Pith. sign in

REVIEW 4 major objections 6 minor 28 references

Pomeron Weights in QCD Processes at High Energy and the $S$-Matrix Unitarity Constraint

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that in the U-matrix unitarization scheme the number of exchanged pomerons is geometrically distributed rather than Poisson, making multi-pomeron exchanges correlated and their fluctuations larger at all energies.

desk verdict The paper's central geometric-distribution claim for the U-matrix does not survive the factor-of-2 inconsistency and failed AGK sum rule, though the formalism review and numerical exploration have some value. read the letter →

arxiv 2412.17267 v1 pith:KSLO4CIP submitted 2024-12-23 hep-ph

Rami Oueslati This is my paper · ORCID
classification hep-ph
keywords pomeronweightsU-matrixunitarizationeikonalmulti-pomeronexchangegeometricdistributioncorrelationsmulti-partoninteractionshigh-energyQCD
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 argues that the choice of unitarization—how one ensures that scattering probabilities respect the unitarity bound—changes the statistics of the soft pomerons exchanged in high-energy proton collisions. In the standard eikonal scheme, the number of exchanged pomerons is Poisson-distributed, so each exchange is statistically independent. In the U-matrix scheme, the paper derives a geometric distribution, meaning one exchanged pomeron makes additional pomeron exchanges more likely. Because pomerons are exchanged between composite, strongly interacting protons, the paper contends that such correlated exchanges are more physical than independent ones. If this is right, the U-matrix scheme is the better unitarization for QCD processes with cross-sections that keep growing with energy.

What carries the argument

The central object is the generalized S-matrix $S[\chi(s,b)]=\int_0^\infty d\tau\,\rho(\tau)e^{i\tau\chi(s,b)}$, where $\chi(s,b)$ is the single-pomeron Born amplitude in impact-parameter space and $\rho(\tau)$ is a spectral density encoding the pomeron weights. The argument works by fixing $\rho(\tau)$: the delta function $\delta(\tau-1)$ gives the eikonal scheme, while the exponential $e^{-\tau/c}/c$ with $c=1/2$ gives the U-matrix scheme. This object carries the derivation because Eq. (26), $\sigma_n=\int d\tau\,\rho(\tau)(2\tau\,\mathrm{Im}\,\chi)^n e^{-2\tau\,\mathrm{Im}\,\chi}/n!$, turns each scheme choice into a concrete pomeron multiplicity distribution, from which the mean, variance, $f_2$, and higher factorial moments are computed.

What would settle it

At collision energies below about $10^4$ GeV, the geometric distribution predicts the number of exchanged pomerons is overdispersed ($\mathrm{Var}>\langle n\rangle$ and $f_2>0$), while the eikonal scheme predicts a Poisson distribution with $\mathrm{Var}=\langle n\rangle$ and $f_2=0$. Measuring the pomeron multiplicity distribution in that energy range and checking whether the variance exceeds the mean, or recomputing $\sigma_n$ from the standard cutting rules with a different answer, would settle the claim.

Watch

Extended reading notes

Core claim

Using the generalized representation of the unitarized elastic amplitude of reference [16], $S[\chi(s,b)]=\int d\tau\,\rho(\tau)e^{i\tau\chi(s,b)}$, the paper derives the pomeron topological cross-section $\sigma_n(s,b)=\int d\tau\,\rho(\tau)\,\frac{(2\tau\,\mathrm{Im}\,\chi)^n}{n!}e^{-2\tau\,\mathrm{Im}\,\chi}$. A delta-function spectral density reproduces the eikonal amplitude and gives the Poisson law $\sigma_n=(2\,\mathrm{Im}\,\chi)^n e^{-2\,\mathrm{Im}\,\chi}/n!$. An exponential spectral density $\rho(\tau)=e^{-\tau/c}/c$ with $c=1/2$ reproduces the U-matrix amplitude $A=\chi/(1-i\chi/2)$ and gives the geometric distribution $\sigma_n=(\mathrm{Im}\,\chi)^n/(1+\mathrm{Im}\,\chi)^{1+n}$. From this distribution the paper concludes that in the U-matrix scheme the variance of the number of exchanged pomerons always exceeds the mean, the two-pomeron correlation $f_2$ stays positive at every energy, and normalized factorial moments grow with rank, so pomeron exchanges are correlated rather than independent. It further uses this geometric weighting in a string-model calculation of proton-proton multiplicities and finds that correlated pomeron exchanges enhance multi-parton collisions, especially double-parton collisions.

