On coalescence as the origin of nuclei in hadronic collisions
Francesca Bellini, Kfir Blum, Alexander Phillip Kalweit, Maximiliano Puccio
CCERN-TH-2020-110
On coalescence as the origin of nuclei in hadronic collisions
Francesca Bellini, ∗ Kfir Blum,
2, 3, † Alexander Phillip Kalweit, ‡ and Maximiliano Puccio § Experimental Physics Department, CERN, CH-1211 Geneve 23, Switzerland Weizmann Institute, Department of Particle Physics and Astrophysics, Rehovot 7610001, Israel Theoretical Physics Department, CERN, CH-1211 Geneve 23, Switzerland
The origin of weakly-bound nuclear clusters in hadronic collisions is a key question to be addressedby heavy-ion collision (HIC) experiments. The measured yields of clusters are approximately consistentwith expectations from phenomenological statistical hadronisation models (SHMs), but a theoreticalunderstanding of the dynamics of cluster formation prior to kinetic freeze out is lacking. The competingmodel is nuclear coalescence, which attributes cluster formation to the effect of final state interactions(FSI) during the propagation of the nuclei from kinetic freeze out to the observer. This phenomenonis closely related to the effect of FSI in imprinting femtoscopic correlations between continuum pairsof particles at small relative momentum difference. We give a concise theoretical derivation of thecoalescence–correlation relation, predicting nuclear cluster spectra from femtoscopic measurements. Wereview the fact that coalescence derives from a relativistic Bethe-Salpeter equation, and recall howeffective quantum mechanics controls the dynamics of cluster particles that are nonrelativistic in thecluster centre of mass frame. We demonstrate that the coalescence–correlation relation is roughlyconsistent with the observed cluster spectra in systems ranging from PbPb to pPb and pp collisions.Paying special attention to nuclear wave functions, we derive the coalescence prediction for hypertritonand show that it, too, is roughly consistent with the data. Our work motivates a combined experimentalprogramme addressing femtoscopy and cluster production under a unified framework. Upcoming pp, pPband peripheral PbPb data analysed within such a programme could stringently test coalescence as theorigin of clusters.
PACS numbers:
Contents
I. Introduction II. The coalescence/femtoscopy framework He. 7
III. Coalescence from correlation functions
IV. Comparison with data. He. 13B. Hypertriton. 15
V. Discussion and summary. Acknowledgments ∗ Electronic address: [email protected] † Electronic address: kfi[email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ nu c l - t h ] J u l A. Corrections to the Gaussian source approximation.
B. Deviations from an isotropic Gaussian source in the phenomenological blast wave model. References I. INTRODUCTION
Loosely-bound nuclei like the D, He, H, H and their anti-particles are detected among the products of high-energyhadronic collisions at the large hadron collider (LHC) and other experiments, and their study is a central objective in heavy-ion collision (HIC) experiments [1–3]. Interestingly, the momentum-integrated yields of these nuclei are roughly consistentwith being drawn from a thermal distribution with the same temperature parameter T ch that fits the yields of mesons andnucleons [4–7]. Taken together from π ± to He, the hadron yields span some ∼ orders of magnitude with only O (1) discrepancies . This has led some authors to speculate that nuclei take part, on equal footing with the “more fundamental”mesons and nucleons, in an equilibrium partition function characterising the high excitation state (HXS) produced in HICs.A recent account of this statistical hadronisation model (SHM) is given in [4].While the SHM is approximately consistent with cluster yields, no first-principle theoretical framework as of yet explainsthe dynamics of cluster formation in the HXS before kinetic freeze out . Clusters are big (several fm) fragile (bindingenergies E b (cid:46) MeV, as low as E b ∼ . MeV for H [10]) objects, while even at kinetic freeze out the HXS does notexceed a few fm and it is a hot state with characteristic particle excitation energies of ∼ MeV. What does it mean fora D with diameter ∼ fm to exist in an equilibrium distribution in an HXS of diameter ∼ fm, produced in pp collisions?What does it mean for H, with effective diameter ∼ fm? This puzzle makes the origin of nuclei uniquely interesting.The kinetic theory analysis of [13], focusing on D formation in high-multiplicity PbPb collisions, may shed some light onthe problem. This analysis demonstrated that, indeed, D observed at the detector must emerge from the kinetic freeze outregion and not from the chemical freeze out region of the HXS, to which the SHM parameter T ch corresponds. However,Ref. [13] treated the D as a point-particle and it is far from clear if and how their analysis could be adapted to smallersystems like pp, pPb or peripheral PbPb collisions .An alternative explanation for the origin of nuclei, bypassing the limitations of kinetic theory, is proposed by coalescence.The basic assumption of the coalescence model is that the expansion of the HXS leads to kinetic freeze out with nucleonsbut – due to their fragility and size – essentially no nuclei. The HXS at kinetic freeze out can be described by a quantummechanical (QM) density matrix. Projecting the density matrix onto particle states at the detector gives the observedparticle spectra. Final state interactions (FSI) mediated by nuclear scattering and Coulomb photon exchange enter thisprojection as they affect the propagation of the particles from the HXS to the detector. FSI manifest themselves in twoways:1. FSI imprint momentum correlations among pairs of continuum particles. The analysis of this phenomenon is knownas femtoscopy or Hanbury Brown-Twiss (HBT) analysis .2. FSI also admit discrete bound-state multi-nucleon solutions, namely nuclei. This is nuclear coalescence.It is important to note that the coalescence model predicts that the yields of nuclei approximately inherit the thermalspectra of their nucleon constituents, up to a dimensionless QM correction factor. In many cases (e.g. low- p t D and Heformation in high multiplicity PbPb collisions) the QM correction factor is close to unity [1, 15]. Thus, the approximatelythermal yield of nuclei need not point to the nuclei taking part in an equilibrium partition function before kinetic freezeout. For some nuclei and systems, however, the QM factor is predicted to be much smaller than unity. An example is theproduction of H in pp collisions, or (especially at high- p t ) in low multiplicity PbPb collisions. These systems offer a key With the discrepancies affecting nuclei [5] at a comparable level to mesons and nucleons [6]. A recent speculation is the formation of compact “preclusters” [8, 9] due to an in-medium modification of the nuclear potential. By this we mainly have in mind the HXS homogeneity radius as revealed by femtoscopy [11, 12], to be discussed later on. But even the totalHXS volume as fitted in the SHM in high multiplicity PbPb collisions is only of diameter ∼ fm, shrinking to ∼ fm in low multiplicity ppand pPb collisions [6]. We thank Urs Wiedemann for pointing out this issue during a workshop at CERN. We will use the terms femtoscopy and HBT analysis interchangeably in this paper, in line with most of the literature. For a comparativediscussion and historical notes, see [14]. discriminator between the coalescence model and the SHM. Our goal in this paper is to study the theoretical prediction ofcoalescence, compare to available experimental data and highlight the path to making this test conclusive.A central feature in this paper is the relation between coalescence and HBT correlations among continuum nucleons.From the perspective of the coalescence model, HBT correlations and nuclei production are closely related. In a clear sense,the successful reconstruction of the imprint of FSI on pair correlations lends credence to the basic framework of coalescence,which deals with the bound state solutions of essentially the same FSI (in different isospin channels). Moreover, once HBTcalibrates the HXS source characteristics, nuclei yields are predicted without free parameters. Over the years the community(experimental and theoretical) developed a habit of considering HBT and nuclei analyses separately, making it cumbersometo combine the information content of the measurements. One of our goals here is to motivate joint experimental analysesof HBT and cluster yields.The plan of the paper, along with a brief summary, are as follows.In Sec. II we briefly review the underlying relation between femtoscopy and cluster formation, defining the coales-cence/femtoscopy framework. The basic formalism was laid out by Lednicky et al [16–18]. We provide a quick reduction ofthis formalism to observationally accessible objects. This is a good starting point for the discussion, because it demonstratesthat coalescence arises in a relativistic quantum field theoretic (QFT) calculation. In Sec. II A we show how, subject to twokey approximations (the smoothness approximation and the equal-time approximation), the model-independent coalescence–correlation relation between deuteron production and two-proton HBT comes about. The main result here is the manifestrelation between the well-known Eq. (14), for HBT, and Eq. (15), for the D coalescence factor. This relation was derivedfirst in [19] starting from the QM limit. Our independent derivation here gives another perspective on this result, showing,for example, that it does not require density matrix factorisation to apply for its validity. In Sec. II B we review how addingthe assumption of density matrix factorisation allows one to connect two-particle HBT analyses to single-particle spectraand three-body coalescence. In Sec. II B 2 we derive the coalescence prediction for H and He. Eqs. (30,31), or theirmomentum space versions of Eqs. (34,35), are the most model-independent versions of these coalescence factors we knowof.None of our results in Sec. II rely on the details of the underlying nucleon emission function, or needs to specify a modelof the dynamical evolution of the HXS: our results simply connect femtoscopy with cluster yields, and the connection shouldapply to any self-consistent HXS model. To make contact with measurements, however, we need to specify the two- andthree-particle source. We turn to this in Sec. III. We start in Sec. III A by appealing to observational HBT parameterisationsof the two-particle source (extended, with some added assumptions, to the three-particle source). Relying on experimentalfits allows us to keep the analysis as model-independent as we can. While this is not an essential requirement of theframework, we stick in this paper to Gaussian or semi-Gaussian parameterisations. In Apps. A and B we give somequantitative model-dependent theoretical examples that suggest that an anisotropic (3D) Gaussian source parameterisationis probably accurate enough for the purpose of testing the origin of clusters via the coalescence–correlations relation.The next ingredient needed is the nuclear wave functions. In Sec. III B we derive the coalescence factors corresponding tothe simplified Gaussian wave functions. The Gaussian wave function is an over-simplification in some cases, but at the costof an O (1) theoretical error it allows us to derive analytic results for the coalescence factors, summarised by Eqs. (49,50,51).We note that the Gaussian wave functions we consider allow for different cluster length scales for three-body states; for Hthis is crucial, as the p-n factor of the wave function is considerably more compact than the Λ -pn factor. Eqs. (49,50,51)also account for the intrinsically anisotropic shape of the two- and three-particle source describing the HXS. As we illustratein App. B, the two-particle source is expected to be truly anisotropic, especially at large p t .In Sec. III C we extend the analysis to more accurate non-Gaussian wave functions. For D we derive an analytic coalescencefactor formula that applies to the Hulthen wave function if the underlying two-particle source is approximated as 1D Gaussian.The more realistic 3D two-particle source can be easily accounted for by numerical integration. For H we consider therecent three-body wave function proposed by [20]. We show that while the theoretically expected wave function is non-Gaussian, exhibiting an extended high- q tail, nevertheless an effective Gaussian wave function fit does a reasonably accuratejob in the coalescence factor calculation (valid to a factor of ∼ ). What does turn out – as already noted above – to bequantitatively important, is the consideration of the two different length scales associated with the p-n and Λ -pn factors ofthe state.In Sec. IV we give a rudimentary comparison to data. In Sec. IV A we recap results from [19] for D and He, adding apPb data point to the PbPb and pp measurements discussed there; we also correct a few typos in [19]. In Sec. IV B wecompare the coalescence prediction for H to PbPb data. We emphasise (as was done before us [21–23]; but here witha robust coalescence calculation) that H data in small systems (pp, pPb, or low multiplicity PbPb) has the potential toconclusively rule out (or support) coalescence as the dominant origin of clusters.In Sec. V we discuss and summarise our results.
