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No Generalized TMD-Factorization in the Hadro-Production of High Transverse Momentum Hadrons

Ted C. Rogers, Piet J. Mulders

arXiv:1001.2977v2hep-ph

TL;DR

Hadro-production requires transverse-momentum-dependent correlators, but standard TMD-factorization lacks the process-independent Wilson-line structures needed for universal PDFs and FFs. The paper uses a non-Abelian spectator model and a double spin asymmetry to test generalized factorization with process-dependent Wilson lines. Two gluons collinear to different hadrons produce contributions inconsistent with generalized TMD-factorization based on separate hadron correlators.

  • Problem

    Hadro-production requires transverse-momentum-dependent correlators, but standard TMD-factorization lacks the process-independent Wilson-line structures needed for universal PDFs and FFs.

    Source: §I · pp. 2–3

  • Method

    The paper uses a non-Abelian spectator model and a double spin asymmetry to test generalized factorization with process-dependent Wilson lines.

    Source: §II · p. 6, §V · p. 13

  • Results

    Two gluons collinear to different hadrons produce contributions inconsistent with generalized TMD-factorization based on separate hadron correlators.

    Source: §VI · p. 14, §VII · pp. 16–17

  • Takeaways & Limitations

    The counterexample shows that separate process-dependent TMD PDFs for each hadron cannot generally describe transverse momentum in back-to-back high-pt hadro-production.

    Source: §VII · p. 16

    The counter-proof shows that naive generalized TMD factorization fails, while the role of Wilson lines and factorization-breaking effects remains open for further research.

    Source: §VII · p. 16

Abstract

from arXiv · show

It has by now been established that standard QCD factorization using transverse momentum dependent parton distribution functions fails in hadro-production of nearly back-to-back hadrons with high transverse momentum. The essential problem is that gauge invariant transverse momentum dependent parton distribution functions cannot be defined with process-independent Wilson line operators, thus implying a breakdown of universality. This has led naturally to proposals that a correct approach is to instead use a type of "generalized" transverse momentum dependent factorization in which the basic factorized structure is assumed to remain valid, but with transverse momentum dependent parton distribution functions that contain non-standard, process dependent Wilson line structures. In other words, to recover a factorization formula, it has become common to assume that it is sufficient to simply modify the Wilson lines in the parton correlation functions for each separate hadron. In this paper, we will illustrate by direct counter-example that this is not possible in a non-Abelian gauge theory. Since a proof of generalized transverse momentum dependent factorization should apply generally to any hard hadro-production process, a single counter-example suffices to show that a general proof does not exist. Therefore, to make the counter-argument clear and explicit, we illustrate with a specific calculation for a double spin asymmetry in a spectator model with a non-Abelian gauge field. The observed breakdown of generalized transverse momentum dependent factorization challenges the notion that the role of parton transverse momentum in such processes can be described using separate correlation functions for each external hadron.

I. INTRODUCTION

The introduction explains why standard TMD-factorization fails for back-to-back high-transverse-momentum hadro-production and why replacing Wilson lines with process-dependent ones may still be insufficient. The paper proposes a non-Abelian counterexample using a double spin asymmetry.

