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A new parton model for the soft interactions at high energies: two channel approximation

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that a two-channel, unitarity-preserving parton model can reproduce the 7 TeV elastic differential cross section, including the dip at $|t|=0.52\,\mathrm{GeV}^2$, once the impact-parameter profile is modified and a QCD…

desk verdict The two-channel model in the title does not actually describe the data it was built for, and the Odderon claim rests on a separate one-channel fit with a new ad hoc profile—still worth a serious referee for the honest negative result and the exact real-part calculation. read the letter →

arxiv 1908.05673 v3 pith:V4ECGMLO submitted 2019-08-15 hep-ph hep-ex

E. Gotsman (Tel Aviv U.) , E. Levin (Tel AvivU./UTFSM) , I. Potashnikova (UTFSM) This is my paper · ORCID
classification hep-phhep-ex PACS 12.38.-t24.85.+p25.75.-q
keywords newpartonmodelPomeronloopstwo-channelapproximationlow-massdiffractionelasticdifferentialcrosssectionproductionOdderonimpact-parameterprofile
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

This paper tries to establish that a two-channel extension of the authors' parton model for soft high-energy scattering, where the proton is a superposition of two states that diagonalize the interaction, can describe the measured total, elastic, and diffractive cross sections while preserving both $t$-channel and $s$-channel unitarity. The central positive result is that the elastic differential cross section at $W=7\,\mathrm{TeV}$ can be reproduced from $|t|=0$ to $1\,\mathrm{GeV}^2$, including the dip at $|t|=0.52\,\mathrm{GeV}^2$, but only after the impact-parameter profile is replaced by a more flexible four-parameter form and a QCD Odderon exchange is added to the real part of the amplitude. The paper also reports a clear limitation: the two-channel mechanism alone yields only about half of the measured single-diffraction cross section, with the other half supplied by large-mass triple-Pomeron production. The paper concludes that LHC elastic data at the dip support the Odderon contribution, and that single-diffraction data show unitarity alone does not determine the hadron's transverse structure.

What carries the argument

The central object is the $S$-matrix of the new parton model, generated by the Hamiltonian $H=-(1/\gamma)\bar P P$ together with the commutation relation $(1-P)(1-\bar P)=(1-\gamma)(1-\bar P)(1-P)$, which sums Pomeron-loop diagrams while preserving both $t$- and $s$-channel unitarity. The two-channel approximation replaces the full diffractive spectrum by a single state $\psi_D$, with the physical states formed from two eigenstates that diagonalize the interaction matrix, so that low-mass diffraction follows from the standard diffraction-dissociation mechanism. The $t$-dependence is generated by the initial impact-parameter profile $S(b,m_i)=m_i b K_1(m_i b)$ and, after the model's first prediction disagreed with the measured dip position, by the modified profile of Eq. (33). The real part of the elastic amplitude is computed from the analytic combination $A(s,t)+A(u,t)$, and the Odderon is added as a crossing-odd real term, giving the elastic amplitude used for the final comparison with the data.

What would settle it

Measure the elastic differential cross section for antiproton-proton scattering at $W=7\,\mathrm{TeV}$ over the same $|t|$ range. The paper's Odderon term changes sign between $pp$ and $\bar p p$, so it predicts a different dip depth and position; identical elastic cross sections would falsify the Odderon contribution. A second check is to recompute the real part of the amplitude at the dip using an independently measured or fitted impact-parameter profile rather than Eq. (33): if the real part is no longer small, the model's need for the Odderon also disappears.

