Generalizing the Soffer Bound: Positivity Constraints on Parton Distributions of Spin-3/2 Particles

Dongyan Fu; Ju-Jun Xie; S. Kumano; Yubing Dong

REVIEW

Generalizing the Soffer Bound: Positivity Constraints on Parton Distributions of Spin-3/2 Particles

Dongyan Fu, Yubing Dong, S. Kumano, Ju-Jun Xie

High Energy Physics - Phenomenology · Submitted 12 Feb 2026 · 26 references

Pith's verdict: Solid first derivation of spin-3/2 PDF positivity bounds; the 'complete set' claim needs a one-line block-diagonalization argument to be fully justified.

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arXiv ID2602.11587 (version 2)

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CitationDongyan Fu, Ju-Jun Xie, S. Kumano, Yubing Dong. "Generalizing the Soffer Bound: Positivity Constraints on Parton Distributions of Spin-3/2 Particles." arXiv:2602.11587 (2026). https://pith.science/paper/LXW74JEW

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Editor's note

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Here's the short version: this is a legitimate generalization of the Soffer bound to spin-3/2, and the main result is almost certainly correct. The 'complete set' claim has a small missing justification, but it's patchable and not load-bearing.

What's actually new: the paper gives the first positivity bounds for the six quark and five gluon leading-twist PDFs of a spin-3/2 hadron, and it provides explicit relations between those PDFs and the forward helicity amplitudes. The logic is standard—build an 8-dimensional amplitude matrix in the product space of parton and hadron helicities, impose positive semidefiniteness, extract the constraints. No parameters are fitted, and the bounds are necessary consistency conditions for any model or lattice QCD extraction.

The soft spots are expositional rather than mathematical. The sufficiency of the six inequalities in Eq. (34) is asserted, not proved. For a generic sparse Hermitian matrix that would be a real gap, because nonnegative diagonals and a few 2x2 minors don't guarantee positivity. But in this case the helicity conservation condition λ'−i=λ−j block-diagonalizes the matrix into 1x1 and 2x2 blocks, as the pattern in Eq. (30) already makes visible. Once the authors state that explicitly, the principal-minor conditions on each block are both necessary and sufficient. They should add this sentence. Similarly, the step from Eq. (22) to the matrices (25)-(27) is compressed—show a bit of the algebra—and the vanishing of h_{1LT} and h_{1LTT} is asserted rather than derived, though it follows from time reversal and the cited earlier work. There are also some typos in the displayed 8x8 matrices that should be cleaned up.

Bottom line: the paper's central argument holds up. It's a solid contribution to hadron structure, specifically for the Delta baryon and other spin-3/2 systems. I'd send it to a serious referee, and I'd expect acceptance after minor revision. For a reading group, it's a nice example of positivity methods, though a bit narrow.

Referee report

Major objections (4)

[Eqs. (22)-(27)] The transition from Eq. (22) to the explicit matrix elements (25)-(27) is asserted without derivation. Because every bound in (35) depends on the numerical coefficients in these tables, the solution of Eq. (22) is load-bearing. Please show the inversion of the spin density matrix rho in the basis listed after Eq. (23) and give at least one complete example, e.g., the (3/2,3/2) diagonal element. Also clarify the 1/2 factor in Eq. (24): are the tabulated M values the entries inside the 2x2 block before the prefactor, and are the A matrices in Eq. (30) the full amplitudes after the prefactor? The two readings lead to different PDF relations in Eqs. (31)-(33).
[Positivity constraints, Eqs. (34a)-(34c)] The 'complete set' claim requires that the listed inequalities are equivalent to positive semidefiniteness of the 8x8 matrix. This is not automatic for a sparse Hermitian matrix; larger principal minors could impose extra conditions. The paper neither states that helicity conservation lambda' - i = lambda - j block-diagonalizes A into 1x1 and 2x2 blocks nor proves that the blocks are exactly of the forms shown in Eqs. (30a)-(30b). Please provide the explicit block decomposition and a short sufficiency argument; this is also needed to justify the counting formula 2s+1+ceil(s)/floor(s) for general s.
[Eq. (34c) versus Eq. (35e)] There is an internal inconsistency in the gluon inequalities. For X = A^g_{-1/2-,3/2+}, Eq. (30b) shows that the 2x2 blocks containing X have diagonals A^g_{3/2+,3/2+} and A^g_{1/2+,1/2+} (rows/columns {1,7} and {2,8}); the diagonal A^g_{-1/2+,-1/2+} does not occur in these blocks. The determinant condition should therefore be A^g_{3/2+,3/2+} A^g_{1/2+,1/2+} >= |X|^2, which is what Eq. (35e) uses. Eq. (34c) as printed uses A^g_{-1/2+,-1/2+} and is inconsistent with the PDF bound that follows from it. Please correct (34c) or, if a different block ordering is intended, explain the ordering and re-derive (35e).
[Time-reversal and h1LT / h1LTT] The text says that the quark h1LT and gluon h1LTT terms violate time reversal and are therefore zero, after noting that the helicity amplitudes are real. This conclusion is used to reduce the number of independent PDFs and affects the final bounds (35c) and (35e), which contain |h1LT|^2 and |h1LTT|^2. Please provide the explicit T-reversal argument (or the precise statement from Refs. [19,20]) and state whether (35c) and (35e) are the final physical bounds with these terms set to zero, or intermediate bounds before imposing T invariance.

