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arxiv: 2601.18206 · v2 · submitted 2026-01-26 · ✦ hep-th · hep-ph

The Regge-Gribov model with odderons

M.A. Braun (Saint-Petersburg State University , Russia) , E.M. Kuzminskii (Petersburg Nuclear Physics Institute , M.I. Vyazovsky (Saint-Petersburg State University This is my paper

Pith reviewed 2026-05-16 11:12 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords Regge-Gribov modelodderonpomeronrenormalization groupfixed pointsphase transitionelastic scatteringFroissart bound
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The pith

The Regge-Gribov model with odderons produces five real fixed points under single-loop renormalization group flow as intercepts reach unity, though the associated phases violate projectile-target symmetry.

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

The paper studies the Regge-Gribov model of interacting pomerons and odderons, incorporating triple reggeon vertices that respect the odderon's negative signature. A zero-dimensional simplification is solved numerically and shows no phase transition when the intercept crosses unity. The full model in two transverse dimensions is analyzed via the renormalization group in the single-loop approximation, with distinct bare intercepts and slopes for the pomeron and odderon. As these intercepts approach their critical values set by the Froissart bound, five real fixed points emerge, each marked by non-trivial branch-point singularities that signal a phase transition. The new phases are nevertheless unphysical because they break the symmetry between projectile and target; near the fixed points the Green functions and elastic amplitude are derived under the Glauber approximation for participant couplings.

Core claim

In the Regge-Gribov model with odderons, five real fixed points appear in the single-loop renormalization-group flow when the bare pomeron and odderon intercepts cross unity. Each fixed point is accompanied by non-analytic branch-point singularities. The asymptotic forms of the Green functions and the elastic scattering amplitude are obtained in the vicinity of these points within the Glauber approximation for the couplings to external participants. The resulting phases, however, violate projectile-target symmetry and are therefore discarded as unphysical.

What carries the argument

Single-loop renormalization-group equations for the triple reggeon vertices that couple pomerons and odderons with independent bare intercepts and slopes.

If this is right

  • Green functions near each fixed point acquire definite power-law or logarithmic behaviors determined by the fixed-point values.
  • The elastic scattering amplitude at high energies follows a specific asymptotic form under the Glauber approximation for participant couplings.
  • No physical phase transition occurs in the model because the new fixed-point solutions break projectile-target symmetry.
  • The zero-dimensional truncation exhibits no transition at unit intercept, consistent with the two-dimensional result that only unphysical phases appear.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The symmetry violation suggests that odderon contributions alone do not generate new physical phases in high-energy scattering within this framework.
  • Higher-loop corrections could be computed to test whether the branch-point singularities persist or are smoothed out.
  • Extensions that restore symmetry by including additional reggeon species might reveal whether physical fixed points can exist.

Load-bearing premise

The single-loop renormalization-group approximation remains valid in the neighborhood of the critical intercepts.

What would settle it

An explicit numerical solution of the model beyond the single-loop approximation that produces a physical phase preserving projectile-target symmetry would disprove the claim that the new phases are unphysical.

Figures

Figures reproduced from arXiv: 2601.18206 by E.M. Kuzminskii (Petersburg Nuclear Physics Institute, M.A. Braun (Saint-Petersburg State University, M.I. Vyazovsky (Saint-Petersburg State University, Russia).

