PaperlayerView exact revision on arXiv

Source-linked AI summary

Color entanglement for azimuthal asymmetries in the Drell-Yan process

M. G. A. Buffing, P. J. Mulders

arXiv:1309.4681v2hep-phnucl-th

TL;DR

The paper addresses how color entanglement from gauge links complicates TMD factorization for Drell–Yan azimuthal asymmetries. It analyzes the tree-level gauge-link structure and finds that factorized terms remain possible but acquire additional color factors, producing suppressed asymmetries and sign changes, especially for double Sivers and double Boer–Mulders observables.

Abstract

from arXiv · show

In the resummation of collinear gluons emitted together with active partons from the hadrons in the Drell-Yan process (DY) effects of color entanglement become important when the transverse directions are taken into account. It is then no longer possible to write the cross section as the convolution of two soft correlators and a hard part. We show that the color entanglement introduces additional color factors that must be taken into account in the extraction of transverse momentum dependent parton distribution functions (TMD PDFs) from azimuthal asymmetries. Examples where such effects matter are the extraction of the double Sivers and double Boer-Mulders asymmetries. Furthermore, we will argue why this color entanglement is a basic ingredient already in the tree-level description of azimuthal asymmetries.

INTRODUCTION

Collinear gluons and gauge links entangle color in hadroproduction, complicating the usual factorization into a hard part and two soft correlators. For Drell–Yan, the paper examines how this changes leading-order azimuthal asymmetries, including double Sivers and double Boer–Mulders effects.

  • Collinear gluons from both target hadrons entangle color and complicate factorization into a hard part and soft correlators.Source: INTRODUCTION · p. 1
  • Gauge links can produce color entanglement already at tree level, preventing a two-hadron process from being written as a product of two correlators.Source: INTRODUCTION · p. 1
  • The paper focuses on Drell–Yan and analyzes asymmetries involving unpolarized or polarized TMD PDFs at leading order in 1/Q.Source: INTRODUCTION · p. 1
  • Additional color effects are relevant to extracting the double Sivers and double Boer–Mulders asymmetries.Source: INTRODUCTION · p. 1

WILSON LINES AT TREE-LEVEL

At integrated qT, collinear gluons enter TMD correlators through simple Wilson lines, but measured qT requires transverse gauge connections spread across the diagram. These connections prevent straightforward color separation, although the tree-level cross section can still be organized using the full gauge-link structure.

  • The leading-order Drell–Yan cross section uses quark and antiquark TMD correlators together with hard scattering that produces a virtual photon or weak vector boson.The transverse momenta satisfy qT = p1T + p2T.Source: WILSON LINES AT TREE-LEVEL · p. 1
  • At integrated qT, collinear gluons produce simple straight Wilson lines, whereas measured qT requires gauge connections involving transverse separations and transverse gluons.Source: WILSON LINES AT TREE-LEVEL · pp. 1–2
  • Figure 1 represents the Drell–Yan process with TMD PDFs in the hadronic blobs and hard scattering producing a virtual photon.Source: Figure 1 · p. 1
  • At measured qT, gauge-link contributions arise in different diagram parts and cannot be trivially absorbed into, or redefined as, separate TMD correlators.Source: WILSON LINES AT TREE-LEVEL · p. 2
  • Staple-like Wilson lines running through minus infinity prevent color separation, with this entanglement appearing already when gauge links are included at tree level.Source: WILSON LINES AT TREE-LEVEL · p. 2

AZIMUTHAL EXPANSION OF THE PARTON CORRELATORS

The azimuthal expansion expresses TMD correlators through definite-rank functions, transverse tensors, and weighted transverse-momentum distributions. Gauge-link-sensitive operator structures introduce gluonic-pole factors and additional color-singlet configurations.

