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New multinucleon knockout model in NuWro Monte Carlo generator

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A two-parameter sampling function for outgoing nucleon pairs lets NuWro reproduce the 2020 Valencia model's momentum correlations after final-state interactions.

desk verdict Solid NuWro MEC update with genuinely new hadronic sampling, but the validation is partly circular and the post-FSI Valencia curve is a reweighted proxy—still worth refereeing. read the letter →

arxiv 2411.11523 v1 pith:OFOYBWBL submitted 2024-11-18 hep-ph hep-exnucl-th

Hemant Prasad , Jan T. Sobczyk , Artur M. Ankowski , J. Luis Bonilla , Rwik Dharmapal Banerjee , Krzysztof M. Graczyk , Beata E. Kowal This is my paper · ORCID
classification hep-phhep-exnucl-th
keywords neutrino-nucleusinteractionsmultinucleonknockout2p2hand3p3hmechanismsmesonexchangecurrentsNuWroMonteCarlogeneratorValenciamodelnucleonmomentumcorrelationsfinal-state
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

Neutrino oscillation experiments infer neutrino energy from Monte Carlo generators, so the way a generator distributes the nucleons knocked out in multinucleon (np-nh) events matters. This paper tries to establish that NuWro, with a two-parameter nucleon sampling function fitted to the 2020 Valencia model, can reproduce that model's correlations in outgoing nucleon momenta closely enough that, after final-state interactions, the leading-proton momentum distribution agrees within FSI uncertainties. The paper also adds a 3p3h component and shows that the choice of hadronic model changes hadronic observables such as the transverse momentum imbalance by tens of percent, affecting comparisons with experiment. The point is that a cheap, tunable hadronic model can stand in for a computationally expensive exact calculation.

What carries the argument

The load-bearing element is the two-parameter nucleon sampling function $f(x)$ with $x=\cos\theta^*$, defined in Appendix B, where $P\in[-1,1]$ sets whether nucleons prefer parallel ($P>0$) or perpendicular ($P<0$) emission relative to the boost direction and $l$ (an integer) sets the sharpness of that preference; $P=0$ gives a flat distribution. The function is evaluated on the interval $[-\kappa,\kappa]$ fixed by a Pauli-blocking condition that keeps the backward nucleon's lab energy above $m_b+E_F$. The parameters are tuned by a reweighting technique that compares NuWro's pp and np+pn nucleon phase spaces with the 2020 Valencia model's binned two-dimensional distributions through the $\tilde\chi^2$ of Eq. (16), minimizing the sum over five neutrino energies.

What would settle it

Generate 2p2h events with the 2020 Valencia code and with NuWro's new hadronic model at neutrino energies outside the calibration set (for example, 1.5, 3, or 10 GeV) and on a target such as argon rather than carbon; if the two-dimensional pp or np+pn momentum distributions disagree by more than the FSI uncertainty band after the cascade, the universal sampling assumption is falsified.

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Extended reading notes

Core claim

The central claim is that correlations in the momenta of outgoing nucleons from 2p2h meson-exchange-current events can be captured by a normalized sampling function $f(\cos\theta^*)$ in the hadronic center-of-mass frame, with one direction parameter $P$ and one strength parameter $l$ for each final-state pair (pp, np, pn). Fitting these parameters to the two-dimensional nucleon phase space of the 2020 Valencia model at five neutrino energies gives $(P,l)_{pp}=(0.77,4)$, $(P,l)_{np}=(0.7,3)$, and $(P,l)_{pn}=(0.8,4)$, all with positive $P$, which pushes the pair toward forward/backward configurations and creates the asymmetric leading/subleading proton pattern of the Valencia model. With this hadronic model inside NuWro, the leading-proton momentum distribution after the intranuclear cascade lies inside the FSI uncertainty band of the Valencia model, whereas the old hadronic model lies outside it. The authors therefore claim that the approximation captures the most important features of the exact exclusive model.

Load-bearing premise

The assumption that a single, energy-independent two-parameter sampling function $f(\cos\theta^*)$ suffices for each final-state pair (pp, np, pn) is load-bearing; if nucleon correlations depend on energy and momentum transfer more strongly than this, the calibrated model will misrepresent events at uncalibrated energies or in other nuclei.

