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Single spin asymmetry in forward $pA$ collisions from Pomeron-Odderon interference

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper establishes that forward $pA$ collisions acquire a single-spin-asymmetry term from Pomeron-Odderon interference in the dense target, with a sign-flip node at $P_{h\perp}\approx1.3\,Q_S(x_0)$ and $A_N\propto A^{-7/6}$ nuclear…

desk verdict A genuine new SSA contribution from Pomeron-Odderon interference in forward pA, honestly labeled as model dependent; the derivation is careful and the A^{-7/6} counting checks out. read the letter →

arxiv 2501.12847 v2 pith:2WZZJFLV submitted 2025-01-22 hep-ph

Sanjin Benić , Eric Andreas Vivoda This is my paper · ORCID
classification hep-ph
keywords singlespinasymmetryPomeronOdderoncolorglasscondensatetwist-3fragmentationfunctiontransversitysaturationscaleforwardpAcollisions
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

The paper identifies a new contribution to the transverse single spin asymmetry (SSA) in forward proton-nucleus collisions, working in the hybrid small-$x$ framework. The needed quantum phase comes from interference between the Pomeron and the Odderon exchanges in the dense nuclear target rather than from loop phases in the scattering kernel. The central formula, Eq. (36), expresses the polarized cross section through the transversity distribution of the polarized proton, the real part of a twist-3 fragmentation function, and the product of the target's Pomeron and Odderon distributions. A model estimate gives a percent-level asymmetry in forward proton-proton scattering at low transverse momentum, a sign-flip node near $P_{h\perp}\approx1.3\,Q_S(x_0)$, and a strong nuclear suppression $A_N\propto A^{-7/6}$ for heavy nuclei. The authors conclude that this mechanism alone cannot explain the current forward-scattering data, but that measuring the predicted node at low $P_{h\perp}$ would provide a direct test.

What carries the argument

The load-bearing object is the off-forward dipole correlator $\langle D(x_\perp,x'_\perp)\rangle_Y=P_Y+iO_Y$, whose real part is the Pomeron and whose imaginary part is the C-odd Odderon exchange; the new asymmetry term arises from the cross term $P_YO_Y$ in the polarized cross section. The derivation also rests on the model ansatz (32) for the Odderon's off-forward shape, $O_Y(r_\perp,b_\perp)\approx R_A T'(b_\perp)O_Y(r_\perp)(\hat r_\perp\cdot\hat b_\perp)$, which fixes the transverse-momentum dependence and the nuclear scaling. Carrying the argument is the chain from the all-order twist-3 fragmentation formula (5), through the identification of the real part of the dynamical fragmentation function $\mathrm{Re}\,\hat E_F(z_1,z)$ as the coefficient of the Odderon contribution, to the compact final formula (43), where the equation-of-motion relation (7) trades the $z_1$ integral for $\hat e_1(z)$.

What would settle it

Measure $A_N$ for forward pion or charged-hadron production in polarized $p+p$, $p+\text{Al}$, and $p+\text{Au}$ collisions at $\sqrt{s}=200$ GeV in the low-$P_{h\perp}$ window around $0.5$--$0.9$ GeV: the mechanism predicts a sign change near $P_{h\perp}\approx1.3\,Q_S(x_0)$ with weak $x_F$ dependence and a heavier-target node shifted to higher $P_{h\perp}$ with $A_N\propto A^{-7/6}$ suppression; observing a fixed sign or a substantially different nuclear scaling in that window would rule out the Pomeron-Odderon contribution as the source.

Watch

Extended reading notes

Core claim

The central discovery is that Pomeron-Odderon interference in the target generates a real, previously missing term in the single spin asymmetry for forward $pA\to hX$. Starting from the all-order twist-3 fragmentation formula and the hybrid framework, the paper derives the polarized cross section (Eq. (36)), proportional to the real part of the dynamical twist-3 fragmentation function, the transversity $h_1(x_q)$, the target Pomeron distribution $F(x_g,k_\perp)$, and the target Odderon distribution $G(x_g,k_\perp)$. Using the QCD equation-of-motion relation (7), the final numerical form (43) depends only on the intrinsic twist-3 fragmentation function $\hat e_1(z)$. The prediction is that $A_N$ reaches about one percent in forward $pp$ collisions for $x_F\gtrsim0.6$ and $P_{h\perp}\sim Q_S$, that it changes sign at $P_{h\perp}\approx1.3\,Q_S(x_0)$ with the node position nearly independent of $x_F$, and that for nuclei $A_N\propto A^{-7/6}$, which the authors point out makes the contribution negligible for heavy targets.

