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Three-baryon femtoscopy as an effective 3$\rightarrow$3 scattering experiment

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Pith's one-line read This paper reports the first measurement of the three-proton correlation function in high-energy proton-proton collisions and shows that it behaves as an effective three-to-three scattering experiment, giving direct access to the…

desk verdict First genuine p-p-p femtoscopy measurement with full three-body continuum calculations, carefully analyzed; main caveat is the unquantified rho0=2r0 source-size relation. read the letter →

arxiv 2608.05708 v1 pith:3S7VOLGV submitted 2026-08-06 nucl-ex hep-exnucl-th

ALICE Collaboration This is my paper · ORCID
classification nucl-exhep-exnucl-th
keywords femtoscopythree-protoncorrelationthree-bodyscatteringisospin-3/2three-nucleonsystempartial-wavedecompositiongrand-angularmomentumeffectivelong-rangeattractionproton-protoncollisions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Femtoscopy—measuring momentum correlations between particles emitted a few femtometers apart—is usually a two-body tool. This paper extends it to three protons produced in high-energy proton-proton collisions, isolating the genuine p-p-p correlation from feed-down and background. It then compares the measurement to full three-body continuum calculations and claims that the correlation function is sensitive to the partial-wave content of the nucleon-nucleon interaction: switching off the p-wave in the AV18 potential makes the theory miss the data by 22.2 standard deviations, while the full calculation agrees within 2.5 standard deviations. The paper also reports the first experimental sign of an effective long-range attraction in a three-proton continuum, tied to the large two-body scattering length. If right, the result makes three-hadron femtoscopy a new kind of scattering experiment for unbound and short-lived systems, extendable to hyperons.

What carries the argument

The load-bearing object is the three-body femtoscopic correlation function $C(Q_3)$, computed from the Koonin–Pratt formula in hyperspherical coordinates with the full three-proton scattering wave function built by hyperspherical adiabatic expansion. The calculation uses the AV18 nucleon-nucleon potential with screened Coulomb interaction, truncates the grand-angular-momentum expansion at $K_{\max}=7$, and fixes the Gaussian source size to $\rho_0 = 2 r_0$ with $r_0$ measured from the two-proton correlation. The analysis-side machinery is the data-driven decomposition of the measured correlation into genuine, feed-down, misidentified, and baseline components (Eq. 2), which lets the genuine $C_{ppp}(Q_3)$ be compared directly to theory; the $K$ truncation and the $\zeta/\rho^3$ asymptotic form are what carry the partial-wave-sensitivity and long-range-attraction conclusions.

What would settle it

Re-measure the p-p-p correlation in a collision system where the three-particle source can be determined without the $\rho_0=2r_0$ assumption—for instance, from three-particle correlations in a channel with no final-state interaction or from an independently modeled source—and check whether removing the p-wave still produces a 22.2σ disagreement when $\rho_0$ is varied over its systematic uncertainty.

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Extended reading notes

Core claim

The paper's central claim is that the measured genuine three-proton correlation function $C_{ppp}(Q_3)$, freed of feed-down and non-femtoscopic contributions, is reproduced by a full three-body scattering calculation only when the nuclear interaction is active in all two-body partial waves up through $d$-waves and the hyperspherical expansion runs to grand-angular momentum $K_{\max}=7$. Removing the $p$-wave interaction degrades the agreement to $22.2\sigma$, whereas the complete calculation matches the data at $2.5\sigma$. The correlation stays above unity as it approaches the uncorrelated limit, which the authors attribute to an effective long-range attraction $\sim \zeta/\rho^3$ with $\zeta$ proportional to the two-body scattering length. Together these observations establish the p-p-p continuum as the first directly measured isospin-3/2 three-nucleon system and three-baryon femtoscopy as an effective $3\to3$ scattering experiment.

Load-bearing premise

The three-proton source size is set to exactly twice the measured two-proton source radius ($\rho_0 = 2 r_0$); if that factor is wrong, the absolute scale of the predicted correlation shifts and the agreement, p-wave sensitivity, and long-range attraction signature could all change.

Editorial extensions

If this is right

  • Three-baryon femtoscopy becomes a working alternative to beam scattering for constraining the partial-wave content of baryon-baryon interactions, including channels that are experimentally inaccessible otherwise.
  • The same method applied to p-p-Λ, p-p-Σ, and p-p-Ξ systems could yield the first continuum three-body data in the strangeness sector, informing hyperon-nucleon interactions relevant to dense matter.
  • The measured long-range attraction provides an experimental anchor for the universal $\zeta/\rho^3$ dynamics used in calculations of three- and four-neutron continuum states.
  • The requirement $K_{\max}\ge7$ for convergence means three-body continuum observables carry information from high grand-angular-momentum components, so future models must retain them to match correlation data.
  • Because the p-wave sensitivity shows up as a 22.2σ effect, correlation functions can now discriminate between NN potentials that agree on two-body scattering observables but differ off-shell.