Editorial extensions

If this is right

  • The 3p3h mechanism contributes about 20% of the total MEC cross section at $E_\nu \gtrsim 1.1$ GeV, so generators that omit it understate multinucleon knockout strength.
  • After FSI, the new NuWro hadronic model and the 2020 Valencia model give leading-proton momentum distributions consistent within FSI uncertainties, while the old hadronic model differs significantly.
  • In MINERvA CC1p0π observables, the new model suppresses the large-$|p_n|$ tail of reconstructed neutron momentum by roughly 40% in the MEC channel and suppresses large $\delta p_T$ and $\delta\alpha_T$ tails by 20–40%, improving $\chi^2$ for most observables but worsening $d\sigma/d|p_n|$.
  • The procedure is not tied to NuWro: the same sampling function and calibration approach can be adopted by other neutrino event generators.

Reading between the lines

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

  • If the universal, energy-independent sampling functions are not truly universal, disagreement should appear at neutrino energies outside {0.5, 0.7, 1, 2, 5} GeV or for nuclei other than carbon; this can be checked directly against the 2020 Valencia code.
  • The fit's failure to improve by splitting the $(\omega,|\mathbf{q}|)$ plane suggests that the two-peak structure (NΔ versus ΔΔ) produces similar angular correlations in both regions; a direct test would fit separate parameters in the two kinematic regions against exclusive proton data.
  • Since FSI masks much of the remaining discrepancy, any future reduction of FSI uncertainties would force the hadronic sampling function to be more exact, likely requiring kinematics-dependent parameters.
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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

4 major / 6 minor

Summary. This paper describes a new hadronic model for the multinucleon (np-nh) contribution in the NuWro event generator. The inclusive part uses the 2020 Valencia model tables; for the 2p2h component the authors introduce a two-parameter sampling function f(cos theta*) for each final-state isospin pair (pp, np, pn), with six parameters globally fitted to reproduce the Valencia two-dimensional outgoing-nucleon momentum distributions at five neutrino energies (0.5-5 GeV), and they add a 3p3h component with combinatorially assigned isospin and three-body phase-space kinematics. The paper compares the resulting pre-FSI phase spaces with the Valencia model and with the old NuWro hadronic model (Figs. 3-5), estimates the post-FSI leading-proton distribution using a reweighting of NuWro events in two-dimensional momentum bins (Fig. 6), and benchmarks the implementation against MINERvA CC1p0pi data (Figs. 7-9, Tables I-II). The central claim is that the new hadronic model approximates the 2020 Valencia correlations well enough that, after FSI, the leading-proton momentum distribution agrees with the Valencia model within FSI-model uncertainties, and that this is an improvement over the previous NuWro hadronic treatment.

Significance. If the central claim is validated, the paper provides a computationally efficient recipe for implementing Valencia-style nucleon correlations in event generators, and its demonstration that multi-nucleon final-state modeling changes hadronic observables such as delta-alpha_T, delta-p_T, and reconstructed neutron momentum is a useful result for the neutrino-oscillation program. The manuscript's strengths are its explicit algorithmic specification (Sect. III B, Appendices A and B), the analytic derivation of the dynamic Pauli-blocking bound in Appendix A, and the quantitative MINERvA benchmarks (Tables I and II), which are external to the fit and thus provide genuinely out-of-sample evidence. The main limitations are that the six fitted parameters carry no reported uncertainties and that the pre-FSI comparisons in Figs. 3-5 are in-sample, leaving the post-FSI reweighted comparison (Fig. 6) and the MINERvA comparison as the independent tests; the former has a methodology caveat discussed in Major Comment 1.

