What exactly is a parton density?

John C. Collins

REVIEW

What exactly is a parton density?

John C. Collins

High Energy Physics - Phenomenology · Submitted 12 Apr 2003 · 28 references

Pith's verdict: A clear, honest diagnostic of the light-cone rapidity divergence in unintegrated pdf's, plus a constructive fix that the author himself labels a conjecture.

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arXiv IDhep-ph/0304122 (version 3)

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CitationJohn C. Collins. "What exactly is a parton density?." arXiv:hep-ph/0304122 (2003). https://pith.science/paper/ETFGZWN2

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Editor's note

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This is Collins writing up a lecture, and it reads like one: patient, first-principles, and unusually candid about what is and isn't nailed down. The part worth your time is the diagnostic. Sections 2.4.2 and 2.4.3 walk through a one-loop calculation (Eqs. 12–14) showing that the light-like Wilson-line definition of an unintegrated pdf has a divergence that survives nonzero quark and gluon masses, so it is not an IR artifact — it comes from the gluon rapidity going to minus infinity, a region with nothing to do with the proton-collinear momenta the pdf is supposed to describe. The observation that real and virtual pieces cancel only for test functions independent of transverse momentum is a real, checkable result, and it explains cleanly why the Brodsky-style kT cutoff definition (Eq. 8) fails. That half of the paper stands on its own and is the reason to keep the preprint on the shelf. The integrated-pdf discussion in Sec. 2.3 is standard Collins–Soper–Sterman, correctly attributed, and the point that UV renormalization kills any literal number interpretation is well made. The constructive half is Eq. (18): a light-like Wilson-line correlator divided by a vacuum expectation value of Wilson lines. Collins introduces it with 'I conjecture', says the form 'appears to follow almost uniquely' from one-loop asymptotics, and in Sec. 4.3 admits that a reader hunting for detailed justification 'will probably be quite frustrated.' So the stress-test's concern is correct but not a discovery — the paper already owns it. What is genuinely unresolved is the matching of divergence structures: the numerator's rapidity divergence lives in the quark–Wilson-line system, the denominator's in pure light-like cusps, and nothing here shows the two agree beyond leading logarithm, in subleading color, or at two loops. Footnote 7 also leaves the relative signs of the u' components open, so even the one-loop-fixed definition is not fully specified. The reliance on [27], listed as in preparation, is a real gap. None of this is overclaiming; it is an honest proposal with an honest evidence base. Worth a referee's time as a proceedings contribution, mainly for students and for anyone building TMD factorisation who wants the divergence story told straight.

Referee report

Significance

The diagnostic half is a genuinely useful piece of systematization. The one-loop computation of Eqs. (12)–(14) is explicit and checkable, and the observation that the rapidity divergence cancels only for transverse-momentum-independent test functions is a sharp, falsifiable statement that clarifies why cutoff-based definitions of integrated densities fail. The treatment of the densities as generalized functions, to be smeared with test functions, is the right technical device and is used consistently. The paper is also candid about the status of its claims: the '??' markers distinguish definitions the author believes fail from those he believes valid, and Sec. 4.3 explicitly warns that detailed justification is often absent from the literature. That candor is a strength and should be preserved. The constructive proposal in Eq. (18) is potentially significant for the transverse-momentum-dependent factorization program, since a definition with an exactly homogeneous rapidity evolution would remove a known obstruction, but its status in this manuscript is that of a conjecture supported by one-loop evidence, not an established result.

Major objections (3)