II. THE COALESCENCE/FEMTOSCOPY FRAMEWORK
The description of femtoscopic correlations between nucleons [24] and the coalescence model for nuclei [25, 26] are twoaspects of the same theoretical framework. The idea is that at kinetic freeze out the HXS can be described by a multi-particle density matrix ˆ ρ HX . Depending on the measured observable, this density matrix can be projected onto final statesof different multiplicities. In what follows we present a concise derivation of coalescence and the coalescence–correlationsrelation [19], aiming to collect different aspects of the problem under the same roof, so to speak. Many of the results werealso derived elsewhere, notably in [16–19] and (albeit with model-dependence) in [15, 27–34]. A. Two-particle correlations and the deuteron.
If the phase space density of nucleons around the time when they last scatter against other particles (mostly pions) inthe HXS is not too high, then the subsequent propagation of pairs of nucleons emitted very near in phase space would bedominated by FSI, while additional interactions with particles other than the pair would be sub-dominant. With this suddenapproximation , the Lorentz-invariant yield of nucleon pairs at total spin s is given by [16, 18] γ γ dN ,s d p d p = 2 s + 1(2 π ) (cid:90) d x (cid:90) d x (cid:90) d x (cid:48) (cid:90) d x (cid:48) Ψ ∗ s,p ,p ( x (cid:48) , x (cid:48) ) Ψ s,p ,p ( x , x ) ρ p ,p ( x , x ; x (cid:48) , x (cid:48) ) , (1)where Ψ s,p ,p ( x , x ) is the continuum Bethe-Salpeter amplitude [37] describing the FSI of the pair. Similarly, the yieldof deuterons at momentum P is given by γ dN d d P = 2 s d + 1(2 π ) (cid:90) d x (cid:90) d x (cid:90) d x (cid:48) (cid:90) d x (cid:48) Ψ ∗ d,P ( x (cid:48) , x (cid:48) ) Ψ d,P ( x , x ) ρ p ,p ( x , x ; x (cid:48) , x (cid:48) ) , (2)where Ψ d,P ( x , x ) is the bound state Bethe-Salpeter amplitude describing the deuteron. The role of the Bethe-Salpeteramplitudes Ψ is to resum soft diagonal (ladder) FSI diagrams, factoring their effect out of an assumed underlying short-distance amplitude forming the density matrix ρ p ,p .Obviously, modelling the spectra Eqs. (1-2) requires proper modelling of the FSI, that are either calculable from firstprinciples (in the case of Coulomb) or measurable (in the case of nuclear scattering amplitudes) and that we assume to beknown. Modelling ρ p ,p from first principles, however, is currently impossible. Our goal in this section, and in the rest ofthe paper, is to demonstrate that even without a-priori knowledge of ρ p ,p , the mere fact that the same ρ p ,p occurs inboth of Eqs. (1) and (2) is enough to allow a model-independent, approximate prediction of the deuteron (and, with someadded assumptions, other clusters) spectrum based on measurements of HBT correlations [19].Let us define c , = ( p , P ) /P , where the pair total momentum is P = p + p ≡ p . In general, the dependence ofthe amplitude Ψ (be it Ψ s,p ,p or Ψ d,P ) on the pair total momentum and centre of mass coordinate X = c x + c x canbe factored out from the dependence on the relative momentum q = c p − c p and relative position x = x − x , via Ψ( x , x ) = e − iP X φ ( x ) . (3)Using Eq. (3) and changing to convenient coordinates, we can rewrite Eqs. (1-2) as: γ γ dN ,s d p d p = 2 s + 1(2 π ) (cid:90) d r (cid:90) d k (2 π ) ˜ D s,q ( k, r ) ˜ S p ,p ( k, r ) , (4) γ dN d d P = 2 s d + 1(2 π ) (cid:90) d r (cid:90) d k (2 π ) ˜ D d ( k, r ) ˜ S p ,p ( k, r ) , (5)where we define the relativistic internal Wigner density ˜ D ( k, r ) = (cid:90) d ζ e ikζ φ (cid:18) r + ζ (cid:19) φ ∗ (cid:18) r − ζ (cid:19) (6) See [1, 25, 35] for nonrelativistic formulations. See [36] for a discussion of corrections to the sudden approximation due to the residual chargeof the HXS, as applied to low energy (80 MeV/nuc) HIC. Note c + c = 1 , qP = 0 and ( p − p ) P = ( c − c ) P = m − m . (with ˜ D s,q and ˜ D d obtained from φ s,q and φ d , respectively) and where ˜ S p ,p ( k, r ) = (cid:90) d x (cid:90) d l e − il ( c P + k ) (cid:90) d l e − il ( c P − k ) × (7) ρ p ,p (cid:18) x + c r + l , x − c r + l x + c r − l , x − c r − l (cid:19) . The Wigner density we would obtain if we could turn off both FSI and quantum statistics is D s,q ( k, r ) = (2 π ) δ (4) ( k − q ) ,independent of r . With this we can define a hypothetical reference uncorrelated pair spectrum, γ γ dN d p d p = (2 s N + 1) (2 π ) (cid:90) d r ˜ S p ,p ( q, r ) , (8)where s N = 1 / is the nucleon spin. The reference pair spectrum is not a real physical object, but we can try to mimicit experimentally by pairing particles from different events with similar event characteristics. With this understanding, thepair correlation function is defined as C ( p, q ) = (cid:80) s γ γ dN ,s d p d p γ γ dN d p d p (9)and the coalescence factor for deuteron formation is defined as B ( p ) = P dN d d P p p dN d p d p ≈ m γ dN d d P γ γ dN d p d p , (10)where we approximated m D ≈ m .At this point it is useful to make two approximations: • Smoothness approximation:
The smoothness approximation was discussed widely in the literature [11, 12, 38].The version of this approximation we take here amounts to replacing S p ,p ( k, r ) ≈ S p ,p (0 , r ) in Eqs. (4) and (5).The k integral of the ˜ D functions can then be done, yielding (cid:82) d k (2 π ) ˜ D ( k, r ) = | φ ( r ) | . Similarly, we replace S p ,p ( q, r ) ≈ S p ,p (0 , r ) in the reference pair spectrum Eq. (8). The accuracy of the smoothness approximationis probably sufficient for our purpose in PbPb collisions; in pp collisions we think that a careful assessment is stillwarranted. • Equal-time approximation:
In the pair rest frame (PRF) we have P = ( M, ) , q = (0 , q ) and x = ( t, x ) . Both fortwo-particle correlations and bound states, we are interested in pairs that are nonrelativistic in the PRF, q (cid:28) m , andcan neglect corrections of O ( q /m ) . A key point, derived clearly in [16, 18], is that the Bethe-Salpeter amplitudein the PRF-nonrelativistic limit is approximately independent of the PRF time: φ ( x ) = φ ( x ) (cid:18) O (cid:18) tm x (cid:19)(cid:19) . (11)At the level of the leading term in the equal-time approximation of Eq. (11), φ s,q ( φ d ) is equal to the QM staticscattering wave (bound state) solution of the Schr¨odinger equation [39].In hadronic collisions we have t ∼ x ∼ few fm, implying a correction ∼ . x to the equal-time approximation.Thus, we are certainly sacrificing some precision when we adopt it: the incurred theoretical error is probably in theballpark of 10% for PbPb collisions, where x ∼ fm, but could be several tens of percent in pp collisions where x ∼ fm. Nevertheless, the equal-time approximation would allow us significant mileage with relatively simplenotation. We think that it probably allows for sufficient accuracy to establish (or exclude) coalescence as the mainorigin of clusters in hadronic collisions. Note that the free solution in the absence of FSI is the plane wave φ ( x ) = e − iqx [(anti-)symmetrised by QS] and the equal-time limit of Eq. (11) is exact in that case, because in the PRF qx = − qx .Adopting the smoothness and equal-time approximations we are led to the definition of the normalised two-particlesource S , a function of the PRF spatial coordinate r , as an integral in PRF time t = r : S ( r ) = (cid:82) dr ˜ S p ,p (0 , r ) (cid:82) d r ˜ S p ,p (0 , r ) . (12) It is common practice (e.g. [40–42]) to drop the explicit mention of p in S ( r ) ; this, despite the fact that S ( r ) does depend on p . As longas we keep this fact in mind, this practice brings no harm. The pair correlation function is then given from Eq. (9) as C ( p, q ) = (cid:88) s w s C s ( p, q ) (13)with the spin weights w s = (2 s + 1) / (2 s N + 1) and C s ( p, q ) ≈ (cid:90) d r | φ s,q ( r ) | S ( r ) . (14)Similarly, the deuteron coalescence factor is given from Eq. (10) as B ( p ) ≈ s d + 1) m (2 s N + 1) (2 π ) (cid:90) d r | φ d ( r ) | S ( r ) . (15)Two general comments are in order. First, the aim of HBT analyses like, e.g., Refs. [40–42] (for reviews, see [11, 12]) isto measure the source S ( r ) by solving the Schr¨odinger equation for the wave functions φ s,q ( r ) [18, 24, 43] and comparingthe observed two-particle spectrum with Eqs. (13-14). In this exercise, obtaining agreement with the experimental dataappears to require the use of the correct set of FSI potentials. While this point is not often stated explicitly, the success ofthe source reconstruction analyses lends some credence to the basic framework leading to Eqs. (13-14). As we have justseen, this is the same framework that stands behind the coalescence model for nuclei, the only difference being that HBTdeals with continuum scattering state solutions of FSI and coalescence deals with discrete bound state solutions of FSI.Thus the mere existence of successful HBT analyses lends some credence to coalescence as the origin of (at least some ofthe) nuclei observed in hadronic collisions.Second, with S ( r ) measured, Eq. (15) predicts the deuteron yield model-independently and with no free parameters.As we show in Sec. IV, this prediction is consistent with experimental data to within a factor of two or so across systemsranging from PbPb to pPb and pp at different multiplicities [19, 34]. This is strong evidence that coalescence contributesto the production of deuterons in hadronic collisions at the O (1) level, at least.A caveat to keep in mind is that although Eq. (14) is commonly used in the literature, we do not know a model-independent way of checking the quantitative corrections due to the smoothness and equal-time approximations, which wehave done to reduce Eq. (4) into Eq. (14); nor to the sudden approximation itself, allowing us to write Eq. (4) in the firstplace. A systematic study of these uncertainties is warranted if one wishes to narrow down the theory uncertainty associatedwith Eq. (15).As a technical aside, it is convenient to carry out some of the analysis in momentum space. To this end it is useful tointroduce the momentum space correlation function C , which is just the Fourier transform of S ( r ) : C ( p, k ) = (cid:90) d r e i kr S ( r ) . (16)(We will usually keep the explicit appearance of p in C .) Defining the momentum space deuteron form factor F d | φ d ( r ) | = (cid:90) d k (2 π ) e i kr F d ( k ) (17)we can rewrite Eq. (15) as B ( p ) ≈ s d + 1) m (2 s N + 1) (cid:90) d k F d ( k ) C ( p, k ) . (18)This is the version of the coalescence–correlation relation derived in Ref. [19]. B. Connecting two-particle states with one- and three-particle states: density matrix factorisation.