major comments (4)
  1. [Sec. IV C, Fig. 6] The central claim that the new hadronic model reproduces the 2020 Valencia leading-proton distribution after FSI within FSI uncertainties is established in Fig. 6 by a reweighting proxy, not by transporting Valencia events through the cascade. The '2020 Valencia model' curve is obtained by rescaling NuWro event weights by the ratio of Valencia-to-NuWro counts in each two-dimensional (leading, subleading) momentum bin and then running NuWro's cascade on the reweighted sample. This is exact only if the within-bin joint distribution of the full nucleon kinematics (momentum magnitudes and orientation relative to q) is identical between the two models. Figure 3 shows this is not the case (the new model smears the NDelta and DeltaDelta peaks along the diagonal), and the angular sampling f(cos theta*) is itself a deliberate modification relative to Valencia. Since the cascade is nonlinear in the full kinematics, the coarse-bin rescaling cannot be assumed to reproduce the Valencia post-FSI distribution, and the band in Fig. 6 (from Ref. [36]) does not include the systematic error of the reweighting. The conclusion that the new model is an improvement over the old model survives this concern, but the specific agreement-within-uncertainties claim needs a validation of the reweighting (for example, a bin-size dependence study, or reweighting that includes angular variables) or a suitably qualified statement.
  2. [Sec. III C, Eqs. (16)-(20)] The fitted parameters (P,l) are obtained by minimizing the chi-tilde-squared of Eq. (16) against the very 2020 Valencia phase-space distributions that are then displayed as the validation targets in Figs. 3 and 4, so the pre-FSI agreement in those figures and in the pre-FSI curve of Fig. 5 is partly in-sample and should be labeled as such in the presentation. More importantly, Eq. (16) sums only over bins where NuWro already has nonzero counts, so the objective function never penalizes the model for failing to populate kinematic regions where the Valencia model has events. The paper should report the coverage fraction (the fraction of Valencia events lying in NuWro-occupied bins) and test whether the best-fit parameters change when zero-count bins are included with a penalty, since this bears directly on the claim of making the NuWro phase space 'as similar as possible' to the Valencia phase space.
  3. [Sec. III C, Eq. (21)] The assumption of a single, energy- and (omega,|q|)-independent sampling function per final-state pair is load-bearing for applying the model at arbitrary beam energies and for other nuclei, but it is tested only via the two-region split of Eq. (21), for which the paper reports 'no significant improvement' without giving the resulting parameters or per-energy chi-tilde-squared values. Because the global fit sums chi-tilde-squared over the five neutrino energies (Eqs. (17)-(18)), a large mismatch at one energy can be masked by good agreement at another. The paper should report per-energy chi-tilde-squared values (or per-kinematic-region agreement) and, ideally, a comparison of the new model and Valencia in bins of (omega,|q|), to substantiate the universality assumption.
  4. [Sec. III A and III B 2, Eq. (11)] The 2020 Valencia code provides no isospin decomposition for the 3p3h contribution (Sec. III A), so the model assigns 3p3h isospin states by combinatorial counts (Eq. (11)) and uses a three-body phase space for kinematics. Both ingredients are untested, and 3p3h contributes roughly 20% of the MEC cross section (Fig. 2). Since the new model worsens the MINERvA dsigma/d|pn| agreement relative to the old one (Table I: 3.21 vs 2.70), the sensitivity of the predicted hadronic observables to the assumed 3p3h isospin composition and phase-space sampling should be quantified before the model is adopted in oscillation analyses.
minor comments (6)
  1. [Sec. III B 2] In the enumeration of antineutrino 3p3h final states, the text lists 'nnn, nnp, nnn'; presumably the last state should be 'npp' (two protons and one neutron), mirroring the neutrino case 'ppp, ppn, pnn'. Please correct the typo and make the mirror symmetry explicit.
  2. [Table II] The column headers in Table II ('-0.2 - 0.2 -0.7 - 0.7') do not identify which columns correspond to the old and new MEC models; please label the four columns explicitly (old/new model times narrow/wide region), as the values are otherwise difficult to interpret.
  3. [Sec. IV B 1] There is a typo in the sentence 'the difference is largly annihilated by FSI effects'; it should read 'largely'.
  4. [Sec. II B 1, Eq. (9)] The text after Eq. (9) refers to the lepton scattering angle as theta_{k'} whereas Eq. (9) uses theta_l; please unify the notation. Also, the term 'W5/5 (E'_l + |k'|)' in Eq. (9) should be checked against the original Valencia derivation, since a misprinted coefficient in a central formula could mislead readers even if the implemented tables are correct.
  5. [Appendix B 2] The five-parameter function f(x) in Eq. (B2) is defined piecewise on [0, kappa]; if it is intended for use on [-kappa, kappa] as a nucleon sampling function, the required symmetrization (or the convention x = |cos theta*|) should be stated explicitly.
  6. [Fig. 4 caption] The caption reads 'The panel from left is for the old hadronic model while the middle panel corresponds to the new hadronic model'; please rephrase this for clarity and consistency with the Fig. 3 caption.