[Sec. 3.3, Eq. (18)] The paper's own framing of Eq. (18) is internally inconsistent in a way that matters for how the result will be cited. The equation is introduced with 'I conjecture that a valid definition is', and the supporting evidence is described as one-loop calculations in Collins and Hautmann [24] and in the unpublished [27]. Yet two paragraphs later the text states 'I propose that either of the last two methods is appropriate and valid, with the third method being the more elegant mathematically.' The second sentence asserts validity where the first asserts a conjecture, and the abstract's 'An improved definition of unintegrated parton densities is proposed' inherits the stronger reading. Nothing in the manuscript exhibits a mechanism by which the numerator's rapidity divergence and the denominator's pure-Wilson-line divergences match beyond the leading logarithm, and the stated motivation for preferring Eq. (18) over Eq. (16), namely that the exact evolution equation for Eq. (16) has an inhomogeneous power-suppressed term, is not shown to be cured by Eq. (18) at any order. The author should either supply an argument (even a sketch at two loops) that the ratio is finite and yields a homogeneous equation, or state plainly that this is an open conjecture and identify the specific calculation that would test it.
[Sec. 3.3, Eq. (18) and footnote 7] As written, Eq. (18) is not a fully specified definition, which prevents a reader from using it. Footnote 7 states that 'there will probably be different signs in its plus and minus components [24, 27], to allow an exactly correct derivation of factorization', so the direction u' of the Wilson lines in the denominator is left undetermined. A definition whose renormalization factor depends on an unspecified sign choice is not yet an operator definition. A second, related specification issue is not addressed anywhere: the denominator is a vacuum expectation value of Wilson lines attached at the same points y and 0 as the numerator, so it is not manifestly a c-number independent of y_T. If it carries transverse-coordinate dependence, dividing by it will modify the y-minus dependence of the correlator and hence the Fourier transform that defines the x-dependence in Eq. (15). The paper should state explicitly whether the denominator is translation invariant in the transverse directions, and if not, how the Fourier transform is to be taken.
[Sec. 2.4.2] The all-orders statement that unintegrated densities defined with light-like Wilson lines are divergent rests on the Collins-Soper equation, and the paper itself records that 'The technical derivation of the Collins-Soper equation was actually only given for unintegrated fragmentation functions. They and Sterman [9] stated the corresponding result for pdf's, but did not give an actual proof.' The explicit one-loop calculation that follows verifies the divergence at first order only, so the general claim is supported by an unproved extension of a result derived in a different context. This is a load-bearing step for the paper's central diagnostic, not a peripheral remark. The author should either restrict the claim to what the one-loop calculation and the fragmentation-function derivation jointly support, or indicate what in the pdf case would differ from the fragmentation case and why the extension is safe.

Minor objections (7)

[Eq. (11)] There is a typographical error in the smearing integral. With P0(x, kT) = delta(1-x) delta^(2-epsilon)(kT) as stated in the preceding sentence, the integral with test function t(x, kT) evaluates to t(1, 0_T), not t(0, 0_T) as printed. Eq. (14) uses t(1, 0_T) consistently, which confirms the intended value.
[Sec. 2.4.1, footnote 4] The argument that parton densities differ from number densities by infinite factors depends on canonical commutation relations holding for the Heisenberg fields as defined perturbatively, and footnote 4 concedes that 'It would be useful to verify that these commutation relations do indeed follow from properties of the Heisenberg fields'. Since the conclusion is also supported independently by the UV renormalization argument, the author could simply note that the commutation-relation route is heuristic and rest the claim on renormalization, which would be cleaner than leaving an acknowledged gap in the main text.
[Sec. 2.2] The remark that a many-body physicist would bring in the concept of a response function, followed by 'whose exploration deserves a separate discussion', is a dangling promise with no citation. Either give one or two references where that comparison is made, or delete the sentence.
[Sec. 3.3, reference [27]] Reference [27] is listed as 'in preparation' and is one of only two sources cited for the one-loop support of Eq. (18). Since an unpublished reference cannot be checked by the reader, the author should either make the argument self-contained or clearly mark which parts of the one-loop evidence rest on the unpublished work.
[Sec. 2.2 and footnote 2] The abstract and title refer to parton densities generally, but the body treats only the quark case, with the gluon distribution relegated to footnote 2. A short paragraph stating whether the constructions of Eqs. (16) and (18) extend to gluons, and where the additional complications lie, would make the scope of the claims unambiguous.
[Sec. 4.2] The bullet stating that the transverse-momentum integral of an unintegrated density should not be expected to equal the corresponding integrated density is correct and important, but it would be more useful with an indication of the size and perturbative order of the mismatch, or a pointer to the specific equation in [16, 20] where the coefficient is given.
[General] The manuscript as supplied contains numerous broken words and spacing artifacts ('Laborato ry', 'co n-ventional', 'de ﬁnitions', 'qu antiﬁcation'), which should be cleaned up in the final version. The '??' markers are intentional and effective, but a single explanatory footnote at first use would help readers who encounter the published version without this context.

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The paper

The paper claims

Parton densities are operator expectation values, not literal quark counts, and the transverse-momentum-dependent ones need a Wilson-line ratio to be finite at all.