While two-particle HBT correlations are directly and model-independently connected to deuteron coalescence, the con-nection to single-particle spectra and to the coalescence of three-body states requires further assumptions. The mainassumption we need is the factorisation of the multi-particle density matrix into the product of single-particle densitymatrices, e.g. [24, 26] ρ p ,p ( x , x ; x (cid:48) , x (cid:48) ) ≈ ρ p ( x ; x (cid:48) ) ρ p ( x ; x (cid:48) ) . (19)Factorisation cannot be exact. For example, in low multiplicity events total momentum conservation inevitably leads tothe breakdown of Eq. (19), seen observationally as “non-femtoscopic correlations” at large q . Nevertheless, keeping thesecaveats in mind we will adopt the factorisation approximation in what follows. Later on we investigate one simple way toparametrise related corrections.
1. Uncorrelated pair spectrum as a product of one-particle spectra.
Considering nucleons (protons and neutrons) with m ≈ m ≈ m , nonrelativistic in the PRF, we have q ≈ p − p andthe factors c ≈ c = + O (cid:0) q /m (cid:1) . Inserting this into Eq. (7) and using Eq. (19) leads to ˜ S p ,p ( q, r ) ≈ (cid:90) d x ˜ S p (cid:16) x + r (cid:17) ˜ S p (cid:16) x − r (cid:17) , (20)with the one-particle emission function ˜ S p ( x ) = (cid:90) d l e − ilp ρ p (cid:18) x + l x − l (cid:19) . (21)This ˜ S p ( x ) coincides (up to constant factors in the definition) with the particle source of [24] and with the emission functionor phase space density of [11, 12, 15, 44, 45]. The reference pair spectrum factorises into the product of single-particlespectra, γ γ dN d p d p ≈ (cid:20) γ dNd p (cid:21) (cid:20) γ dNd p (cid:21) , (22) γ dNd p = (2 s N + 1)(2 π ) (cid:90) d x ˜ S p ( x ) . (23)Finally, the two-particle source S is constructed from single-particle emission functions as S ( r ) = (cid:82) dr (cid:82) d x ˜ S p (cid:0) x + r (cid:1) ˜ S p (cid:0) x − r (cid:1)(cid:104)(cid:82) d x ˜ S p ( x ) (cid:105) . (24)It can be more convenient to calculate C ( p, q ) , by inserting Eq. (24) into Eq. (16), giving the prescription: C ( p, q ) = (cid:12)(cid:12)(cid:12)(cid:82) d x e iqx ˜ S p ( x ) (cid:12)(cid:12)(cid:12) (cid:104)(cid:82) d x ˜ S p ( x ) (cid:105) . (25)In evaluating Eq. (25), recall that we require q = (0 , q ) as specified in the PRF.As a slight detour, consider the proton pair correlation with FSI turned off but quantum statistics still on, in the spin-asymmetric or spin-symmetric state where φ s,q ( r ) = √ (cid:0) e i qr ± e − i qr (cid:1) , respectively. Using Eq. (24) and noting that q = 0 in the PRF, the pair correlation of Eq. (14) would be C s ( p, q ) ≈ (cid:90) d r | φ s,q ( r ) | S ( r ) = 1 ± (cid:12)(cid:12)(cid:12)(cid:82) d x e iqx ˜ S p ( x ) (cid:12)(cid:12)(cid:12) (cid:104)(cid:82) d x ˜ S p ( x ) (cid:105) , (26)consistent with the usual expression in the literature [11, 12, 45, 46] (note that q as defined in, e.g. [46] is equal to q inour notation).
2. Hypertriton and He.
The starting point in the coalescence calculation for hypertriton H (pn Λ ) is similar to Eq. (2) for the deuteron: γ dN H d P = 2 s H + 1(2 π ) (cid:90) d x p (cid:90) d x n (cid:90) d x Λ (cid:90) d x (cid:48) p (cid:90) d x (cid:48) n (cid:90) d x (cid:48) Λ × Ψ ∗ H ,P ( x (cid:48) p , x (cid:48) n , x (cid:48) Λ ) Ψ H ,P ( x p , x n , x Λ ) ρ p p ,p n ,p Λ (cid:0) x p , x n , x Λ ; x (cid:48) p , x (cid:48) n , x (cid:48) Λ (cid:1) , (27)where Ψ H ,P ( x p , x n , x Λ ) is the bound state Bethe-Salpeter amplitude describing the H . The total momentum is P = p p + p n + p Λ ≡ p . We also define P pn = p p + p n , c I = ( p I P ) /P with I = p, n, Λ , and ˜ c J = ( p J P pn ) /P pn with J = p, n .The centre of mass coordinate is then X = c n x n + c p x p + c Λ x Λ , and useful relative coordinates are r pn = x p − x n and r Λ = x Λ − ˜ c p x p − ˜ c n x n . With these definitions the Bethe-Salpeter amplitude factorises into Ψ H ,P = e − iP X φ H ( r pn , r Λ ) .The calculation for He (ppn) is similar, with the replacements r pn → r pp and r Λ → r n , etc.Following the same steps as in Sec. II A and adding to that the smoothness approximation, the equal-time approximation,as well as density matrix factorisation a-la Eq. (19) extended to three particles, we are led – after the dust settles – to thenormalised three-particle source expressed as integrals of single-particle emission functions, S ( r pn , r Λ ) = (cid:82) dr pn (cid:82) dr (cid:82) d x ˜ S p (cid:0) x + r pn − r Λ (cid:1) ˜ S p (cid:0) x − r pn − r Λ (cid:1) ˜ S (Λ) p (cid:0) x + r Λ (cid:1)(cid:104)(cid:82) d x ˜ S p ( x ) (cid:105) (cid:82) d x ˜ S (Λ) p ( x ) , (28) S ( r pp , r n ) = (cid:82) dr pp (cid:82) dr n (cid:82) d x ˜ S p (cid:0) x + r pp − r n (cid:1) ˜ S p (cid:0) x − r pp − r n (cid:1) ˜ S p (cid:0) x + r n (cid:1)(cid:104)(cid:82) d x ˜ S p ( x ) (cid:105) . (29)The coalescence factors are then found as B ( p ) ≈ m s H + 1(2 s N + 1) (2 π ) (cid:90) d r pn (cid:90) d r Λ (cid:12)(cid:12)(cid:12) φ H ( r pn , r Λ ) (cid:12)(cid:12)(cid:12) S ( r pn , r Λ ) , (30) B ( p ) ≈ m s He + 1(2 s N + 1) (2 π ) (cid:90) d r pp (cid:90) d r n | φ He ( r pp , r n, ) | S ( r pp , r n ) . (31)We highlight that in the S calculation one emission function corresponds to the emission of a Λ , rather than a nucleon.For simplicity we approximated m Λ ≈ m in the prefactor, at the cost of an error of about ∼ .Again, we can define the Fourier transform C ( p, q , q ) = (cid:90) d r (cid:90) d r e i q r + i q r S ( r , r ) (32)and the momentum space form factor F H , (cid:12)(cid:12)(cid:12) φ H ( r pn , r Λ ) (cid:12)(cid:12)(cid:12) = (cid:90) d k pn (2 π ) e i k d r pn (cid:90) d k Λ (2 π ) e i k Λ r Λ F H ( k pn , k Λ ) . (33)In terms of these (and their equivalents for He) the coalescence factors read B ( p ) ≈ m s H + 1(2 s N + 1) (cid:90) d k pn (cid:90) d k Λ F H ( k pn , k Λ ) C ( p, k pn , k Λ ) , (34) B ( p ) ≈ m s He + 1(2 s N + 1) (cid:90) d k pp (cid:90) d k n F He ( k pp , k n ) C ( p, k pp , k n ) . (35)The coalescence of hypertriton and He can be connected to HBT analyses, to the extent that the three-particle normalisedsources S and S can be accessed by correlation measurements. This connection would be crucial for the attempt to use H , due to its large nucleus size, as a test of the coalescence framework. However, the connection is not as direct as it isfor deuteron formation. We discuss this connection further in the next section. III. COALESCENCE FROM CORRELATION FUNCTIONSA. HBT source parameterisation.