Circularity Check

1 steps flagged · score 5.0 of 10

The new model's pre-FSI agreement with the 2020 Valencia phase space is a calibration check, because the sampling-function parameters were fit to those same distributions; post-FSI and MINERvA tests provide partial independent grounding.

  1. fitted input called prediction [Sec. III C (Eqs. 16-20); Sec. IV B, Figs. 3-5]
    "We found best-fit values of the parameters (P,l)pp, (P,l)np, and (P,l)pn for two parameter nucleon sampling function ... The optimization was done by demanding that two-dimensional momentum distributions of outgoing nucleons for 2p2h in NuWro are made as similar as possible to the corresponding distributions in the 2020 Valencia model."

    The same 2020 Valencia two-dimensional momentum distributions that enter the chi2 in Eq. (16) and the global minimization in Eqs. (17)-(18) are later presented as the benchmark for the new hadronic model in Figs. 3-5, including the leading-proton projection in Fig. 5. The close pre-FSI agreement between the 'new hadronic model' and the '2020 Valencia model' is therefore largely a consequence of fitting the six parameters (P,l) to those very distributions, not an independent prediction. The paper is transparent that the parameters are calibrated, and the post-FSI leading-proton comparison and MINERvA benchmark provide some independent grounding, so this is partial circularity rather than complete construction.

full rationale

The central derivation chain is a Monte Carlo implementation: the inclusive Valencia response functions are taken from an external code, and the hadronic part is a tunable ansatz with parameters (P,l) chosen by minimizing chi2 against Valencia's exclusive two-nucleon phase spaces. The pre-FSI agreement shown in Figs. 3-5 is therefore a fit diagnostic rather than a genuine prediction; the abstract itself acknowledges that the parameters 'are calibrated by comparing our results to those of the Valencia model.' This is the main circular element. The post-FSI comparison in Fig. 6 is not fully circular: the '2020 Valencia model' curve is produced by reweighting NuWro events in coarse leading/subleading momentum bins to match Valencia's bin counts and then applying NuWro's cascade, so it is a proxy rather than a direct transport of Valencia events, but the agreement is not a pure identity. The MINERvA comparisons in Figs. 7-9 use external data not involved in the fit and thus provide independent content. No load-bearing self-citation or uniqueness-import circularity is present: Ref. [18] is external theoretical work, and the FSI uncertainty band from Ref. [36], though authored by one of the present authors, is a separate uncertainty estimate rather than a forced result. Overall the paper is transparent about its calibration, but because the headline pre-FSI 'benchmark' reduces to the fitting target and the post-FSI proxy weakens the otherwise independent check, a moderate partial-circularity score of 5 is appropriate.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the 2020 Valencia model as reference (code from Ref. [18]), on the local Fermi gas description, on a universal sampling function assumption, on a combinatorial isospin model for 3p3h, and on NuWro's FSI cascade. No new physical entities are introduced. The fitted sampling parameters are the only free parameters calibrated to a benchmark.

free parameters (3)
  • (P,l)_pp = (0.77, 4)
    Fitted to minimize chi2 between NuWro and Valencia pp nucleon phase space (Eq. 17, 19).
  • (P,l)_np = (0.7, 3)
    Fitted to minimize chi2 for the np contribution in the combined np+pn phase space (Eq. 18, 20).
  • (P,l)_pn = (0.8, 4)
    Fitted to minimize chi2 for the pn contribution in the combined np+pn phase space (Eq. 18, 20).
assumptions (4)
  • domain assumption Initial nucleon momenta are sampled from a local Fermi gas distribution f(p) = 3 p^2 / p_F^3 with p_F from the nuclear density profile.
    Used in Sect. III B 1 for both 2p2h and 3p3h; a standard but approximate description of the nuclear ground state.
  • ad hoc to paper A universal sampling function f(cos theta*) for each final-state nucleon pair is independent of (omega, |q|).
    Stated as an important assumption in Sect. III C; the attempted kinematic split found no significant improvement, but universality is assumed for all energies and nuclei.
  • ad hoc to paper 3p3h final-state isospin probabilities are proportional to combinatorial counts (Z choose k)(A-Z choose ...).
    Sect. III B; the 2020 Valencia model provides no isospin decomposition for 3p3h, so the authors impose a counting-based model.
  • domain assumption Pauli blocking of outgoing nucleons is enforced through the dynamic condition Elab_b > m_b + E_F, giving cos theta* in [-kappa, kappa].
    Derived in Appendix A and applied in Sect. III B 1; it is a modeling choice for the hadronic final state.

