The gluon Sivers function and its process dependence from RHIC data

Carlo Flore; Cristian Pisano; Francesco Murgia; Pieter Taels; Umberto D'Alesio

REVIEW 3 major objections 4 minor 9 references

The gluon Sivers function and its process dependence from RHIC data

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read RHIC pion and D-meson data put the first constraints on the two gluon Sivers functions

desk verdict A short, honest proceedings paper that takes a small but real step toward constraining the two gluon Sivers functions; treat the numeric ranges as Ansatz-dependent illustrations, not measured constraints. read the letter →

arxiv 1908.07312 v1 pith:KXLOQ4ZU submitted 2019-08-20 hep-ph

Cristian Pisano , Umberto D'Alesio , Carlo Flore , Francesco Murgia , Pieter Taels This is my paper · ORCID

classification hep-ph

keywords gluonSiversfunctiontransversesingle-spinasymmetrymomentumdependentdistributionscolorgaugeinvariantgeneralizedpartonmodelprocessdependenceRHICDmesonproductionneutralpion

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 tries to establish that existing RHIC measurements of transverse single-spin asymmetries in neutral pion and D-meson production can already constrain the gluon Sivers function, the transverse-momentum distribution of gluons inside a transversely polarized proton. It works in a version of the parton model that includes initial- and final-state interactions, where two independent gluon Sivers functions enter, an f-type and a d-type. The combined data give, for the first time, preliminary ranges for their normalizations: $N_g^{(f)}$ between $+0.05$ and $-0.01$, and $N_g^{(d)}$ between $-0.15$ and $+0.15$, with the f-type strongly suppressed and a slight preference for a positive d-type. The paper also compares with the simpler generalized parton model and notes that current data cannot yet discriminate between the two frameworks.

What carries the argument

The load-bearing object is the pair of modified hard-scattering functions $H^{\mathrm{Inc}(f)}$ and $H^{\mathrm{Inc}(d)}$, which replace the universal hard functions when an unpolarized gluon from the polarized proton undergoes initial- or final-state interactions with the remnant. They encode the process dependence: the $f$-type combination enters with different signs in channels such as $gq\to gq$ versus $g\bar q\to g\bar q$, and the $d$-type combination vanishes for $gg\to gg$ and cancels between quark and antiquark channels, which is why the $d$-type contribution is naturally small. The second moving part is the Ansatz of Eqs. (3.1)-(3.2), which fixes the shape of both Sivers functions and leaves only the normalizations $N_g^{(f)}$ and $N_g^{(d)}$ free.

What would settle it

A measurement of single-spin asymmetries in $p^\uparrow p \to J/\psi\, X$ at RHIC, where the gluon-gluon fusion channel dominates and the $f$- and $d$-type modified hard functions weight differently than in pion and $D$-meson production, would settle the central claim: if the data demand an $f$-type normalization outside $+0.05$ to $-0.01$, the combined constraint from pion and $D$-meson data is contradicted.

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Extended reading notes

Core claim

The paper's central claim is that, at the level of the adopted parametrization, the RHIC pion and D-meson data put the first quantitative limits on the two independent gluon Sivers functions of the color gauge invariant generalized parton model. In that model, initial- and final-state interactions break the universality of the gluon Sivers function and generate two color-singlet combinations: an $f$-type, even under charge conjugation, and a $d$-type, odd under charge conjugation. The neutral-pion data alone would allow a large negative $d$-type contribution, but the muon-from-$D$ data limit $N_g^{(d)}$ to $|N_g^{(d)}|\le 0.15$ with a slight preference for positive values; combining both data sets then forces $N_g^{(f)}$ into the narrow range $+0.05\ge N_g^{(f)}\ge -0.01$. The paper stresses that this is not a true fit and that the generalized parton model description still agrees with the data, so the main result is a first, model-dependent constraint rather than a measurement.

Load-bearing premise

The paper assumes the $x$- and transverse-momentum dependence of both gluon Sivers functions is fixed by the Ansatz of Eqs. (3.1)-(3.2), so the data constrain only the overall normalizations $N_g^{(f)}$ and $N_g^{(d)}$; if the true shape differs, the quoted ranges do not apply to the actual functions.

Editorial extensions

If this is right

If the extracted ranges hold, the $f$-type gluon Sivers function is close to zero at RHIC kinematics, so future analyses of gluon-initiated single-spin asymmetries should use a small $f$-type normalization rather than the large values allowed by positivity alone.
The $d$-type function, although small, is not forced to vanish and is slightly preferred positive; processes sensitive to the $d$-type color combination should show small but nonzero asymmetries with a definite sign pattern.
Because the two functions enter different partonic channels with different signs, the gluon Sivers effect is genuinely process-dependent; no single universal function can describe both pion and open-heavy-flavor data.
The comparison with the generalized parton model shows that the simpler framework still fits the data, so additional channels are needed before process dependence itself is established.