The reading

Collins asks what a parton density actually is: not a heuristic picture of quarks rattling around inside a proton, but the expectation value of a specific operator in the proton state, defined precisely enough that a factorization theorem can be proved with it. For densities integrated over all transverse momentum, he argues the question is settled: a quark-antiquark correlator joined by a light-like Wilson line, renormalized in the ordinary way, whose scale dependence is exactly the DGLAP equations. For unintegrated, transverse-momentum-dependent densities the obvious definitions break, and he demonstrates this with a one-loop calculation in a superrenormalizable Abelian model with massive gluon and massive quark, where a divergence appears from gluons whose rapidity runs to minus infinity, a region of momentum with nothing to do with the proton-collinear momenta the density is meant to describe. He then proposes two repairs, one using Wilson lines in a non-light-like direction and one using light-like lines divided by a vacuum expectation value of Wilson lines, and argues the second is mathematically cleaner. The payoff, if it holds, is that the transverse-momentum-dependent densities used for Drell-Yan at low transverse momentum and for semi-inclusive deep-inelastic scattering become well-defined operators obeying a homogeneous evolution equation, giving the factorization program a rigorous starting point and a clean interface to models of non-perturbative hadron structure.

What carries the argument

The load-bearing object is the Wilson line, the path-ordered exponential of the gluon field that joins the quark and antiquark fields and makes the correlator gauge invariant. Its direction is the whole story: a light-like direction $n^mu$ makes the density match the collinear approximation used in factorization but introduces a divergence from the region where the gluon's rapidity goes to minus infinity, while a non-light-like direction $u^mu$ removes the divergence but leaves a parameter $zeta=(p cdot u)^2/u^2$ on which the density depends. The proposed repair is a ratio of two such Wilson-line objects, a proton matrix element in the numerator and a vacuum expectation value in the denominator, so that the divergent factor is cancelled multiplicatively rather than cut off. The auxiliary rapidity cut-off and the renormalization scale $mu$ are then traded for universality through the Collins-Soper-Sterman evolution equations in $zeta$ and the DGLAP equations in $mu$.

Extended reading notes

Core claim

The paper's central claim is that the natural definition of an unintegrated parton density, a quark correlator with a light-like Wilson line (or, equivalently, the non-gauge-invariant definition evaluated in light-cone gauge $A^+=0$), is divergent, and that the divergence is not an ordinary ultraviolet or soft divergence that renormalization can absorb. Working with a massive gluon in an Abelian model in $4-2$ dimensions so that no wave-function renormalization is needed, Collins computes the one-loop real-emission density and finds a $4/(1-x)$ term whose integral diverges at $x=1$; the virtual graphs contribute minus exactly that integral, so the divergence cancels only when the density is smeared against a test function independent of transverse momentum. Any definition that cuts off the transverse-momentum integral, or that keeps transverse-momentum dependence, inherits an uncancelled divergence traced to gluons of infinitely negative rapidity. He therefore proposes a replacement: keep the light-like Wilson line but divide by the vacuum expectation value of a Wilson-line operator built from the same light-like direction and an auxiliary non-light-like direction, Eq. (18), a generalized renormalization that is conjectured to cancel the divergence while preserving the factorization constraints on Wilson-line directions. A second, more conservative option is to define the density with Wilson lines in a non-light-like direction $u^mu$, which is a gauge-invariant transcription of the earlier planar-gauge definition, at the cost of an evolution equation with a power-suppressed inhomogeneous term.

Load-bearing premise

The repair in Eq. (18) is assumed to cancel the rapidity divergence at every order in perturbation theory, but the paper supports this only with one-loop calculations, and the author states plainly in Sec. 4.3 that a reader seeking detailed justification for many of the statements will be frustrated.

Editorial extensions

If this is right

Unintegrated parton densities become finite operator matrix elements defined without a transverse-momentum cut-off, so the factorization theorems for low-transverse-momentum Drell-Yan and for semi-inclusive deep-inelastic scattering have a well-defined non-perturbative input.
The generalized renormalization factor gives a subtraction-based recipe that can be implemented in Feynman-graph calculations, rather than requiring the awkward non-light-like Wilson lines of the earlier definition.
The evolution equation in the rapidity cut-off becomes exactly homogeneous, removing a power-suppressed inhomogeneous term that had to be discarded in applications.
The direction of the Wilson line, future- or past-pointing, is tied to the process, which fixes the relative signs of time-reversal-odd densities such as the Sivers function between DIS-type and Drell-Yan-type reactions.
Momentum and quark-number sum rules remain exact for the integrated densities under a suitable renormalization scheme, even though the densities themselves cannot be read as literal parton counts.