Load-bearing premise

The result rests on one model assumption for the off-forward shape of the Odderon, that its dependence on the dipole size and impact parameter factorizes with the specific angular form $\hat r_\perp\cdot\hat b_\perp$, and on the approximation that replaces the $z_1$ integral by its value at $z_1=z$; if either gives way, the central formula's momentum dependence and $A^{-7/6}$ nuclear scaling change.

Editorial extensions

If this is right

  • $A_N$ in forward $pA\to hX$ acquires a new term proportional to the real part of the twist-3 fragmentation function (equivalently $\hat e_1(z)$), the transversity $h_1(x_q)$, and the Pomeron-Odderon product $F\cdot G$; complete small-$x$ SSA computations must include it.
  • The predicted node in $A_N$ at $P_{h\perp}\approx1.3\,Q_S(x_0)$ has a weak $x_F$ dependence, and the node shifts to higher $P_{h\perp}$ for heavier targets, so a low-transverse-momentum measurement can test the mechanism directly.
  • Parametrically $A_N\propto A^{-7/6}$ for large nuclei after small-$x$ evolution, giving a per-mille asymmetry in $pA$ collisions; this mechanism alone cannot account for the observed nuclear suppression.
  • The same Pomeron-Odderon interference structure is expected to contribute to polarized $\Lambda$ production in $ep$ and $eA$ collisions, offering a possible search channel at a future electron-ion collider.

Reading between the lines

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

  • Editorial inference: because the node position is set mainly by the Odderon shape while the overall magnitude is set by $\hat e_1(z)$, an experimental determination of the node would constrain the Odderon contribution even before better fragmentation-function input is available.
  • Editorial inference: the combination of a sign change and a steep $A^{-7/6}$ suppression distinguishes this mechanism from the imaginary-part twist-3 fragmentation contribution, whose nuclear scaling near $P_{h\perp}\sim Q_S$ is milder; a joint fit to the $A$-dependence and the sign behavior could separate the two contributions in existing data.
  • Editorial inference: if the factorized Odderon ansatz (32) were replaced by a different off-forward shape, both the node location and the $A^{-7/6}$ scaling would change; running the same derivation with alternative $b_\perp$ dependences would map how robust the predictions are.
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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 / 4 minor

Summary. The paper derives a new contribution to the transverse single-spin asymmetry (SSA) in forward pA collisions in the hybrid factorization framework. The phase for the SSA is generated by interference between the Pomeron and the Odderon in the dense target, and the contribution is sensitive to the real part of the twist-3 fragmentation function, in contrast to the conventional imaginary-part mechanism. The central result, Eq. (36), expresses the polarized cross section in terms of the transversity distribution, the real twist-3 fragmentation function e1(z), the Pomeron distribution F, and the Odderon distribution G. Using model inputs (a BK fit for F, a numerical Odderon fit for G, and a chiral quark model for e1), the authors estimate AN to be at the percent level for pp collisions at forward xF and low P_h⊥, find a strong nuclear suppression AN ∝ A^{-7/6}, and predict a node in AN near the initial saturation scale. The derivation is checked by reproducing the double-Pomeron result of Ref. [42] as a special case.

Significance. If the result is correct, the paper identifies a previously missing contribution to SSA in pA collisions that arises from the real part of the twist-3 fragmentation function and the Pomeron-Odderon interference in the target. It is a clean derivation with a non-trivial cross-check, and it makes a falsifiable prediction for the node position. The authors are transparent about the model dependence of the numerics, and they provide reproducible fits for the input distributions. The main caveats are the reliance on the factorized Odderon ansatz and an inconsistency in the parametric A-scaling that is flagged below.