We now want to make practical contact with observational information on the HXS source size, available from HBTstudies. Experimental analyses commonly fit the measured two-particle source using a Gaussian approximation [40–42], thesimplest version of which is the isotropic (or 1D) Gaussian [16] S D ( r ) = 1(4 πR ) e − r R . (36)The radius parameter R inv depends on p [40–42].From the theoretical perspective [11, 12, 38] Eq. (36) could arise, for example, if the emission function ˜ S p ( x ) receivesmost of its support near some configuration space location R s and is also sharply peaked around kinetic freeze out time t f , as measured in the emitted particle rest frame, allowing the approximation ˜ S p ( x ) ∝ e − ( r − Rs ) R δ ( t − t f ) . (37)Inserting Eq. (37) into Eq. (24) immediately yields Eq. (36). While more difficult to test experimentally [47], Eq. (37) alsopredicts S of Eqs. (28) and (29), S D ( r , r ) = (cid:82) d x e − ( x + r − r − R s ) ( x − r − r − R s ) ( x + 2 r − R s ) R (cid:20)(cid:82) d x e − ( x − Rs ) R (cid:21) = 1(12 π R ) e − r
21+ 43 r R . (38)The momentum space versions of the isotropic Gaussian source model are C D ( p, q ) = e − q R , (39) C D ( p, q , q ) = e − R ( q + q ) . (40)Note that here we have considered the correlation of particles (e.g. protons) with the same underlying emission function ˜ S p ( x ) . Soon, however, we will use Eq. (40) to analyse H production which involves both ˜ S p ( x ) and ˜ S (Λ) p ( x ) , and there isno guarantee that these emission functions involve the same values of R s and R inv in Eq. (37). Experimental results in ppcollisions [42] suggest that R inv as extracted from p Λ correlations actually differs from R inv extracted from pp correlationsby ∼ or so . It must then be understood that any effective value we use for R inv in Eqs. (38) or (40) cannot be moreaccurately determined than the aforementioned ∼ , without introducing model-dependent assumptions concerning thebehaviour of the emission functions. We will comment on this point again in Sec. IV B when we discuss the comparisonbetween the H coalescence prediction and experimental data.The isotropic Gaussian source model is quite unrealistic. Even if the HXS “fire ball” was somehow isotropic in the labframe (which it generally isn’t; e.g. [11, 46]), it would be seen as anisotropic in the PRF due to Lorentz contraction alongthe direction of p . In addition, the beam line, of course, is a special direction in the initial state forming the HXS. As asimple generalisation that can capture some of these effects (and others), one can consider an anisotropic (3D) Gaussiansource with a free normalisation : C D ( p, q ) = λ e − q l R l − q o R o − q s R s , (41) C D ( p, q , q ) = λ e − R l ( q l + q l ) − R o ( q o + q o ) − R s ( q s + q s ) . (42)Here, we split the 3-vector q into its three components: q o (“out”) along the direction of the mean transverse momentum p t ; q l (“longitudinal”) along the beam axis; and q s (“side”) along the third orthogonal direction p t × ˆ z .Like the HBT radii R o,s,l , the normalisation factors (“intercept” or chaoticity parameter [17, 48]) λ , should best bemeasured directly from the data. For two-nucleon correlations this is sometimes done [40]. For three-nucleon correlations,as far as we know there is as yet no precedence.In App. A we briefly review how weak and strong resonance decays can distort the Gaussian shape of the source. InApp. B we give numerical examples of C calculated in the phenomenological blast wave model, illustrating how anisotropicflow and Lorentz contraction at p t > give rise to a true 3D source. We find that while the 1D Eq. (39) can fail quitebadly, the 3D Eq. (41) as a phenomenological parameterisation is flexible enough to capture the true physical source togood accuracy.Given the source parameterisation of Eqs. (41-42), the final ingredient we need to evaluate Eqs. (18), (34) and (35) forthe coalescence factors are the nucleus wave functions, encoded by the form factors F d, He , H . We turn to that next. This is attributed in [42] to the different contributions of strong resonance decays to the p and Λ spectra. From the point of view of ourdiscussion, however, the cause of the difference in R inv is not essential. Adding off-diagonal components like q o q l R ol , for example, into our analysis would be straightforward, but we avoid it here for simplicity. B. Nucleus wave functions: Gaussian wave function approximation.
The results are particularly tractable if we make the simplifying approximation of Gaussian wave functions. This is usefulfor analytic insight and is also reasonable if one wants to test coalescence at the O (1) level, so we report the results in thissection. In the next section we consider more accurate parameterisations of the wave functions.For D we consider φ d ( r ) = (cid:18) πb d (cid:19) e − r b d , (43)with momentum space form factor F d ( k ) = e − b d k . (44)For He we consider a Gaussian that is isotropic in the normalised Jacobi coordinates, related to our natural kinematiccoordinates via η pp = √ r pp , η n = (cid:113) r n : φ He ( r pp , r n ) = (cid:18) π b He (cid:19) e − r pp + 43 r n b . (45)With this φ He the momentum space form factor is F He ( k pp , k n ) = (cid:90) d r pp e − i k pp r pp (cid:90) d r n e − i k n r n | φ He ( r pp , r n ) | = e − b ( k pp + k n ) . (46)For H , as a first approximation we consider a product of Gaussians, φ H ( r pn , r Λ ) ≈ (cid:18) π b pn b (cid:19) e − r pn b pn − r b . (47)The momentum space form factor is then F H ( k pn , k Λ ) ≈ e − ( b pn k pn + b k ) . (48)We need to match the b parameters to nuclear data. For D we take b d = 3 . fm, corresponding to the RMS chargeradius r rms = (cid:113) b d = 2 . fm [49]. For an isotropic three-body Gaussian wave function, the parameter b as we definedit is directly the RMS charge radius. For He this is b He ≈ . fm [49]. For H , Hildenbrand & Hammer [20] reported (cid:113) (cid:104) r pn (cid:105) = √ b pn ≈ fm and (cid:112) (cid:104) r (cid:105) = b Λ ≈ . +3 − . fm. It is important to note, however, that Ref. [20] reported thefull numerical momentum space form factors and the (cid:112) (cid:104) r (cid:105) parameters they quote refer only to the small- k expansion ofthese form factors. A more accurate treatment of the wave function is obtained by using the full form factors, which we doin Sec. III C. Based on that analysis, for the Gaussian approximation we set b pn = 1 . fm (as opposed to b pn = 1 . fmthat would be read from the low- k fit) and b Λ = 7 . +2 − fm.With the Gaussian nuclear wave functions described in this section, assuming the HBT source parameterisation ofEqs. (41-42) and recalling the spins s N = s He = s H = , s d = 1 , the coalescence factors evaluated from Eqs. (18), (34) One can verify that (cid:82) d r pp (cid:82) d r n | φ He ( r pp , r n ) | = (3) (cid:82) d η pp (cid:82) d η n (cid:18) π b (cid:19) e − η pp + η nb = 1 . Please note that (cid:113) (cid:104) r pn (cid:105) in [20] refers to the distance between the n and the p; that is, the diameter, not the radius of that subsystem. Namely, (cid:104) r pn (cid:105) = (cid:82) d r Λ (cid:82) d r pn r pn (cid:12)(cid:12)(cid:12) φ H ( r pn , r Λ ) (cid:12)(cid:12)(cid:12) = 3 b pn . Similar goes for the pn- Λ distance (cid:104) r (cid:105) = (cid:82) d r Λ (cid:82) d r pn r (cid:12)(cid:12)(cid:12) φ H ( r pn , r Λ ) (cid:12)(cid:12)(cid:12) = b . : B ≈ π λ m ( b d + 4 R l ) ( b d + 4 R o ) ( b d + 4 R s ) , (49) B ≈ π λ √ m (cid:0) b He + 2 R l (cid:1) (cid:0) b He + 2 R o (cid:1) (cid:0) b He + 2 R s (cid:1) , (50) B ≈ π λ √ m (cid:0) b pn + 2 R l (cid:1) ( b + 2 R l ) (cid:0) b pn + 2 R o (cid:1) ( b + 2 R o ) (cid:0) b pn + 2 R s (cid:1) ( b + 2 R s ) . (51)We stress again that the analytic results obtained with Gaussian wave functions are brought here as means for an easy,rough assessment of the coalescence factor. More accurate calculations should use more accurate wave functions, especiallyfor the D and H . We consider this refinement next. C. Nucleus wave functions: more accurate parameterisation.
Here we consider more accurate parameterisations for the wave functions of the D and the H . For He we maintainthe Gaussian ansatz of Sec. III B.
1. Deuteron wave function.
A more accurate parametrisation of the D wave function, that should be used instead of the Gaussian ansatz forquantitative analyses, is given by the Hulthen formula: φ d ( r ) = (cid:115) αβ ( α + β )2 π ( α − β ) e − α | r | − e − β | r | | r | . (52)The RMS radius is given by r rms = β ( α + β )8 α ( α − β ) (cid:18) − α ( α + β ) + α β (cid:19) = 18 α (cid:18) αβ + O (cid:18) α β (cid:19)(cid:19) . (53)For this parameterisation it is assumed that β > α , so above we expanded in the ratio ( α/β ) . For the numerical evaluationwe set α = 0 . fm − and β = 1 . fm − , reproducing r rms = 2 . fm [49]. We have checked that using the slightlydifferent values of α and β quoted in Ref. [50] gives results that are equal to ours to 5% accuracy. The form factor Eq. (17)can only be obtained numerically. Amusingly, in the 1D Gaussian source limit Eq. (39) we do not need it because thecoalescence factor itself can be obtained analytically: B = 32 m (cid:90) d k e − k R (cid:90) d r | φ d ( r ) | e − i kr (54) = 3 π mR αβ ( α + β )( α − β ) (cid:16) e α R erfc (2 αR inv ) − e ( α + β ) R erfc (( α + β ) R inv ) + e β R erfc (2 βR inv ) (cid:17) . The coalescence factor B is shown in Fig. 1. Solid line shows the prediction of Eq. (54), obtained for the Hulthen wavefunction. For comparison, dashed line shows the less accurate Gaussian wave function prediction of Eq. (49). For simplicityhere we use the isotropic 1D Gaussian source model with R o = R s = R l = R inv and λ = 1 . Note that measurements atsmall R inv ∼ are possible in pp or pPb collisions, and may be sensitive to effect of the wave function seen in Fig. 1.To use the 3D source Eq. (41), the coalescence factor needs to be calculated numerically from Eq. (18) or (15). Our Eq. (49) is consistent with Eq. (28) of [19]. Our Eq. (50) corrects a typo in Eq. (31) of [19], where one should replace ( d A / → ( d A / √ . Gaussian Hulthen × - × - R inv [ fm ] B [ G e V ] FIG. 1: Deuteron coalescence factor B , calculated in the isotropic (1D) Gaussian source model, showing the difference between theGaussian and Hulthen wave function parameterisations.