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Extended reading notes

Core claim

The core claim, stated on the paper's own terms, is that a two-channel new parton model, using two orthogonal states $\psi_1,\psi_2$ that diagonalize the interaction and forming the physical proton as $\psi_h=\alpha\psi_1+\beta\psi_2$, generates an elastic amplitude whose $t$-dependence agrees with the measured $d\sigma_{el}/dt$ at $W=7\,\mathrm{TeV}$ across $0\leq |t|\leq 1\,\mathrm{GeV}^2$. The dip appears at $|t|_{\min}=0.52\,\mathrm{GeV}^2$, and the value of the cross section and its behavior for larger $|t|$ are reproduced. To achieve this, the original impact-parameter profile $S(b,m_i)=m_i b K_1(m_i b)$ is replaced by the four-parameter form of Eq. (33), and the real part of the amplitude, computed exactly from $A(s,t)+A(u,t)$ rather than by approximate formulas, is supplemented by an Odderon exchange with intercept one, $f(s,t)=f_{\mathrm{Eq.(21)}}(s,t)\pm\sigma_{\mathrm{odd}} e^{B_{\mathrm{odd}}t}$, with $\sigma_{\mathrm{odd}}\approx0.5$ mb and $B_{\mathrm{odd}}=5.6\,\mathrm{GeV}^{-2}$. Without the Odderon the model's elastic cross section at the dip is roughly an order of magnitude too small. For single diffraction, the paper reports that the two-channel mechanism accounts for only about half of the measured cross section and attributes the remainder to large-mass production via triple-Pomeron diagrams.

Load-bearing premise

The premise that carries the most weight is that a fitted curve for the proton's transverse interaction profile (Eq. 14, then replaced by Eq. 33) is the true geometric structure; every $t$-dependent prediction, including the real part at the dip and the need for the Odderon, is computed from that curve.

Editorial extensions

If this is right

  • If the model is right, the measured position and depth of the elastic dip at $W=7\,\mathrm{TeV}$ provide direct evidence for Odderon exchange in proton-proton scattering.
  • The model predicts a sign-flipped Odderon contribution in antiproton-proton elastic scattering, so the dip structure should differ between $pp$ and $\bar p p$ at the same energy.
  • The impact-parameter amplitudes in the two-channel model show $A_{11}$ saturated at $b=0$ already at $W=0.5\,\mathrm{TeV}$ while $A_{12}$ develops an energy-dependent maximum at larger $b$; this structure implies that geometric features of the proton remain visible in soft scattering at LHC energies.
  • Single diffraction requires both mechanisms: low-mass dissociation and large-mass triple-Pomeron production each contribute roughly half of the cross section, so a model that omits either mechanism will miss the data by approximately a factor of two.
  • Near the elastic minimum, derivative-based approximations for the real part fail; future determinations of the $\rho$ parameter at the dip must use the full analytic form of the amplitude.

Reading between the lines

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

  • The need to switch from Eq. (14) to Eq. (33) after seeing the data suggests the dip position is largely carried by the fitted geometric profile; testing the Odderon conclusion with an independently constrained proton profile would show how robust the inference is.
  • Since the modified profile introduces four fitted parameters and the Odderon two, a sharper test would be to determine both from QCD-inspired inputs and see whether the same dip position emerges without additional tuning.
  • If the Odderon interpretation is correct, the dip should move with collision energy in a computable way, and elastic $\bar p p$ data at 7 TeV should exhibit a visibly shifted or shallower dip; existing or future data can settle this.
  • The fact that both this unitarity-preserving model and the earlier model without full unitarity reproduce only half of single diffraction suggests the missing half is not an artifact of the unitarity constraints but a common dynamical ingredient, most plausibly the large-mass component.
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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 extends a previously proposed 'new parton model' for soft high-energy interactions to a two-channel (Good-Walker) approximation with the aim of describing diffractive production while preserving both s- and t-channel unitarity. The two-channel model is fitted to total, elastic, slope, and low-mass single/double diffraction data using the initial impact-parameter profile of Eq. (14). The authors report that the model describes the total, elastic, and slope data, but reproduces only about half of the single-diffraction cross section and fails on double diffraction; adding large-mass (triple-Pomeron) contributions does not cure this. In the second half of the paper, the initial profile is replaced by a more elaborate form, Eq. (33), in a one-channel fit, and the elastic differential cross section is computed. The predicted dip at |t| ≈ 0.3 GeV^2 is moved to the TOTEM value |t| = 0.52 GeV^2, the real part of the amplitude at the dip is found to be small, and an Odderon contribution, Eq. (36), is added from external QCD estimates to obtain agreement with TOTEM data. The paper also presents impact-parameter profiles of the amplitudes and a prediction for proton-antiproton scattering.