Reading between the lines

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

If the ratio in Eq. (18) cancels the rapidity divergence at every order, the same subtraction pattern should apply to unintegrated fragmentation functions and to generalized parton densities, which carry the same light-cone singularity structure.
The proposal makes a sharp perturbative target: the two-loop coefficient of the rapidity divergence in the light-like correlator should be exactly reproduced by the two-loop vacuum Wilson-line correlator, so the conjecture is checkable order by order without any non-perturbative input.
Because the density is defined as a ratio of matrix elements rather than a number operator expectation, the frequent statement that structure functions are not parton probabilities becomes a structural fact about the definition rather than a dynamical statement about binding.
A clean operator definition also gives lattice calculations of transverse-momentum-dependent densities a definite continuum target, since the rapidity cut-off and its evolution can be matched to the lattice scheme.

Original abstract

I give an account of the definitions of parton densities, both the conventional ones, integrated over parton transverse momentum, and unintegrated transverse-momentum-dependent densities. The aim is to get a precise and correct definition of a parton density as the target expectation value of a suitable quantum mechanical operator, so that a clear connection to non-perturbative QCD is provided. Starting from the intuitive ideas in the parton model that predate QCD, we will see how the simplest operator definitions suffer from divergences. Corrections to the definition are needed to eliminate the divergences. An improved definition of unintegrated parton densities is proposed.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,217 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.30 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

I give an account of the definitions of parton densities, both the conventional ones, integrated over parton transverse momentum, and unintegrated transverse-momentum-dependent densities. The aim is to get a precise and correct definition of a parton density as the target expectation value of a suitable quantum mechanical operator, so that a clear connection to non-perturbative QCD is provided. Starting from the intuitive ideas in the parton model that predate QCD, we will see how the simplest operator definitions suffer from divergences. Corrections to the definition are needed to eliminate the divergences. An improved definition of unintegrated parton densities is proposed.

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	I give an account of the definitions of parton densities, both the conventional ones, integrated over parton transverse momentum, and unintegrated transverse-momentum-dependent densities.	contribution 0.82	0.70	0.08	asserted 0.95
s1	The aim is to get a precise and correct definition of a parton density as the target expectation value of a suitable quantum mechanical operator, so that a clear connection to non-perturbative QCD is provided.	contribution 0.83	0.39	0.26	asserted 0.80
s2	Starting from the intuitive ideas in the parton model that predate QCD, we will see how the simplest operator definitions suffer from divergences.	background 0.78	0.40	0.03	asserted 0.97
s3	Corrections to the definition are needed to eliminate the divergences.	background 0.72	0.17	0.08	asserted 1.00
s4	An improved definition of unintegrated parton densities is proposed.	contribution 0.97	0.89	0.46	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.59 · evidence from argument 0.74.

What was examined, and what was not

sources read

arXiv listing abstract v3

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

Figures

**Figure 1.** Deep-inelastic scattering. 2. Development of definitions of pdf ’s 2.1. Parton model The basic ideas of pdf’s and the parton model are due to Feynman [4] and predate QCD. Consider deep-inelastic scattering (DIS) in the e–p center-ofmass frame —

**Figure 2.** (a) Handbag diagram for DIS. (b) Structure of general lea

**Figure 3.** Diagrammatic interpretation of Eq. (18). This is ordinary pdf

Figures reproduced from arXiv:hep-ph/0304122 by the authors.

References

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Non-perturbative contributions to vector-boson transverse momentum spectra in hadronic collisions
hep-ph 2020-02 conditional novelty 5.0 of 10

Global and LHC-only fits to low-transverse-momentum Drell-Yan data show that the intrinsic transverse-momentum distribution and the non-perturbative Sudakov kernel are correlated, and that current data do not constrai...
Limits on TMD Evolution From Semi-Inclusive Deep Inelastic Scattering at Moderate $Q$
hep-ph 2014-01 conditional novelty 5.0 of 10

