{"id":"c7e3922b-1bff-4e14-b9d1-d8c5f75f6686","arxiv_id":"2505.22190","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The ppp correlation function converges only when the interacting wave function includes all channels with grand angular momentum up to K=7, and with that the calculation matches ALICE data.","lead":"This paper tests how many quantum states are needed to compute the three-proton correlation function measured at the LHC, and finds that several more than earlier work used are required for the high-momentum tail. With those states included, the theoretical calculation matches the ALICE data, supporting three-body femtoscopy as a tool for studying proton interactions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The K=7 sufficiency claim is extrapolated from cheap high-J channels; no K0=8/9 calculation is shown for the intermediate J^pi states that dominate the correlation function, so the central convergence claim is not yet demonstrated.","rationale":"The reader's verdict is CONDITIONAL, and the conditional status is appropriate. My stress-test pass converges on the same main vulnerability: the K=7 convergence claim is supported by explicit K0=8/9 checks only for high-J channels, while intermediate-J channels are extrapolated. I therefore agree with the reader's primary weakest-assumption statement. I do not elevate the hyperangular-averaged Coulomb approximation to the same level of concern, because the convergence claim concerns the K-truncation of the strong interaction; the Coulomb approximation is a systematic uncertainty that affects all K0 values and would need a separate benchmark, not a targeted check of the K=7 sufficiency claim. The abstract's statement about J^pi=21/2^- is internally inconsistent with the body's K=7 conclusion, but it is a presentation error rather than a flaw in the numerical argument. Given that the central computational claim is not yet fully demonstrated for intermediate channels, the conditional verdict should stand. If the proposed K0=8 test for 5/2^+ shows a negligible shift, the verdict could move to ACCEPT; if it shows a non-negligible shift, the conclusion would need to be weakened or revised.","tokens_in":10957,"tokens_out":7177,"duration_ms":78006,"concrete_test":"Compute the ppp correlation function with K0=8 for J^pi=5/2^+ using the same AV18 potential, K_max=130 and source parameters as in Fig. 3, and compare the resulting total C_ppp(Q3) with the K0=7 result at Q3=0.3, 0.5 and 0.7 GeV/c. If the shift is smaller than the width of the quoted theory band at those points, the K=7 convergence claim is confirmed. If the shift exceeds roughly one standard deviation of the data or 1-2% of C_ppp, the conclusion that K=7 is sufficient would need to be revised. A complementary run for 5/2^- with K0=9 would test the negative-parity side of the same extrapolation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that channels with K<=7 capture the entire strong-interaction part of the ppp wave function and that K>7 can be treated as free. The paper does not actually compute K0=8 or K0=9 for most of the intermediate partial waves that contribute most to C_ppp. Checks are limited to high-J states where the calculation is cheap: for 17/2^-, 19/2^- and 21/2^- the K0=9 correction is ~10^-4, and for 15/2^+, 17/2^+ and 19/2^+ the K0=8 correction is at most ~10^-3. From this the paper concludes in Section 4 that 'increasing K0 up to 8 would give rise to a negligible contribution' and that K0=8 is 'not worth it.' But the positive-parity intermediate states 3/2^+, 5/2^+ and 7/2^+ (31 channels at K0=8) and the negative-parity states 5/2^-, 7/2^- and 9/2^- (39-41 channels at K0=9) are precisely the ones for which no K0=8/9 result is presented. The convergence of these channels is inferred from the trend seen in lower-K0 calculations and from the behavior of high-J states, not computed. This is load-bearing because the agreement with ALICE data in Fig. 3 and the statement that 'above K=7 the wave function can be considered as free' depend on the K0 truncation. If any of the omitted intermediate channels develops a K=8 or K=9 correction of order a few percent in the tail region Q3>0.3 GeV/c, the central convergence conclusion and the quoted 2-sigma agreement would both be affected. The abstract/body J^pi=21/2^- mismatch is a wording issue, not the main computational vulnerability.