Significance. If the central t-dependence and Odderon claim were robust, the paper would be significant: it would give a unitarity-satisfying Reggeon-field-theory description that simultaneously produces a realistic elastic dip and a quantitative argument for an Odderon contribution at LHC energies, together with a falsifiable ppbar prediction. The paper is honest about its failures, explicitly conceding that only half of the single-diffraction cross section is described and that the impact-parameter structure might be an artifact of the simple two-channel ansatz. Those negative results are valuable. However, the positive t-dependence claim does not currently test the two-channel model advertised in the title: it is obtained after switching to a one-channel fit with a new seven-parameter profile and after importing an Odderon amplitude with parameters from another reference. As presented, the evidence is not strong enough to establish the Odderon conclusion, though the framework and the stated limitations make the manuscript a reasonable candidate for major revision.

major comments (3)
  1. [Section V, Eq. (33) and Table II] The central t-dependence claim is not a test of the two-channel model of Sections II-III: after the two-channel calculation placed the dip at |t| ≈ 0.3 GeV^2, the paper switches to a one-channel fit (p02 = 0) with a new impact-parameter profile Eq. (33) and seven free parameters (Table II), and this one-channel amplitude is then used for the real-part calculation and the Odderon inference. Because the dip depth near |t|min is essentially Re^2, no sensitivity study of Eq. (33) is given, and no demonstration is provided that the two-channel model with the modified profile would produce the same dip or the same small real part, the claimed agreement with TOTEM and the inferred need for an Odderon could be artifacts of the ad hoc profile rather than consequences of the model's two-channel dynamics.
  2. [Section V, Eq. (36)] The Odderon contribution is imported with sigma_odd ≈ 0.5 mb and B_odd = 5.6 GeV^-2 from Ref. [32] rather than determined by a fit within the model. The no-Odderon curve is roughly an order of magnitude below the data at the dip, so the conclusion that an Odderon is needed depends entirely on the computed real part being reliable. A fit with free Odderon parameters, a goodness-of-fit statistic, and a check that the two-channel model or an alternative profile yields a comparable real part are required before this conclusion can be considered load-bearing.
  3. [Section V and Eq. (21)] The 'exact' real part is introduced only by the statement that the authors consider the sum Aik(s, +i epsilon t) + Aik(u - i epsilon, t), without specifying the variable u, the continuation procedure, or how this sum is evaluated from the b-space amplitude of Eq. (21). Since Eqs. (34) and (35) are shown to be inadequate specifically at the dip, this exact procedure is the sole basis for the small Re^2 value that drives the Odderon conclusion; the procedure needs to be written out fully and checked for sensitivity to the profile choice.
minor comments (5)
  1. [Abstract and Section II.A] 'Dependance' should be 'dependence' in the abstract and Conclusions, and item 1 of Section II.A says 'GCC approach' where 'CGC approach' is meant.
  2. [Section IV] The second paragraph of Section IV refers to 'Fig. 5-a' for the comparison of the elastic amplitudes in the one- and two-channel models; the relevant figure appears to be Fig. 8-a.
  3. [Eq. (14)] The quantity zm = e^(Delta(1-p01)Y) is appended to the initial-condition line without any definition or explanatory sentence at that point; it should be introduced before it is used.
  4. [Section III.B and figure captions] The caption text 'The blog represents the triple Pomeron vertex' should read 'blob', and the figure captions for Figs. 9 and 10 contain the typographical artifact 'sqrt(s) - = 7 TeV' instead of 'sqrt(s) = 7 TeV'.
  5. [Table I] The entry 'm1 = 1.03 ± 012' is missing a decimal point and should read '1.03 ± 0.12'.