Load-bearing premise

The whole comparison rests on an unproved counting rule for the number of cut pomerons, taken without derivation from reference [16], plus the specific exponential weighting that defines the U-matrix scheme; if either differs from the correct physics, the geometric-versus-Poisson conclusion does not follow.

Editorial extensions

If this is right

  • If the U-matrix scheme is right, pomeron multiplicity in proton-proton collisions is overdispersed at every collision energy, so hadron multiplicity distributions should be broader than a Poisson distribution even below the $10^4$ GeV threshold where the eikonal predicts Poissonian behavior.
  • The geometric distribution implies that the exchange of one pomeron increases the chance of further exchanges, which changes the interpretation of double and multi-parton collisions and enhances double-parton scattering in the U-matrix scheme.
  • The unitarity-limit decrease of each multi-pomeron topological cross-section occurs at a slightly higher energy in the U-matrix than in the eikonal, giving an energy-dependent signature that data on total, elastic, and inelastic cross-sections can distinguish.
  • Monte Carlo event generators built on eikonal or quasi-eikonal unitarization should swap the Poissonian weight for the geometric distribution to obtain more realistic high-energy and cosmic-ray predictions.
  • The factorial moments $F_q$ grow with rank $q$ and depend on impact parameter, so correlation measurements in central versus peripheral collisions can test which unitarization scheme is realized.

Reading between the lines

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

  • A testable extension the paper does not pursue is to fit $\rho(\tau)$ as a free function to total, elastic, and inelastic cross-section data; if the preferred density is far from exponential, the geometric distribution would be an artifact of the $c=1/2$ choice.
  • Because compounding a Poisson with an exponential rate is what produces the geometric distribution, the same formalism could accommodate other rate distributions, for example a gamma distribution, yielding negative-binomial-like pomeron multiplicities that still satisfy unitarity; the paper does not explore these alternatives.
  • The correlated exchanges imply a non-factorizable two-parton distribution $F(x_1,x_2)$ inside the proton, which could be probed in same-event double parton scattering; the paper flags this question but leaves the calculation for future work.
  • The paper compares schemes using separately fitted parameters for each; a stricter test would fix one scheme's parameters and evaluate the other scheme's predictions with the same input, isolating the effect of unitarization from parameter fitting.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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. The paper applies Kancheli's generalized representation of the unitarized elastic amplitude, S[χ] = ∫ dτ ρ(τ) e^{iτχ}, to derive what it calls the pomeron topological cross-section σ_n(s,b) for two unitarization schemes: the eikonal, with ρ(τ)=δ(τ−1), and the U-matrix, with ρ(τ)=e^{−τ/c}/c and c=1/2. The resulting expressions, σ_n = (2 Im χ)^n e^{−2Im χ}/n! and σ_n = (Im χ)^n/(1+Im χ)^{n+1}, are interpreted as a Poisson and a geometric distribution of the number of exchanged pomerons, respectively. On this basis the paper computes pomeron multiplicity distributions, mean and variance of the number of pomerons, the two-particle correlation f2, normalized factorial moments, and a hadron multiplicity model for pp collisions, concluding that the U-matrix scheme inherently produces larger pomeron fluctuations and positive higher-order correlations, and that correlated pomeron exchanges enhance multi-parton collisions. The eikonal results reproduce known Poisson behavior, while the U-matrix results are advertised as a qualitative distinction between unitarization schemes.