2. Hypertriton wave function.
Hildenbrand & Hammer [20] reported a three-body theoretical calculation of the H wave function. Green squares in the left panel of Fig. 2 show the projected pn − Λ form factor obtained in that work. It is well reproduced by the two-bodycalculation of Congleton [51], shown by solid red line, provided we adjust the Q Λ parameter of [51] from Q Λ = 1 . fm − in the original paper to Q Λ = 2 . fm − . For simplicity we therefore consider the effective pn − Λ wave function from [51],given in momentum space by ˆ φ H(Λ d ) ( q ) = A e − q Q q + α = (cid:90) d r Λ e − i r Λ q Λ φ H(Λ d ) ( r Λ ) (55)where the normalisation constant A is defined such that (cid:82) d r Λ (cid:12)(cid:12)(cid:12) φ H(Λ d ) ( r Λ ) (cid:12)(cid:12)(cid:12) = (cid:82) d q (2 π ) (cid:12)(cid:12)(cid:12) ˆ φ H(Λ d ) ( q ) (cid:12)(cid:12)(cid:12) = 1 . Using the pn − Λ wave function of [51], with Q Λ = 2 . fm − and α = 0 . fm − , we can calculate the integrals in Eq. (34) or (30)numerically.In the right panel of Fig. 2 we show numerical calculations from [20], delimiting the uncertainty due to the H bindingenergy . We find that the lower (upper) range for the form factor is well fitted again by Congleton’s formula, with Q Λ = 2 . fm − and α = 0 . (0.082) fm − , respectively.Ref. [20] also provided the effective form factor for the pn subsystem. Their numerical result is shown by black squaresin Fig. 3. This form factor is reasonably well reproduced by a Gaussian of the form given by Eq. (48), with b pn =1 . fm (orange solid line). This can be compared with the low- k expansion of the form factor, which would lead to b pn (low − k fit) = 1 . fm as noted in Sec. III B. Given this discussion we can maintain the Gaussian ansatz of the pnfactor in Eq. (48), setting b pn = 1 . fm.In Fig. 4 we plot B vs R inv , showing the difference between the Gaussian (dashed grey) and the more realistic numerical(red) wave function parameterisations. Shaded band around the numerical result reflects the wave function uncertainty, asdepicted in the right panel of Fig. 2. We are grateful to Fabian Hildenbrand for providing this calculation to us. GaussianCongleton '92 ( Q Λ = - ) Congleton '92 ( Q Λ = - ) H&H 2019 e - R k , R = ( top ) , 5 fm ( bottom ) × - k [ fm - ] F pn - Λ Congleton '92 ( Q Λ = - , α = - ) Congleton '92 ( Q Λ = - , α = - ) Congleton '92 ( Q Λ = - , α = - ) H&H 2019H&H 2019 ( low ) H&H 2019 ( high ) × - k [ fm - ] F pn - Λ FIG. 2: Hypertriton pn − Λ form factor. Green squares show the result from the three-body calculation of Hildenbrand & Hammer [20]. Left:
Dotted and solid red line shows the effective two-body calculation of Congleton [51] for different values of their Q Λ parameter.Blue dashed line shows a Gaussian approximation with the same charge radius as found in [20]. For comparison, thick solid blacklines show exponential factors with scale radius R = 1 fm (top) and fm (bottom), respectively. Right:
Cyan and magenta markersshow a numerical calculation delimiting the uncertainty due to the H binding energy, as implemented in [20].
Gaussian ( b pn = ) Gaussian ( b pn = ) H&H 2019 k [ fm - ] F NN ' FIG. 3: Hypertriton pn form factor. Black squares show the result from the three-body calculation of Hildenbrand & Hammer [20].Solid orange and dashed blue lines show a Gaussian approximation a-la Eq. (48) with b pn = 1 . fm and b pn = 1 . fm, respectively. IV. COMPARISON WITH DATA.A. Deuteron and He.
Fig. 5 shows the theoretical prediction for B ( left ) and B ( right ), calculated as function of the 1D HBT parameter R inv using Eq. (54) (based on the Hulthen wave function) for D and Eq. (50) (Gaussian wave function) for He, with R o = R s = R l = R inv . The calculation, shown by a grey shaded band, uses an estimate of the experimentally measured valueof λ . To define the upper edge of the bands, we interpolate between λ = { , . , . } defined at R inv = { . , . , } .To define the lower edge we interpolate between λ = { . , . , . } defined at R inv = { . , . , } . This range of λ is estimated as follows. For large R inv values we have a measurement of λ in PbPb collisions [40], obtained as thesum λ ≈ λ ( pp ) + λ ( p Λ) in the notation of that paper. For small R inv values, corresponding to pp collisions, we have nomeasurement of λ ; Refs. [41, 52], which could in principle measure λ , effectively fixed λ → in their fit. As a nextbest solution we use λ measured from kaon femtoscopy [52]. This is a potentially reasonable estimate because Ref. [40]demonstrated HBT parameters that were the same, within measurement uncertainties, for kaon and proton final states at4 Gaussian numeric × - × - × - × - × - × - R inv [ fm ] B Λ [ G e V ] FIG. 4: Hypertriton coalescence factor B , calculated in the isotropic (1D) Gaussian source model, showing the difference betweenthe Gaussian (dashed grey) and the more realistic numerical (red) wave function parameterisations. the same m t . In addition to this attempt to estimate λ from data we also show the result obtained fixing λ = 1 (blacksolid line).Based on our analysis in App. B we can expect that for the p t (cid:46) . GeV values, in which the cluster data in Fig. 5 aregiven, our use of the simplistic 1D source parameterisation should cause us to over-estimate a more accurate 3D prediction(not available to us, as the experiments reported 1D HBT fits only) by (cid:46) or so. We do not include this uncertaintyin the plot. It adds up other sources of systematic uncertainty, expected to be roughly at a similar level (with unknownsigns), due to the smoothness, equal-time, and factorisation approximations (the latter relevant for He only).The comparison to experimental data is as follows. The red horizontal bands in Fig. 5 show experimental coalescencefactor measurements for PbPb at (0-10%) (for B ) and (0-20%) (for B ) centrality classes [53]. Each of the three redbands corresponds to a different bin in m t , among the three bins shown in the HBT R inv measurement [40]. The bluehorizontal bands show the result for the (20-40%) (for B ) and (20-80%) (for B ) events, respectively, again from [53].The green band shows the result for p-p collisions [54] .For He we can also add a crudely estimated data point for pPb collisions. To do so, we combine the 1D HBT R inv measurement of kaon femtoscopy reported in [55] with the He measurement of [56]. To approximately match m t betweenthe data sets, we use here the highest k t bin in [55] and the lowest p t bin in [56]. We use the (0-20%) multiplicity classfrom [55], joining together the (0-10%) and (10-20%) B data from [56]. The result is shown in purple in the right panelof Fig. 5. We thank Bhawani Singh and the Fabbietti TUM group for pointing out a typo in the plot of the B data for pp collisions in Ref. [19]. R inv [ fm ] B [ GeV ] R inv [ fm ] - - - - B [ GeV ] FIG. 5: Summary of D and He data, reproduced from Ref. [19] with some improvements (see text).
Left: B vs. R inv . Right: B vs. R inv . As should be clear by now, a main uncertainty in the theory prediction shown in Fig. 5 is related to the determinationof the chaoticity parameters λ and λ . A measurement of λ is missing for the pp and pPb systems, and we had tocomplete this information by using results from kaon femtoscopy. While the HBT results for kaons and protons come closeto each other where they are available at the same value of m t , it is clear that a proton HBT result is more suitable for thederivation of nuclear coalescence. For λ we have no data, as such data would require three-proton femtoscopy. In Fig. 5we bypassed this by assuming λ = λ . Addressing this issue experimentally would be challenging, and we do not knowof a model-independent way to estimate the associated theory uncertainty. Assessing the uncertainty within specific HXSmodels, along the lines of App. B, may be warranted in future work.Another obvious difficulty is due to the need to construct Fig. 5 patch-wise from data at different, often only partiallyoverlapping, multiplicity class and p t or m t bins. A dedicated experimental analysis combining HBT and cluster yieldswould solve this problem.Altogether, Fig. 5 shows that coalescence is roughly consistent with the D and He data for systems ranging from ppto pPb and PbPb at different regions of p t and at different multiplicity classes. This comparison spans a dynamical rangeof about a factor of 30 for B and a factor of for B . While, as we discussed, there are theoretical and experimentaluncertainties, there are no free parameters once HBT calibrates the computation. From this point of view, the usual claimto fame of the SHM [4], to describe the yields of nuclei across many orders of magnitude, is seen to be comparably wellapplicable to coalescence.Having said that, it is worth highlighting that for pp collisions the experimental coalescence factors for both D and Heare found to be higher than the coalescence prediction. Depending mainly on the poorly determined value of λ , but alsoon possible systematic uncertainties related to different event classes entering the HBT and cluster measurements, thediscrepancy could be as much as a factor of 2 for D and a factor of 4 for He. We think that this situation is strongmotivation for a joint experimental analysis of coalescence and HBT in small systems.