I. INTRODUCTION

  • TMD-factorization aims to describe transverse-momentum-sensitive hadro-production using transverse-momentum-dependent PDFs and FFs.Such observables can probe hadron structure and parton transverse momentum.Source: §I · p. 1
  • Standard TMD-factorization requires reusable, process-independent correlation functions, but hadro-production generally produces process-dependent Wilson-line structures.Extra gluons connecting spectators to initial- and final-state partons obstruct the usual Ward-identity argument.Source: §I · p. 2
  • Unlike collinear factorization, cancellations after transverse-momentum integration do not generally hold point-by-point for TMD-factorization.This prevents the cancellations used in standard hadron-hadron factorization proofs from resolving the TMD problem.Source: §I · p. 2
  • Generalized TMD-factorization assumes that separate hadron correlators with process-dependent Wilson lines preserve the basic factorized structure.Different subprocesses and color routings may require different TMD PDFs and FFs, without introducing joint matrix elements involving multiple hadrons.Source: §I · pp. 2–3
  • Previous studies considered extra gluons from one hadron at a time, leaving simultaneous non-Abelian gluons from different hadrons unresolved.The paper targets whether such gluons can jointly eikonalize and factorize after summing graphs.Source: §I · p. 3
  • The paper shows that generalized TMD-factorization breaks down at two extra gluons using a double Sivers contribution to a double transverse spin asymmetry.A spectator model limits the relevant graphs and tests whether graph-level violations cancel in their sum.Source: §I · p. 3

II. SETUP

The setup uses a spectator model with a massless non-Abelian SU(Nc) gauge field and a colorless hard interaction to isolate the color structure relevant to factorization breaking. It focuses on Glauber-region virtual gluons and the conditions needed for a counterexample.

II. SETUP

  • The model replaces the Abelian gauge field of earlier studies with a massless non-Abelian SU(Nc) gauge field.Quarks are scalar fields, hadrons and spectators are Dirac spinors, and QCD corresponds to Nc = 3.Source: §II · pp. 3–4
  • A massive color-neutral U(1) boson X mediates the hard subprocess, simplifying the color flow while retaining colored initial- and final-state partons.The authors state that the general conclusions do not depend on this specific hard interaction.Source: §II · p. 4
  • At zeroth order, the model has one subprocess and one color flow, so generalized factorization would require a single separately factorized term.The basic graph describes high-pt hadro-production with no extra gluon attachments.Source: §II · p. 4, Figure 1 · p. 4
  • The analysis adopts collinear approximations for struck partons and treats final-state high-transverse-momentum hadrons or jets as on-shell final-state quarks.The incoming hadrons move in opposite large-rapidity directions and split into active quarks and spectators.Source: §II · p. 4
  • Factorization requires uncanceled soft or collinear singularities to be absorbed into nonperturbative correlation functions, including Wilson-line contributions.The discussion leaves aside several technical TMD-definition issues, including light-cone divergences, soft factors, and overlap subtraction.Source: §II · pp. 5–6
  • Glauber-region virtual gluons prevent the usual Ward-identity approximations unless contours can be deformed or graph cancellations occur.The required contour deformations differ according to initial- or final-state attachments, while transverse-momentum integration can restore cancellations in collinear factorization.Source: §II · p. 6
  • The factorization-breaking interactions arise from gluons whose momentum components are all small, associating them with large-distance scales.Many graphs vanish through TrC[ta] = 0, restricting the relevant graph classes for the counterexample.Source: §II · p. 6

III. ONE EXTRA GLUON

The one-extra-gluon analysis reviews how suitable contour deformations turn gluon attachments into eikonal factors associated with standard Wilson lines. It uses a single-spin-asymmetry setup to identify the relevant attachments and phases.

III. ONE EXTRA GLUON

  • The analysis begins with extra gluons attaching near the hard part on the side of their parent hadron.The zeroth-order graph has no spin asymmetry because its traces contain too few Dirac matrices for a nonzero γ5 contribution.Source: §III · pp. 6–7
  • The corresponding attachments across the cut generate the eikonal structures needed to analyze the imaginary part of the single-spin asymmetry.The figure distinguishes gluons from H1 and H2 through the shading of the eikonal attachments.Source: Figure 3 · p. 7
  • For a gluon collinear to H1, deforming its momentum integral out of the Glauber region permits the intermediate struck-quark line to be replaced by an eikonal factor.The contour-deformation direction determines the sign of the eikonal iε prescription.Source: §III · p. 7

2 TrD

At one extra gluon, the non-Abelian model reproduces the standard Wilson-line contributions while candidate anomaly terms vanish through trivial color factors.