Minor objections (4)

[Abstract] The abstract promises that a Cauchy-Schwartz inequality determines constraints between the PDFs and generalized parton distributions, but this is not derived or even mentioned in the body. Please either add the derivation or remove the sentence.
[Eqs. (20), (24), (25)] The notation M_{lambda'+,lambda+} is overloaded: in Eq. (24) it denotes an entry inside the 1/2-weighted 2x2 block, while in the tables it seems to denote the unweighted combination of PDFs. Please define the notation once and use it consistently, or introduce separate symbols for the block entries and the full amplitude.
[Throughout] The paper says 'positive definiteness' but the inequalities allow equality and the relevant matrices are positive semidefinite. Please use 'positive semidefinite' consistently.
[Introduction/Summary] There are a few grammatical and typographical issues, e.g., 'the definitions is the same' and the inline display after Eq. (2) where the integral measure is broken. A careful proofread would improve clarity.

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The paper

The paper claims

The paper derives, for the first time, the complete set of positivity inequalities that the leading-twist quark and gluon parton distribution functions (PDFs) of a spin-3/2 hadron must obey, generalizing the Soffer bound from spin-1/2 nucle

The reading

This paper aims to generalize the Soffer bound—the classic inequality for spin-$\frac{1}{2}$ nucleons that ties transversity to unpolarized and helicity distributions—to spin-$\frac{3}{2}$ hadrons such as the Delta resonance. The authors show that demanding positive semidefiniteness of the forward antiparton–hadron scattering amplitude, written in the combined helicity space of parton and hadron, forces the six quark and five gluon leading-twist PDFs of a spin-$\frac{3}{2}$ particle to satisfy a specific set of five inequalities. These bounds constrain any model calculation or extraction of high-spin hadron structure and complete a program that had been carried out for spin-1 particles. If correct, any future lattice or experimental determination of these distributions will have to pass these consistency checks.

What carries the argument

The key object is the helicity-amplitude matrix $A_{\lambda' i,\lambda j}$ in the eight-dimensional tensor-product space of the parton helicity (up/down for quarks, left/right for gluons) and the four spin projections of the spin-$\frac{3}{2}$ hadron (from $-\frac{3}{2}$ to $+\frac{3}{2}$). Because the forward scattering amplitude must be the square of a transition amplitude, this matrix must be positive semidefinite; the paper's move is to write every nonzero matrix element as a known linear combination of the PDFs, turning matrix positivity into explicit inequalities on the PDFs. This is accomplished by projecting the light-cone correlation function onto parton helicity states and relating the result to the hadron spin d

Extended reading notes

Core claim

The central claim is that the complete positivity bounds for a spin-$\frac{3}{2}$ hadron are given by Eqs. (35a)–(35e): two absolute-value bounds relating the unpolarized and longitudinal quark/gluon PDFs, a product inequality for the quark transversity combinations, and a product inequality for gluon transversity. These inequalities follow from requiring the $8\times 8$ helicity-amplitude matrix in the tensor product of parton helicity and hadron spin projection to be positive semidefinite, after expressing each nonzero amplitude as a linear combination of the six quark PDFs ($f_1$, $g_1$, $h_1$, $f_{1LL}$, $g_{1LLL}$, $h_{1LLT}$) or five gluon PDFs ($f_1$, $g_1$, $f_{1LL}$, $h_{1TT}$, $g_{1LLL}$). The paper also defines the PDFs directly in terms of h

Load-bearing premise

The derivation assumes that the diagonal entries and the $2\times 2$ principal minors listed in Eq. (34) are sufficient, not merely necessary, for the full $8\times 8$ helicity-amplitude matrix to be positive semidefinite; if a larger principal minor can turn negative while (34) holds, the bounds in (35) would be necessary but not complete.