Significance. If the framework were internally consistent, the paper would offer a clean, falsifiable separation between unitarization schemes: in the eikonal case f2 is predicted to be negative below roughly 10^4 GeV and positive above it, whereas in the U-matrix case f2 is predicted to be positive at all energies, with normalized factorial moments growing with rank. The manuscript is transparent in several respects: the Poisson-exponential mixture algebra leading to the geometric distribution is shown explicitly (Eqs. 33–34), the fit parameters are tabulated, and the conflict with the AGK-based results of Luna and Ryskin [22] is acknowledged rather than hidden. The generalized-representation formalism follows Kancheli [16] in a readable way, and the eikonal limit is textbook-correct. However, the central results are analytic consequences of the assumed exponential spectral density and of the unproven input formula (26), and the paper's own equations demonstrate that the U-matrix σ_n violates the cut-unitarity sum rule that any topological cross-section must satisfy.

major comments (4)
  1. [§III.A, Eqs. (31) and (42)] The U-matrix amplitude in Eqs. (31) and (42) is inconsistent with the spectral representation (17)–(30) used in the same paper. With ρ(τ)=e^{−τ/c}/c and c=1/2, Eq. (17) gives S(s,b)=1/(1−iχ/2). Combined with the standard definition A=i(1−S) used in Eqs. (21)–(28) and (38), this yields A=χ/(2−iχ), not A=χ/(1−iχ/2) as stated in (31) and (42). The discrepancy is a factor of two in the numerator of the U-matrix amplitude; it propagates into the inelastic cross-section (46) and affects the fitted parameters and all impact-parameter and energy-dependent results of Section III. The revision must correct the amplitude or redefine the Born term consistently and recompute the derived quantities.
  2. [§III.A, Eqs. (46)–(47)] The proposed pomeron topological cross-section (32) fails the cut-unitarity sum rule against the paper's own U-matrix amplitude. For a purely imaginary Born term χ=iγ, Eq. (46) gives G_inel = 2γ/(1+γ/2)^2, whereas the sum in Eq. (47) gives Σ_{k≥1} σ_k = γ/(1+γ); at γ=1 these are 8/9 and 1/2, respectively, and for γ→∞ they tend to 0 and 1. The manuscript acknowledges the disagreement but does not resolve it: the claim that the discrepancy arises from the omission of diffractive-state contributions in the AGK application of [22] is an assertion, not a calculation, and the sentence stating that G_inel→1 is 'closer to the expected asymptotic behavior of G_inel→0' is self-contradictory. Because W_n in Eq. (52) is defined by normalizing σ_n with Σσ_n, the identification of W_n as the probability of n cut pomerons requires Σ_{n≥1}σ_n = G_inel; without this identity, Eq. (32) is not a valid topological cross-section for the amplitude used in this paper, and the geometric-distribution and correlation claims of Sections III.B and III.C lack a valid basis. The surrounding text is also internally inconsistent, stating first that [22]'s result differs by 'an additional multiplicative factor of 2' and then that Eq. (45) is 'exactly the same as our result (32)'.
  3. [§II, Eq. (26)] Equation (26), the pomeron topological cross-section expressed as a superposition of Poisson distributions with mean 2τ Im χ, is quoted from [16] without derivation, and it is the only input that separates the statistics of the two schemes. Combined with the assumed exponential density (30), Eq. (26) yields the geometric distribution (32) through the elementary Poisson-exponential mixture identity demonstrated in Eqs. (33)–(34). The central results of the paper—geometric pomeron statistics, variance exceeding the mean at all energies, positive f2, and growing factorial moments—are therefore analytic consequences of the chosen spectral density rather than derived consequences of U-matrix unitarization. Moreover, the manuscript itself cites [22] as having shown that U-matrix unitarization is inconsistent with the AGK rules for a pomeron with intercept greater than 1. The revision must derive Eq. (26) from the AGK cutting rules, including the diffractive-state contributions invoked in the text, and demonstrate that the resulting σ_n satisfies the unitarity sum rule; otherwise the comparison between the eikonal and U-matrix pomeron weights reduces to a comparison of two assumed ρ(τ) functions.
  4. [§III.D, Eqs. (59)–(62)] The hadron multiplicity model assumes a Poisson distribution for the number of particles produced in n showers, with mean n⟨N1⟩, even though the manuscript acknowledges in the same paragraph that this ignores the known violation of KNO scaling. This is acceptable for an illustrative toy model, but the abstract and conclusion advertise as a finding that 'correlated pomeron exchanges within the U-matrix summation play a key role in enhancing multi-parton collisions.' That claim is drawn from the Poisson-shower model combined with the disputed σ_n; the revision should either strengthen the model (for example, by using a negative-binomial or KNO-violating shower distribution) or explicitly qualify the multi-parton-enhancement claim as a model-dependent illustration.
minor comments (6)
  1. [§II and §III.A] Several cross-references to equation numbers are wrong: the text near Eq. (15) refers to 'the series' structure in (38)', and §III.A refers to 'the generalized representation of the S matrix 38', but Eq. (38) is defined only later and is the same object as Eq. (15). Renumber or fix the citations.
  2. [§III.C, Fig. 10 and surrounding text] The text says the factorial moments are computed 'specifically at 13 GeV and 57 GeV', while Fig. 10 and Figs. 13–14 use 13 TeV and 57 TeV. The GeV/TeV discrepancy should be corrected.
  3. [§III.A, Eq. (37)] Equation (37) equates σ_n(s,b) with the probability P(X=n) before the normalization W_n = σ_n/Σσ_n is introduced in Eq. (52). This is only consistent because σ_n in (32) sums to 1 when n=0 is included; the distinction between the n=0 diffractive term and the n≥1 inelastic terms should be stated explicitly.
  4. [§III.A, footnotes [20] and [23]] Both footnotes say only 'There appears to be a typo in the formula in [16]' without specifying the typo or the corrected formula. Since Eq. (30) is load-bearing, the corrected spectral density should be stated explicitly.
  5. [§III.B, Table I] The table reports best-fit parameters and χ²/d.o.f. for the eikonal and U-matrix fits, but neither the data sets nor the energy range used in the fits are specified. This information is needed to assess whether the two schemes are fitted on the same footing.
  6. [Throughout] There are numerous typos and grammatical slips, including 'Poisonian' for 'Poissonian' (twice in §I), 'with the coefficients ... are :' in Eq. (42) context, and 'the odds' style phrasing in §III.B. These should be cleaned up.

