Coalescence and flow in ultra-relativistic heavy ion collisions

Ruediger Scheibl; Ulrich Heinz

REVIEW

Coalescence and flow in ultra-relativistic heavy ion collisions

Ruediger Scheibl, Ulrich Heinz

Nuclear Theory · Submitted 30 Sep 1998 · 71 references

Pith's verdict: The core result — B2 expressed in terms of HBT homogeneity lengths — is sound and genuinely useful; the data comparison is weaker than the abstract implies because the NA44 spectra are not shown.

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CitationRuediger Scheibl, Ulrich Heinz. "Coalescence and flow in ultra-relativistic heavy ion collisions." arXiv:nucl-th/9809092 (1998). https://pith.science/paper/6DYRXEVP

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Editor's note

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The thing worth knowing: Eqs. (4.12) and (6.3) turn the coalescence factor from a fitted number into something computable from pion HBT radii plus the deuteron size. That is the real contribution, and it holds up. The derivation chain from (3.5) through (3.17) to (4.12) is careful, the numerical checks are honest (0.79 vs 0.81; two wave functions in Table I), and the Hagedorn overlap-volume reading (Cd ≈ 0.15 against Hagedorn's 0.17) is a nice payoff. The out-of-sample structure — source parameters from pion HBT, applied to deuteron yields — is the strongest thing about the paper and the reader is right to flag it as such. On the stress-test concern, I would push back. Eq. (6.4) is heuristic, yes; it is just the ratio of two exponentials once the slopes differ. But the actual data comparison in Sect. VII B is not done with (6.4). The box-profile fits come from numerically evaluating the full integral (2.12) with a box H(ρ); (6.4) is a summary of the resulting mt-dependence of B2, not the engine of the fit. So the claim that the headline formula and the data fit are different objects is technically true but less damaging than it sounds. Where it is actually soft: Sect. VII B compares to preliminary NA44 spectra that the paper explicitly does not show. The reader sees curves and no points. The box-profile parameter sets 1–4 are hand-picked, with no fit statistic and no combined p+d fit; the authors say so. That was 1999 phenomenology norm, but it means 'successfully describes' in the abstract is doing more work than the figure supports. The NA52 comparison (Fig. 1) is one point per species at 4% centrality, so it tests normalization only. Also acknowledged and correctly placed: the saddle-point expansion (2.13) is unreliable at large Pt and eta_f; causality restrictions on the r-integration are imposed by hand with details deferred to [45]; Sect. V's three-body number is a quick estimate, not a derivation. None of these are load-bearing failures. The central claim — coalescence governed by the same homogeneity lengths as HBT — is well supported. Who should read it: anyone using the coalescence model who wants to know what B2 means, and HBT people who want a second handle on source shape. The box-versus-Gaussian finding, confirming Polleri et al., is a useful side result. Recommendation: send it out. It is a solid, citable paper; the weaknesses are in the phenomenology, not the formalism.

Referee report

Significance

The paper's structural strength is that the deuteron prediction is genuinely out of sample: the source parameters (T, eta_f, Delta_eta, Delta_rho, tau_0) are taken from a prior fit to pion spectra and two-pion correlations [9,44], not from deuteron data, so the absolute normalization of B_2 in Eq. (7.1) is a parameter-free test, and it lands on the NA52 value. The reduction of the coalescence factor to a product of HBT homogeneity lengths (Eqs. 4.12, 6.3) is a clean and reusable result, and it gives a natural interpretation of both the phenomenological coalescence-model parameter and Hagedorn's overlap volume (the authors' <C_d> ~ 0.15 for pp agrees with Hagedorn's 0.17). The checks against two deuteron wave functions (Table I) and the numerical check <C_d>(0)=0.79 versus C^0_d=0.81 are the right kind of validation. The prediction that B_2 rises with M_t under a box profile (Eq. 6.4) is falsifiable and was subsequently borne out. This is a substantive contribution to the interpretation of cluster yields as a phase-space correlation observable.