B. Hypertriton.
Ref. [57] reported a measurement of H in PbPb collisions. After some gymnastics we extract the measured B fromthe three p t bins in the left panel of their Fig. 7. For the bins p t ≈ (0 . − . , (1 . − , and (2 − . GeV we find B ≈ (7 . − × − , (1 − . × − , and (1 . − × − GeV , respectively. The multiplicity class is (0-10%).For HBT data, we have Ref. [40] with R inv and λ measured in the same multiplicity class, but binned in m t rather than p t (see Figs. 7-8 in [40]). We can match the first low p t bin of [57] into the m t range covered in [40]. We also consider thesecond p t bin of [57] that somewhat overshoots the coverage of the last m t bin in [40]. With some interpolation (and, forthe second bin, extrapolation) of the results from [40], we obtain the corresponding estimated ranges in R inv . The resultis shown by markers in Fig. 6.To plot the theory prediction, we need an estimate of λ . On the left panel of Fig. 6 we set λ = λ , with λ taken inthe range (0 . − . , estimated from the data [40]. The theory prediction is shown by the red band, with the width of the Our p t is corresponds to p t /A as defined in [57]. λ . In this exercise we use the numerical wave function of Sec. III C 2. For reference,the result for λ = 1 is shown by a black line. On the right panel we illustrate the wave function uncertainty, as well as theuncertainty (mentioned earlier in Sec. III A) associated with determining R inv from pp vs. p Λ or ΛΛ correlations. Setting λ = 0 . (in between 0.3 and 0.7), the result of varying the numerical wave function in the range corresponding to the rightpanel of Fig. 2 is shown by the red band. To implement the R inv uncertainty, an orange band shows the effect of shiftingthe R inv argument, entering the B computation, by ± w.r.t. to the R inv value on the x-axis. For comparison wealso show the Gaussian wave function Eq. (51) in grey dashed. Black line for reference is same as on the left. ■■ ■■ - - - - R inv [ fm ] B Λ [ G e V ] ■■ ■■ × - × - × - × - × - × - R inv [ fm ] B Λ [ G e V ] FIG. 6: B vs. the 1D HBT radius R inv . Left:
The coalescence prediction for the numerical wave function of Sec. III C 2, using λ = λ taken from data (red band). The width of the band derives from the uncertainty on λ . For reference, the result for λ = 1 is shown by solid black line. Right:
Impact of wave function and R inv uncertainties. Red band: numerical wave function uncertainty.Orange band: R inv uncertainty (see text). Grey dashed: Gaussian wave function approximation. Black line is same as on the left. Comparing between the left and right panels of Fig. 6 we can see that the estimated uncertainty due to the chaoticityparameter λ (or λ ) is larger than that due to the H wave function, again motivating an experimental effort to extractthe chaoticity from data. We can also try to bypass some of this uncertainty as follows. If Nature is kind, and λ ≈ λ ,then the uncertainty associated with λ may cancel in the ratio : S = B B . (56)For the first p t bin of [57] we find B ≈ (1 . − . × − GeV , and for the second bin B ≈ (3 . − . × − GeV .We could, in principle, combine this directly with B to extract the measured S . However, this would ignore the fact thatsome of the experimental uncertainty involved in deriving B and B could cancel out in the ratio. Instead, we thereforeadopt the following procedure. Taking the hypertriton spectrum from [57] (averaging the H and H results), we divideby the He-spectrum from [53] and scale by the Λ /p ratio taken from [58, 59], evaluated in the corresponding p t interval.The result is shown by markers in Fig. 7.The theoretical prediction for S , using the numerical H wave function of Sec. III C 2, is shown by the red band in Fig. 7with the band width determined by the wave function uncertainty. The effect of varying R inv by ± is shown by theorange band. For comparison we also show the Gaussian approximation with a grey dashed line. The coalescence predictionis somewhat below the data, with tension at the ∼ σ level . It is clear from Fig. 7 that a more precise experimentalmeasurement of S in conjunction with HBT is a promising observable to exclude (or support) the framework. At small R inv , S is predicted to be much below unity. Neither the H wave function uncertainty, nor the details of working witha 1D vs. 3D HBT source parameterisation are expected to enable, e.g., S > . at R inv ≈ fm, characteristic for ppcollisions. Not to be confused with the three-particle normalised source, S , defined in Sec. II B 2. To quantify the significance of the tension more precisely we would need to combine the two data points, which very much overlap in R inv .We prefer to leave these details to a dedicated experimental analysis. ■■■■ R inv [ fm ] S = B Λ / B FIG. 7: S vs. R inv predicted in the coalescence model compared with data. V. DISCUSSION AND SUMMARY.
The formation of nuclei by coalescence and femtoscopic [or Hanbury Brown-Twiss (HBT)] correlations between continuumproton pairs are two manifestations of final state interactions (FSI) that “dress” an underlying high-excitation state (HXS)produced in hadronic collisions. In Sec. II, starting from a relativistic quantum field theoretic (QFT) computation, werecalled how the effective quantum mechanical (QM) description of the dynamics of pairs that are nonrelativistic in the pairrest frame (PRF) gives rise to the usual HBT formalism, with which a two-particle source, characterising the HXS, can bemeasured. The same two-particle source enters the coalescence formula for deuteron formation. Much of our analysis wasbased on the formalism reviewed by Lednicky [18]; our contribution was to reduce this formalism to observationally accessiblelanguage in the context of nuclear clusters. In both cases – HBT and coalescence – effective QM holds for q (cid:28) m ,where q is the PRF momentum difference and m is the nucleon mass; and up to corrections of order ∼ . x , where x is the average PRF characteristic distance between the nucleons in the HXS. Although it should not be trusted beyond O (10%) accuracy , the QM limit is useful for deriving simple physically-transparent formulae and should be sufficient forruling out (or supporting) coalescence as the origin of nuclei in hadronic collisions. In principle, while we did not pursuethis route, the QM approximation can be avoided if one works directly with the relativistic FSI Bethe-Salpeter amplitudes.Coalescence after kinetic freeze out must occur at some level in hadronic collisions, but this does not mean that it isnecessarily the dominant origin of clusters. The key assumption made by the coalescence model is that the long-rangeaction of FSI can be factored out of an underlying short-range HXS dynamics. The level of accuracy of this factorisationis not obvious. To some extent, the apparent success of HBT analyses [40–42] in reconstructing a fit of the two-nucleonsource, using sophisticated physical FSI calculations, supports the coalescence framework. But we feel that this is not yetfully convincing: while the HBT analyses [40–42] reported measurements of the two-nucleon source with a stated precisionof O (10%) , they did so using a naive 1D Gaussian source fit . However, the true underlying source is not expected to be1D; in fact, at least for mean lab frame momentum p (cid:38) m , it must be anisotropic at the O (1) level – as seen in the PRF– due to Lorentz contraction. We can add to this the chaoticity parameter λ (denoted just λ in most of the literature),which was found to be significantly smaller than 1 in [40], but was held fixed to 1 in the fit of [41, 42]. Altogether, itis clear that higher statistics HBT analyses have an important future role to play [62] in establishing to what extent truephysical HXS information is revealed in femtoscopy. Our work highlights the importance of this question also to the originof clusters.Keeping the caveats above in mind, the main point of our work is to establish the coalescence/femtoscopy framework asa means to test coalescence by grounding the coalescence predictions with HBT information. The coalescence–correlationsrelation between HBT and deuteron formation, summarised by Eqs. (14,15) [or equivalently
Eqs. (16,18) in momentum Making some interesting proposed O (10%) tests potentially challenging [60, 61]. This is true also for [42], which assumed a 1D core source and modulated it by strong resonance decays. p t ). It is also consistent with the body of work by Mrowczynski [27–33].The connection of HBT with deuteron formation does not rely on density matrix factorisation. To derive formulaefor three-body clusters, though, factorisation is needed. Assuming factorisation we derived the formulae for H and He,summarised in
Eqs. (30,31) [or
Eqs. (34,35) in momentum space]. As an aside, we note that the derivation does not leaveroom for the confusion between “pn Λ channel” and “D Λ channel” advocated in [21, 22] for H. Three-body coalescencecomes from three-body FSI and is captured by a single formula, involving the three-body nucleus wave function.In Sec. III we combined formulae for the two-nucleon source, constrained by experimental HBT fits, with nuclear wavefunctions to obtain expressions for coalescence factors that can be compared to data. Exploring different forms for thewave function of H, we showed that although a full numerical representation of the wave function deviates significantlyfrom a Gaussian, the numerical impact on the coalescence factor is modest, about a factor of two (with the more realisticnumerical wave function predicting a higher coalescence factor). Our calculations allow for different scales in the wavefunctions of three-body states. This is particularly important for H where the pn- Λ factor is significantly more extendedthan the effective pn factor. Some earlier implementations [23] of the coalescence–correlations relation for H did notaccount for this fact.Our calculations also account for the anisotropic shape of the two- and three-nucleon source describing the HXS. As weillustrate in App. B, the two-particle source is expected to be truly anisotropic in nature, especially at large p t .In Sec. IV we compared our theoretical predictions to data. Fig. 5 shows that coalescence, calibrated by HBT, is consistentat the O (1) level with the p t -differential yields of D and He for systems ranging from pp, pPb to PbPb and across differentcentralities. This comparison spans a dynamical range of ∼ for B and ∼ for B . Some tension, at the ∼ σ levelor so, is seen for pp in both of the D and He yields. This situation gives strong motivation for a dedicated experimentalanalysis, studying HBT and cluster yields side by side in the same data set under the same kinematic conventions and cuts.On the HBT side, we urge the experiments to report HBT measurements allowing the chaoticity parameter λ to float inthe fit. In addition, as much as statistics permits, measurements of the 3D source (in the out-side-long parameterisation)would be preferred over 1D measurements of R inv . Such combined experimental analysis could zoom in on the coalescence–correlations prediction beyond the O (1) level by which it can currently be tested. This could, in principle, sharpen tensionswith D and/or He data where they are currently difficult to establish conclusively.Hypertriton H is known as a sensitive test of coalescence because of its large size, suggesting that the QM factordiscriminating coalescence from the statistical hadronisation model (SHM) should be small and discernible. Our calculation,depicted in
Fig. 6 , shows that the current measurements of B in PbPb collisions are consistent with the coalescenceprediction. The observable S = B / B may be more robust than B , because some experimental and theoreticaluncertainties may cancel in the ratio. We compare our calculation of S with the data in Fig. 7 , finding some tension: the S data tends to be somewhat higher than coalescence predicts. This discrepancy is not (yet) very significant, around σ .A higher statistics measurement, and measurements in small systems like pp, pPb, or low multiplicity PbPb collisions – inshort, systems with small HBT radius – would provide a critical test of coalescence. Acknowledgments
We thank Fabian Hildenbrand for providing us with numerical calculations of the H wave function, the Fabbietti TUMgroup, Urs Wiedemann and the participants of the CERN workshop “Origin of nuclear clusters” for insightful discussionsand Nitsan Bar and Yossi Nir for comments on the manuscript. KB is incumbent of the Dewey David Stone and HarryLevine career development chair at the Weizmann Institute of Science.
Appendix A: Corrections to the Gaussian source approximation.
The Gaussian source considered in Sec. III A cannot be exact. It is therefore important to estimate the uncertainty inthe coalescence calculation, that comes about if this approximation is used. In this section we consider mechanisms thatviolate the Gaussian source approximation and estimate their quantitative impact.