2 TrD

  • Graphs with an extra gluon radiated from H2 do not contribute to its unpolarized TMD PDF because their real eikonal parts cancel between graphs.The H2 analysis uses n2 and the minus-direction momentum component l−.Source: §2 · p. 9
  • Single-gluon anomaly contributions vanish because they contain the trivial color factor Tr_C[t^a] = 0.Thus, only the eikonal propagators from the standard Wilson-line expansion contribute to the SSA at this order.Source: §2 · p. 9
  • The non-vanishing eikonal factors match the order-g expansion of the standard Wilson lines in the two TMD PDFs.The Wilson-line insertions are gauge-invariant operators with transverse links at light-cone infinity and contracted color indices.Source: §2 · p. 9
  • Unlike the Abelian case, the model’s one-gluon anomaly vanishes only because of its simple hard-process color structure; violations therefore first appear at two gluons or higher.When color is exchanged in the hard part, standard TMD-factorization can break already at one extra gluon.Source: §2 · p. 9
  • The one-gluon graphs suggest a standard TMD-factorization formula, but the paper marks that formula as ultimately incorrect and tests it at two gluons.The displayed construction includes separate TMD PDFs for the two hadrons and the same hard factor as at zeroth order.Source: §2 · p. 9

IV. TWO EXTRA GLUONS

The two-gluon analysis classifies which graphs reproduce standard Wilson lines and isolates the remaining graphs that can generate factorization anomalies.

IV. TWO EXTRA GLUONS

  • The structure of the two-gluon anomaly constrains possible generalizations of TMD factorization.The paper uses this structure to motivate the later analysis of generalized factorization.Source: §IV · pp. 9–10
  • Figure 6 represents the standard-Wilson-line contributions at two-gluon order, including Hermitian conjugates and configurations with both gluons collinear to p2.Figure 5 supplies a three-point interaction contributing to a single-gluon Wilson-line attachment.Source: Figure 6 · p. 10, Figure 5 · p. 10
  • Graphs with internal second-gluon attachments contribute only further terms to the single-gluon standard Wilson lines, while many others vanish through Tr_C[t^a] = 0.The surviving order-g^2 Wilson-line graphs are those represented in Figs. 6, 7, and 8.Source: §IV · p. 10
  • At order g^2, the analysis separates graphs consistent with standard factorization, graphs violating it, and graphs reserved for testing generalized factorization.Graphs of the last type are deferred to Sect. VI.Source: §IV · p. 10

A. Graphs Contributing to Standard Factorization

Graphs of the Fig. 6 type correspond one-to-one with the separate Wilson-line expansions in the standard TMD-factorization formula.

A. Graphs Contributing to Standard Factorization

  • Figures 6(a,b) give the order-g Wilson-line contribution from each TMD PDF, while Figures 6(c,d) and related graphs give order-g2 contributions in one PDF with the other at zeroth order.Source: A. Graphs Contributing to Standard… · p. 11
  • Because the Fig. 6 graphs exhaust the standard Wilson-line expansion through order g^2, any remaining uncanceled Fig. 7 or Fig. 8 terms violate standard TMD factorization.The classification therefore distinguishes Wilson-line contributions from anomalous contributions.Source: A. Graphs Contributing to Standard… · p. 11

B. Violation of Standard TMD-Factorization

Two-gluon graphs produce uncanceled contributions beyond the standard Wilson lines, yielding factorization anomalies in unpolarized scattering while some SSA graphs still vanish by color.

