Extraction of transverse-momentum-dependent helicity distributions of the nucleon

Bo-Qiang Ma; Ke Yang; Peng Sun; Tianbo Liu; Yiyu Zhou; Yuxiang Zhao

REVIEW

Extraction of transverse-momentum-dependent helicity distributions of the nucleon

Ke Yang, Tianbo Liu, Bo-Qiang Ma, Peng Sun, Yuxiang Zhao, Yiyu Zhou

High Energy Physics - Phenomenology · Submitted 22 Oct 2026 · 88 references

Pith's verdict: Technically careful helicity-TMD extraction whose consistency checks pass; the main caveat is an error budget that stops at experimental uncertainties.

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arXiv ID2610.31290 (version 1)

Pith IDKM3E4JNL, Pith's permanent record of this paper and its review

CitationBo-Qiang Ma, Ke Yang, Peng Sun, Tianbo Liu, Yiyu Zhou, Yuxiang Zhao. "Extraction of transverse-momentum-dependent helicity distributions of the nucleon." arXiv:2610.31290 (2026). https://pith.science/paper/KM3E4JNL

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Editor's note

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Read this if you work on TMD phenomenology or on EIC/EicC spin projections. This is the TNT group's follow-up to their first helicity-TMD extraction, adding NNLO small-bT matching, target- and hadron-mass corrections, a T0 treatment of correlated normalizations, and a set of new predictions. What is genuinely new: the WW-type prediction of the cos phih modulation of ALL from the extracted TMDs, tested against the 66 excluded in-region points at chi2/N = 0.71; the CLAS12 ALL predictions; the W+/- single-spin-asymmetry predictions with decay kinematics and lepton acceptance, compared qualitatively with preliminary STAR data; and the EicC pseudodata impact study.

The central claims hold up. The fit is clean, chi2/N = 1.05 on 246 points; u- and d-quark helicity TMDs are clearly nonzero, sea and gluon compatible with zero. The Fig. 7 agreement with collinear helicity PDFs is a real self-consistency check: the deformation parameters are free, so agreement is not automatic, even though the baseline is NNPDFpol1.1. Comparing against a second collinear set would have been stronger, but that is minor.

The soft spot the stress-test identifies is real. Equation (19) fits g1L normalized by SV19's f1 and DSS's D1, with NNPDFpol1.1 as the longitudinal baseline, and none of those uncertainties enter the 100-replica bands. The kT-shape parameters lambda1, lambda2, lambda5 absorb the transverse profile of f1 and D1, so a shift in SV19's nonperturbative widths would masquerade as a change in the Fig. 6 claim that polarization decreases with kT. The deuteron effective polarization is fixed at 0.925, and the acknowledged neglect of nuclear smearing is not quantified. This is standard practice for TMD fits, so I would not call it fatal: the qualitative picture is robust. But "within uncertainties" in the abstract is narrower than it reads, because the bands are conditional on external inputs. An explicit statement plus a crude sensitivity scan would fix most of it.

Two smaller notes. chi2/N = 0.71 for the excluded cos phih data is supportive but soft: a value that far below 1 means the test is not demanding much of the theory. And the "world data" base is two experiments, HERMES and CLAS; the COMPASS exclusion is explained, but the label is generous. No fit outputs or code are shipped, so the replica bands cannot be independently reproduced; MAP ships grids, and this group should too.

Who gets value: people building on helicity TMDs for predictions and projections. The W+/- and EicC sections are clearly framed as impact studies, so read them as such.

Recommendation: yes, send to serious refereeing, expecting minor-to-moderate revision. This is the most technically complete helicity-TMD extraction to date, and the requested changes concern the honesty of the error budget, not the physics claims.

Referee report

Significance

If the results hold, the paper provides the most detailed current map of g1L from world SIDIS data, and it deserves credit for several genuinely supportive features: the main fit is of good quality (chi2/N = 1.051), the cos phi_h modulation was excluded from the fit and is reproduced under the stated WW-type approximation (chi2_uncor/N = 0.710 on 66 in-region points), the STAR W+- comparison is an independent-process probe, and the authors are disciplined about reporting that sea and gluon components are not constrained rather than overinterpreting them. The integrated-to-collinear comparison of Fig. 7 is a legitimate self-consistency check. The main limitation is that the uncertainty budget is conditional on fixed external inputs: the SV19 unpolarized TMDs, the NNPDFpol1.1 reference helicity, and the deuteron effective polarization. This does not invalidate the central results, but it requires either sensitivity studies or a more careful qualification of the 'within uncertainties' claims.