Circularity Check

1 steps flagged · score 6.0 of 10

Central t-dependence and dip claim reduces to a 7-parameter fit: Eq. (33)/Table II are fitted to the same TOTEM dσel/dt data that the Conclusions claim to reproduce; the Odderon inference inherits that fitted amplitude.

  1. fitted input called prediction [Section V (Eq. (33), Table II) and Section VI (Conclusions)]
    "TABLE II: Fitted parameters for dσel/dt dependence.∆dressed = ∆ (1−p01). ... Our attempt to describe the t-dependence of the elastic cross section shows that we can reproduce the main features of the t-dependence that are measured experimentally: the slope of the elastic cross section at smallt, the existence of the minima in t-dependence which is located at|t|min = 0.52GeV 2 at W= 7 TeV; and the behaviour of the cross section at|t| > |t|min."

    The parameter set in Eq. (33) (m1, μ1, ν1, ν2, κ1 plus p01 and Δ_dressed) is introduced only after the original Eq. (14) profile predicted the dip at |t|≈0.3 GeV^2 while TOTEM measured 0.52 GeV^2 (Section V). Table II is explicitly 'Fitted parameters for dσel/dt dependence', i.e. fitted to the same TOTEM data that the Conclusions present as reproduced. The dip position and the t-behaviour are therefore fit outputs, not predictions of the two-channel model. The exact-real-part calculation and the resulting claim that an Odderon is needed (Eq. (36)) use this same fitted one-channel amplitude; since no sensitivity study of the ad hoc profile is given, that inference is contingent on the fitted ansatz. The central positive claim thus reduces, at least partially, to its fitted input.

full rationale

The paper is mostly an honest phenomenological application: Table I parameters are fitted to σtot, σel, Bel, SD/DD, and the paper openly reports that the two-channel model describes only about half of the single-diffraction data and fails double diffraction. The original Eq. (14) model's prediction of a dip at |t|≈0.3 GeV^2, disagreeing with TOTEM's 0.52 GeV^2, is a genuine (failed) prediction, which is non-circular. The circular content enters in Section V: after this failure, the authors introduce a new seven-parameter profile Eq. (33), fit it to the TOTEM dσel/dt data (Table II, 'Fitted parameters for dσel/dt dependence'), and then in the Conclusions present the resulting dip at 0.52 GeV^2 and t-behaviour as something the model 'reproduces'. That is a fitted input presented as a model outcome. The Odderon inference is built on the real part computed from this same fitted one-channel amplitude; because the profile was chosen ad hoc to fix the t-dependence, the smallness of Re (and hence the need for an Odderon) is not independent of the fit. The paper gives no sensitivity study of Eq. (33) and no demonstration that the original two-channel model with Eq. (14) yields the same dip or same real part, so the central positive claim is partially circular: it reduces to a fit. The self-citations for the Hamiltonian and S-matrix (Refs. [1,2]) are not scored as circular because they are theory results with stated assumptions, not fitted data. Overall score 6.

Assumptions & free parameters 9 free parameters · 7 assumptions · 2 invented entities

The model's predictions rest on the reggeon-field-theory machinery imported from Refs. [1,2] (Hamiltonian, commutation relations, S-matrix), the two-channel Good-Walker truncation (Eq. 10), six parameters fitted in Table I, the hand-chosen triple-Pomeron mass m_IP = 3 GeV, the replacement b-profile of Eq. (33) with four new fitted parameters, and a QCD Odderon amplitude with sigma_odd near 0.5 mb and B_odd = 5.6 GeV^-2 imported from Ref. [32]. No genuinely new entity is invented: psi_D is a standard modeling construct, and the Odderon is inherited from perturbative QCD with an independent predictive handle (the pp versus pbar-p asymmetry).