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript analyzes the convergence of the three-proton correlation function computed in the hyperspherical adiabatic basis. The authors extend a previous calculation that used K0=2 to include grand-angular quantum numbers up to K0=7, adding channels for the heavier negative- and positive-parity states. They argue that contributions from K>7 are negligible, based on checks for selected high-J states, and that with K0=7 the computed C_ppp agrees with ALICE data within about 2 sigma after including feed-down and source-size uncertainties. The work is presented as the first systematic study of the convergence of the interacting three-body wave function with relative energy for this observable.","tokens_in":11304,"tokens_out":5908,"duration_ms":58470,"significance":"If the K0=7 convergence claim is accepted, the paper provides an important practical result: realistic three-body femtoscopy calculations must include partial waves up to K=7, not just the low-K states that dominate the correlation peak. The calculation uses the realistic Argonne v18 potential, a Gaussian source whose radius is fixed by a resonance source model, and ALICE data as an external benchmark; no parameters are fitted to the ppp correlation function itself. The detailed channel counts in Table 1 are a useful resource for future three-body femtoscopy calculations. The main caveat is that convergence is not directly demonstrated for the intermediate partial waves that dominate the correlation function, so the central claim rests partly on extrapolation rather than on explicit computation at the next cutoff.","major_comments":[{"comment":"The abstract states that 'it is necessary to consider three-body states up to J^pi=21/2^-', but Table 1 shows that K=7, the value concluded in Section 5, supports at most J^pi=17/2^- for negative parity. The 19/2^- and 21/2^- channels appear only at K=9, and the paper's own K0=9 checks show their interacting contribution is negligible. The abstract should be corrected to 'K up to 7 (i.e., J^pi up to 17/2^-)' or the wording should be changed to clarify that higher J states are checked only to establish that they can be treated as free.","section":"Abstract; Section 5"},{"comment":"The statement that 'increasing K0 up to 8 would give rise to a negligible contribution' is an extrapolation. The paper reports K0=8/9 checks only for the high-J states 15/2+,17/2+,19/2+ and 17/2-,19/2-,21/2-, whose centrifugal barriers make them cheap. No K0=8 result is shown for the positive-parity intermediate states 3/2+,5/2+,7/2+ (which have up to 31 channels at K0=8), and no K0=9 result is shown for the negative-parity states 5/2-,7/2-,9/2- (39-41 channels at K0=9). Since these intermediate states are the ones that dominate the correlation function, the central conclusion that 'above K=7 the wave function can be considered as free' is not directly demonstrated. A calculation of at least one of these channels at the next K0, or a quantitative bound on its contribution, is needed to support the claimed convergence.","section":"Section 4"},{"comment":"The hyperangular-averaged Coulomb potential V_Coul(rho)=16 sqrt(2/pi) e^2/rho is introduced as an alternative to screening, but no benchmark or error estimate is given for its accuracy in the tail region (Q3 > 0.3 GeV/c). The quoted agreement with ALICE data depends on this approximation, so the paper should either cite a validation study or quantify the systematic uncertainty of the averaging procedure.","section":"Section 2; Eq. (7)"}],"minor_comments":[{"comment":"The caption contains a stray word 'Fabbiettithe' in the last sentence ('Fabbiettithe expansion given in Eq. (3)'), which appears to be a typesetting artifact and should be removed.","section":"Table 1 caption"},{"comment":"There is a typo 'Futhermore' in the paragraph after Fig. 3; it should be 'Furthermore'.","section":"Section 4"},{"comment":"The statement 'we can foresee that increasing K0 up to 8 would give rise to a negligible contribution' is ambiguous because K0=8 adds even-K channels only; the statement should specify that this applies to positive parity and that negative parity requires K0=9.","section":"Section 4"},{"comment":"The bottom panel label 'n σ' is not defined in the caption; the text explains it, but the figure would be self-contained if the caption defined it as 'number of standard deviations'.