B. Violation of Standard TMD-Factorization

  • Some anomalous SSA graphs vanish because they contain Tr_C[t^a] = 0, but the non-cancellation of the unpolarized anomaly remains.Thus, the color suppression of selected SSA terms does not restore standard factorization.Source: B. Violation of Standard… · p. 11
  • The unpolarized anomaly comes from two gluons radiated by the same hadron and attached near the opposite side of the hard subgraph.The calculation includes all attachment patterns and replaces off-shell quark propagators by eikonal propagators.Source: B. Violation of Standard… · p. 11
  • The mismatch between opposite-side and same-side cut denominators prevents I1(k1T) + I2(k1T) from vanishing point-by-point, so standard TMD factorization fails.The resulting uncanceled terms are not accounted for by the standard Wilson lines.Source: B. Violation of Standard… · p. 12
  • Both hadrons’ TMD PDFs acquire factorization anomalies at the two-gluon level in the unpolarized cross section.For H2, the analogous eikonal factors use n2 and delta functions involving l−.Source: B. Violation of Standard… · p. 12

V. GENERALIZED TMD-FACTORIZATION

The proposed generalized TMD-factorization modifies separate hadron TMDs with process-dependent Wilson-line structures, including color-traced Wilson loops. Low-order consistency constrains these modifications, but the resulting formula cannot accommodate certain cross-hadron two-gluon contributions.

V. GENERALIZED TMD-FACTORIZATION

  • The generalized formula retains separate, process-dependent TMD PDFs and FFs while adding non-standard Wilson-line structures to the individual hadron correlators.The construction contains no matrix element involving both external hadrons together.Source: §V · pp. 12–13
  • Two-gluon factorization-anomaly terms require each modified TMD PDF to contain an extra color-traced Wilson loop.The loop’s color trace reproduces the required color factor, while its eikonal legs match the relevant gluon attachments.Source: §V · pp. 12–13
  • The proposed generalized formula ultimately fails because its separate TMD Wilson lines cannot consistently represent contributions with one gluon radiated from each hadron.Adding the required separate order-g anomaly terms would contradict the absence of a one-gluon violation established earlier.Source: §V · p. 13
  • Consistency between order-g and order-g2 terms requires the color trace, while alternative Wilson-line redefinitions are strongly constrained by the same low-order analysis.This excludes simply contracting the added Wilson lines directly with the quark fields and constrains modifications such as tilting the Wilson-line direction.Source: §V · p. 13
  • The Wilson-loop contribution vanishes at order g because a single attached gluon produces TrC[t^a] = 0, so disagreement first appears when at least two gluons are collinear to one hadron.This preserves agreement with standard factorization for graphs containing only one extra gluon.Source: §V · p. 13

VI. BREAKDOWN OF GENERALIZED TMD-FACTORIZATION

The breakdown is tested by calculating anomalous two-gluon contributions to a double Sivers effect. Graphs involving one gluon collinear to each hadron cannot be absorbed into separate modified TMDs and therefore yield a direct counterexample when their sum is nonzero.

VI. BREAKDOWN OF GENERALIZED TMD-FACTORIZATION

  • The paper targets the remaining two-gluon graphs by calculating their anomalous contribution to a double Sivers effect.Earlier graph classes either match standard TMD Wilson lines or motivate the color-traced Wilson-loop modifications.Source: §VI · p. 13
  • Figure 8 comprises 16 graph variants generated by all relevant attachments of l1 and l2, together with their Hermitian conjugates.These graphs include attachments to the lines associated with both hadrons.Source: Figure 8 · p. 14
  • Graphs with one gluon radiated collinear to each hadron cannot be incorporated by modifying Wilson lines in separate correlation functions.Such terms would require separate single-gluon anomaly contributions, contradicting the one-gluon analysis.Source: §VI · p. 14
  • A nonzero sum of the Figure 8 graphs provides the direct counterexample establishing violation of generalized TMD-factorization.The calculation must also rule out cancellations among the contributing graphs.Source: §VI · p. 14

A. Same Side of the Cut

For graphs with both extra gluons on the same side of the cut, the spectator-attachment factors combine to produce a nonzero delta-function contribution relevant to the double spin asymmetry.