Editorial extensions

If this is right

Any phenomenological or lattice extraction of spin-3/2 quark PDFs must satisfy inequalities (35a)–(35d), and gluon PDFs must satisfy (35a), (35b) and (35e); a violation at any x signals an unphysical parametrization.
In the limit where the spin-3/2-specific PDFs f1LL, g1LLL, and h1LLT vanish, the bounds reduce exactly to the spin-1/2 Soffer bound, showing the generalization is continuous.
The counting rule derived—2s+1+ceil(s) constraints for quarks and 2s+1+floor(s) for gluons—extends the method to any spin-s hadron.
The same positivity logic, via a Cauchy-Schwarz step, links PDFs to generalized parton distributions, so the inequalities constrain off-forward quantities as well.

Reading between the lines

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

If the completeness claim is right, satisfying (35) is also sufficient for the amplitude matrix to be positive semidefinite, which would make (35) a practical all-purpose admissibility test for model builders—something the paper does not explicitly assert.
The same tensor-space positivity construction could be carried over to transverse-momentum-dependent distributions or to spin-2 hadrons, where the principal-minor question for larger helicity matrices will need the same scrutiny.
A concrete numerical test of the framework: fit a simple toy model for Delta PDFs and check whether inequalities (35a)-(35e) become tight at small x, where positivity is usually the most restrictive.
The bounds as written drop the time-reversal-odd distributions h1LT and h1LTT; if a future analysis ever finds evidence they are nonzero, the full set of inequalities would need to be re-derived with those terms included.

Original abstract

We derive the complete set of positivity bounds for the leading-twist parton distribution functions (PDFs) of a spin-3/2 hadron for the first time. This work generalizes the Soffer bound, a fundamental constraint for spin-1/2 nucleons, to quark and gluon distribution functions in higher-spin systems. Expressing the antiparton-hadron scattering amplitudes in terms of the PDFs and the spin density matrix, we establish the connections between the PDFs and the scattering amplitudes in the tensor product space of the parton and hadron spins. Moreover, we obtain the definitions of the PDFs in terms of the helicity amplitudes. Positive definiteness of the scattering amplitude matrix yields a set of inequalities that define the physically allowed parameter space for the helicity amplitudes and a set of constraints for the PDFs. The Cauchy-Schwartz inequality determines the constrains between the PDFs and generalized parton distributions.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,994 input tokens.

What the abstract itself states, as probabilities: claims first 0.96 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.33 · compares with prior results 0.61 · mentions code or data availability 0.02.

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We derive the complete set of positivity bounds for the leading-twist parton distribution functions (PDFs) of a spin-3/2 hadron for the first time. This work generalizes the Soffer bound, a fundamental constraint for spin-1/2 nucleons, to quark and gluon distribution functions in higher-spin systems. Expressing the antiparton-hadron scattering amplitudes in terms of the PDFs and the spin density matrix, we establish the connections between the PDFs and the scattering amplitudes in the tensor product space of the parton and hadron spins. Moreover, we obtain the definitions of the PDFs in terms of the helicity amplitudes. Positive definiteness of the scattering amplitude matrix yields a set of inequalities that define the physically allowed parameter space for the helicity amplitudes and a set of constraints for the PDFs. The Cauchy-Schwartz inequality determines the constrains between the PDFs and generalized parton distributions.

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 derive the complete set of positivity bounds for the leading-twist parton distribution functions (PDFs) of a spin-3/2 hadron for the first time.	contribution 0.82	0.97	0.80	asserted 1.00
s1	This work generalizes the Soffer bound, a fundamental constraint for spin-1/2 nucleons, to quark and gluon distribution functions in higher-spin systems.	contribution 0.92	0.90	0.46	asserted 1.00
s2	Expressing the antiparton-hadron scattering amplitudes in terms of the PDFs and the spin density matrix, we establish the connections between the PDFs and the scattering amplitudes in the tensor product space of the parton and hadron spins.	method 0.90	0.93	0.07	asserted 1.00
s3	Moreover, we obtain the definitions of the PDFs in terms of the helicity amplitudes.	contribution 0.60	0.94	0.06	asserted 1.00
s4	Positive definiteness of the scattering amplitude matrix yields a set of inequalities that define the physically allowed parameter space for the helicity amplitudes and a set of constraints for the PDFs.	result 0.63	0.92	0.38	asserted 1.00
s5	The Cauchy-Schwartz inequality determines the constrains between the PDFs and generalized parton distributions.	result 0.67	0.82	0.13	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v2

not examined

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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