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The abstract-body mismatch on J^pi=21/2^- should be fixed before acceptance. The convergence extrapolation for the intermediate channels is the main technical risk; if the authors can compute one representative heavy intermediate channel at K0=8/9 or provide a quantitative bound, the paper's central claim would be much stronger. The paper is within scope for a nuclear physics journal focused on few-body systems and femtoscopy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper before reading it. First, the numerical result is real and useful: pushing the grand-angular cutoff from K0=2 to K0=7 in the hyperspherical adiabatic calculation lifts the ppp correlation tail into agreement with ALICE data, and it demonstrates that three-body femtoscopy needs many more partial waves than two-body intuition suggests. Second, the abstract overstates the requirement as J^pi up to 21/2^-, while the body's own Table 1 and conclusion say K up to 7, which for negative parity is J^pi up to 17/2^-. That mismatch is worth fixing but not fatal.\n\nWhat the paper does well: it carefully separates the computed (interacting) part of the wave function from the free part, gives a clean counting of adiabatic channels for each J^pi and K0, and shows quantitatively why K0=2, the value used in the earlier Ref. [7], was not enough for the tail. The comparison with ALICE data includes feed-down from ppLambda and the source-size uncertainty, and the lower panel of Fig. 3 showing deviations in sigma is a nice touch. The conclusion that 'above K=7 the wave function can be considered free from the strong interaction' is the kind of statement that is actually actionable for future three-body femtoscopy calculations.\n\nThe soft spot is the convergence evidence for the intermediate partial waves. The paper states that K0=8/9 calculations are too expensive for states like 5/2^+, 7/2^+, 5/2^-, 7/2^-, and 9/2^-, and that convergence is inferred from the trend at lower K0 and from the negligible corrections in high-J states. That is a reasonable inference, but it is not a computation. If one of those intermediate channels develops a K=8 or K=9 correction of order a few percent in the tail, the central convergence claim and the quoted 2-sigma agreement would both be affected. The paper would be stronger with at least one representative intermediate channel computed at K0=8 or K0=9, or with a quantitative bound on the missing contribution. The Coulomb treatment is also an approximation (hyperangular-averaged potential), but that is standard in this line of work and not a new concern.\n\nOverall, this deserves a serious referee. The main numerical result is likely right, the presentation is clear, and the discrepancy it resolves is real. I would recommend sending it to peer review, with a request to correct the abstract and to show explicit convergence checks (or bounds) for the intermediate partial waves. I would not cite it in my own work until the abstract/body inconsistency is cleaned up and the convergence question for the omitted channels is addressed.\n\nRead this at your reading group if you want a concrete example of why three-body femtoscopy is harder than two-body.","headline":"Solid numerical convergence study that fixes a real discrepancy with ALICE data, but the abstract overstates the J requirement and the K=7 sufficiency claim is partly extrapolated rather than fully computed.","tokens_in":11917,"tokens_out":2894,"would_cite":false,"duration_ms":27958,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that the three-proton correlation function converges in the hyperspherical adiabatic basis only when the interacting part of the wave function includes grand angular momentum up to $K=7$, and that with this cutoff…","keywords":["ppp correlation function","femtoscopic correlation function","hyperspherical adiabatic expansion","grand angular momentum","three-body scattering","Coulomb interaction","nuclear femtoscopy"],"falsifier":"Compute the $K_0=8$ radial equations for the $5/2^+$ and $7/2^+$ states and compare their interacting contribution to the free-wave contribution over $Q_3=0.3$--$0.8$ GeV/c: a difference above about $10^{-3}$ anywhere in that range would show that the $K=7$ cutoff is not sufficient. Alternatively, replace the hyperangular-averaged Coulomb potential with a numerically exact three-body Coulomb asymptotic treatment and check whether the $2\\sigma$ agreement survives.","tokens_in":10703,"feed_emoji":"⚛️","tokens_out":16077,"duration_ms":149415,"temperature":0.7,"pith_summary":"The paper establishes a convergence threshold for the three-proton correlation function in the hyperspherical adiabatic basis. The interacting part of the three-proton scattering wave function must include channels up to grand angular momentum $K=7$; once those channels are solved dynamically, channels with $K>7$ can be treated as free waves. Only with this cutoff does the computed $ppp$ correlation function reproduce the measured high-momentum tail, approaching unity from above and staying within about two standard deviations of the data. The earlier cutoff $K_0=2$ was sufficient only near the low-energy peak, which explains the previous theory-data discrepancy in the tail. The test matters because three-proton scattering cannot be measured directly, so femtoscopic correlation functions are the only window on this process.","feed_headline":"K=7 is the cutoff that fixes the ppp correlation tail","feed_subtitle":"With up to 23 coupled channels, the calculated ppp tail matches the measured one within about two standard deviations.","key_machinery":"The carrying mechanism is the two-level truncation of the hyperspherical adiabatic expansion. At fixed hyperradius $\\rho$, the adiabatic basis functions $\\Phi^{JM}_n(\\rho,\\Omega_\\rho)$ are eigenfunctions of the hyperangular part of the three-body Schrödinger equation, labeled at $\\rho=0$ by the grand angular momentum $K$, which acts as the three-body analogue of the partial wave and controls the centrifugal barrier. The scattering wave function is split as $\\Psi_s=\\Psi^{\\rm comp}_s+\\Psi^{\\rm free}_s$: for channels with $K\\le K_0$, the radial functions are computed from a coupled set of differential equations with the two-body potential; for $K>K_0$, they are replaced by the analytic free-wave form. The Coulomb interaction among three protons is handled by hyperangular averaging, $V_{\\rm Coul}(\\rho)=16\\sqrt{2/\\pi}\\,e^2/\\rho$, which turns the asymptotic Bessel functions into regular and irregular Coulomb functions with Sommerfeld parameter $\\eta=16me^2/(\\pi\\hbar^2 Q)$. The same basis is converged at the angular level ($K_{\\max}=130$) and at the channel level ($K_0=7$), and this two-level convergence is what keeps the number of coupled equations manageable.","core_discovery":"The claim, stated in the conclusions, is that grand angular quantum numbers up to $K=7$ are required to describe the interacting part of the three-proton wave function, and above $K=7$ the wave function can be considered free from the strong interaction. In practice this means solving the coupled radial equations for all adiabatic channels with $K\\le 7$, including up to 23 coupled channels for the $5/2^-$ and $7/2^-$ states, while including the remaining channels analytically as free waves with Coulomb functions. The heavy channels that could be tested with $K_0=8$ or $9$, such as $19/2^-$ and $21/2^-$, contribute at the level of $10^{-4}$, and interpolation leads the authors to expect the untested $5/2^+$ and $7/2^+$ $K_0=8$ channels to be equally negligible. With $K_0=7$ the correlation function converges over $Q_3$ up to about 0.8 GeV/c, reproduces the data within about $2\\sigma$, and matches the experimental behavior of approaching 1 from above.","pith_inferences":["If the $K=7$ threshold holds, the strong-interaction content of the three-proton wave function is negligible beyond $K=7$, so the correlation tail at $Q_3$ above about 0.8 GeV/c is essentially a Coulomb-plus-source effect; this could be tested by comparing the calculation with a pure Coulomb version at high momentum.","The paper's interpolation that $K_0=8$ for the $5/2^+$ and $7/2^+$ states is negligible is the part of the convergence claim not yet computed; carrying out those two calculations would turn the claim from an extrapolation into a direct proof.","The hyperangular-averaged Coulomb potential is the main model commitment; connecting the calculation to an exact three-body Coulomb asymptotic form would say whether the $2\\sigma$ agreement reflects the nuclear dynamics or a deliberate smoothing of the three-body Coulomb tail.","The same $K_0$-convergence analysis, applied to different source sizes $\\rho_0$, could sharpen the extraction of the proton source radius from the tail of the correlation function, where the $K_0=7$ curves separate more strongly between $\\rho_0$ values than at the peak."],"forward_implications":["The cutoff $K_0=2$ used previously reproduces only the low-energy peak; the tail above $Q_3\\approx 0.2$ GeV/c requires the full set of channels up to $K=7$.","With $K_0=7$, the computed correlation function approaches unity from above at large $Q_3$, matching the measured behavior; the old $K_0=2$ result approached from below.","Including feed-down from $\\Lambda$ decay and the source-radius uncertainty, the $K_0=7$ curve stays within about $2\\sigma$ of the data for $Q_3>0.3$ GeV/c.","The cost remains tolerable: the largest systems at $K_0=7$ involve 23 coupled channels, whereas a direct