Circularity Check

1 steps flagged · score 7.0 of 10

U-matrix geometric pomeron distribution is the exponential spectral function rewritten as σ_n; Eq. (32) is the Poisson–exponential mixture by construction.

  1. self definitional [Section III A, Eqs. (26), (30), (32)]
    "the pomeron topological cross-section: ... σn(s, b) = R ∞ 0 dτ ρ(τ ) (2τ Im(χ))n n! e−2τ Im(χ) (26) ... if we take for the spectral function, the expression : ρ(τ ) = e−τ /c c (30) ... and for the pomeron topological cross-section, Eq. 26 gives : σn(s, b) = (Im(χ))n (1 + Im(χ))1+n (32)"

    Eq. (32) is literally the integral in Eq. (26) evaluated with the exponential density in Eq. (30): with c=1/2, ∫ dτ e^{-τ/c}/c (2τγ)^n/n! e^{-2τγ} = γ^n/(1+γ)^{1+n}. The geometric distribution, Var>mean, positive f2, and rising factorial moments are all properties of this Poisson–exponential mixture. The paper itself says that unitarization schemes arise from the choice of the spectral function, and concludes that the pomeron distribution is fixed by the chosen scheme. Thus the central 'prediction' that the U-matrix yields correlated, overdispersed pomeron exchange is the assumed ρ rewritten as σ_n; it is not derived from the U-matrix amplitude or from unitarity.