Major objections (3)

[Sec. III D; Eq. (19)] The uncertainty budget of the extraction is conditional on external inputs that are held fixed. The 100-replica ensemble of Sec. III D resamples only the experimental A_LL data; the unpolarized TMD PDF f1 (SV19, Sec. III A) and the TMD FF D1 (DSS) enter Eq. (19) as exact denominators, and no uncertainty from these inputs is propagated into g1L or into the bands of Figs. 6-8 and 12-14. Because the fitted asymmetry is the ratio B0[g1L,D1]/B0[f1,D1], a systematic error in the SV19 nonperturbative parameters (Table I) shifts the extracted g1L directly, and the fitted transverse parameters lambda1, lambda2, lambda5 (Table IV) partly compensate the kT profile of f1. A within-SV19 variation of the f1 widths would therefore change the transverse shape of g1L and could affect the qualitative claim that the quark polarization rate decreases with kT (Fig. 6; Sec. VI). I recommend repeating the fit with f1/D1 inputs varied within their quoted uncertainties (or with an alternative unpolarized input set) and reporting the induced spread on the extracted TMDs and on the Fig. 7 comparison; at minimum, the 'within uncertainties' claims in the abstract and Sec. VI should specify that only experimental uncertainties are propagated.
[Eq. (40) and Fig. 7] The x-dependence of g1L is anchored to NNPDFpol1.1 by construction in Eq. (40), so the agreement in Fig. 7 is partly a test of the deformation factors Nf, alpha, beta, epsilon and of the kT-integral normalization rather than of the underlying x-profile. This is a legitimate self-consistency check, but the claim that the integrated TMDs 'agree with collinear helicity distributions within uncertainties' (Sec. VI) would be considerably strengthened by refitting with an independent helicity PDF reference (for example, a DSSV-type or JAM set) and verifying that the comparison in Fig. 7 is reproduced. Without such a test, the possibility that the agreement reflects the input anchor rather than the data cannot be excluded.
[Sec. III C, Eq. (44)] The deuteron effective polarization is fixed to P = 0.925 with no uncertainty, and nucleon momentum smearing is neglected, as the authors state after Eq. (45) (sentence b168). Since 154 of the 246 fitted points come from the deuteron target, an error in P propagates nearly linearly into the extracted d-quark g1L and into the d-quark panel of Fig. 7. Assigning an uncertainty to P (or freeing it in the fit) and, where feasible, estimating the impact of momentum smearing would complete the error budget of the d-quark extraction.

Minor objections (4)

[Sec. IV A] The consistency check for the cos phi_h modulation reports only chi2_uncor/N = 0.710; reporting the chi2 computed with the correlated systematic uncertainties as well would match the treatment used in the main fit and make the check directly comparable to the fit quality.
[Sec. IV C and Fig. 13] The W+- predictions extend to qT of about 16 GeV, where qT/MW is about 0.2 and the leading-power TMD description is only marginally valid; the paper should either restrict the displayed range or quantify the power-suppressed corrections, and it should note that the rapidity limits in Eq. (58) do not include the transverse recoil that is nevertheless used in the decay kinematics.
[General] There are several typographical errors (for example, 'suﬀicient' in Sec. III D), and the Table II caption ('numbers outside (inside) parentheses') would be easier to parse if written as 'A_LL (A^cos_phi_h_LL) data points'.
[Sec. V] The EicC impact statement would be more precise with an explicit sentence noting that the 5-7% relative uncertainties are conditional on the world-data fit central values used to generate the pseudodata and on the assumed beam-polarization and nuclear-correction systematics.

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The paper

The paper claims

This paper shows existing SIDIS spin data determine how up- and down-quark helicity spreads in transverse momentum, with sea quarks and gluons still unconstrained, and verifies the result two independent ways.

The reading

This paper claims that the existing world data on the longitudinal double-spin asymmetry in semi-inclusive deep-inelastic scattering (SIDIS) determine how the helicity of up and down quarks inside a longitudinally polarized nucleon is distributed in transverse momentum. From the HERMES and CLAS measurements it extracts the transverse-momentum-dependent helicity distribution $g_{1L}(x, k_T^2)$ — the difference between quark densities polarized parallel and antiparallel to the nucleon spin — and finds the up and down components constrained away from zero, while sea quarks and gluons stay compatible with zero. Two checks anchor the result: integrating the extracted TMDs over transverse momentum up to the hard scale reproduces the collinear helicity parton distributions within uncertainties, and the same distributions, fed through a Wandzura-Wilczek-type approximation, predict the $\cos\phi_h$ modulation of the asymmetry that was deliberately excluded from the fit. If the paper is right, the transverse-momentum profile of valence-quark spin is no longer an open question but a determined input for future spin physics.