free parameters (9)
  • dressed Pomeron intercept Delta_dressed = 0.488 +/- 0.002 (set I); 0.499 +/- 0.01 (set II)
    Controls the energy growth of the amplitudes; fitted to sigma_tot, sigma_el, Bel in Section III.A (Table I). The bare Delta near 1 follows from Delta_dressed = Delta(1-p01).
  • p01 (dipole amplitude for state 1 at low energy) = 0.748 +/- 0.002 (set I); 0.972 +/- 0.02 (set II)
    Determines the initial amplitude size and hence the strength of shadowing corrections; fitted in Table I.
  • p02 (dipole amplitude for state 2) = 0.005 +/- 0.001 (set I); 0.166 +/- 0.001 (set II)
    Second state's low-energy dipole amplitude; fitted in Table I.
  • m1 (inverse radius for state 1) = 1.03 +/- 0.12 GeV (set I); 1.05 +/- 0.01 GeV (set II)
    Sets the b-dependence S(b,m1) = m1 b K1(m1 b) in Eq. (14); fitted.
  • m2 (inverse radius for state 2) = 0.49 +/- 0.08 GeV (set I); 1.44 +/- 0.02 GeV (set II)
    Sets the b-dependence of the second state; fitted.
  • beta^2 (Good-Walker mixing weight) = 0.134 +/- 0.003 (set I); 0.2 +/- 0.01 (set II)
    Mixes the diffractive state psi_D with the physical proton in Eq. (10); directly controls the size of low-mass diffraction; fitted.
  • m_IP (triple-Pomeron vertex mass scale) = 3 GeV (by hand)
    Chosen by hand in Section III.B to set the triple-Pomeron vertex radius ('We choose the typical mass mIP = 3GeV'); the authors assert insensitivity without showing the study.
  • t-fit profile parameters mu1, nu1, nu2, kappa1 = mu1 = 7.6344 GeV, nu1 = 0.9, nu2 = 0.1, kappa1 = 0.48 (plus Delta_dressed = 0.48, p01 = 0.8, m1 = 0.860 GeV)
    Four new parameters of the modified initial condition Eq. (33), fitted to the TOTEM dsigma_el/dt in the one-channel model (Table II); introduced after the predicted dip position failed.
  • Odderon amplitude sigma_odd and slope B_odd = sigma_odd near 0.5 mb (20.6 alpha_bar_S^3 mb with alpha_bar_S = 0.3); B_odd = 5.6 GeV^-2
    Imported from QCD estimates (Ref. [32]) to fill the dip after the exact real part came out too small; not fitted in this paper but load-bearing for the final conclusion (Eq. 36).
assumptions (7)
  • domain assumption NPM Hamiltonian H_NPM = -(1/gamma) P-bar-P and the commutation relation (1-P)(1-P-bar) = (1-gamma)(1-P-bar)(1-P), Eqs. (2)-(3), define the dynamics; the S-matrix Eq. (5) is taken from Ref. [2].
    The entire amplitude machinery of Section II rests on this model from the authors' earlier work; it is not derived in this paper.
  • domain assumption Two-channel truncation of the diffractive spectrum: psi_h = alpha psi_1 + beta psi_2 and psi_D = -beta psi_1 + alpha psi_2 (Eq. 10), with psi_1 and psi_2 diagonalizing the interaction.
    Replaces the rich diffractive final-state spectrum by a single state; the paper itself calls this a simplification (Section II.C).
  • standard math Unitarity constraint 2 Im A_ik = |A_ik|^2 + G^in_ik (Eq. 12), with the solution form A_ik = 1 - S_ik (Eq. 13).
    Standard s-channel unitarity in impact-parameter space; the constraint is standard, while the nonperturbative solution is model-specific.
  • ad hoc to paper Initial conditions p_i(b) = p0i m_i b K1(m_i b), Eq. (14), and the modified profile of Eq. (33) with mu1, nu1, nu2, kappa1.
    Phenomenological Ansatz for the proton's transverse structure; the paper changes this Ansatz in Section V to move the dip, showing how load-bearing it is.
  • domain assumption Non-perturbative QCD fixes the dipole size gamma, replacing the saturation scale as the source of the interaction scale (Section II.B).
    Central physical premise distinguishing NPM from CGC; stated but not derived.
  • domain assumption Large-mass diffraction is computed from triple-Pomeron diagrams with a survival probability, Eqs. (28)-(30).
    Standard reggeon calculus; the paper uses its own model vertices for the triple-Pomeron coupling.
  • domain assumption QCD Odderon with intercept alpha_odd(0) = 1, contributing only to the real part, f_odd = sigma_odd exp(B_odd t), with negative sign for pp (Eq. 36).
    Imported from Refs. [30,32]; needed to reach agreement with TOTEM at the dip.
invented entities (2)
  • Single diffractive state psi_D in the two-channel approximation
    purpose: Generates low-mass single and double diffraction through Good-Walker interference (Eqs. 26-27).
    A modeling construct replacing the full diffractive spectrum; the paper acknowledges it as a simplification, and the fitted beta^2 to SD data is its only handle.
  • Odderon exchange term (Eq. 36) independent evidence
    purpose: Supplies the missing real part of the elastic amplitude at the dip (|t| near 0.52 GeV^2).
    Not invented here: parameters taken from QCD estimates (Ref. [32]); independent handle is the predicted pp versus pbar-p asymmetry shown in Fig. 11, consistent with the empirical evidence of Ref. [31].