Major objections (4)

[Sect. VII B, Fig. 2] The central data comparison of the paper is presented in Fig. 2 without the data. The text states that the analysis 'grew out of an analysis of preliminary data by the NA44 collaboration ... but have so far not been published (which is the reason why no data are shown)'. As a result the reader cannot check the claim of abstract:s5 that the model 'successfully describe[s] deuteron and antideuteron production data', nor judge the quality of the fits labelled 1-7 in panels (b) and (c). Either the preliminary points should be plotted (with the collaboration's permission) or, at minimum, the fitted M_t bins and values should be tabulated so that the residuals of curves 1-4 can be assessed. This is load-bearing for the paper's only quantitative confrontation with deuteron spectra.
[Sect. VI A, Eq. (6.4), and Sect. VII B] Eq. (6.3) is derived for the Gaussian transverse profile of Eq. (2.8b), which the paper itself rejects in Sect. II D and Sect. VII B as phenomenologically inadequate. The comparison actually made to data therefore rests on Eq. (6.4), whose additional factor exp[2(m_t-m)(1/T*_p - 1/T*_d)] is introduced as an amendment and is not derived from the density-matrix formalism of Sects. III-IV. Two things are needed: (i) a derivation or at least a first-principles justification of the amendment at the same level as Eq. (6.3), and (ii) a statement of whether the correction factor <C_d> entering Eq. (6.4) is still the Gaussian-source value (~0.8) or was recomputed for the box profile, together with an estimate of the error introduced by using the Gaussian value. As written, the headline formula and the formula used for the data fit are not the same object, and the reader cannot tell how much of the agreement in Fig. 2 comes from the derived part and how much from the heuristic factor.
[Sect. II C, Eq. (2.13); Sect. VII B] The saddle-point expansion (2.13) underlies essentially every analytic result in the paper, including the effective volume (2.15), the HBT radii (2.17), and hence B_2 in Eq. (6.3). The paper states that the approximation 'becomes unreliable for large P_t and/or eta_f' and, in Sect. II D, restricts the Gaussian analysis to v_t <= 0.6c. However, the box-profile fits of Sect. VII B, which are the evidence for the box conclusion, extend over M_t - M up to 0.3-0.7 GeV, where v_t can exceed that bound, and Eq. (6.9) explicitly states that 'the same saddle-point approximation as in the derivation of (6.1)' was used. A numerical evaluation of the unsimplified integral (2.12) for one representative parameter set (e.g. curve 2 or 3) at the largest M_t bin, compared with the saddle-point result, would settle whether the slope conclusion survives. Without it the box-profile claim carries an unquantified systematic.
[Sect. IV C and Sect. V] The numerical checks that underwrite the analytic formula are delegated to Ref. [45], a PhD thesis: the statement that <C_d>(P_d=0)=0.79 versus C^0_d=0.81, the implementation of the causality restriction on the d^3r domain (Sect. IV B, where the text says 'we refrain from giving a detailed account ... referring instead to Ref. [45]'), and the claim that <C_d> is essentially momentum-independent. Since these checks are what justify replacing the exact Eq. (3.20) by the closed form (4.12), at least a short summary of the numerical procedure and a small table of <C_d> versus P_t (or versus v_t) should appear in the paper itself. The same applies to the three-body values C^0_t = 0.78 and 0.67 quoted in Sect. V, which are given without any indication of the integration method or of the sensitivity to the three-body wave function, in contrast to the careful Table I treatment for the deuteron.

Minor objections (8)