Reading between the lines

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

A natural next step would be to promote the fixed exponents in the Ansatz to free parameters and refit the combined data; this would test whether the narrow $N_g^{(f)}$ range is an artifact of the assumed $x$-shape or a stable feature.
Because the $d$-type function is odd under charge conjugation, processes that select the $gg$-initiated subprocess with no antiquark compensation, such as quarkonium production, should provide a sharper handle on the $d$-type function than the channels analyzed here.
The same modified hard-function machinery could be applied to future electron-ion collider data, where TMD measurements access gluon Sivers effects with different kinematic coverage and could break the remaining degeneracy between the $x$-shape and the normalization.
The quoted ranges come from a specific choice of $\rho=2/3$; varying this transverse-momentum parameter would move the extracted normalizations, so the ranges should be read as conditional on that choice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a compact proceedings contribution from a group that has done the heavy lifting elsewhere. The genuinely new step is combining existing PHENIX pi0 and D-meson data within the CGI-GPM framework to get simultaneous ranges for the f-type and d-type gluon Sivers normalizations, Eq. (3.3). That is a legitimate extension of their own program, and the qualitative conclusion—f-type strongly suppressed, d-type small—looks consistent with the plotted curves. The GPM comparison provides a useful cross-check, and the authors are candid that the data 'do not allow for a true fit.' I give them credit for that honesty and for stating the preliminary nature of the result in the abstract.

The soft spots are real, though. The central numbers come from manual parameter scans, with no chi-square or error bars. More importantly, the quoted ranges depend on the assumed x-shape in Eqs. (3.1)–(3.2), including fixed exponents and rho = 2/3 for pions. If the true x-dependence differs, the ranges in (3.3) are not constraints on the actual functions. The stress-test note is right that the endpoint scenarios are not displayed against both data sets: Fig. 1 uses Ng(d) = -1, outside the final d-type range, and Fig. 2 shows only the d-type contribution with Ng(f) fixed at 0.1. So the claim that a 'very good description' of both pi0 and muon data is possible is asserted rather than demonstrated.

That said, these are soft spots in a proceedings-length paper, not fatal flaws. The framework is established, the external data are real, and the authors openly flag the limitations. If this were submitted as a proper research letter, I would send it to a referee—the topic is important for TMD spin physics and the analysis is a plausible first step. But the referee should ask for either a genuine fit or a prominent caveat that the numeric ranges are conditional on the adopted parametrization. I would not myself cite the numbers in (3.3) as extracted values without checking whether a later, more complete analysis has superseded them.

Referee Report

3 major / 4 minor

Summary. This proceedings paper analyzes PHENIX single-spin asymmetries for p↑p → π0X at midrapidity and for muons from D-meson decays, as functions of pT and xF. In the color-gauge-invariant generalized parton model (CGI-GPM), two independent gluon Sivers functions, f-type and d-type, enter with modified hard functions. The authors adopt a factorized Gaussian ansatz for the Sivers x- and k⊥-dependence, saturate positivity for orientation, and then scan the normalizations Ng(f) and Ng(d). They conclude that the f-type gluon Sivers function is strongly reduced (+0.05 ≥ Ng(f) ≥ −0.01), while the d-type is small (−0.15 ≤ Ng(d) ≤ +0.15), and they compare these results with the simpler GPM.

Significance. The qualitative finding, if correct, is useful: it suggests that initial- and final-state interactions suppress the f-type gluon Sivers function relative to the GPM extraction, and it informs predictions for J/ψ and direct-photon processes at RHIC. The paper is transparent about its model assumptions, applies positivity constraints, and displays the relevant hard functions; the explicit GPM comparison is a valuable cross-check. The numerical ranges in Eq. (3.3) are, however, illustrative constraints within a fixed ansatz rather than a measured extraction, a limitation the authors acknowledge in the text.

major comments (3)

[Section 3, Eq. (3.3) and Figs. 1-2] The endpoint pairs in Eq. (3.3) are never actually shown against both data sets. The right panel of Fig. 1 uses Ng(d) = −1, which lies outside the final d-type range, and Fig. 2 shows only the d-type contribution. The statement that a very good description of both the μ± and π0 data is possible for the pairs in Eq. (3.3) is therefore asserted rather than demonstrated. Please add plots, or a table of χ² values, in which the final endpoint scenarios are overlaid on the PHENIX π0 and μ± data.
[Section 3, Eqs. (3.1)-(3.3)] The constraints refer only to the normalizations Ng(f) and Ng(d) within the assumed x- and k⊥-shapes, with fixed α, β, and ρ, and with Ng(x) taken constant for the D-meson analysis. The PHENIX data cover only a few pT or xF bins, and the paper itself states that the number and precision of the available data points do not allow for a true fit. If the true x-dependence differs from the ansatz, the quoted ranges change. The abstract's phrase constraints on the two independent gluon Sivers functions should be explicitly qualified as constraints within the adopted parametrization.
[Section 3, Eq. (3.3)] No quantitative goodness-of-fit measure is given. The only uncertainty estimate is the ±20% band on Ng(f) in Fig. 1 at fixed endpoints, and no error bars are attached to the ranges in Eq. (3.3). It is therefore unclear whether Eq. (3.3) delimits an allowed region or merely a pair of manually chosen points. A χ² or profile-χ² scan over Ng(f) and Ng(d) would substantiate the word constraints.