1. Feed-down from weak decays and source chaoticity.
In comparing theory to data one should account for the effect of feed-down from weak decay, where a particle that isemitted from the HXS as Λ or Σ decays into a proton in the detector (see [48] for a parallel discussion for pions). Thedecay vertex is displaced by O (1 cm) from the HXS, which means that the FSI relevant for two-particle correlation are9those involving Λ or Σ . Ref. [42] estimated that a fraction α p ≈ . of detected protons in their analysis originate from agenuine emitted proton, while a fraction α Λ ≈ . and a fraction α Σ ≈ . of detected protons originate from an emitted Λ or Σ , respectively . Neglecting particle misidentification (which add up to about 1% in [42]) we have α p ≈ − α Λ − α Σ .Thus, if HBT correlations are ignored , a fraction λ ( pp ) ≈ (1 − α Λ − α Σ ) ≈ . of detected pp pairs come from genuineemitted pp, while, for example, a fraction λ ( p Λ) ≈ α Λ (1 − α Λ − α Σ ) ≈ . of detected pp pairs come from an emitted p Λ pair, etc.In constructing the observable two-proton correlation, Refs. [40–42] divided the measured proton pair spectrum by thespectrum of pairs from uncorrelated mixed events, without feed-down subtraction. Then, in fitting a model of the correlationto the data, the correlation function was split into a part coming from genuine emitted pp pairs and parts coming fromemitted p Λ , ΛΛ , p Σ and ΣΣ pairs, using the relevant FSI for each part : C model ( p , q ) = 1 + (cid:88) i =pp , pΛ ,... λ ( i ) (cid:104) C ( i ) ( p , q ) − (cid:105) , (A1)with the channel-specific weights and correlation functions C ( i ) ( p , q ) = (cid:88) s w s (cid:90) d r S ( i )2 ( r ) (cid:12)(cid:12)(cid:12) φ ( i ) s,q ( r ) (cid:12)(cid:12)(cid:12) . (A2)Here, for example, for i = p Λ the wave function at spin channel s is given by φ ( p Λ) s,q ( r ) , etc. In Ref. [42], the nucleonsource functions S ( i )2 ( r ) were modelled differently for different i , to account for strong resonances that have a differentcharacteristic decay range for p and Λ daughters. In contrast, Refs. [40, 41] assumed a common nucleon source S ( i )2 = S .As long as the λ ( i ) parameters are correctly calibrated to account for weak decays, a quick check verifies that thecorrelation functions C ( i ) in Eq. (A2) do indeed match with the theoretical definition of the same objects as derivedin Sec. II A. Refs. [41, 42] assumed that this was the case and fixed the numerical values of the λ ( i ) according to theexperimentally determined single proton purity. In contrast, Ref. [40] did not fix the value of λ ( pp ) and λ ( p Λ) , but ratherconsidered these parameters as part of the experimental fit. Interestingly, the fit resulted in λ ≡ λ ( pp ) + λ ( p Λ) ≈ . − . ,significantly less than unity .From the theoretical point of view, the approach of [40] is beneficial over that of [41, 42]. The reason has to do with themodelling of the source function S ( r ) [12, 19]. Refs. [40–42] assumed in their fit the 1D Gaussian form Eq. (36). Thissource function integrates to unity, (cid:82) d r S ( r ) = 1 , corresponding to C ( q = 0) = 1 . However, there is no a-priori reasonto assume that the true C (or S ) satisfy this normalisation exactly. In fact, observing a departure from this normalisationin the data could hint, for example, at a violation of the factorisation assumption of Sec. II B. In addition, it is well knownthat if the experimental fit assumes a Gaussian form for C ( q ) , but the true C ( q ) is non-Gaussian, then adding an interceptparameter λ to the fit, as in Eq. (41), can absorb some of the difference.We can thus interpret the λ measurement of [40] as a measurement of the parameter λ , via λ ≈ λ . In contrast, theanalysis of Refs. [41, 42] effectively forced λ → in the fit procedure.Considering now the coalescence factor, a direct comparison to the HBT analysis is possible if the single proton spectrumused in the experimental definition of B A (e.g. Eq. (10)) is obtained subtracting the weak decay feed-down contributions.With this the coalescence–correlation relation of Eq. (15) or Eq. (18) applies as is. For B , , a fully data-driven analysiswould require an experimental measurement of C . In the absence of that, we can roughly estimate that for λ (cid:54) = 1 theGaussian source expressions for B , should be modified by B , → λ B , [19].Unrelated to weak decays, a correction to the expressions of Sec. III arises from the fact that Ref. [42] actually founddifferent fit results for the sources S ( r ) deduced from pp and p Λ pairs. One physically motivated reason for the differencestems from the different decay range of the strong resonances, that feed into p and Λ states in the HXS. The same effectalso predicts a non-Gaussian form for S ( r ) (and C ( q ) ), as we discuss next. The feed-down fractions in [42] were allowed to vary by 20% as part of the systematic uncertainty estimate. Indeed, these fractions weredifferent by ∼ in Ref. [41], although this may be in part due to different experimental kinematical cuts and selection criteria. Ref. [63] also used a similar formalism in studying baryon–anti-baryon correlations. Useful details can be found in [64]. For comparison, the single-particle purity estimates of [41, 42] read λ ( pp ) + λ ( p Λ) ≈ . and . ,respectively; the remaining probability being associated mostly with p Σ pairs.
2. Strong resonance decays.
Even if the underlying emission function was an exact Gaussian, the decay of strong resonances with lifetimes of the orderof a few fm would distort the effective source. Ref. [42] studied this problem for pp and p Λ correlations in pp collisions. Anisotropic source model including strong resonance decay, which was found in [42] to give an adequate fit to the correlationdata, is reproduced here in the left panel of Fig. 8. The full (non Gaussian) source is shown by circles, compared with theGaussian source in dotted lines. Strong resonances lead to a non Gaussian tail of S ( r ) . In the right panel of Fig. 8 we -3 -2 -1 -1 -2 -1 FIG. 8:
Left:
The source S ( r ) , reproduced from the experimental fit result of Fig. 4 in Ref. [42]. Right: C , the Fourier transformof the sources on the left. show the C ( q ) curves corresponding to the S ( r ) curves on the left.Quantitatively, despite the strong resonance contribution, the Gaussian approximation can be seen to give a ratheraccurate description of the source: the difference between the full non Gaussian C and the Gaussian approximation issmaller than ∼ throughout the range where C > . . While Ref. [42] dealt with pp initial state, we might expectthat in larger systems like PbPb the relative impact of strong resonance decay could be even less significant.We emphasise that Ref. [42], which provided the example for our discussion, assumed an underlying Gaussian sourcewhich they then deformed by strong resonances. What we learn from this exercise, then, is that if the underlying “genuine”emission function is Gaussian, then the effective two-particle source after strong decays is also consistent with Gaussian to10% accuracy, in the range of r (or, in momentum space, q ) that is relevant for coalescence. We stress, however, thatwhile the assumed 1D Gaussian source of [42] was experimentally consistent with HBT data, there is no guarantee thatthe true physical source is Gaussian. In App. B we recall theoretical reasons to expect otherwise. Appendix B: Deviations from an isotropic Gaussian source in the phenomenological blast wave model.
Relativistic expansion of the HXS “fire ball” is expected to proceed differently along and transverse to the beam line.This unisotropic flow predicts that the source S ( r ) should depend on the direction of r w.r.t. the beam line and also w.r.t.the pair mean momentum vector p . Even if the nucleon source was somehow isotropic in the lab frame, it would not beseen as isotropic in the PRF for p (cid:54) = 0 , due to Lorentz contraction. These effects were not modelled in the experimentalanalyses of Refs. [40–42], that performed fits to the 1D source Eqs. (39). Here we consider these effects from the point ofview of the phenomenological blast wave model (BWM). Our goal is not to argue either in favour or against the validityof the model, but simply to gain some feeling for the possible systematic error associated with adopting the 1D Gaussiansource approximation, if that is applied to coalescence calculations via the coalescence–correlation relation.To recall the BWM, we take the beam line to be along the ˆ z axis. The space-time coordinates are chosen as τ = √ t − z , ρ = (cid:112) x + y , η = arctanh( z/t ) , and the azimuthal angle φ . The position 4-vector is then R µ = ( τ cosh η, ρ cos φ, ρ sin φ, τ sinh η ) and d R = dτ τ dρρ dη dφ . The 4-momentum vector of a particle emitted atrapidity Y , azimuthal angle Φ , and transverse momentum p t , is p µ = ( m t cosh Y, p t cos Φ , p t sinh Φ , m t sinh Y ) , with m t = (cid:112) m + p t . The single-particle emission function is parameterized using 6 parameters, τ , ∆ τ, R , β S , n, T (we1neglect chemical potentials), as follows ( θ ( x ) is the Heaviside step function): ˜ S p ( R ) = (cid:114) π m t m cosh ( η − Y ) J ( τ ) θ ( R − ρ ) e − puT , (B1) u µ ( R ) = (cosh η cosh η t , sinh η t cos φ, sinh η t sin φ, sinh η cosh η t ) , (B2) η t ( ρ ) = tanh − (cid:18) ρ n R n β S (cid:19) , (B3) J ( τ ) = 1 √ π ∆ τ e − ( τ − τ τ . (B4)With these definitions we can evaluate C ( p, q ) numerically via Eq. (25). In doing so, recall that we require q = (0 , q ) specified in the PRF, while the emission function ˜ S p ( x ) depends on space-time coordinates in the lab frame. Defining b = p/m , we can write q (cid:48) in the lab frame as q (cid:48) = bq , q (cid:48) = q + bq b b [15]. For simplicity, in the examples below wechoose p = ( m t , p t , , . With this choice pu = m t cosh η cosh η t − p t