Reading between the lines

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

  • If the assumed $\rho_0=2r_0$ relation carries a hidden systematic error, the absolute scale of $C_{ppp}$ shifts, so the cleanest follow-up is a source-size determination internal to the three-body system; until then the quoted significances should be read with that caveat.
  • The 22.2σ p-wave sensitivity suggests the same data could rank NN potentials by how well they reproduce three-body correlations, effectively an off-shell benchmark beyond two-body data.
  • The long-range attraction should appear generically in other three-body systems with large two-body scattering lengths, so p-p-Λ femtoscopy offers a direct test of whether this signature persists when one baryon is strange.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper reports the first measurement of the three-proton correlation function C_ppp(Q3) in pp collisions at sqrt(s)=13.6 TeV with ALICE Run 3 data. The authors develop a data-driven decomposition (Eq. 2) with lambda parameters, feed-down constraints from a dedicated p-p-Lambda measurement, and a cumulant-expansion baseline cross-check, then compare the extracted genuine C_ppp with continuum three-body calculations using the AV18 potential and a screened Coulomb interaction. They find that the full model agrees with the data within n_sigma = 2.4-2.5, while removing the p-wave component from the nuclear interaction produces a 22.2 sigma disagreement. They further find that the nuclear interaction must be retained up to grand-angular momentum K=7 and that the correlation approaches unity from above at large Q3, which they interpret as the first experimental signature of an effective long-range attraction in a three-proton continuum system. The paper's central claim is framed as establishing three-baryon femtoscopy as an effective 3-to-3 scattering experiment.

Significance. Should the central claims hold, this is a substantial experimental advance: it extends femtoscopy from two-body to unbound three-baryon systems, provides a controlled benchmark for three-body continuum calculations in the isospin-3/2 channel, and offers a path toward p-p-Lambda, p-p-Sigma, and p-p-Xi measurements. The analysis has real strengths: the lambda parameters are data-driven, the feed-down is constrained by a dedicated p-p-Lambda measurement, the non-femtoscopic baseline is cross-checked by a cumulant expansion from the two-body measurement, and the theory comparison includes checks against the Norfolk potential and K_max truncation. The reporting of n_sigma with correlated systematics is useful. However, the quantitative claims (2.5 sigma agreement, 22.2 sigma p-wave sensitivity, and the long-range-attraction signature) depend on the assumed relation rho0 = 2r0 and on the self-consistency of the baseline extraction; these are the main points that need attention before publication.