Works cited (26)

[1]

John C. Collins and Davison E. Soper. Parton Distribu- tion and Decay Functions.Nucl. Phys. B, 194:445–492, 1982

View on Pith 1982
[2]

Jacques Soffer. Positivity constraints for spin dependent parton distributions.Phys. Rev. Lett., 74:1292–1294, 1995

View on Pith 1995
[3]

Werner Vogelsang. Next-to-leading order evolution of transversity distributions and Soffer’s inequality.Phys. Rev. D, 57:1886–1894, 1998

View on Pith 1998
[4]

Y. Goto et al. Polarized parton distribution functions in the nucleon.Phys. Rev. D, 62:034017, 2000

View on Pith 2000
[5]

Umberto D’Alesio, Carlo Flore, and Alexei Prokudin. Role of the Soffer bound in determination of transver- sity and the tensor charge.Phys. Lett. B, 803:135347, 2020

View on Pith 2020
[6]

C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl. First simultaneous global QCD analysis of dihadron fragmentation functions and transversity par- ton distribution functions.Phys. Rev. D, 109(3):034024, 2024

View on Pith 2024
[7]

Akbari Jahan and Diptimonta Neog. Comparative in- sights into gluon and proton structure through parton distribution functions.Phys. Essays, 38:195, 2025

View on Pith 2025
[8]

B. Pire, Jacques Soffer, and O. Teryaev. Positivity con- straints for off - forward parton distributions.Eur. Phys. J. C, 8:103–106, 1999

View on Pith 1999
[9]

Tanmay Maji and Dipankar Chakrabarti. Parton distri- bution functions of proton in a light-front quark-diquark model.Springer Proc. Phys., 203:151–154, 2018

View on Pith 2018
[10]

Martha Constantinou et al. Parton distributions and lattice-QCD calculations: Toward 3D structure.Prog. Part. Nucl. Phys., 121:103908, 2021

View on Pith 2021
[11]

Peng Cheng, Yang Yu, Hui-Yu Xing, Chen Chen, Zhu- Fang Cui, and Craig D. Roberts. Perspective on polarised parton distribution functions and proton spin.Phys. Lett. B, 844:138074, 2023

View on Pith 2023
[12]

C´ edric Lorc´ e, Andreas Metz, Barbara Pasquini, and Pe- ter Schweitzer. Parton Distribution Functions and their Generalizations. 7 2025

View on Pith 2025
[13]

A. Bacchetta and P. J. Mulders. Positivity bounds on spin one distribution and fragmentation functions.Phys. Lett. B, 518:85–93, 2001

View on Pith 2001
[14]

Sabrina Cotogno. Parametrization of the Transverse Mo- mentum Dependent Light-Front Correlator for Gluons. Few Body Syst., 58(2):92, 2017

View on Pith 2017
[15]

D. Keller. The Transverse Structure of the Deuteron with Drell-Yan. 5 2022

View on Pith 2022
[16]

S. Kumano. Parton distribution functions and fragmen- tation functions of spin-1 hadrons.Eur. Phys. J. A, 60(10):205, 2024

View on Pith 2024
[17]

Shunzo Kumano and Kenshi Kuroki. Tensor-polarized twist-3 parton distribution functionsf LT (x) for the spin- 1 deuteron by using twist-2 relations. 9 2025

View on Pith 2025
[18]

William Rarita and Julian Schwinger. On a theory of particles with half integral spin.Phys. Rev., 60:61, 1941

View on Pith 1941
[19]

Dongyan Fu, Bao-Dong Sun, and Yubing Dong. Gener- alized parton distributions in spin-3/2 particles.Phys. Rev. D, 106(11):116012, 2022

View on Pith 2022
[20]

Dongyan Fu, Yubing Dong, and S. Kumano. Transver- sity generalized parton distributions in spin-3/2 particles. Phys. Rev. D, 109(9):096006, 2024

View on Pith 2024
[21]

Yahui Chai et al. Parton distribution functions of ∆ + on the lattice.Phys. Rev. D, 102(1):014508, 2020