[Sect. II E, Eq. (2.19)] Please verify the A-scaling in Eq. (2.19) against the surrounding text. The text states that 'for deuterons this implies an effective volume which is about 1/3 that of the nucleons', which corresponds to V_eff(A) proportional to A^{-3/2}, whereas the displayed equation appears to give A^{+3/2} and (2 pi A)^{3/2} V_hom. The two cannot both be right, and the sign matters for the interpretation of Eq. (6.2).
[Sect. IV C, Eqs. (4.11b,c)] The last terms under the square roots in gamma_perp and gamma_parallel are stated to be negligible for the parameters of Sect. II F. Giving their numerical values for the quoted parameter set would let the reader see how much margin there is, and would make the check of Eq. (3.21) against Eq. (3.20) quantitative rather than qualitative.
[Table I] The caption lists tau_0, T and eta_f for each column but not Delta_rho and Delta_eta, which also enter the homogeneity radii and hence C^0_d. Please state the full parameter set used, or state explicitly that the values of Sect. II F were held fixed.
[Sect. III C, Eqs. (3.15)-(3.17)] The cancellation of the inverse scattering time against the time integration assumes that the distribution functions and H(R) are constant over the interval tau_scat/2. Quoting a representative value of tau_scat for a nucleon in the SPS fireball would justify this step, which is otherwise the least transparent approximation in the derivation.
[Sect. VI C, Eq. (6.8)] The extracted freeze-out temperature depends strongly on the exponent chi of Eq. (6.10): T = (144 +/- 5) MeV for chi = -0.5 versus (124 +/- 4) MeV for chi = 1. Since chi is not determined by the data, the temperature should be quoted as a range reflecting this model dependence rather than as a central value with a small error bar, and the chi-dependence should be stated in the abstract-level summary of the result.
[Sect. II F and Sect. II B] The non-zero emission duration Delta_tau = 1.5 fm/c is described as having 'considerable uncertainty [44]' and is then set to zero for cluster formation on the basis of estimates in [45]. A short quantitative statement of the expected size of the effect on <C_d> and on V_eff would make this truncation defensible rather than asserted.
[Fig. 1] The caption notes a 40% systematic error on the vertical scale, and Eq. (7.1a) quotes an asymmetric uncertainty from the (T, eta_f) variation. The prediction band in Fig. 1 should be drawn with that uncertainty so that the reader can see whether the claimed agreement with NA52 is meaningful at the stated precision.
[Sect. VII A, Sect. VII B, Sect. VI C, Sect. VI] Several typographical errors should be corrected: 'sytematically' (Sect. VII A), 'somewat' (Sect. VII B), 'temparature' (Sect. VIII), 'find find' (Sect. VI C), and 'of of' (Sect. VI).

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The paper

The paper claims

The deuteron yield in a heavy-ion collision is not a free parameter: it is fixed by the same source sizes that pion interferometry measures.

The reading

Nuclear clusters such as deuterons are too loosely bound to exist inside a hot fireball; they can only form at the very end, when a proton and a neutron happen to emerge close together with almost the same velocity. The standard way to describe this coalescence is a phenomenological formula with a fitted constant $B_2$ that has resisted a clean physical interpretation for decades. This paper argues that the constant is not a new property of the collision at all: it is fixed by the same lengths of homogeneity — the effective source sizes along and across the beam — that two-pion Hanbury Brown–Twiss interferometry already extracts. If that is right, cluster yields become a cross-check on the source reconstructed from pion data rather than an independent measurement, and the long-standing fitted parameter of the coalescence model acquires a definite meaning in terms of source geometry and the deuteron's own radius.

What carries the argument

The load-bearing object is the quantum-mechanical correction factor $C_d$, defined as an integral of the deuteron Wigner density over the region of the source around the deuteron's centre of mass where the nucleon phase space is still homogeneous. It measures how much of the deuteron's internal wave function fits inside one homogeneity volume: for a large, static source it tends to unity and the classical coalescence formula is recovered exactly; for a rapidly expanding source it drops below one and suppresses the cluster yield. The same two lengths $R_��$ and $R_��$ appear both in the effective volume of the yield and in $C_d$, which is what ties the absolute normalization of $B_2$ to interferometry.

Extended reading notes

Core claim

The authors start from the quantum-mechanical definition of the deuteron spectrum as a projection of the two-nucleon density matrix at freeze-out onto the deuteron's internal wave function, and use the Danielewicz–Schuck treatment to keep energy conservation honest while one of the two nucleons is momentarily off-shell. Evaluating this with a hydrodynamically motivated source — Bjorken longitudinal flow, a power-law transverse flow profile, Gaussian longitudinal and transverse density profiles — they find the spectrum factorizes into the naive thermal coalescence formula times a correction factor $C_d$ that accounts for the deuteron's finite size. That factor reduces to a simple closed form, $C_d = 1/[(1+(d/2R_��)^2)^{1/2}(1+(d/2R_��)^2)]$, built only from the deuteron size parameter $d$ and the longitudinal and transverse homogeneity lengths at the nucleon mass. The effective volume in the cluster yield turns out to be the same homogeneity volume measured by HBT, scaled by $A^{3/2}$, so the invariant coalescence factor becomes $B_2 = 3π^{3/2}⟨C_d⟩/(2m_t R_��^2 R_��)$ — a quantity computable from interferometry rather than fitted. Feeding in source parameters previously extracted from NA49 pion spectra and two-pion correlations, the authors reproduce deuteron and antideuteron yields and $B_2$ values from NA44 and NA52 in central Pb+Pb collisions at the SPS, and confirm that the different transverse-mass slopes of protons and deuterons require a transverse density profile closer to a box than to a Gaussian.