major comments (3)
  1. [Sec. 2, source-size assumption rho0 = 2r0] The relation rho0 = 2r0 in Sec. 2 is treated as exact: the only propagated uncertainty is delta_r0 = 0.02 fm from the two-proton fit, giving rho0 = (2.54 +/- 0.04) fm. Because r0 is an effective radius that absorbs short-lived resonance feed-down and experimental selection effects, there is no reason for the three-proton effective radius to be exactly twice the two-proton one, and the factor 2 enters the absolute scale of every C_ppp(Q3) calculation used in Fig. 4. A 10-20% error in rho0 would rescale the correlation over the full Q3 range and would likely shift the quoted n_sigma values (full model about 2.5, no-p-wave 22.2) as well as the magnitude of the high-Q3 tail interpreted as long-range attraction. Please add a systematic estimate for the rho0-r0 relation (e.g., from alternative source parametrizations, event-generator templates, or a direct two-body-constrained three-body source model), or demonstrate explicitly that the conclusions are stable under such variations.
  2. [Sec. 3, Fig. 4 and Eq. (2)] The 22.2 sigma no-p-wave significance in Sec. 3 (Fig. 4, left) is obtained by comparing the 'genuine' C_ppp, which is extracted using the full-model baseline and feed-down decomposition of Eq. (2), with a calculation in which the p-wave is switched off. Because the baseline B(Q3) is fitted to the measured spectrum using the full model, the extracted C_ppp is not fully model-independent, and part of the disagreement may reflect that the baseline has been optimized to the full model. I request a self-consistent test: re-fit the baseline (and, where applicable, rho0 and the lambda parameters) under the no-p-wave hypothesis and recompute the n_sigma, or otherwise show that the no-p-wave discrepancy survives with a baseline determined without using C_ppp.
  3. [Sec. 3, Fig. 2 inset and Fig. 4] The claim of an effective long-range attraction rests on the genuine C_ppp remaining above unity as Q3 approaches 1 GeV/c (Sec. 3 and Fig. 4). The baseline B(Q3) = a + b Q3^2 + c Q3^3 is the only quantity fitted in the three-body case, and its uncertainty is not propagated into the high-Q3 tail in an explicit way; the inset of Fig. 2 shows agreement with the cumulant-based baseline only qualitatively. Please quantify the stability of the extracted C_ppp tail under (i) alternative baseline functional forms, (ii) variations of the fit range, and (iii) the minimum-bias-sample boundary near Q3 about 1.1-1.4 GeV/c, and report binned residuals of the cumulant-based baseline comparison.
minor comments (5)
  1. [Fig. 4, bottom panels] The tick labels in the bottom panels appear as '10-0' and '20-10-0'; this is likely a rendering or formatting error and should be corrected.
  2. [Sec. 2 and Sec. 3] The text quotes n_sigma = 2.4 for the full model in Sec. 2 and n_sigma = 2.5 for the genuine C_ppp comparison in Sec. 3; please clarify explicitly which data/model comparison each value refers to.
  3. [Fig. 3] The figure is informative but dense; please define K, l_x, l_y, and the relation K = l_x + l_y + 2n in the caption itself rather than only in the main text.
  4. [Appendix A.5, Eq. (A.2)] Please state in one place the sign convention for zeta (units of length and whether zeta < 0 corresponds to attraction), since the text and the f(rho) plots use the same quantity with opposite presentations.
  5. [Appendix A.3, Eq. (A.1)] The symbols P_i and F_i used in the definition of lambda_ij are introduced only in Appendix A.2; please define all symbols at first use in Eq. (A.1) or add a brief reminder.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the p-p-p comparison is driven by an external two-body potential and an independently calibrated source; no fitted quantity is relabeled as a prediction.

full rationale

The derivation chain is self-contained and non-circular. The only quantity fitted to the three-body data is the non-femtoscopic baseline B(Q3)=a+bQ3^2+cQ3^3, and the paper explicitly cross-validates it against the cumulant expansion built from the two-body measurement: 'Since the baseline is the only quantity fitted in the three-body case, and its behavior is independently confirmed in a data-driven way by the two-body measurement, the remaining agreement between model and data constitutes a direct comparison to the interaction models.' The genuine C_ppp is then compared with calculations from Ref. [47] using the AV18 potential, which is fixed by external two-nucleon scattering data, not by the p-p-p correlation function. The three-body source radius is calibrated from the measured p-p correlation within triplets via the stated relation rho0=2r0; this is an input assumption rather than a predicted output, and while the factor-of-two carries no quoted systematic uncertainty, that is a robustness gap, not circularity. No parameter is fitted to C_ppp itself, and the partial-wave sensitivity claim (22.2 sigma without p-wave versus 2.5 sigma full) is a comparison of two fixed theory variants to the same residual, so the result is not forced by construction or by a self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central comparison rests on a Gaussian three-body source whose size is fixed by the two-proton source radius through rho0 = 2 r0, on a fitted polynomial baseline, and on theoretical approximations (screened Coulomb, no three-nucleon force, Kmax = 7 truncation). These are calibration and modeling choices rather than free parameters tuned to the three-body data.