Figure 1
Figure 1. Figure 1: The solid curves show pomeron (left panel) and odderon (r [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The pomeron ground state energies E vs their theoretical values without odderon EP for different ρ in the model with odderon. that the ground state energy becomes lower due to interaction with odderons, being 2/3 of EP at high ρ (low λ). Note that at ρ < 3 Eq. (10) poorly describes the actual ground state energy level in absence of odderons, the latter being considerably greater. This has to be taken into … view at source ↗
Figure 3
Figure 3. Figure 3: Self masses for Γ1 (a+ b) and Γ2 (c). Pomerons and odderons are shown by solid and dashed lines, respectively. 4 Two transverse dimensions Passing to the real world we are to consider the model with D = 2. However, the renormaliza￾tion technique that we are going to apply requires a model with D = 4. So as usual in such cases we start with the theory in D = 4−ǫ dimensions, study it in the vicinity of small… view at source ↗
Figure 4
Figure 4. Figure 4: Diagrams for Γ10,20 (a + b), Γ10,02 (c + d) and Γ01,11 (e + f). Inverse diagrams for Γ 10,20 and Γ10,02 are identical. For Γ01,11 inverse diagrams are different and shown as (e ′ ) and (f ′ ). Pomerons and odderons are shown by solid and dashed lines, respectively. In the lowest order in small ǫ the β-functions are found in [29] to be the following. If u 6= 0 the four β-functions are β1 = − 1 4 ǫg1 + 3 2 g… view at source ↗
Figure 5
Figure 5. Figure 5: Elasic amplitude with a given number of pomerons (solid lines) a [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Elasic scattering for effective pointlike particles (protons [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

The Regge-Gribov model describing interacting pomerons and odderons is proposed with triple reggeon vertices taking into account the negative signature of the odderon. Its simplified version with zero transverse dimensions is first considered. No phase transition occurs in this case at the intercept crossing unity. This simplified model is studied without more approximations by numerical techniques. The physically relevant model in the two-dimensional transverse space is then studied by the renormalization group method in the single loop approximation. The pomeron and odderon are taken to have different bare intercepts and slopes. The behaviour when the intercepts move from below to their critical values compatible with the Froissart limitation is studied. Five real fixed points are found with singularities in the form of non-trivial branch points indicating a phase transition as the intercepts cross unity. The new phases, however, are not physical, since they violate the projectile-target symmetry. In the vicinity of fixed points the asymptotical behaviour of Green functions and elastic scattering amplitude is found under Glauber approximation for couplings to participants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript extends the Regge-Gribov model to incorporate odderons with negative signature in the triple-reggeon vertices. A zero-dimensional simplified version is solved numerically and exhibits no phase transition when intercepts reach unity. The two-dimensional transverse-space model is then analyzed via single-loop renormalization-group flow with distinct bare pomeron and odderon intercepts and slopes. Five real fixed points are identified, accompanied by non-trivial branch-point singularities that signal a phase transition as the intercepts cross their critical values. The new phases are discarded as unphysical on the grounds that they violate projectile-target symmetry. Asymptotic forms of the Green functions and elastic amplitude are derived near the fixed points under the Glauber approximation for couplings to participants.

Significance. If the one-loop fixed-point structure and branch-point singularities prove robust, the work would contribute to the understanding of high-energy scattering by furnishing a concrete dynamical mechanism for phase transitions in Regge theory and by illustrating how discrete symmetries can eliminate unphysical solutions. The exact numerical treatment of the zero-dimensional model provides a controlled benchmark, while the allowance for independent intercepts and slopes adds phenomenological realism. The Glauber-based asymptotics offer explicit predictions for the elastic amplitude that could be tested against data once higher-order corrections are controlled.

major comments (3)
  1. [RG analysis] RG flow and fixed-point section: The central claim of five real fixed points with non-trivial branch-point singularities is obtained from the single-loop beta functions. No quantitative estimate of higher-loop corrections or non-perturbative effects is supplied, yet the expansion parameter is expected to grow near the critical intercepts; this truncation is load-bearing for both the number of fixed points and the character of the singularities.
  2. [Phase discussion] Phase interpretation section: The assertion that the new phases are unphysical because they violate projectile-target symmetry is invoked after the fixed points are located. The manuscript should demonstrate explicitly how the fixed-point values break this symmetry in the definitions of the vertices or propagators, rather than appealing to it post hoc.
  3. [Zero-dimensional model] Zero-dimensional versus two-dimensional comparison: The zero-dimensional numerical solution finds no transition, while the two-dimensional RG flow does. A clearer discussion is needed of how the dimensional reduction affects the fixed-point structure and why the single-loop approximation can be trusted in 2D when the zero-dimensional case is treated exactly.
minor comments (2)
  1. [Notation] The notation for bare intercepts, slopes, and triple vertices would benefit from an early summary table or explicit definitions to improve readability.
  2. [References] A few standard references on odderon phenomenology and prior Regge-Gribov analyses with negative signature appear to be omitted; adding them would place the work in clearer context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions that will be incorporated in the next version.

