Event selection
The results are based on the analysis of a dataset comprising inelastic pp collisions at √s = 13 TeV, recorded with the ALICE detector45,46 during the LHC run 2 (2015–2018). The events are selected using a high-multiplicity (HM) trigger, which captures the highest multiplicity events—specifically, the top 0.17% of all inelastic collisions that include at least one charged particle within the pseudorapidity interval |η| < 1 (denoted as 0.17% INEL > 0). This approach ensures a statistically rich sample, as a five-fold increase in the production of (anti)deuteron candidates has been observed in HM pp collisions compared with minimum bias pp collisions47. The sample of HM-triggered collisions considered for this analysis corresponds to 1 × 109 events. On average, 31 charged tracks are found within |η| < 0.5 (ref. 48) for the HM-triggered collisions. Detailed descriptions of the event selection criteria, pileup rejection techniques, primary-vertex reconstruction methods and the HM trigger procedure are provided in ref. 49.
Tracking and particle identification
Particle identification and momentum measurement of charged particles are performed using the inner tracking system (ITS)50, time projection chamber (TPC)51 and time-of-flight (TOF)52 detectors of ALICE covering the whole azimuthal angle and the pseudorapidity interval |η| < 0.9. These detectors are located within a uniform magnetic field of 0.5 T along the beam axis, generated by the ALICE solenoid magnet, which causes the trajectories of particles to bend. The curvature of the charged-particle tracks is used to measure the particle momenta. The transverse momentum for pion and deuteron candidates is determined with a resolution ranging from approximately 2% for tracks with p T ≈ 10 GeV c−1 to below 1% for p T < 1 GeV c−1. Particle identification is performed by measuring the energy loss per unit track length (dE/dx) in the TPC detector and the particle velocity (β) in the TOF detector. For tracks in the TPC detector, the signal is obtained from the nσ TPC distribution, where nσ TPC represents the deviation of the measured signal from the expected value for a given particle hypothesis, normalized by the detector resolution. Similarly, for the TOF detector, the resolution is defined by nσ TOF , which quantifies the difference between the measured and expected time of flight, also normalized by the resolution. Further experimental details are discussed in ref. 46. The selection criteria for pion and deuteron tracks used in this work are described in refs. 30,41.
Pions are identified by the measurement of the specific energy loss within |nσ TPC | < 3 in a transverse momentum range p T ∈ [0.14, 4.0] GeV c−1. This information is combined with the TOF measurement by taking the geometric sum, \(\sqrt{n{\sigma }_{{\rm{TPC}}}^{2}+n{\sigma }_{{\rm{TOF}}}^{2}} < 3\), for track momentum p > 0.5 GeV c−1. Similarly, the deuteron candidates are selected within a transverse momentum range p T ∈ [0.8, 2.4] GeV c−1. They are identified by using |nσ TPC | < 3 for candidate tracks with momentum p < 1.4 GeV c−1, whereas both TPC and TOF information are required, \(\sqrt{n{\sigma }_{{\rm{TPC}}}^{2}+n{\sigma }_{{\rm{TOF}}}^{2}} < 3\), for candidates with p > 1.4 GeV c−1. Moreover, for (anti)deuteron candidate selections, electrons are rejected by the condition nσ TPC,e > 6 for p < 1.4 GeV c−1 and pions are rejected by the condition nσ TPC,π > 3 for the tracks with momentum p > 1.4 GeV c−1. Overall, using these methods, a purity of 99% for π± and 100% for (anti)deuterons is achieved.
The selection criteria of pions and deuterons constitute the primary source of systematic uncertainties associated with the measured correlation function. All particle selection criteria are varied from their default values. To account for the effect of possible correlations, the analysis of π+–d and π−–d pairs is repeated 44 times using random combinations of these selection criteria. The total systematic uncertainties are extracted by first randomly selecting a correlation function from the 44 systematic variations. For each sampled function, a bootstrap method is applied by randomly varying the C(k*) values in the individual k* bins according to their statistical uncertainties, assuming Gaussian errors. This results in a distribution of values for each k* bin, which is then fitted to determine the total uncertainty. As the statistical and systematic uncertainties are independent, the total uncertainty is obtained by adding them in quadrature. The systematic component is then determined by subtracting the known statistical uncertainty. The systematic uncertainties are largest at low k* ≈ 10 MeV c−1, reaching 1%. The same procedure is applied to extract the uncertainties of the fitted parameters and propagated to the final results on the fraction of deuterons stemming from resonance-assisted fusion processes.
Characterization of the particle-emitting source
A standard approach to evaluate the source function, used by ALICE in pp collisions, is the resonance source model (RSM)41,44. In these publications, the ALICE Collaboration measured the source size for baryon–baryon, meson–baryon and meson–meson pairs, demonstrating a common emission source of all particles and resonances produced directly in the collision. These are described as primordial particles, whereas the short-lived resonances that decay into the pairs of interest on the timescale of fm c−1 will lead to an increase in the effective source size. If this increase in the source size is properly modelled by Monte Carlo simulations, the underlying primordial source has a Gaussian profile of width r core , and scales as a function of the pair transverse mass \({m}_{{\rm{T}}}=({k}_{{\rm{T}}}^{2}+{m}^{2}{)}^{1/2}\), where m is the average mass, the average of the masses of the two particles constituting the pair and k T = |p T,1 + p T,2 |/2 is the average transverse momentum of the pair41,44. The scaling of the primordial source size follows a power law \({r}_{{\rm{core}}}=a{\langle {m}_{{\rm{T}}}\rangle }^{b}+c\), where the parameters for the high-multiplicity pp collisions at √s = 13 TeV used for the present π–d analysis are provided in ref. 44. The knowledge of both the pair average m T and the cocktail of contributing resonances allows us to evaluate both the r core and subsequently the total source distribution S(r*). The present analysis incorporates the resonances decaying into pions from the ThermalFIST model34,37, as already performed in the ALICE π–π and p–π analyses38,41. From the study of p–d and K+–d correlations in pp collisions at √s = 13 TeV (ref. 30), it has been shown that in pp collisions, the hadron–deuteron pairs follow the same transverse mass scaling as other hadron–hadron pairs, allowing to constrain the π–d emission source using the RSM. The deuterons are not produced directly by resonances. Nevertheless, the present work demonstrates that resonances decaying into nucleons are an important step in the production mechanism. This will lead to an effective delay in the deuteron production, an effect already described in a previous analysis of the K+–d analysis53. The present analysis adopts a conservative approach and integrates two extreme scenarios for the deuteron production as part of the systematic uncertainties, namely, assuming either that all deuterons are primordial or that the deuteron formation is delayed based on the amount of emission delay by which their constituent nucleons are affected44. This variation, which affects the effective source size, r eff of up to 0.08 fm, is included in the systematic uncertainties on the modelling of the correlation functions. The final values for the r eff , after the inclusion of resonances, are summarized in the Extended Data Table 1, along with the total uncertainties.
Corrections of the correlation function
The experimental correlation function, defined as \(C({k}^{* })=\)\({\mathcal{N}}\,[{N}_{{\rm{same}}}({k}^{* })/{N}_{{\rm{mixed}}}({k}^{* })]\) is only corrected by a normalization constant \({\mathcal{N}}\), by ensuring that the correlation becomes unity for k* ∈ (400, 600) MeV c−1. The remaining corrections are included in the fit function
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