions. This close resem-blance between primary Λ /p and H / He indicates thatthe hypernuclei formation occurs from the primordial nu-clei and hyperons.
D. Tritium to He ratio The ratio of particles related to the same isospin mul-tiplet helps to understand the isospin variation in theheavy-ion collision. In this context, the neutron to protonand π − /π + are the representatives of the isospin asym-metry at the hadronic sector. The detected spectra of theneutron are not available in most of the √ s NN , so theratio of π − and π + represents the variation of net isospin.The neutron to proton ratio remains 1 . √ s NN . Net negative isospin willfavor an abundance of π − than its antiparticle. This ef-fect will decrease at higher RHIC and LHC energies andthe ratio π − /π + becomes 1. The rato H / He representsthe isospin asymmetry in the light nuclei sector. Tritium( H) is composed of n n p, whereas He is a n p p boundstate. Therefore the tritium ( H) to He ratio shouldreveal the neutron to proton ratio [3].The experimental data for tritium to He is availablein SPS energy[3]. We have plotted data of π − /π + ratioalongside tritium to Helium-3 in fig.5a. The H / He ratiohas a close similarity with the pion ratio. The tritium and He differ only in isospin and charge, like the chargedpions. So the isospin asymmetry of the thermal sourceshould be observed in H / He.In a thermal model, this isospin asymmetry generatesa non-zero value of the corresponding chemical composi-tion ( µ I ). Considering the Gell-MannNishijima relation,we have used µ Q instead of µ I . In fig.5b we have plottedmodel prediction for both π − /π + and H / He. The µ Q guides the √ s NN variation of these ratios. The neutronand proton asymmetry of the colliding nuclei will dynam-ically propagate in the final state and induce an abun-dance of hadrons and nuclei with negative isospin value.Baryon stopping amplifies this asymmetry via large nu-cleon deposition in lower √ s NN and increases these ra-tios. It is indeed interesting to observe that both theratio π − /π + and H / He resemble each other, thoughtheir respective masses are widely different. This behav-ior proposes that the light nuclei share the same chemicalfreezeout surface with that of the hadrons.Here we want to mention that, the double ratio N t N p /N d from the thermal model will be importantin this context. But individual yields for tritium (t)yields in all the relevant experiments are still preliminary(HADES, STAR, ALICE). IV. SUMMARY AND OUTLOOK
The description of light nuclei in a thermal model holdsdifficulties due to their small binding energy. In thismanuscript, we have revisited the light nuclei equilibra-tion at the chemical freezeout of the heavy-ion collision.We have performed the parametrization with ratios of thenet baryon charge to total baryon charge and total mul-
1 6 8 10 12 14 16 18 20 H / H e , π - / π + √ s NN in GeV Data π - / π + Data H/ He (a)
1 10 100 1000 H / H e , π - / π + √ s NN in GeV Data π - / π + ModelData H/ HeModel (b)
FIG. 5. (Upper panel) Experimental data of π − /π + (red)and H / He (black) in SPS [3]. (Lower panel) Variation ofthermal model predictions for π − /π + (blue) and H / He (vi-olet). tiplicity. We have verified the efficiency of our parameterset by comparing thermal model predictions with avail-able experimental data.We have addressed separate ratios to check the lightnuclei equilibration in baryon, strangeness, and isospinsector. We have represented light nuclei to proton ra-tio to discuss the equilibrium in the baryon sector. Onthe other hand, a proper agreement between the ther-mal model and data for the ratio H / He signifies the strangeness-baryon equilibrium in light nuclei. In thecontext of isospin, we have shown that the H / He ra-tio has resemblance with π − /π + . Both these ratios carrythe information of isospin asymmetry, in a thermal modelprescription.An essential outcome of the present work is a properthermal model description of the strangeness populationfactor S . We have found a good agreement with dataat both RHIC-200 and LHC-2.76 TeV. The equilibriumin the hypernuclei sector is apparent from the agreementbetween the thermal model and data. The successfuldescription from the thermal model emphasizes the factthat the light nuclei exist in equilibrium with the hadronsat the chemical freezeout boundary.We have especially examined the relationship betweenthe light nuclei ratios and their hadronic counterpart ¯d / d,(¯p / p) and Λ /p , H / He to discuss the formation andfreezeout of the light nuclei and hypernuclei. First, wehave reviewed the individual ratios with the standardthermal model prescription. Then we have proposed thata better resemblance between light nuclei ratios and theirhadronic counterpart can be found without the decaycontribution in the final yields of hadrons. These resultsdenote that the formation of light nuclei and hypernucleitakes place long before the decay of hadronic resonancesoccurs to constituting hadrons. In that case, the ratioof the primordial yields of the hadronic constituents is agood estimation of the light nuclei and hypernuclei ratios.To summarize, in this study we have introduced a newapproach to investigate the relationship among the lightnuclei to their hadronic constituents at freezeout. Withturning on and off the decay feed-down in the hadrons,we have shown that a better correlation between lightnuclei ratios and corresponding hadronic ones can befound when we exclude the decay feed-down into hadrons.These results indicate that the light nuclei ratios are fixednear the standard chemical freezeout surface and beforethe decay of the hadronic resonance occurs. This methodwill serve as a benchmark to discuss the formation oflight nuclei and the inclusion of decay into their con-stituents. We also want to mention that our introducedmethod is applicable only for the ratio of mass clusterswith the same mass number. We shall address this issuewith other light nuclei and hypernuclei yields from theexpected results from the RHIC-BES and SPS, CBM atFAIR.
ACKNOWLEDGEMENTS
This work is funded by UGC and DST of the Gov-ernment of India. The author wants to take this op-portunity to thank Sumana Bhattacharyya, Dr. SanjayK. Ghosh, and Dr. Rajarshi Ray regarding the variousdiscussion regarding the statistical thermal Model. DBthanks Samapan Bhadury and Pratik Ghosal for criticalreading of the manuscript. [1] L. Ahle et al. (E802), Phys. Rev.
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