full rationale

The claimed derivation chain reduces to an identity. The paper imports Eq. (26), a mixed-Poisson ansatz, from Kancheli [16], and then identifies the U-matrix scheme with the exponential spectral density Eq. (30). Inserting Eq. (30) into Eq. (26) produces Eq. (32) algebraically; every subsequent result (W_n, mean, variance, f2, factorial moments, correlations) is an analytic property of that geometric distribution. The paper even states that different schemes are distinguished only by the spectral function and that the pomeron distribution is fixed by the chosen unitarization scheme, confirming that the main claim is self-definitional. The numerical energy/impact-parameter dependence uses χ(s,b) with parameters fitted in an earlier paper [24]; that part is externally benchmarked and not circular, but it does not supply an independent derivation of the geometric distribution. Separately, the paper acknowledges an internal consistency problem: its Eq. (47) gives Σσ_k = Imχ/(1+Imχ), while its Eq. (46) gives G_inel = 2Imχ/|1 - iχ/2|^2, which disagree (e.g., as χ→∞, Eq. (46)→0 while Eq. (47)→1). This reinforces that σ_n is not actually tied to the U-matrix amplitude used in the paper; it is the chosen spectral density restated as a pomeron distribution. Hence the central claim is equivalent to its input by construction, warranting a score of 7.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The statistical results are not independent predictions: they are consequences of the chosen spectral functions and of quoted formulas from [16]. The numerical energy dependence uses parameters fitted in [24] by the same group. No new entities are introduced.

free parameters (6)
  • epsilon (pomeron intercept shift) = 0.11 +/- 0.01 (eikonal), 0.10 +/- 0.01 (U-matrix)
    Fitted to total, elastic, and inelastic cross-section data in [24]; controls the energy growth of chi(s,b).
  • alpha_prime (pomeron trajectory slope) = 0.31 +/- 0.19 (eikonal), 0.37 +/- 0.28 (U-matrix)
    Fitted to cross-section data in [24]; sets the t-dependence of the pomeron trajectory.
  • gp (pomeron-proton coupling) = 7.3 +/- 0.9 (eikonal), 7.5 +/- 0.8 (U-matrix)
    Fitted to cross-section data in [24]; sets the overall strength of the Born amplitude.
  • t0 (dipole form-factor scale) = 1.9 +/- 0.4 (eikonal), 2.5 +/- 0.6 (U-matrix)
    Fitted to cross-section data in [24]; sets the proton form-factor falloff.
  • a and b in single-shower mean multiplicity = a = -7.3, b = 2.56
    Taken from [27] and used in Eq. (61); not refitted here but assumed in the pp multiplicity model.
  • c (spectral-density scale in U-matrix) = 1/2
    Chosen in Eq. (30) to reproduce the U-matrix amplitude; this choice forces the geometric pomeron distribution.
assumptions (7)
  • domain assumption The unitarized S-matrix has the generalized expansion S(s,b) = sum_n beta_n^2 / n! (i chi)^n with real coefficients beta_n >= 1 (Eq. 15).
    Taken from Kancheli [16]; underlies all subsequent pomeron-counting formulas.
  • domain assumption The pomeron topological cross-section is given by the mixed-Poisson formula sigma_n(s,b) = integral d tau rho(tau) (2 tau Im chi)^n / n! exp(-2 tau Im chi) (Eq. 26).
    Quoted from [16] without derivation; if this cut-counting rule is wrong, the geometric distribution and correlation results fail.
  • standard math There exists a spectral density rho(tau) such that beta_n = integral tau^n phi(tau) d tau and rho(tau) = integral (d tau1/tau1) phi(tau1) phi(tau/tau1), with beta_0 = beta_1 = 1.
    Mellin-type transform representation; existence and positivity are assumed without proof.
  • ad hoc to paper For the U-matrix scheme, rho(tau) = exp(-tau/c)/c with c = 1/2 (Eq. 30).
    This choice is the definition of the U-matrix unitarization and is what forces the geometric pomeron distribution.
  • domain assumption The Born amplitude is a(s,t) = gp^2 F1(t)^2 (s/s0)^alpha(t) xi(t) with dipole form factor F1 = 1/(1-t/t0)^2 and linear trajectory alpha(t) = 1 + epsilon + alpha' t (Eq. 48).
    Standard phenomenological inputs from [24]; used to compute chi(s,b) by Fourier transform.
  • ad hoc to paper The multiplicity in n showers is Poisson with mean n times the single-shower mean, despite known KNO scaling violation (Eqs. 59-61).
    The paper explicitly states this simplifies the analysis and contradicts the known KNO violation; it affects the pp multiplicity comparison.
  • ad hoc to paper The discrepancy between the U-matrix inelastic cross-section and the sum of cut pomeron cross-sections is due to the omission of diffractive-state contributions in the AGK application of [22].
    This assertion is not proven; it is used to reconcile the U-matrix with the unitarity equation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Pomeron Weights in QCD Processes at High Energy and the $S$-Matrix Unitarity Constraint." pith.science (2026). https://pith.science/paper/KSLO4CIP