read point-by-point responses

  1. Referee: [RG analysis] RG flow and fixed-point section: The central claim of five real fixed points with non-trivial branch-point singularities is obtained from the single-loop beta functions. No quantitative estimate of higher-loop corrections or non-perturbative effects is supplied, yet the expansion parameter is expected to grow near the critical intercepts; this truncation is load-bearing for both the number of fixed points and the character of the singularities.

    Authors: We agree that the single-loop truncation is a limitation, as the expansion parameter grows near the critical intercepts where the fixed points are located. The five real fixed points and branch-point singularities are derived within this approximation. While a quantitative estimate of higher-loop corrections lies beyond the present scope, we will add an explicit discussion of the truncation's limitations and its expected range of validity to the revised manuscript. revision: partial

  2. Referee: [Phase discussion] Phase interpretation section: The assertion that the new phases are unphysical because they violate projectile-target symmetry is invoked after the fixed points are located. The manuscript should demonstrate explicitly how the fixed-point values break this symmetry in the definitions of the vertices or propagators, rather than appealing to it post hoc.

    Authors: We will revise the phase interpretation section to include an explicit demonstration. At the additional fixed points we will show how the specific coupling values break projectile-target symmetry by examining the resulting triple-reggeon vertices and the symmetry properties of the propagators, making the argument self-contained. revision: yes

  3. Referee: [Zero-dimensional model] Zero-dimensional versus two-dimensional comparison: The zero-dimensional numerical solution finds no transition, while the two-dimensional RG flow does. A clearer discussion is needed of how the dimensional reduction affects the fixed-point structure and why the single-loop approximation can be trusted in 2D when the zero-dimensional case is treated exactly.

    Authors: The zero-dimensional model is solved exactly numerically and shows no transition at unit intercept. In two dimensions the transverse-momentum loop integrals generate the non-trivial branch-point singularities absent in zero dimensions. The single-loop approximation supplies the leading perturbative beta functions, and the zero-dimensional solution benchmarks the numerical methods. We will expand the comparison section to clarify the role of transverse dimensions and the status of the one-loop analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity: fixed points obtained by solving derived RG beta functions

full rationale

The paper first solves the zero-dimensional model numerically with no approximations and finds no phase transition. It then derives the single-loop RG beta functions for the two-dimensional transverse model from the Regge-Gribov Lagrangian with distinct pomeron and odderon intercepts and slopes. The five real fixed points and associated branch-point singularities are located by setting these beta functions to zero as the bare intercepts approach unity; this is a direct algebraic/numerical solution of the derived flow equations rather than a fit or redefinition of inputs. The subsequent check that the new phases violate projectile-target symmetry is performed by inspecting the symmetry properties of the fixed-point couplings, which is an independent verification step. No self-citation chain, ansatz smuggling, or renaming of known results is required for the central claims, and the derivation remains self-contained against the model's own equations.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The model introduces no new particles but relies on standard Regge-theory assumptions plus the single-loop truncation; bare intercepts and slopes are varied as external parameters rather than derived.

free parameters (3)
  • bare pomeron intercept
    Varied from below to the critical value compatible with the Froissart bound; central to locating the fixed points.
  • bare odderon intercept
    Taken different from the pomeron intercept and varied independently.
  • trajectory slopes
    Pomeron and odderon slopes treated as distinct input parameters.
axioms (2)
  • domain assumption Single-loop renormalization group approximation suffices near the critical intercepts
    Invoked for the two-dimensional transverse-space analysis.
  • domain assumption Glauber approximation captures the leading asymptotic coupling to external particles
    Used to extract the elastic amplitude behavior near fixed points.

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