Load-bearing premise

The chain rests on the saddle-point expansion of the flow factors, which the authors themselves note becomes unreliable at large transverse momentum and large transverse flow and produces an artificial transverse-mass scaling of the HBT radii; the factorization of $B_2$ inherits that approximation directly.

Editorial extensions

If this is right

Cluster yields and the invariant coalescence factor $B_2$ can be predicted from pion HBT radii measured in the same centrality class, with the deuteron radius as the only extra input.
Hagedorn's overlap probability for cluster production in proton-proton collisions, previously taken as a fixed number of order 0.17, is reinterpreted as the same correction factor evaluated with homogeneity lengths of about 1 fm.
Ratios of cluster yields at zero transverse momentum become a route to the freeze-out temperature and to separate neutron and proton fugacities, since the correction factors vary weakly across the relevant parameter range.
The measured rise of $B_2$ with transverse mass, and the flatter deuteron than proton spectra, are evidence for a transverse density profile closer to a box than a Gaussian.
Three-nucleon clusters (triton and helium-3) are predicted to have correction factors of similar size, around 0.7, to the deuteron's.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the correction factor depends on $R_��$ and $R_��$ only through the ratio $d/2R$, deuteron formation is a direct probe of source sizes comparable to the deuteron radius itself; this suggests scanning $B_2$ across centrality as a calibrated ruler for homogeneity lengths in systems where pion interferometry is statistics-limited.
The same logic should apply to any loosely bound composite — hypertriton, light hypernuclei, or exotic bound states — with the internal size as the only species-specific input, which would let cluster yields test whether such states are really formed by coalescence at freeze-out rather than by other routes.
The authors' own remark that the saddle-point expansion behind the factorization degrades at large transverse momentum and large transverse flow implies a measurable diagnostic: the ratio of $B_2$ to the HBT-predicted value should drift at high $p_t$ if the approximation is failing, and that drift is itself a probe of the flow profile.
If cluster yields really are fixed by HBT lengths, then any experiment reporting a $B_2$ inconsistent with its own pion radii at the same $m_t$ and centrality points to either a non-coalescence production mechanism or a breakdown of the thermal freeze-out picture.