@misc{pith2026241217267,
  author       = {Pith},
  title        = {Pith review of: Pomeron Weights in QCD Processes at High Energy and the $S$-Matrix Unitarity Constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSLO4CIP}},
  note         = {Machine review of arXiv:2412.17267}
}
abstract

The pomeron topological cross-section is derived for the eikonal and the $U$-matrix unitarization schemes using a generalized expansion of the unitarized elastic amplitude in an effort to examine pomeron characteristics, namely the multiplicity distribution, fluctuation, and correlation, and to reveal the impact of pomeron weights on the $pp$ multiplicity distribution. The results demonstrate that the U-matrix inherently incorporates a larger amount of diffraction production into the multi-pomeron vertices, yielding a larger pomerons' variability regardless of the energy range, while such fluctuations become significant only beyond a specific high-energy threshold in the eikonal and quasi-eikonal schemes. Most importantly, our findings indicate that within the $U$-matrix scheme, an increase in exchanged pomerons results in more pronounced higher-order pomeron correlations, which are affected by the energy and the impact parameter. Interestingly, our outcomes also highlight that the correlated pomeron exchanges within the U-matrix summation play a key role in enhancing multi-parton collisions. In light of these results, we can argue that the U-matrix is fundamentally more valid for theories with growing cross-sections with energy, such as QCD at high energies.

Figures

Figures reproduced from arXiv: 2412.17267 by the authors.

Figure 2
Figure 2. FIG. 2. Impact parameter evolution of the Pomeron Topo [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Impact parameter evolution of the Pomeron Topo [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Energy evolution of the Pomeron topological cross [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Pomeron multiplicity distribution in both cases, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Mean and variance of the number of pomerons in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The two-pomeron correlation parameter versus the [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Normalized factorial moment of pomeron exchanges [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Impact parameter evolution of the normalized [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Impact parameter evolution of the normalized [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Topological cross sections [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Topological cross sections [PITH_FULL_IMAGE:figures/full_fig_p016_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 27 canonical work pages

  1. [22]

    N. L. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete Distributions, 2nd ed. (John Wiley & Sons, New York, 1992)

    work page 1992

  2. [16]

    O. V. Kancheli, (2013), arXiv:1309.5860 [hep-ph]

    work page Pith review arXiv 2013

  3. [1]

    Andersson, G

    B. Andersson, G. Gustafson, G. Ingelman, and T. Sj¨ ostrand, Physics Reports97, 31 (1983)

    work page 1983

  4. [2]

    V. N. Gribov, Zh. Eksp. Teor. Fiz. 53, 654 (1967)

    work page 1967

  5. [3]

    Riehn, R

    F. Riehn, R. Engel, A. Fedynitch, T. K. Gaisser, and T. Stanev, Phys. Rev. D 102, 063002 (2020)

    2020

  6. [4]

    Ostapchenko, EPJ Web Conf

    S. Ostapchenko, EPJ Web Conf. 208, 11001 (2019)

    work page 2019

  7. [5]

    K. G. Boreskov, A. B. Kaidalov, V. A. Khoze, A. D. Mar- tin, and M. G. Ryskin, The European Physical Journal C 44, 523–531 (2005)

    work page 2005

  8. [6]

    Martynov and G

    E. Martynov and G. Tersimonov, Phys. Rev. D 101, 114003 (2020)

    work page 2020

Show all 28 references
  1. [7]