minor comments (4)

[Section 3, text after Eq. (3.4)] The sentence with Ng(f) = 0.05 and Ng(f) = 0.15 appears to contain a typo; the second normalization should presumably be Ng(d) = 0.15 for the d-type curve.
[Abstract and Section 3] The abstract says for the first time, while the text is more cautious and calls the analysis a first attempt; the abstract should include the model-dependence qualification.
[Figs. 1 and 2] The figure captions do not fully specify all curves. Fig. 1 should state the exact values of Ng(f) and Ng(d) in the right panel, and Fig. 2 should clarify the mapping of the thin solid and dash-dotted lines to Ng(d) = +1 and Ng(d) = −1.
[Notation] The notation such as N( f )g is visually ambiguous; consistent use of Ng(f) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed constraints are fits to external PHENIX data under an explicit parametrization, not self-referential derivations.

full rationale

The paper's central claim is an extraction of preliminary ranges for the two gluon Sivers normalizations from RHIC data. The asymmetries for p↑p→π0X and p↑p→D→μX are taken from external PHENIX measurements (Refs. [8,9]), and the quoted ranges in Eq. (3.3) are obtained by varying Ng(d) and requiring Ng(f) to reproduce those data. This is a fit to independent data, not a prediction derived from the model's own output. The hard functions HInc(f,d) are imported from the authors' previous papers [5,6], but they serve as fixed model inputs, not as the target of the extraction; the extracted Sivers normalizations could in principle disagree with the data, and the paper explicitly shows the comparison. The x-dependence is fixed by the Ansatz in Eqs. (3.1)-(3.2), which is model dependence rather than circularity: the extracted normalizations are conditional on that Ansatz, but they are not defined in terms of the data they are meant to explain. The GPM comparison uses a gluon Sivers function previously extracted from the same neutral pion data [6], but that comparison is presented as an illustration and does not feed back into the CGI-GPM constraints. No equation is shown to reduce to its own input by construction, and no fitted parameter is renamed as a prediction. The self-citations to [5,6] supply formalism and hard-scattering coefficients, but they do not substitute for the external benchmarks. Therefore the analysis is self-contained against external data and exhibits no meaningful circularity.

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

The quantitative results rest on five model assumptions: TMD factorization, the CGI-GPM one-gluon-exchange hard functions from the authors' previous papers, a fixed functional form for the Sivers x and kT dependence, neglect of Collins fragmentation for pions, and LO dominance for D-meson production. The first two are standard in the field but not directly tested here; the third is an ad hoc choice that the paper acknowledges makes the constraints preliminary. No invented entities are introduced.

free parameters (5)

Ng(f): f-type gluon Sivers normalization = +0.05 to -0.01 (combined); +0.1 (pion-only scenario)
Central results in Eq. (3.3) are obtained by scanning this parameter to match RHIC pion and D-meson SSA data; it is a fit parameter, not derived.
Ng(d): d-type gluon Sivers normalization = -0.15 to +0.15 (combined); -1 (saturated pion scenario)
Same scanning procedure; constrained by D-meson muon data; no underlying derivation.
alpha, beta: x-shape exponents in Ng(x) = 0.6 for GPM fit in Ref. [6]; not fully specified for CGI-GPM combined scenario
The extraction is conditional on this choice; changing these exponents changes the inferred normalizations.
rho = M'^2/(<kT^2> + M'^2) = 2/3 for pion analysis; 0.1 for GPM comparison
Sets the transverse momentum shape of the Sivers Ansatz; chosen by hand.
<kT^2>_g (unpolarized gluon TMD width) = 1 GeV^2 (GPM comparison)
Taken from prior fit in Ref. [6].

assumptions (5)

domain assumption TMD factorization with soft and hard parts factorized is valid for p+p to h+X at RHIC energies.
Invoked in Sec. 1: 'we adopt the generalized parton model, which assumes the validity of factorization of soft and hard parts'.
domain assumption The one-gluon-exchange modified hard functions H_inc^(f,d) computed in Refs. [5,6] correctly capture initial- and final-state interactions.
Eqs. (2.4)-(2.6) are taken from those papers; errors in these hard functions would propagate directly into the extracted Sivers normalizations.
ad hoc to paper Both gluon Sivers functions are parametrized by Eqs. (3.1)-(3.2) with |Ng| <= 1 and a fixed functional form.
This Ansatz is introduced in Sec. 3 solely to make the scan feasible; it is not derived from QCD.
domain assumption For pion production the Collins fragmentation contribution is negligible at RHIC kinematics.
Asserted in Sec. 2 with reference to Ref. [6].
domain assumption For D-meson production, LO gg to c cbar and q qbar to c cbar subprocesses dominate after transverse-momentum integration.
Sec. 2 states all other contributions are strongly suppressed; NLO corrections are not estimated.