major comments (3)
  1. [II.B, Eqs. (36)-(39)] The parametric nuclear scaling in Eq. (39) is internally inconsistent. With R_A ∝ A^{1/3} and Q_S^2 ∝ A^{1/3}, the expression for G in Eq. (38) scales as A^{23/6}, not as A^{-1/6} as stated in Eq. (39). Furthermore, even if one accepts G ∝ A^{-1/6} and ∫F ∝ A^{2/3}, Eq. (36) has an explicit prefactor 1/(P_{h⊥}R_A^3), which yields dΔσ ∝ R_A^{-3} G ∫F ∝ A^{-1} A^{-1/6} A^{2/3} = A^{-1/2}, and hence AN ∝ A^{-5/6} for dσ ∝ A^{1/3}, not A^{-7/6} as claimed. This discrepancy should be resolved because the A^{-7/6} suppression is a central quantitative claim of the paper.
  2. [II.B, Eq. (32)] The factorized Odderon ansatz in Eq. (32), OY(r⊥,b⊥) ≈ R_A T'(b⊥) OY(r⊥)(r̂⊥·b̂⊥), is load-bearing in two ways: the angular factor r̂⊥·b̂⊥ is what prevents the Pomeron-Odderon interference from averaging to zero in the angular integrals of Appendix A, and the T'(b⊥) dependence produces the 1/(P_h⊥ R_A) factor in Eq. (36) that controls the nuclear suppression. The quantitative predictions (the A-dependence and the node position) are therefore outputs of this model assumption. A robustness check with a different angular harmonic or a non-factorizable b⊥ dependence, or a clearly delimited statement of which results are model-independent, would substantially strengthen the paper.
  3. [III, after Eq. (43)] The statement that the numerical suppression is 'very close' to A^{-7/6} is not supported by any displayed analysis. Given the inconsistency in the parametric estimate, the numerical check should be shown explicitly (e.g., a table of AN for different A at fixed kinematics) to confirm the claimed exponent.
minor comments (4)
  1. [Title and Abstract] Please fix the missing space in 'forwardpA' in the title and rephrase the abstract's 'complete formula' to 'a new contribution to the polarized cross section', since Eq. (36) gives only the Pomeron-Odderon term.
  2. [Eq. (38) and Fig. 2] The units for G in Fig. 2 (GeV^{-4}) appear inconsistent with the definition in Eq. (33), where G is a two-dimensional Fourier transform of a dimensionless amplitude; please verify the normalization and units.
  3. [III, Eq. (42)] The approximation (42) is central to obtaining the numerical formula (43). Footnote 4 reports that the first-order term is small, but no quantitative estimate is given; providing the error estimate would make the approximation more convincing.
  4. [II.B, Eq. (39) and Ref. [40]] The paper cites Ref. [40] for the same A^{-7/6} parametric estimate; the relation to the present derivation should be clarified, especially in light of the discrepancy raised in Major Comment 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central cross-section formula is derived from QCD diagrammatics; model inputs are openly fitted to independent computations, not to SSA data.

full rationale

The central claim does not reduce to its inputs. Eq. (36) is obtained by inserting the all-order twist-3 expression (5) into the quark-target scattering amplitudes (13)-(19), evaluating the Dirac and color traces (17)-(21), decomposing the target correlator into Pomeron and Odderon via (9), passing to momentum space (25)-(26), and performing the angular integrals in App. A. At no point is A_N or the node position used as an input or a fit parameter. The quantitative estimates are presented as model computations: the Pomeron F uses the GBW/BK-inspired forms (37)/(40), the Odderon O_Y uses the factorized form (32) with G from the JV/BK model (38)/(41), and the fragmentation function e1 comes from the chiral quark model (44). These inputs are openly labeled as models, and the text explicitly says the result depends on them; a different off-forward Odderon shape would change the A-dependence and the node. That is model dependence, not circularity. The self-citations [45] and [64] are not load-bearing in a circular sense: the hard amplitudes from [45] are re-derived in the text, and [64] provides a numerical Odderon solution independently anchored to the Jeon-Venugopalan model [89] and to recent microscopic proton computations [63]. The A^{-7/6} statement is a parametric estimate from the stated inputs, not an output fitted to any asymmetry data. A separate check of the A-counting in Eqs. (36) and (39) is warranted because the displayed combination may give A^{-5/6} rather than A^{-7/6} outside a specific P_{h\perp} regime, but that is an arithmetic-consistency question, not circularity.