free parameters (4)
  • effective two-proton source radius r0 = 1.27 ± 0.02 fm
    Extracted from a fit to the p-p correlation function measured within p-p-p triplets (Section 2, Fig. 1). Used to set the three-body source size rho0 = 2 r0.
  • three-body source radius rho0 = 2.54 ± 0.04 fm
    Derived from r0 via rho0 = 2 r0 [45]; enters the computation of C_ppp(Q3) (Section 2).
  • two-body baseline coefficients a,b,c = not quoted
    Coefficients of B(k*) = a + b k*^2 + c k*^3 fitted simultaneously with r0 to the p-p correlation function (Appendix A.3).
  • three-body baseline coefficients a,b,c = not quoted
    Coefficients of B(Q3) = a + b Q3^2 + c Q3^3 fitted to the measured p-p-p correlation function in [0,1.6] GeV/c (Section 2).
assumptions (5)
  • domain assumption The three-proton source function is Gaussian with rho0 = 2 r0, where r0 is the two-proton effective source radius.
    Section 2: 'The following assumption links the three- and two-particle source descriptions... rho0 = 2r0 [45]'. This links the measured p-p source to the p-p-p source and is load-bearing for the absolute scale of C_ppp.
  • domain assumption Independent emission of the three protons, so the three-body source factorizes into a Gaussian in hyperradius rho.
    Section 2, Eq. (1) and text: S(rho,Omega) = 1/(pi^3 rho0^6) exp(-rho^2/rho0^2). Assumed independent emission.
  • domain assumption The three-nucleon force is negligible in the p-p-p system (effect below 0.1%).
    Section 2: 'The three-nucleon force is not considered, since in the p-p-p system it is suppressed by Pauli blocking...' This relies on Refs [45,47].
  • ad hoc to paper Coulomb interaction in the p-p-p system can be treated with a screened Coulomb potential; no exact three-body Coulomb solution exists.
    Section 2 and Appendix A.4: 'the Coulomb interaction in the p-p-p system is treated approximately' via the screened Coulomb formalism of Refs [45,61]. The paper acknowledges this as an approximation.
  • domain assumption The non-femtoscopic baseline B(Q3) is a third-order polynomial without a linear term.
    Section 2: B(Q3) = a + b Q3^2 + c Q3^3, fitted in [0,1.6] GeV/c. This functional form is an assumption; the paper cross-checks it with a cumulant expansion from the two-body baseline.

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Pith. "Pith review of Three-baryon femtoscopy as an effective 3$\rightarrow$3 scattering experiment." pith.science (2026). https://pith.science/paper/3S7VOLGV

@misc{pith2026260805708,
  author       = {Pith},
  title        = {Pith review of: Three-baryon femtoscopy as an effective 3$\rightarrow$3 scattering experiment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3S7VOLGV}},
  note         = {Machine review of arXiv:2608.05708}
}
abstract

Scattering experiments have long been the gold standard for constraining hadron$-$hadron interactions, providing direct information on the angular momentum and spin dependence over a wide range of kinematic configurations. However, experimental constraints on three-body dynamics remain limited, specifically for unbound systems and systems involving short-lived hadrons. In this work, the three-proton correlation function is measured in pp collisions at $\sqrt{s}=13.6$ TeV with ALICE at the LHC and presented as a novel approach to access hadronic interactions in three-body systems. A new analysis strategy is employed to isolate the p$-$p$-$p contribution to the correlation function by correcting for background channels and experimental effects, and enabling a direct comparison with state-of-the-art three-body continuum calculations. The extracted correlation function provides the first direct access to the isospin $3/2$ three-body system. The measured observable is found to be sensitive to the partial-wave structure of the nucleon$-$nucleon interaction and indicates that the nuclear interaction acts even at high angular momentum and parity states of the three-body system, revealing an effective long-range attractive component, observed experimentally for the first time in a three-proton continuum system. Hence, three-hadron femtoscopy emerges as an effective 3$\rightarrow$3 scattering experiment with three unbound hadrons in initial and final states. The copious production of hyperons at the modern high-energy colliders ensures the possibility of extending such measurements beyond nucleons, opening a new avenue for future precision studies of three-body dynamics in the strangeness sector.

Figures

Figures reproduced from arXiv: 2608.05708 by the authors.

Figure 1
Figure 1. The p–p correlation function for proton pairs within p–p–p triplets measured in pp collisions at √ s = 13.6 TeV is presented in black markers. The vertical lines represent the statistical uncertainty, while the shaded boxes correspond to the systematic uncertainty. The correlation function is normalized to unity in the range k ∗ ∈ [150,200] MeV/c. The red band corresponds to the correlation function fit as described… view at source ↗
Figure 2
Figure 2. The p–p–p correlation function measured in pp collisions at √ s = 13.6 TeV, normalized to unity in the range Q3 ∈ [1.0,1.2] GeV/c. The vertical bars indicate the statistical uncertainties, while the shaded boxes represent the systematic ones. The magenta band represents the genuine p–p–p contribution, and the other colored bands correspond to the different components used in the modeling of the correlation function … view at source ↗
Figure 3
Figure 3. Possible configurations of protons in two- and three-body systems shown for correspondence to their quantum numbers, the angular momentum, l, for the two-body system and the grand-angular momentum, K, for the three-body system. The K = 0 configuration is forbidden for three protons due to antisymmetrization arguments. The green color here encodes particles in l = 0, red in l = 1 and blue in l = 2 states. orbital ang… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Genuine p–p–p correlation function with statistical (bars) and systematic uncertainties (boxes). Left: The magenta and green bands depict the calculations from Ref. [47] obtained by using the AV18 nuclear potential with and without including the p-wave component of the…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.