    F. M. Liu, J. Aichelin, M. Bleicher, H. J. Drescher, S. Ostapchenko, T. Pierog, and K. Werner, Phys. Rev. D 67, 034011 (2003)

    2003

  2. [8]

    Oueslati and A

    R. Oueslati and A. Trabelsi, JHEP 07, 100, arXiv:2403.02263 [hep-ph]

    arXiv

  3. [9]

    Oueslati, JHEP 08, 087, arXiv:2305.03424 [hep-ph]

    R. Oueslati, JHEP 08, 087, arXiv:2305.03424 [hep-ph]

    arXiv

  4. [10]

    Blankenbecler and M

    R. Blankenbecler and M. L. Goldberger, Phys. Rev. 126, 766 (1962)

    1962

  5. [11]

    S. M. Troshin and N. E. Tyurin, Journal of Physics G: Nuclear and Particle Physics 29, 1061 (2003)

    2003

  6. [12]

    Ohashi, H

    K. Ohashi, H. Menjo, Y. Itow, T. Sako, and K. Kasahara, Progress of Theoretical and Experimental Physics 2021, 033F01 (2021), https://academic.oup.com/ptep/article- pdf/2021/3/033F01/36647718/ptab013.pdf

    2021

  7. [13]

    Vanthieghem, A

    A. Vanthieghem, A. Bhattacharya, R. Oueslati, and J.-R. Cudell, JHEP 09, 005, arXiv:2104.12923 [hep-ph]

    arXiv

  8. [14]

    Feofilov, V

    G. Feofilov, V. Kovalenko, and A. Puchkov, EPJ Web Conf. 171, 18003 (2018), arXiv:1711.00842 [nucl-th]

    2018 arXiv

  9. [15]

    Bleibel, L

    J. Bleibel, L. V. Bravina, A. B. Kaidalov, and E. E. Zabrodin, Phys. Rev. D 93, 114012 (2016), arXiv:1011.2703 [hep-ph]

    2016 arXiv

  10. [17]

    V. N. Gribov, Soviet Physics JETP 100 (1983)

    1983

  11. [18]

    V. N. Gribov and A. A. Migdal, Sov. J. Nucl. Phys. 8, 1002 (1968)

    1968

  12. [19]

    Cudell, E

    J.-R. Cudell, E. Predazzi, and O. V. Selyugin, Phys. Rev. D79, 034033 (2009), arXiv:0812.0735 [hep-ph]

    2009 arXiv

  13. [20]

    and with c = 1 2 then we get for the unitarised elastic scattering amplitude, this form : A(s, b) = χ(s, b) 1 − iχ(s, b)/2 (31) which is the U -Matrix form of the unitarisation scheme [19], and for the pomeron topological cross-section, Eq. 26 gives : σn(s, b) = (Im(χ))n (1 + ...

  14. [23]

    E. G. S. Luna and M. G. Ryskin, Phys. Rev. D 110, 094007 (2024), arXiv:2408.00194 [hep-ph]

    2024 arXiv

  15. [24]

    There appears to be a typo in the formula in [16]

  16. [25]

    Bhattacharya, J.-R

    A. Bhattacharya, J.-R. Cudell, R. Oueslati, and A. Vanthieghem, Phys. Rev. D 103, L051502 (2021), arXiv:2012.07970 [hep-ph]

    2021 arXiv

  17. [26]

    Bodnia, D

    E. Bodnia, D. Derkach, G. Feofilov, V. Kovalenko, and A. Puchkov, PoS QFTHEP2013, 060 (2013), arXiv:1310.1627 [hep-ph]

    2013 arXiv

  18. [27]

    I. M. Dremin, Phys. Usp. 37, 715 (1994), arXiv:hep- ph/9406231

    1994

  19. [28]

    S. G. Matinyan and W. D. Walker, Phys. Rev. D 59, 034022 (1999), arXiv:hep-ph/9801219

    1999 arXiv

  20. [29]

    Z. Koba, H. Nielsen, and P. Olesen, Nuclear Physics B 40, 317 (1972)

    1972

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.