Assumptions & free parameters 10 free parameters · 5 assumptions · 0 invented entities

The calculation introduces no new particles or forces. It relies on existing objects: Pomeron, Odderon, transversity PDF, and twist-3 fragmentation functions. All free parameters are from fits to theoretical or numerical inputs (BK evolution, JV model, chiral quark model), not fitted to the SSA data being predicted.

free parameters (10)
  • Pomeron fit c0 = 0.087 GeV^2
    Fit parameter in Eq. (40) adjusted to reproduce HERA-constrained BK numerical solutions from [102]; sets the initial saturation scale.
  • Pomeron fit c1 = 0.80 GeV
    Fit parameter in Eq. (40) in the exponent controlling the r⊥ and rapidity dependence.
  • Pomeron fit c2 = 1 GeV
    Scale parameter in Eq. (40).
  • Pomeron fit c3 = -0.77
    Power parameter in Eq. (40).
  • Odderon fit B = 0.15
    Fit parameter in Eq. (41) controlling the exponent power for the Odderon.
  • Odderon fit c = 1.21 GeV
    Fit parameter in Eq. (41).
  • Odderon fit gamma = -0.04
    Fit parameter in Eq. (41).
  • Odderon fit c'_0 = 0.10 GeV^2
    Fit parameter in Eq. (41).
  • Odderon normalization lambda = 2.2e-4
    Overall normalization fixed to match the quark light-front model [63]; directly sets the magnitude of A_N.
  • Constituent quark mass M_q = M_N/3
    Input to the chiral quark model for e1(z), Eq. (44), from [65,66]; controls the size of the real twist-3 fragmentation function.
assumptions (5)
  • domain assumption The all-order twist-3 fragmentation formula (5) from Ref. [30] extends to forward pA collisions in the hybrid framework as used in Ref. [42].
    Section II, Eq. (5); the starting point is not rederived.
  • domain assumption Hybrid framework: the target is treated to all twists via eikonal Wilson lines with BK evolution, valid for forward production.
    Section II, text above Eq. (2).
  • domain assumption Large-Nc factorization of the double-dipole correlator: ⟨D D⟩_Y → ⟨D⟩_Y ⟨D⟩_Y.
    Section II, after Eq. (23).
  • ad hoc to paper The Odderon decomposes as O_Y(r⊥,b⊥) ≈ R_A T'(b⊥) O_Y(r⊥)(r⊥·b⊥) with a directed-flow angular correlation.
    Eq. (32); this is an ansatz from the literature [40,46,84,89-91], load-bearing for Eq. (36).
  • domain assumption The z1 integral in Eq. (36) can be replaced by its value at z1=z, Eq. (42), allowing use of the QCD equation-of-motion relation (7).
    Section III, Eq. (42); the authors state they numerically confirmed the first-order term is small.

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Cite this review

Pith. "Pith review of Single spin asymmetry in forward $pA$ collisions from Pomeron-Odderon interference." pith.science (2026). https://pith.science/paper/2WZZJFLV

@misc{pith2026250112847,
  author       = {Pith},
  title        = {Pith review of: Single spin asymmetry in forward $pA$ collisions from Pomeron-Odderon interference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WZZJFLV}},
  note         = {Machine review of arXiv:2501.12847}
}
abstract

Working in the hybrid framework of the high energy $pA$ collisions we identify a new contribution to transverse single spin asymmetry (SSA). The phase necessary for the SSA is provided by the Pomeron-Odderon interference in the dense nuclear target. The complete formula for the $pA \to h X$ polarized cross section also contains the transversity distribution for the polarized projectile as well as the real part of the twist-3 fragmentation function. We numerically estimate the asymmetry $A_N$ and its nuclear dependence. Based on a model computation we find that $A_N$ can be a percent level in the forward and low-$P_{h\perp}$ region. For large nuclei we find significant suppression, with $A_N \propto A^{-7/6}$ parametrically. As a notable feature we find a node of $A_N$ as a function of the $P_{h\perp}$ around the values of the initial saturation scale that could be used to test this mechanism experimentally.

Figures

Figures reproduced from arXiv: 2501.12847 by the authors.

Figure 1
Figure 1. FIG. 1. Diagrams contributing to polarized cross section in the leading order. Left diagram corresponds to the two-body [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Fourier transform of the Odderon exchange for proton target as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Forward citations

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Reference graph

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