What carries the argument

The object carrying the argument is the azimuthally averaged double-spin asymmetry $$A_{LL} = \sqrt{1-\$varepsilon^{2}$}\,\frac{B_0[g_{1L},D_1]}{B_0[f_1,D_1]},$$ in which the hard matching coefficient cancels, so each data point constrains the flavor-weighted convolution of the helicity TMD $g_{1L}$ with the unpolarized fragmentation function $D_1$, normalized by the known unpolarized TMDs. The helicity TMD is built as a perturbative matching convolution over a deformed collinear helicity input — a shape-modified NNPDFpol1.1 — times the exponential nonperturbative transverse factor $\exp[-(\lambda_1(1-x) + \lambda_2 x + x(1-x)\lambda_5) b_T^2]$, and the fit determines the deformation and the transverse widths. For the excluded observable, the Wandzura-Wilczek-type approximation drops genuine twist-3 contributions and reduces the $\cos\phi_h$ structure function to the first $b_T$-derivative of the helicity TMD, $F_{LL}^{\cos\phi_h} \approx -(2M^2/Q)\,B_1[g_{1L}^{(1)}, D_1]$. Target- and hadron-mass corrections, NNLO small-$b_T$ matching, NNLL evolution, and a $T_0$-prescription $\chi^2$ tie the framework to the SV19 unpolarized analysis, which supplies $f_1$, $D_1$, and the evolution kernel.

Extended reading notes

Core claim

On its own terms the paper establishes that the HERMES and CLAS data constrain the helicity TMDs of up and down quarks away from zero over the probed transverse-momentum range, with magnitudes that generally decrease as $k_T$ grows. After integrating the extracted TMDs over transverse momenta up to $k_T = Q$, the results agree with the NNPDFpol1.1 collinear helicity distributions within uncertainties, tying the three-dimensional extraction back to the established one-dimensional spin structure. The fit uses the 246 $A_{LL}$ data points satisfying $\delta = q_T/Q < 0.5$ and reaches $\chi^2/N = 1.051$ with a 12-parameter ansatz that deforms the collinear helicity input in $x$ and adds a flavor-independent nonperturbative transverse factor. As a consistency check on data not used in the fit, the Wandzura-Wilczek-type approximation applied to the extracted TMDs reproduces the $\cos\phi_h$ modulation measured by HERMES and CLAS with $\chi^2_{\mathrm{uncor}}/N = 0.710$ for the 66 in-region points. The paper further reports that the sea-quark and gluon helicity TMDs are compatible with zero, and that an EicC pseudodata projection would shrink the uncertainties on all quark and antiquark components to roughly 5–7 percent.

Load-bearing premise

Everything that sets the size of the measured asymmetry is treated as exactly known: the unpolarized TMDs $f_1$ and $D_1$ from the SV19 and DSS fits, the NNPDFpol1.1 reference helicity distribution whose deformation defines the $g_{1L}$ input, and the deuteron effective polarization of 0.925, so any systematic error in those inputs shifts the extracted distributions and every prediction directly.

Editorial extensions

If this is right

The transverse-momentum profile of valence-quark polarization becomes a known quantity: in the extracted up and down distributions, the polarization rate $g_{1L}/f_1$ decreases as $k_T$ grows, consistent with the Wigner-rotation expectation in the valence region.
The agreement with NNPDFpol1.1 after integration up to $k_T = Q$ gives future TMD fits a built-in closure target: an extraction whose transverse integral disagrees with the collinear helicity PDFs would be internally suspect.
The successful $\cos\phi_h$ prediction from the Wandzura-Wilczek-type approximation means a leading-twist extraction can describe a subleading-power observable without a full twist-3 framework, within present experimental precision.
The CLAS12 and STAR predictions are concrete and testable, including sign-definite longitudinal single-spin asymmetries, negative for $W^+$ and positive for $W^-$, at RHIC kinematics.
The EicC impact study indicates the sea-quark helicity TMD, currently compatible with zero, would become measurable at intermediate $x$ with relative uncertainties near 5–7 percent.

Reading between the lines

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

Because the normalization inputs are held fixed, the quoted bands and the 'compatible with zero' verdict on sea quarks and gluons are lower bounds on the true uncertainty; propagating the errors of $f_1$, $D_1$, and the reference helicity PDF would widen them.
The world-data fit imposes one flavor-independent transverse width, so the extracted $k_T$ shape of $g_{1L}/f_1$ is partly an ansatz; the EicC projection relaxes this restriction, and flavor-separated widths from that fit would be a sharper test of the Wigner-rotation picture.
The closure test uses the single cutoff $k_T = Q$; repeating the integral at cutoffs between $Q/2$ and $2Q$ and checking that the deviation from the collinear PDFs scales as a power correction would upgrade the agreement from one curve to a quantitative scaling law.