Original abstract

Using a density matrix approach to describe the process of coalescence, we calculate the coalescence probabilities and invariant momentum spectra for deuterons and antideuterons. We evaluate our expressions with a hydrodynamically motivated parametrization for the source at freeze-out which implements rapid collective expansion of the collision zone formed in heavy ion collisions. We find that the coalescence process is governed by the same lengths of homogeneity which can be extracted from HBT interferometry. They appear in the absolute cluster yield via an effective volume factor as well as in a quantum mechanical correction factor which accounts for the internal structure of the deuteron cluster. Our analysis provides a new interpretation for the parameters in the popular phenomenological coalescence model and for the effective overlap volume in Hagedorn's model for cluster production in pp collisions. Using source parameters extracted from a recent HBT analysis of two-pion correlations, we successfully describe deuteron and antideuteron production data from Pb+Pb collisions at the CERN SPS as measured by the NA44 and NA52 collaborations. We also confirm the recent finding by Polleri et al. that the different measured slopes of nucleon and deuteron transverse mass spectra require a transverse density profile of the source which is closer to a box than to a Gaussian shape.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,792 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Using a density matrix approach to describe the process of coalescence, we calculate the coalescence probabilities and invariant momentum spectra for deuterons and antideuterons. We evaluate our expressions with a hydrodynamically motivated parametrization for the source at freeze-out which implements rapid collective expansion of the collision zone formed in heavy ion collisions. We find that the coalescence process is governed by the same lengths of homogeneity which can be extracted from HBT interferometry. They appear in the absolute cluster yield via an effective volume factor as well as in a quantum mechanical correction factor which accounts for the internal structure of the deuteron cluster. Our analysis provides a new interpretation for the parameters in the popular phenomenological coalescence model and for the effective overlap volume in Hagedorn's model for cluster production in pp collisions. Using source parameters extracted from a recent HBT analysis of two-pion correlations, we successfully describe deuteron and antideuteron production data from Pb+Pb collisions at the CERN SPS as measured by the NA44 and NA52 collaborations. We also confirm the recent finding by Polleri et al. that the different measured slopes of nucleon and deuteron transverse mass spectra require a transverse density profile of the source which is closer to a box than to a Gaussian shape.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Using a density matrix approach to describe the process of coalescence, we calculate the coalescence probabilities and invariant momentum spectra for deuterons and antideuterons.	method 0.61	0.96	0.07	asserted 0.99
s1	We evaluate our expressions with a hydrodynamically motivated parametrization for the source at freeze-out which implements rapid collective expansion of the collision zone formed in heavy ion collisions.	method 0.99	0.94	0.04	asserted 0.96
s2	We find that the coalescence process is governed by the same lengths of homogeneity which can be extracted from HBT interferometry.	result 0.97	0.94	0.47	asserted 1.00
s3	They appear in the absolute cluster yield via an effective volume factor as well as in a quantum mechanical correction factor which accounts for the internal structure of the deuteron cluster.	result 0.86	0.82	0.13	asserted 0.99
s4	Our analysis provides a new interpretation for the parameters in the popular phenomenological coalescence model and for the effective overlap volume in Hagedorn's model for cluster production in pp collisions.	interpretation 0.98	0.88	0.17	asserted 1.00
s5	Using source parameters extracted from a recent HBT analysis of two-pion correlations, we successfully describe deuteron and antideuteron production data from Pb+Pb collisions at the CERN SPS as measured by the NA44 and NA52 collaborations.	result 0.89	0.92	0.21	asserted 1.00
s6	We also confirm the recent finding by Polleri et al. that the different measured slopes of nucleon and deuteron transverse mass spectra require a transverse density profile of the source which is closer to a box than to a Gaussian shape.	result 0.97	0.66	0.12	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from analysis of existing data 0.43.

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sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** FIG. 2. Fits to the proton

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References

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Forward citations

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Antideuteron-net-Lambda correlations are consistent with statistical hadronization, while antideuteron-antiproton correlations require an additional coalescence step, indicating deuterons form via nucleon fusion.
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How later papers used it

86 later papers cite it

Evidence: substantial. 85 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 163 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

5 later papers by other authors are listed below: 5 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 6 of 10 papers with this label · 1 by the same authors

In the case of (anti)deuterons, the following wave functions are used: single and double Gaussian [ref] , Hulthen [ref] , and a function obtained from chiral Effective Field Theory ($χEFT$) of order N4LO with a cutoff at 500 MeV [ref] .

Production of light (anti)nuclei in pp collisions at $\sqrt{s} = 13$ TeV (2021) · Coalescence parameter as a function of transverse momentum · 0.86
Relied on the formalism proposed in this paper.

Testing coalescence and statistical-thermal production scenarios for (anti-)(hyper-)nuclei and exotic QCD objects at energies available at the CERN Large Hadron Collider (2018) · 1.00 · paraphrased: the licence does not allow quoting
Wrote the n-body Wigner density of the source calculated from the transport model as given in this paper.

Production of the $\Xi N$ dibaryon as a weakly bound system in $pp$ collisions (2023) · 0.93 · paraphrased: the licence does not allow quoting
In the coalescence approach, deuterons form via $n+p→d$ at the kinetic freeze-out, and its number ($N_d$) is given by [ref] [equation] where the summation is over all pairs of neutron and proton in each collision event, $g_d = 3/4$ is the statistical factor for forming a deuteron of spin 1 from two spin 1/2 proton and neutron, the deuteron Wigner function $W_d$ is taken to be Gaussian in spatial and momentum coordinates for simplicity.