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Pith. "Pith review of A new parton model for the soft interactions at high energies: two channel approximation." pith.science (2026). https://pith.science/paper/V4ECGMLO

@misc{pith2026190805673,
  author       = {Pith},
  title        = {Pith review of: A new parton model for the soft interactions at high energies: two channel approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4ECGMLO}},
  note         = {Machine review of arXiv:1908.05673}
}
abstract

The primary goal of this paper is to describe the diffraction production using the model that takes into account the Pomeron interaction, and satisfies both $t$ and $s$ channel unitarity. We hope that these features will allow us to describe the diffraction production in a more convenient way than in CGC motivated models, that do not satisfy these unitarity constraints. Unfortunately, we show that both approaches are only able to describe half of the cross section for the single diffraction production, leaving the second half to be estimate of the large mass production in the Pomeron approach.The impact parameter dependance of the scattering amplitudes show that soft interactions at high energies measured at the LHC, have a much richer structure than presumed. We discuss the $t$-dependence of the elastic cross section in wide range of $|t|=0 \div 1 \,GeV^2 $. We show that in the kinemati region of the minimum, we cannot use approximate formulae to calculate the real part of the amplitude. The exact calculation in our model, shows that the real part is rather small, and it is necessary to include the Odderon contribution in order to describe the experimental data.

Figures

Figures reproduced from arXiv: 1908.05673 by the authors.

Figure 1
Figure 1. FIG. 1: The energy behaviour of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The single diffraction production of large masses. Fig. 2-a and Fig. 2-b present the first diagrams for the diffraction [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The single diffraction production of large masses including the survival probability. The wavy lines denote the Pomeron, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: σdd = (31) 4 Z Y 0 dy0 Z y 0 0 d y00 Z d 2 b 0 Z d 2 b 00 A˜ i,IP Y − y 0 , b − b 0  A˜2 IP ,IP y 0 − y 00 , b 0 − b 00 A˜ k,IP y 00 , b − b 0   1 − Ai,k (Y, b) 2 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 4
Figure 4. Figure 4: FIG. 4: The double diffraction production of large masses including the survival probability. The double wavy lines describe [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The single diffraction cross section as function of energy [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The double diffraction cross section as function of energy [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The scattering amplitudes versus impact parameter [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The scattering amplitudes versus impact parameter [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

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Works this paper leans on

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