Original abstract

We present an updated global analysis of the transverse-momentum-dependent (TMD) helicity distributions of the nucleon from measurements of the longitudinal double-spin asymmetry $A_{LL}$ in semi-inclusive deep-inelastic scattering (SIDIS). The analysis incorporates target- and detected-hadron-mass corrections, next-to-next-to-leading-order matching in the small-$b_T$ operator product expansion, and an improved treatment of correlated systematic uncertainties. The present data constrain the up- and down-quark helicity TMDs, whereas the sea-quark and gluon distributions remain weakly constrained. After integration over transverse momenta up to the hard scale, the up- and down-quark distributions agree with collinear helicity distributions within uncertainties. Using the extracted TMDs and the Wandzura-Wilczek-type approximation, we predict the $\cos\phi_h$ modulation of $A_{LL}$ and find agreement with the HERMES and CLAS measurements. We also provide predictions for SIDIS $A_{LL}$ at CLAS12 and for longitudinal single-spin asymmetries in $W^\pm$ production with leptonic decays at STAR kinematics. An impact study with pseudodata for the proposed Electron-Ion Collider in China (EicC) indicates substantial potential to improve constraints on helicity TMDs at intermediate $x$, including the presently weakly constrained sea-quark sector.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-23: 1 request, 5,862 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.92 · human participants 0.02 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.93 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present an updated global analysis of the transverse-momentum-dependent (TMD) helicity distributions of the nucleon from measurements of the longitudinal double-spin asymmetry $A_{LL}$ in semi-inclusive deep-inelastic scattering (SIDIS). The analysis incorporates target- and detected-hadron-mass corrections, next-to-next-to-leading-order matching in the small-$b_T$ operator product expansion, and an improved treatment of correlated systematic uncertainties. The present data constrain the up- and down-quark helicity TMDs, whereas the sea-quark and gluon distributions remain weakly constrained. After integration over transverse momenta up to the hard scale, the up- and down-quark distributions agree with collinear helicity distributions within uncertainties. Using the extracted TMDs and the Wandzura-Wilczek-type approximation, we predict the $\cos\phi_h$ modulation of $A_{LL}$ and find agreement with the HERMES and CLAS measurements. We also provide predictions for SIDIS $A_{LL}$ at CLAS12 and for longitudinal single-spin asymmetries in $W^\pm$ production with leptonic decays at STAR kinematics. An impact study with pseudodata for the proposed Electron-Ion Collider in China (EicC) indicates substantial potential to improve constraints on helicity TMDs at intermediate $x$, including the presently weakly constrained sea-quark sector.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present an updated global analysis of the transverse-momentum-dependent (TMD) helicity distributions of the nucleon from measurements of the longitudinal double-spin asymmetry $A_{LL}$ in semi-inclusive deep-inelastic scattering (SIDIS).	contribution 1.00	0.95	0.13	asserted 1.00
s1	The analysis incorporates target- and detected-hadron-mass corrections, next-to-next-to-leading-order matching in the small-$b_T$ operator product expansion, and an improved treatment of correlated systematic uncertainties.	method 0.96	0.88	0.04	asserted 1.00
s2	The present data constrain the up- and down-quark helicity TMDs, whereas the sea-quark and gluon distributions remain weakly constrained.	result 0.89	0.80	0.31	asserted 1.00
s3	After integration over transverse momenta up to the hard scale, the up- and down-quark distributions agree with collinear helicity distributions within uncertainties.	result 0.96	0.87	0.14	asserted 0.98
s4	Using the extracted TMDs and the Wandzura-Wilczek-type approximation, we predict the $\cos\phi_h$ modulation of $A_{LL}$ and find agreement with the HERMES and CLAS measurements.	result 0.97	0.95	0.13	asserted 1.00
s5	We also provide predictions for SIDIS $A_{LL}$ at CLAS12 and for longitudinal single-spin asymmetries in $W^\pm$ production with leptonic decays at STAR kinematics.	contribution 0.62	0.91	0.06	asserted 1.00
s6	An impact study with pseudodata for the proposed Electron-Ion Collider in China (EicC) indicates substantial potential to improve constraints on helicity TMDs at intermediate $x$, including the presently weakly constrained sea-quark sector.	result 0.38	0.86	0.18	hedged possibility 0.99

What kind of paper the classifier reports this is: theoretical result 0.30 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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