View on Pith 2020
[22]

Jing Zhao, Zhe Zhang, Zuo-tang Liang, Tianbo Liu, and Ya-jin Zhou. Inclusive and semi-inclusive production of spin-3/2 hadrons in e+e- annihilation.Phys. Rev. D, 106(9):094006, 2022

View on Pith 2022
[23]

S. Kumano and Qin-Tao Song. Transverse-momentum- dependent parton distribution functions up to twist 4 for spin-1 hadrons.Phys. Rev. D, 103(1):014025, 2021

View on Pith 2021
[24]

P. J. Mulders and J. Rodrigues. Transverse momen- tum dependence in gluon distribution and fragmentation functions.Phys. Rev. D, 63:094021, 2001

View on Pith 2001
[25]

Dani¨ el Boer, Sabrina Cotogno, Tom van Daal, Piet J. Mulders, Andrea Signori, and Ya-Jin Zhou. Gluon and Wilson loop TMDs for hadrons of spin≤1.JHEP, 10:013, 2016. 8

View on Pith 2016
[26]

M. Diehl. Generalized parton distributions.Phys. Rept., 388:41–277, 2003

View on Pith 2003

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Octupole parton structure of spin-3/2 targets
hep-ph 2026-10 accept novelty 7.0 of 10

The paper constructs the complete leading-twist gluon TMD basis for spin-3/2 targets with 32 independent coefficients per Wilson-line structure, of which 14 describe octupole polarization.
Twist-2 relations for the twist-3 tensor-polarized distribution function $f_{LT}$ of a spin-1 hadron by the operator-product-expansion method
hep-ph 2026-05 unverdicted novelty 4.0 of 10

Using the local operator product expansion, the authors derive a Wandzura-Wilczek-like relation and a Burkhardt-Cottingham-like sum rule linking the twist-3 tensor-polarized distribution f_LT to the twist-2 function f...

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

There are recent theoretical developments on polarized parton distribution functions (PDFs) of spin-1 [ref] and spin-3/2 [ref] hadrons.

Twist-2 relations for the twist-3 tensor-polarized distribution function $f_{LT}$ of a spin-1 hadron by the operator-product-expansion method · Introduction · 1.00

What it cites · 17 sentences without a confident label

Cites as background 11 papers

A critical theory constraint on these functions is the Soffer bound, $|h_1(x)| ≤1/2 [f_1(x) + g_1(x)]$ [ref] , which is derived from the positivity of probability densities and provides essential constraints for global QCD analyses of the polarized deep-inelastic scattering and provides a fundamental test method for the self-consistency of the parton model [ref] .

Positivity Constraints for off-Forward Parton Distributions · Introduction · 0.81
A critical theory constraint on these functions is the Soffer bound, $|h_1(x)| ≤1/2 [f_1(x) + g_1(x)]$ [ref] , which is derived from the positivity of probability densities and provides essential constraints for global QCD analyses of the polarized deep-inelastic scattering and provides a fundamental test method for the self-consistency of the parton model [ref] .

Parton distributions and lattice QCD calculations: toward 3D structure · Introduction · 0.81
A critical theory constraint on these functions is the Soffer bound, $|h_1(x)| ≤1/2 [f_1(x) + g_1(x)]$ [ref] , which is derived from the positivity of probability densities and provides essential constraints for global QCD analyses of the polarized deep-inelastic scattering and provides a fundamental test method for the self-consistency of the parton model [ref] .

Polarised parton distribution functions and proton spin · Introduction · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

John C. Collins and Davison E. Soper. Parton Distribu- tion and Decay Functions.Nucl. Phys. B, 194:445–492, 1982

View on Pith 1982

[2] [2]

Jacques Soffer. Positivity constraints for spin dependent parton distributions.Phys. Rev. Lett., 74:1292–1294, 1995

View on Pith 1995

[3] [3]

Werner Vogelsang. Next-to-leading order evolution of transversity distributions and Soffer’s inequality.Phys. Rev. D, 57:1886–1894, 1998

View on Pith 1998

[4] [4]

Y. Goto et al. Polarized parton distribution functions in the nucleon.Phys. Rev. D, 62:034017, 2000

View on Pith 2000

[5] [5]

Umberto D’Alesio, Carlo Flore, and Alexei Prokudin. Role of the Soffer bound in determination of transver- sity and the tensor charge.Phys. Lett. B, 803:135347, 2020

View on Pith 2020

[6] [6]