hyperspherical-harmonic expansion would require thousands of coupled equations.","The convergence protocol transfers to other three-hadron correlation functions, such as $pp\\Lambda$, where three-body scattering is not directly measurable."],"supporting_citations":[{"why":"The preceding calculation of the ppp correlation function with the same method; it defined the K0=2 baseline and the Coulomb-averaging prescription.","marker":"[7]"},{"why":"The measured ppp correlation function used for comparison and for the feed-down uncertainty band.","marker":"[6]"},{"why":"The hyperspherical adiabatic expansion method that defines the basis used to build the scattering wave function.","marker":"[13]"},{"why":"The derivation of the three-body scattering wave function form and its asymptotic normalization in Eqs. (4)-(6).","marker":"[15]"},{"why":"The realistic two-nucleon potential used as the strong interaction in the coupled-channel equations.","marker":"[17]"},{"why":"The Koonin-Pratt correlation formula at the two-body level, generalized here to three particles.","marker":"[10]"},{"why":"The three-particle version of the Koonin-Pratt formula and the source-function averaging used in Eq. (1).","marker":"[11]"},{"why":"Establishes the conversion between the three-body momentum Q used in the paper and the Lorentz-invariant Q3 of the data.","marker":"[12]"}],"fun_headline_variants":["K=7 cutoff resolves ppp correlation tail","23 coupled channels tame ppp correlation tail","ppp correlation converges for K≤7 up to 0.8 GeV/c","Hyperspherical K=7 basis reproduces ppp data within 2σ","No strong interaction above K=7 in ppp system"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion depends on the assumption that the few channels too expensive to compute at the next cutoff really do contribute nothing visible, and that the angle-averaged Coulomb force used here is a fair stand-in for the true three-proton repulsion.","fun_headline_variants_meta":{"raw":{"variants":["K=7 cutoff resolves ppp correlation tail","23 coupled channels tame ppp correlation tail","ppp correlation converges for K≤7 up to 0.8 GeV/c","Hyperspherical K=7 basis reproduces ppp data within 2σ","No strong interaction above K=7 in ppp system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":3078,"prompt_tokens":1033,"completion_tokens":2045,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":649,"completion_tokens_details":{"reasoning_tokens":1958}},"tokens_in":649,"tokens_out":2045,"duration_ms":15447,"temperature":1.0,"reasoning_tokens":1958,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:13:12.067915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $K_0=8$ radial equations for the $5/2^+$ and $7/2^+$ states and compare their interacting contribution to the free-wave contribution over $Q_3=0.3$--$0.8$ GeV/c: a difference above about $10^{-3}$ anywhere in that range would show that the $K=7$ cutoff is not sufficient. Alternatively, replace the hyperangular-averaged Coulomb potential with a numerically exact three-body Coulomb asymptotic treatment and check whether the $2\\sigma$ agreement survives.","supporting_citations":[{"cited_title":"Kievsky, E","cited_arxiv_id":null,"evidence_quote":"The preceding calculation of the ppp correlation function with the same method; it defined the K0=2 baseline and the Coulomb-averaging prescription."},{"cited_title":"Acharyaet al.(ALICE Collaboration), Eur","cited_arxiv_id":null,"evidence_quote":"The measured ppp correlation function used for comparison and for the feed-down uncertainty band."},{"cited_title":"Nielsen, D.V","cited_arxiv_id":null,"evidence_quote":"The hyperspherical adiabatic expansion method that defines the basis used to build the scattering wave function."},{"cited_title":"Garrido, A","cited_arxiv_id":null,"evidence_quote":"The derivation of the three-body scattering wave function form and its asymptotic normalization in Eqs. (4)-(6)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The realistic two-nucleon potential used as the strong interaction in the coupled-channel equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Koonin-Pratt correlation formula at the two-body level, generalized here to three particles."},{"cited_title":"Pratt, T","cited_arxiv_id":null,"evidence_quote":"The three-particle version of the Koonin-Pratt formula and the source-function averaging used in Eq. (1)."},{"cited_title":"Del Grande, L","cited_arxiv_id":null,"evidence_quote":"Establishes the conversion between the three-body momentum Q used in the paper and the Lorentz-invariant Q3 of the data."}],"review_version":1}