A. Same Side of the Cut

  • The same-side graph sum yields the extra-gluon factor 4π2δ(l+_1)δ(l-_2) after combining the two spectator-attachment factors.The two factors of i give i2 = −1, and Hermitian-conjugate graphs produce the same result.Source: A. Same Side of the Cut · p. 14

B. Opposite Side of Cut

For graphs with one extra gluon on each side of the cut, the attachment phases instead multiply to +1, but the resulting eikonal contribution is the same as in the same-side case.

B. Opposite Side of Cut

  • Opposite-side graphs also yield 4π2δ(l+_1)δ(l-_2) because the attachment phases i and −i multiply to +1.The eikonal-factor result therefore matches the same-side graph class.Source: B. Opposite Side of Cut · pp. 14–15

C. Together

The calculation shows that graphs with gluons collinear to both hadrons produce a nonzero contribution absent from the generalized TMD-factorization formula. Thus, separately modifying Wilson lines in each TMD PDF cannot recover factorization.

C. Together

  • At two-gluon order, generalized TMD-factorization breaks down because simultaneous radiation from both hadrons produces a nonzero contribution not represented by its Wilson-line expansion.The relevant integral is generally nonzero, while no corresponding factorization-anomaly term appears in the generalized formula.Source: C. Together · p. 15
  • The double spin asymmetry receives no generalized-factorization contribution beyond standard TMD-factorization through two gluons, except same-hadron graphs affecting the unpolarized cross section.Spin dependence requires at least one extra gluon from each hadron, so the violating graphs are precisely those omitted by the separate-TMD construction.Source: C. Together · p. 15
  • Allowing one-gluon factorization anomalies in each TMD PDF would contradict the single-extra-gluon treatment established for the single spin asymmetry.Therefore, the missing double-spin contribution cannot be repaired by independently modifying the Wilson lines.Source: C. Together · pp. 15–16
  • The factorized color flow permits only color-singlet gluon exchange with each traced Wilson loop, making the single-gluon TMD contributions vanish.In the unfactorized graph, interlaced two-gluon color flow permits non-singlet exchange and yields a nonzero color factor.Source: C. Together · p. 16

VII. DISCUSSION AND CONCLUSION

The discussion concludes that generalized TMD-factorization fails in high-pT back-to-back hadro-production because non-Abelian gluons collinear to both hadrons cannot be assigned to separate TMD PDFs. The result is a counterexample to any general proof based only on process-dependent Wilson-line modifications.

VII. DISCUSSION AND CONCLUSION

  • Generalized TMD-factorization does not generally exist for high-pT back-to-back hadro-production because gluons collinear to both hadron directions interact nontrivially.The counterexample shows that separate TMD PDFs cannot consistently encode these effects, even with process-dependent Wilson lines.Source: §VII · p. 16
  • The specific breakdown is a consequence of non-Abelian color: Abelian theories can factor the discussed double-spin graphs into nonzero single-gluon Wilson-loop contributions.The paper does not establish a full generalized-factorization proof for the Abelian case.Source: §VII · p. 16
  • The breakdown occurs in the regime of hard processes with large pT and is not specific to the double or single spin asymmetries used in the illustration.The authors use color-singlet boson exchange to simplify the argument while emphasizing its broader origin.Source: §VII · p. 16
  • At higher orders, factorization-breaking attachments correspond to soft and collinear divergences rather than negligible corrections, and graphs involving both hadrons become increasingly prevalent.A viable treatment would need to account for these contributions in an all-orders reorganization.Source: §VII · p. 16
  • The counter-proof creates practical difficulties but motivates alternatives that retain Wilson lines or directly model factorization-breaking phases.The authors mention small-x methods and perturbative calculations with explicit infrared and collinear cutoffs as possible directions.Source: §VII · p. 16
  • Nonlinear phase effects prevent assigning the A+ and A− contributions independently to modified Wilson lines for the two hadrons.In the non-Abelian theory, the A− gluons from one hadron are affected by the presence of A+ gluons from the other.Source: §VII · pp. 16–17
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