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Cite this review

Pith. "Pith review of New multinucleon knockout model in NuWro Monte Carlo generator." pith.science (2026). https://pith.science/paper/OFOYBWBL

@misc{pith2026241111523,
  author       = {Pith},
  title        = {Pith review of: New multinucleon knockout model in NuWro Monte Carlo generator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFOYBWBL}},
  note         = {Machine review of arXiv:2411.11523}
}
abstract

We present the implementation and results of a new model for the n-particle n-hole ($\it{np-nh}$) contribution in the NuWro event generator, grounded in the theoretical framework established by the Valencia group in 2020. For the $\it{2p2h}$ component, we introduce a novel nucleon sampling function with tunable parameters to approximate correlations in the momenta of outgoing nucleons. These parameters are calibrated by comparing our results to those of the Valencia model across a range of incoming neutrino energies. In addition, our model incorporates a distinct contribution from the $\it{3p3h}$ mechanism. We discuss the differences between the new NuWro implementation, the original Valencia model, and the previous NuWro version, focusing on the distribution of outgoing nucleon momenta. Finally, we assess the impact of the hadronic model on experimental analyses involving hadronic observables.

Figures

Figures reproduced from arXiv: 2411.11523 by the authors.

Figure 1
Figure 1. FIG. 1. Nucleon-sampling function (normalized to unit area [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. We first discuss the nucleon phase space from the 2020 Valencia model. The rightmost panel in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Outgoing nucleon distribution [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Outgoing nucleon distribution [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Distribution of momentum of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. ( [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Same as the figure in Fig [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Nucleon-sampling function (not normalized) as rep [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 10
Figure 10. Figure 10: FIG. 10. A schematic representation of the Pauli blocking [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

74 extracted references · 52 canonical work pages · cited by 3 Pith papers

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    6 GeV • DIS: inelastic reactions with W > 1. 6 GeV • HYP: quasi-elastic antineutrino hyperon produc- tion ¯νl + N → l+ + Λ/ Σ (3) Neutrino-nucleus scattering is described in the impulse approximation as a two-step process where primary neutrino-bound nucleon reaction is followed by final state 1 Note that this work does not incorporate the latest calculat ...

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    This part is referred to as the inclusive part of the model

    Generation of kinematics for outgoing lepton using information about neutrino energy and nuclear re- sponse functions. This part is referred to as the inclusive part of the model

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    Generation of nucleons outgoing from primary in- teraction by modeling their isospin and momenta This part is referred to as the hadronic part of the model

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    Af- ter some algebra, the expression for the cross-section in Eq.(

    Inclusive part For the inclusive reaction on a nucleus A νl(k) + A → l−(k′) + X (7) with the remnant hadronic system denoted as X, the double differential cross section, concerning the outgoing lepton kinematical variables is given by a general expres- sion d2σνl dΩ( ˆk′)dE′ l = G2 F cos2 θc 4π 2 |k′| |k| Lαβ W αβ (8) with k and k′ the incoming and outgoin...

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    Here, m3 and m4 are the rest masses of the final-state nucleons

    Move to the hadronic center-of-mass frame (a) Check whether the hadronic condition is met for a possibility of ejection of two nucleons: p2 sys ≥ (m3 + m4)2 (13) If it’s not the case, steps (3-5) are repeated. Here, m3 and m4 are the rest masses of the final-state nucleons. At this stage, both values are determined. (b) Produce two outgoing nucleons in bac...