At momentum transfers of a few GeV, the coordinate-space region dominating semi-inclusive deep inelastic scattering extends to transverse distances of several GeV^-1, so the nonperturbative part of the Collins-Soper k...
Scheme dependence and Transverse Momentum Distribution interpretation of Collins-Soper-Sterman resummation
hep-ph 2015-05 conditional novelty 4.0 of 10

Three TMD factorization schemes yield identical non-perturbative form factors when fitted to Drell-Yan data, consistent with standard CSS resummation.
Intrinsic transverse parton momenta in deeply inelastic reactions
hep-ph 2010-03 conditional novelty 4.0 of 10

The Gauss model describes intrinsic transverse parton momenta in SIDIS and Drell-Yan well, and the widths extracted from the two processes are compatible, supporting TMD factorization.
TMD Handbook
hep-ph 2023-04 accept novelty 2.0 of 10

A multi-author reference volume that fixes conventions, gives field-theoretic definitions of TMD parton distributions and fragmentation functions, and surveys factorization, evolution, lattice methods, models, and phe...
The x-dependence of hadronic parton distributions: A review on the progress of lattice QCD
hep-lat 2020-10 accept novelty 2.0 of 10

Lattice QCD has reached the point of producing x-dependent PDFs, first GPDs and exploratory TMD PDFs with quantified systematics, but reliable hadron boosts near or above 2 GeV remain out of reach.
Azimuthal and Single Spin Asymmetries in Hard Scattering Processes
hep-ph 2007-12 conditional novelty 2.0 of 10

Large left-right asymmetries in pion production off polarized protons are best described by a transverse-momentum-dependent Sivers function, but the factorization underpinning single-inclusive hadroproduction remains ...
Generalized Parton Distributions
hep-ph 2003-07 accept novelty 2.0 of 10

A comprehensive 2003 review consolidating the theory of generalized parton distributions: their definitions, symmetries, scale evolution, models, and the exclusive processes that measure them.
Parton Distribution Functions and their Generalizations
hep-ph 2025-07 unverdicted novelty 1.0 of 10

A textbook-style introduction to the definitions, QCD properties, experimental access, and current phenomenological status of PDFs and their generalizations.
Transverse-Spin and Transverse-Momentum Effects in High-Energy Processes
hep-ph 2010-11 accept novelty 1.0 of 10

The review consolidates the theory and the HERMES, COMPASS, Belle and RHIC measurements of transverse-spin asymmetries, and the global fits that extract transversity and the Collins function from them.

How the literature uses it

46 arXiv papers cite this in 52 sentences

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 · 22 sentences without a confident label

Compares with it 1 paper

This is different than for example the inclusive jet substructure observables discussed in [ref] where an additional rapidity regulator needs to be introduced (thus subtleties could arise [ref] ) which corresponds to SCET$_II$.

Jet angularity measurements for single inclusive jet production · Integrating the semi-inclusive angularity jet function · 0.91

Cites it as support 3 papers

Such a gauge link is process dependent [ref] , so the resulting augmented LFWFs are not universal [ref] .

Novel QCD Phenomena · Diffractive Deep Inelastic Scattering · 0.98
Such a gauge link is process dependent [ref] , so the resulting augmented LFWFs are not universal [ref] .

Novel High Transverse Momentum Phenomena in Hadronic and Nuclear Collisions · Diffractive Deep Inelastic Scattering · 0.96
Such a gauge link is process dependent [ref] , so the resulting augmented LFWFs are not universal [ref] .

Novel QCD Effects from Initial and Final State Interactions · Diffractive Deep Inelastic Scattering · 0.97

Cites it as background 26 papers

In the nonsingular gauge, in which the gauge potential vanishes at the space-time infinity, the transverse-momentum parton distribution is defined in the literature as [ref] [equation] where [equation] and all fields are evaluated at equal $y^+=0$.

Derivation of the Gauge Link in Light Cone Gauge · introduction · 0.80
Intrinsic transverse momentum distributions enter as boundary conditions to the renormalization group evolution equations for TMDs, while non-perturbative Sudakov effects are taken into account via non-perturbative contributions to the kernel of the evolution equations associated with TMD rapidity divergences [ref] .

Non-perturbative contributions to vector-boson transverse momentum spectra in hadronic collisions · Introduction · 0.80
This 3D structure is fundamentally encoded in transverse momentum-dependent parton distribution functions (TMD PDFs) [ref] .

Polarized jet anisotropy at the Electron-Ion Collider · Introduction · 0.99

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.

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