sinh η t cos φ and using the q l , q o , q s decomposition wehave qx = p t m q o τ cosh η − m t m q o ρ cos φ − q s ρ sin φ − q l τ sinh η . We consider separately the cases q = ( q o , , , q = (0 , q s , ,and q = (0 , , q l ) .In Fig. 9 we plot C ( p, q ) , projected onto the out, side, and long directions (solid blue, orange, and green, respectively).Independent Gaussian fits to each projection are shown by dotted lines. Black dashed line shows the 1D Gaussian sourcecomputed with R inv = ( R o R s R l ) . For definiteness we take T = 100 MeV, n = 1 , β s = 0 . and τ = 7 fm, ∆ τ = 1 . fm, R = 7 fm, chosen to roughly represent PbPb collisions [53]. Note that the phenomenological BWM is based on classicalintuition and satisfies by construction – effectively – the density matrix factorisation assumption. Thus the chaoticityparameters λ = λ = 1 in this computation, and we are guaranteed that C ( p, ) = 1 .The source depicted in Fig. 9 was chosen to resemble PbPb collisions. From the fits in the plot, the radii of homogenietyfor this source at p t = 0 , for example, are R o = R s ≈ . fm, R l ≈ . fm, extended compared to the deuteron RMSradius r rms = (cid:112) / b d ≈ . fm. To see the size of the effect for smaller systems, e.g. pp collisions, we recalculate C fordifferent values of R = τ = 2 fm, leading to R o = R s ≈ . fm, R l ≈ fm at p t = 0 . (The other BWM parameters areunchanged.) The results are shown in Fig. 10.Figs. 9-10 demonstrate how unisotropic flow leads to an unisotropic source. Notably, the width along the out directionshows Lorentz contraction in the PRF at p t > . This perspective calls into question the attempt in [42] to fit the detailsof S ( r ) to O (10%) precision, based on the 1D isotropic Gaussian model.Our main interest is to check to what extent the deviation from 1D Gaussian source affects the coalescence calculation.Again from Figs. 9-10 (as well as from the discussion in Sec. A 2), we expect that a 3D Gaussian approximation may doa reasonably accurate job describing C ; thus we will use the 3D Gaussian parameterisation as a standard for comparison.For a rough estimate of the error, incurred by using the 1D fit of Eq. (39) to describe a 3D source, we calculate R o , R s , R l in the BWM and define R inv = ( R o R s R l ) . (B5)We can now compare the results of using Eq. (49) with R o = R s = R l → R inv , to the results of the original Eq. (49): B D B D ∼ (cid:115) ( b d + 4 R l ) ( b d + 4 R o ) ( b d + 4 R s )( b d + 4 R ) . (B6)Note that in both limits b d → and b d → ∞ this correction factor is equal to 1. The results are summarised in Tab. I.We give a rough estimate of the correction to B in a similar way: B D B D ∼ (cid:118)(cid:117)(cid:117)(cid:116) (cid:0) b pn + 2 R l (cid:1) ( b + 2 R l ) (cid:0) b pn + 2 R o (cid:1) ( b + 2 R o ) (cid:0) b pn + 2 R s (cid:1) ( b + 2 R s ) (cid:0) b pn + 2 R (cid:1) (cid:0) b + 2 R inv (cid:1) . (B7)The calculation for B is the same up to b pn → b Λ → b He . The results are also summarised in Tab. I.We have also done a numerical calculation of the ratio B D / B D using the Hulthen D wave function. In this exercisewe calculated B D from Eq. (15), comparing that with the result of Eq. (54). The results we find for PbPb and pp and forall values of p t are numerically very close to those found in Tab. I using the Gaussian wave function.The conclusion from Tab. I is that the 1D Gaussian parameterisation tends to slightly over-estimate the coalescencefactor, in comparison to the more accurate 3D parameterisation. The effect is more pronounced in small systems than inPbPb, and increases with increasing p t . Using the 1D R inv parameterisation results with O (10%) error at p t ∼ GeV,2 p t = outsidelong [ GeV ] C p t = outsidelong [ GeV ] C p t = outsidelong [ GeV ] C p t = outsidelong [ GeV ] C FIG. 9: The two-particle source C ( p, q ) calculated in the blast wave model and projected onto the out, side, long directions. Solidlines show the numerical result, while dotted lines show independent Gaussian fits to each projection. Black dashed line shows the1D Gaussian source with R inv = ( R o R s R l ) . The BWM parameters are chosen to mimic PbPb collisions at the LHC (see text). Top left, top right, bottom left, bottom right: p t = 0 , . , , GeV. p t = outsidelong [ GeV ] C p t = outsidelong [ GeV ] C p t = outsidelong [ GeV ] C p t = outsidelong [ GeV ] C FIG. 10: Same as Fig. 10, with BWM model parameters chosen to mimic pp collisions at the LHC. TABLE I: The corrections of Eqs. (B6-B7), comparing between the coalescence factor obtained from a 1D and a 3D Gaussian fit to C computed in the BWM. We round the result to two significant digits. p t m t B D / B D B D / B D B D / B D B D / B D B D / B D B D / B D [GeV] [GeV] PbPb pp PbPb pp PbPb pp rising to as much as a factor of 2 for He at p t = 2 GeV in pp collisions. It should be stressed that these conclusionsare drawn from simple, model-dependent calculations in the BWM, and should only be taken as crude estimates of thetheoretical uncertainty in the coalescence calculation. [1] L. P. Csernai and J. I. Kapusta, Phys. Rept. , 223 (1986).[2] A. Andronic, P. Braun-Munzinger, J. Stachel, and H. Stocker, Phys. Lett. B , 203 (2011), 1010.2995.[3] J. Cleymans, S. Kabana, I. Kraus, H. Oeschler, K. Redlich, and N. Sharma, Phys. Rev. C , 054916 (2011), 1105.3719.[4] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Nature , 321 (2018), 1710.09425.[5] V. Vovchenko, B. Dnigus, and H. Stoecker, Phys. Lett. B , 171 (2018), 1808.05245.[6] V. Vovchenko, B. Dnigus, and H. Stoecker, Phys. Rev. C , 054906 (2019), 1906.03145.[7] S. Acharya et al. (ALICE), Nucl. Phys. A , 1 (2018), 1710.07531.[8] E. Shuryak and J. M. Torres-Rincon, Phys. Rev. C , 024903 (2019), 1805.04444.[9] E. Shuryak and J. M. Torres-Rincon, Phys. Rev. C , 034914 (2020), 1910.08119.[10] M. Juric et al., Nucl. Phys. B , 1 (1973).[11] M. A. Lisa, S. Pratt, R. Soltz, and U. Wiedemann, Ann. Rev. Nucl. Part. Sci. , 357 (2005), nucl-ex/0505014.[12] U. W. Heinz and B. V. Jacak, Ann. Rev. Nucl. Part. Sci. , 529 (1999), nucl-th/9902020.[13] D. Oliinychenko, L.-G. Pang, H. Elfner, and V. Koch, Phys. Rev. C , 044907 (2019), 1809.03071.[14] R. Lednicky, Braz. J. Phys. , 939 (2007), nucl-th/0702063.[15] R. Scheibl and U. W. Heinz, Phys. Rev. C59 , 1585 (1999), nucl-th/9809092.[16] R. Lednicky and V. L. Lyuboshits, Sov. J. Nucl. Phys. , 770 (1982), [Yad. Fiz.35,1316(1981)].[17] S. Akkelin, R. Lednicky, and Y. Sinyukov, Phys. Rev. C , 064904 (2002), nucl-th/0107015.[18] R. Lednicky, Phys. Part. Nucl. , 307 (2009), nucl-th/0501065.[19] K. Blum and M. Takimoto, Phys. Rev. C , 044913 (2019), 1901.07088.[20] F. Hildenbrand and H. W. Hammer, Phys. Rev. C100 , 034002 (2019), 1904.05818.[21] Z. Zhang and C. M. Ko, Phys. Lett. B , 191 (2018).[22] K.-J. Sun, C. M. Ko, and B. Dnigus, Phys. Lett. B , 132 (2019), 1812.05175.[23] F. Bellini and A. P. Kalweit, Phys. Rev. C , 054905 (2019), 1807.05894.[24] S. E. Koonin, Phys. Lett. , 43 (1977).[25] R. Bond, P. Johansen, S. Koonin, and S. Garpman, Phys. Lett. B , 43 (1977).[26] H. Sato and K. Yazaki, Phys. Lett. , 153 (1981).[27] S. Mrowczynski, J. Phys. G13 , 1089 (1987).[28] S. Mrowczynski, Phys. Lett.
B248 , 459 (1990).[29] S. Mrowczynski, Phys. Lett.
B277 , 43 (1992).[30] S. Mrowczynski, Phys. Lett.
B308 , 216 (1993).[31] S. Mrowczynski, Phys. Lett.
B345 , 393 (1995), hep-ph/9502215.[32] R. Maj and S. Mrowczynski, Phys. Rev.
C71 , 044905 (2005), nucl-th/0409061.[33] S. Mrowczynski, Acta Phys. Polon.
B48 , 707 (2017), 1607.02267.[34] K. Blum, K. C. Y. Ng, R. Sato, and M. Takimoto, Phys. Rev.
D96 , 103021 (2017), 1704.05431.[35] S. Pratt and M. Tsang, Phys. Rev. C , 2390 (1987).[36] L. Martin, C. Gelbke, B. Erazmus, and R. Lednicky, Nucl. Phys. A , 69 (1996).[37] S. S. Schweber, H. A. Bethe, and F. de Hoffmann (1955).[38] S. Pratt, Phys. Rev. C , 1095 (1997).[39] L. D. Landau and E. Lifshits, Quantum Mechanics: Non-Relativistic Theory, vol. v.3 of Course of Theoretical Physics(Butterworth-Heinemann, Oxford, 1991), ISBN 978-0-7506-3539-4.[40] J. Adam et al. (ALICE), Phys. Rev. C92 , 054908 (2015), 1506.07884.[41] S. Acharya et al. (ALICE), Phys. Rev. C , 024001 (2019), 1805.12455. [42] S. Acharya et al. (ALICE) (2020), 2004.08018.[43] D. Mihaylov, V. Mantovani Sarti, O. Arnold, L. Fabbietti, B. Hohlweger, and A. Mathis, Eur. Phys. J. C , 394 (2018),1802.08481.[44] D. Anchishkin, U. W. Heinz, and P. Renk, Phys. Rev. C , 1428 (1998), nucl-th/9710051.[45] U. A. Wiedemann, B. Tomasik, and U. W. Heinz, Nucl. Phys. A , 475C (1998), nucl-th/9801017.[46] S. Chapman, J. R. Nix, and U. W. Heinz, Phys. Rev. C52 , 2694 (1995), nucl-th/9505032.[47] B. B. Abelev et al. (ALICE), Phys. Lett. B , 139 (2014), 1404.1194.[48] U. A. Wiedemann and U. W. Heinz, Phys. Rev.
C56 , 3265 (1997), nucl-th/9611031.[49] I. Sick (2015), 1505.06924.[50] L. Lamia, M. La Cognata, C. Spitaleri, B. Irgaziev, and R. Pizzone, Phys. Rev. C , 025805 (2012).[51] J. G. Congleton, J. Phys. G18 , 339 (1992).[52] B. Abelev et al. (ALICE), Phys. Rev.
D87 , 052016 (2013), 1212.5958.[53] J. Adam et al. (ALICE), Phys. Rev. C , 024917 (2016), 1506.08951.[54] S. Acharya et al. (ALICE), Phys. Rev. C97 , 024615 (2018), 1709.08522.[55] S. Acharya et al. (ALICE), Phys. Rev. C , 024002 (2019), 1903.12310.[56] S. Acharya et al. (ALICE), Phys. Rev. C , 044906 (2020), 1910.14401.[57] J. Adam et al. (ALICE), Phys. Lett. B , 360 (2016), 1506.08453.[58] B. B. Abelev et al. (ALICE), Phys. Rev. Lett. , 222301 (2013), 1307.5530.[59] B. Abelev et al. (ALICE), Phys. Rev. C , 044910 (2013), 1303.0737.[60] S. Mrowczynski and P. Slon (2019), 1904.08320.[61] S. Bazak and S. Mrowczynski (2020), 2001.11351.[62] Z. Citron et al., Working Group 5: Future physics opportunities for high-density QCD at the LHC with heavy-ion and proton beams(2019), vol. 7, pp. 1159–1410, 1812.06772.[63] J. Niedziela (ALICE), EPJ Web Conf.177