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Pith. "Pith review of The gluon Sivers function and its process dependence from RHIC data." pith.science (2026). https://pith.science/paper/KXLOQ4ZU

@misc{pith2026190807312,
  author       = {Pith},
  title        = {Pith review of: The gluon Sivers function and its process dependence from RHIC data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXLOQ4ZU}},
  note         = {Machine review of arXiv:1908.07312}
}

abstract

We present a phenomenological analysis of available data on single spin asymmetries for pion and $D$ meson production in proton-proton collisions at RHIC within the color gauge invariant generalized parton model (CGI-GPM), which includes initial (ISIs) and final (FSIs) state interactions in a transverse momentum dependent formalism. This study allows us, for the first time, to put preliminary constraints on the two independent gluon Sivers functions entering the model. We also present a comparison with the simpler generalized parton model (without ISIs and FSIs).

Figures

Figures reproduced from arXiv: 1908.07312 by the authors.

**Figure 1.** Maximized gluon Sivers contributions to AN for p ↑ p → π 0 X as a function of pT , together with the quark Sivers effect (left panel); results for AN obtained with a reduced f-type and a negative saturated d-type gluon Sivers functions, i.e. with N (f) g = 0.1 and N (d) g = −1 (right panel). The shaded area describes a ±20% uncertainty on N (f) g . Data are from Ref. [8]. where we have introduced the invariants s˜≡ … view at source ↗

**Figure 2.** Contributions of f ⊥g(d) 1T to AN for p ↑ p → µ + X (left panel) and p ↑ p → µ − X (right panel) from D-meson decay as a function of xF. The maximized effects correspond to N (d) g = +1 (thin solid lines) and N (d) g = −1 (thin dash-dotted lines). Predictions within the GPM are also presented. Data are from Ref. [9]. a relative cancellation between the f- and d-type Sivers functions occurs, with a strongly reduced a… view at source ↗

**Figure 3.** Values of the first transverse moments of the resulting gluon Sivers functions at Q 2 = 2 GeV2 . To conclude, we have performed a first attempt towards a quantitative study of the process dependence of the gluon Sivers function. Although we are not able yet to clearly discriminate between the GPM and the CGI-GPM frameworks, our results are encouraging; they have been used to provide predictions for AN in p ↑ p → J/ψ… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 4 canonical work pages

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[9]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

D’Alesio and F

U. D’Alesio and F. Murgia, Azimuthal and Single Spin Asymmetries in Hard Scattering Processes, Prog. Part. Nucl. Phys.61 (2008) 394 [arXiv:0712.4328 [hep-ph] ]

arXiv 2008

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D. W. Sivers, Single Spin Production Asymmetries from the Hard Scattering of Point-Like Constituents, Phys. Rev. D 41 (1990) 83

work page 1990

[3] [3]

Process dependent Sivers function and implications for single spin asymmetry in inclusive hadron production

L. Gamberg and Z. B. Kang, Process dependent Sivers function and implication for single spin asymmetry in inclusive hadron production, Phys. Lett. B 696 (2011) 109 [arXiv:1009.1936 [hep-ph]]

work page Pith review arXiv 2011

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D’Alesio, L

U. D’Alesio, L. Gamberg, Z. B. Kang, F. Murgia and C. Pisano, Testing the process dependence of the Sivers function via hadron distributions inside a jet, Phys. Lett. B 704 (2011) 637 [arXiv:1108.0827 [hep-ph] ]

arXiv 2011

[5] [5]

D’Alesio, F

U. D’Alesio, F. Murgia, C. Pisano and P. Taels,Probing the gluon Sivers function in p↑ p→ J/ψ X and p↑p→ DX , Phys. Rev. D 96 (2017) 036011 [arXiv:1705.04169 [hep-ph] ]

arXiv 2017

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D’Alesio, C

U. D’Alesio, C. Flore, F. Murgia, C. Pisano and P. Taels,Unraveling the Gluon Sivers Function in Hadronic Collisions at RHIC, Phys. Rev. D 99 (2019) 036013 [arXiv:1811.02970 [hep-ph] ]

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