Shedding Light on (Anti-)nuclei Production with Pion-Nucleus Femtoscopy (2025) · Introduction · 0.92
The Wigner function of the $p̅-n̅$ state is denoted by $W_pn$, while $𝒟$ is the Wigner function of the antideuteron, defined as [ref] : [equation] The antideuteron Wigner function is normalized such that: [equation]

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By the same authors

Employed this paper's Wigner-function coalescence model, which has no free parameters.

Role of bulk viscosity in deuteron production in ultrarelativistic nuclear collisions (2022) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

70 arXiv papers cite this in 129 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 72 sentences without a confident label

Builds on it 2 papers

Starting from the model described in [ref] , we have obtained in [ref] a generalised expression for $B_A$

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Following the discussion in [ref] , Eq. [ref] may be generalised as

Testing coalescence and statistical-thermal production scenarios for (anti-)(hyper-)nuclei and exotic QCD objects at energies available at the CERN Large Hadron Collider · Coalescence approach · 0.89

Uses its method or tool 7 papers

In the case of (anti)deuterons, the following wave functions are used: single and double Gaussian [ref] , Hulthen [ref] , and a function obtained from chiral Effective Field Theory ($χEFT$) of order N4LO with a cutoff at 500 MeV [ref] .

Production of light (anti)nuclei in pp collisions at $\sqrt{s} = 13$ TeV · Coalescence parameter as a function of transverse momentum · 0.86
We rely on the formalism proposed in [ref] .

Testing coalescence and statistical-thermal production scenarios for (anti-)(hyper-)nuclei and exotic QCD objects at energies available at the CERN Large Hadron Collider · Coalescence approach · 1.00
In both cases, the yields of produced are determined employing the coalescence model that builds antinuclei from antineutrons and antiprotons that are close-by in phase space [ref] .

Measurement of of $^{3}\overline{\rm He}$ nuclei absorption in matter and impact on their propagation in the galaxy · Propagation of antinuclei in the interstellar medium · 1.00

Uses its data or values 2 papers

The $σ_ρ$ in coalescence of two-body triton is set to 2.155 fm [ref] .

Effects of Initial Nucleon-Nucleon Correlations on Light Nuclei Production in Au+Au Collisions at $\sqrt{s_\mathrm{NN}} = 3\ $ GeV · Introduction · 0.99
The primordial ratio $0.984 ±0.015$ is extrapolated from available measurements [ref] and, under the assumption that the deuteron yield is proportional to the squared yield of protons [ref] , the primary ratio amounts to $0.968 ±0.030$.

Measurement of the low-energy antideuteron inelastic cross section · Introduction · 0.99

Compares with it 1 paper

The prediction of [ref] qualitatively describes the data if the rise in the nucleon abundance dominates over the increase in the volume size in pp collisions.

Multiplicity dependence of (anti-)deuteron production in pp collisions at $\sqrt{s}$ = 7 TeV · Deuteron-to-proton ratio · 0.81

Cites it as support 1 paper

Both lattice-QCD calculations and phenomenological models indicate that near-threshold $Y N$ and $Y Y$ systems can exhibit attractive interactions sufficiently strong to transiently form correlated di-hyperon states or molecular precursors to the $S$ [ref] .

Non-Thermal Production of Sexaquark Dark Matter · Increasing the coalescence probability · 0.99

Cites it as background 36 papers

In the coalescence picture, nucleons produced in the collision coalesce into nuclei if they are close in space and have similar velocities [ref] .

Testing production scenarios for (anti-)(hyper-)nuclei with multiplicity-dependent measurements at the LHC · The coalescence approach · 0.80
Such a process is commonly described by coalescence models [ref] , which describe the (anti)nuclei production as the phase-space overlap of the (anti)nucleus with the (anti)nucleons.

Light Antinuclei Coalescence: Femtoscopic Constraints via Neural-Flow Surrogates · Introduction · 0.87
On the other hand, the production of light (anti-)nuclei can be explained via the coalescence of protons and neutrons which are close by in phase space at the kinetic freeze-out and thus form a nucleus [ref] .

(Anti-)nuclei production and flow in pp, p--Pb and Pb--Pb collisions with ALICE · Production of light (anti-)nuclei in ion collisions · 0.87

Discussion

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