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    W ij do not contain information regarding corre- lations among the outgoing nucleon momenta and their corresponding isospin

    Hadronic part The individual components W ij describe inclusive elec- troweak nuclear responses on a pair of correlated nucle- ons. W ij do not contain information regarding corre- lations among the outgoing nucleon momenta and their corresponding isospin. However, in MC generators one has to explicitly model the kinematics for the outgo- ing nucleon stat...

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    can be written down as d2σ dE′ l d cos θl = 2G2 F cos2 θcE′ l|k′| π { 2W1 sin2 θl 2 + W2 cos2 θl 2 ± W3(E + E′ l) sin2 θl 2 + m2 µ (E′ l + |k′|)E′ l · [ W1 cos θl − W2 2 cos θl ± W3 2 (E′ l(1 − cos θl) + |k′| − E cos θl) + W4 2 (m2 µ cos θl + 2E′ l(E′ l + |k′|) sin2 θl) − W5 5...

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    FSI The hadronic model describes nucleons after primary interaction but before FSI. The nucleons arising from the primary interactions must be propagated through the nucleus. For carbon, there is ∼ 40% probability (see e.g. [ 35]) that each one of them interacts at least once....

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    For 3p3h, further decisions must be taken about the isospin of the final state. With no information in this respect, it is decided based on a combinatorial model by calcu- lating a number of possible sets of three nucleons in the final state with different isospin i.e ppp, ppn, p...

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    2p2h The algorithm is defined as follows. For given values of energy ω and momentum transfer q:

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    Set the direction of each initial nucleon uniformly chosen within the unit sphere

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    (e) Sample the scattering angle of the outgo- ing nucleon cos θ∗ from the probability dis- tribution function f(cos θ∗) described in Ap- pendix B with cos θ∗ ∈ [−κ, κ ]

    is given in the Appendix A. (e) Sample the scattering angle of the outgo- ing nucleon cos θ∗ from the probability dis- tribution function f(cos θ∗) described in Ap- pendix B with cos θ∗ ∈ [−κ, κ ]. The az- imuthal angle is sampled uniformly in the range [0, 2π]

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    At this stage outgoing nucleons are assumed to be on-shell

    Boost back to the lab frame. At this stage outgoing nucleons are assumed to be on-shell. Later on, the NuWro FSI module accounts for the fact that they are immersed in a nuclear potential. The method we use to assign momenta to final state nu- cleons is an approximation of the ...

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    At the initial steps, we proceed in exactly the same way as in Sect III B 1 (steps (1-5) differ only in the sense that three nucleons are sampled in the initial state)

    3p3h If a particular isospin configuration within 3p3h mechanism is selected, then modeling of final state nucleons is done using a three-body phase space model for the kinematics of the outgoing nucleons. At the initial steps, we proceed in exactly the same way as in Sect III B...

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    (15) If it is not a case, steps (3-5) are repeated

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    If not, the steps 6b and 7 are re- peated

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    77, 4)pp (0

    5 (0. 77, 4)pp (0. 7, 3)np (0. 8, 4)pn cos θ∗ f (cos θ∗) FIG. 1. Nucleon-sampling function (normalized to unit area ) as represented in Eq. ( B1), obtained for the optimized values of the parameters ( ˆP , ˆl) at κ = 1. The black-solid curve represents f (cos θ∗ ) for the outg...

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    (21) Events within the nucleon phase space correspond- ing to the peak at lower energy transfer (dominated by the N∆ mechanism) were assigned one set of parame- ters, (P, l )reg

    2 · |q|[GeV/c ]. (21) Events within the nucleon phase space correspond- ing to the peak at lower energy transfer (dominated by the N∆ mechanism) were assigned one set of parame- ters, (P, l )reg. I . Alternatively, events from higher energy transfer (dominated by the ∆∆ mechan...

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    3 shows the pp nucleon phase space from the 2020 Valencia model

    pp nucleon phase space The rightmost panel in Fig. 3 shows the pp nucleon phase space from the 2020 Valencia model. The x-axis represents the proton with higher momentum denoted by |p1| (the leading proton), while the y-axis corresponds to the one with lower momentum denoted b...

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    np + pn nucleon phase space In this case, the contribution to the cross section is significantly smaller compared to the pp part, see Fig

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