C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl. First simultaneous global QCD analysis of dihadron fragmentation functions and transversity par- ton distribution functions.Phys. Rev. D, 109(3):034024, 2024

View on Pith 2024

[7] [7]

Akbari Jahan and Diptimonta Neog. Comparative in- sights into gluon and proton structure through parton distribution functions.Phys. Essays, 38:195, 2025

View on Pith 2025

[8] [8]

B. Pire, Jacques Soffer, and O. Teryaev. Positivity con- straints for off - forward parton distributions.Eur. Phys. J. C, 8:103–106, 1999

View on Pith 1999

[9] [9]

Tanmay Maji and Dipankar Chakrabarti. Parton distri- bution functions of proton in a light-front quark-diquark model.Springer Proc. Phys., 203:151–154, 2018

View on Pith 2018

[10] [10]

Martha Constantinou et al. Parton distributions and lattice-QCD calculations: Toward 3D structure.Prog. Part. Nucl. Phys., 121:103908, 2021

View on Pith 2021

[11] [11]

Peng Cheng, Yang Yu, Hui-Yu Xing, Chen Chen, Zhu- Fang Cui, and Craig D. Roberts. Perspective on polarised parton distribution functions and proton spin.Phys. Lett. B, 844:138074, 2023

View on Pith 2023

[12] [12]

C´ edric Lorc´ e, Andreas Metz, Barbara Pasquini, and Pe- ter Schweitzer. Parton Distribution Functions and their Generalizations. 7 2025

View on Pith 2025

[13] [13]

A. Bacchetta and P. J. Mulders. Positivity bounds on spin one distribution and fragmentation functions.Phys. Lett. B, 518:85–93, 2001

View on Pith 2001

[14] [14]

Sabrina Cotogno. Parametrization of the Transverse Mo- mentum Dependent Light-Front Correlator for Gluons. Few Body Syst., 58(2):92, 2017

View on Pith 2017

[15] [15]

D. Keller. The Transverse Structure of the Deuteron with Drell-Yan. 5 2022

View on Pith 2022

[16] [16]

S. Kumano. Parton distribution functions and fragmen- tation functions of spin-1 hadrons.Eur. Phys. J. A, 60(10):205, 2024

View on Pith 2024

[17] [17]

Shunzo Kumano and Kenshi Kuroki. Tensor-polarized twist-3 parton distribution functionsf LT (x) for the spin- 1 deuteron by using twist-2 relations. 9 2025

View on Pith 2025

[18] [18]

William Rarita and Julian Schwinger. On a theory of particles with half integral spin.Phys. Rev., 60:61, 1941

View on Pith 1941

[19] [19]

Dongyan Fu, Bao-Dong Sun, and Yubing Dong. Gener- alized parton distributions in spin-3/2 particles.Phys. Rev. D, 106(11):116012, 2022

View on Pith 2022

[20] [20]

Dongyan Fu, Yubing Dong, and S. Kumano. Transver- sity generalized parton distributions in spin-3/2 particles. Phys. Rev. D, 109(9):096006, 2024

View on Pith 2024

[21] [21]

Yahui Chai et al. Parton distribution functions of ∆ + on the lattice.Phys. Rev. D, 102(1):014508, 2020

View on Pith 2020

[22] [22]

Jing Zhao, Zhe Zhang, Zuo-tang Liang, Tianbo Liu, and Ya-jin Zhou. Inclusive and semi-inclusive production of spin-3/2 hadrons in e+e- annihilation.Phys. Rev. D, 106(9):094006, 2022

View on Pith 2022

[23] [23]

S. Kumano and Qin-Tao Song. Transverse-momentum- dependent parton distribution functions up to twist 4 for spin-1 hadrons.Phys. Rev. D, 103(1):014025, 2021

View on Pith 2021

[24] [24]

P. J. Mulders and J. Rodrigues. Transverse momen- tum dependence in gluon distribution and fragmentation functions.Phys. Rev. D, 63:094021, 2001

View on Pith 2001

[25] [25]

Dani¨ el Boer, Sabrina Cotogno, Tom van Daal, Piet J. Mulders, Andrea Signori, and Ya-Jin Zhou. Gluon and Wilson loop TMDs for hadrons of spin≤1.JHEP, 10:013, 2016. 8

View on Pith 2016

[26] [26]

M. Diehl. Generalized parton distributions.Phys. Rept., 388:41–277, 2003

View on Pith 2003