AZIMUTHAL EXPANSION OF THE PARTON CORRELATORS

  • The expansion applies to both TMD distributions and the Drell-Yan cross section through transverse moments.The transverse separations in the gauge connections correspond to the momentum variables entering the azimuthal expansion.Source: AZIMUTHAL EXPANSION OF THE PARTON… · p. 2, Figure 2 · p. 2
  • Transverse-momentum weightings relate definite-rank TMDs to azimuthally integrated TMD PDFs weighted with symmetric traceless transverse tensors.The angular dependence is carried by transverse tensors constructed from p_T or q_T.Source: AZIMUTHAL EXPANSION OF THE PARTON… · p. 2
  • The correlator expansion uses quark or gluon matrix elements containing additional derivatives or gluonic fields determined by the inserted operator.Only particular combinations of these correlators are needed for TMD PDFs.Source: AZIMUTHAL EXPANSION OF THE PARTON… · p. 2
  • For a gauge link U, transverse weighting separates a derivative-related correlator from a gluonic-pole matrix element multiplied by the gauge-link coefficient C[U].The gluonic-pole matrix element represents emission of a zero-momentum collinear gluon, while the resulting functions are collinear and gauge-link independent.Source: AZIMUTHAL EXPANSION OF THE PARTON… · p. 2
  • Multiple color-singlet constructions require an extra index c, while staple-like links have one relevant configuration with C[±]GG,2 = 0.Weighted results provide a finite parametrization in terms of definite-rank TMDs, azimuthal tensors, and gluonic-pole factors.Source: AZIMUTHAL EXPANSION OF THE PARTON… · p. 3
  • The azimuthal expansion is finite-rank: quark correlators in a nucleon reach rank 2, whereas gluon correlators can reach rank 3.Time-reversal symmetry constrains the highest-rank structures, and the tensors and correlators are made traceless for unique identification.Source: AZIMUTHAL EXPANSION OF THE PARTON… · p. 3

AZIMUTHAL EXPANSION FOR THE CROSS SECTION

At measured qT, transverse-momentum weightings expose color entanglement in gluonic-pole contributions, adding process-dependent color factors beyond the basic 1/Nc factor. These factors can suppress azimuthal asymmetries and, in some cases, reverse their signs.

Color entanglement

  • Mixed transverse-momentum contributions acquire color-entangled Wilson-line factors when gluonic-pole correlators are inserted.The inserted gluon field carries a color charge at the gauge connection, while some terms retain the usual 1/Nc trace splitting.Source: AZIMUTHAL EXPANSION FOR THE CROSS… · p. 3

Color factors

  • Higher-weighted contributions use ratios of symmetrized color charges, which generally suppress multiple gluonic-pole contributions in azimuthal asymmetries.The color factors depend on gluonic rank and are insensitive to partial-derivative terms.Source: AZIMUTHAL EXPANSION FOR THE CROSS… · p. 3

Azimuthal expansion

  • For the two-gluonic-pole term, the additional color factor causes both suppression and a sign change relative to naive parton calculations.This is the concrete effect of the color trace involving four color generators.Source: AZIMUTHAL EXPANSION FOR THE CROSS… · p. 3
  • The resulting tree-level cross section is a sum of terms with characteristic azimuthal dependences and process-dependent factors tied to the gluonic ranks of the TMDs.For Drell–Yan, the relevant quark-correlator factors are tabulated in Table I.Source: AZIMUTHAL EXPANSION FOR THE CROSS… · pp. 3–4, Table I · p. 4

Cross-section structure

  • The squared amplitude remains a transverse-momentum convolution but receives an additional color factor beyond the basic 1/Nc.For Drell–Yan, these factors depend on the gluonic pole ranks and gauge-link structures of both TMDs.Source: AZIMUTHAL EXPANSION FOR THE CROSS… · p. 4

DISCUSSION AND CONCLUSIONS

The discussion identifies transverse directions as the source of color-entanglement complications and applies them to leading azimuthal asymmetries. It concludes that additional color factors modify Drell–Yan asymmetries, while emphasizing that the tree-level analysis does not establish full factorization.

Affected asymmetries

  • The first leading-order asymmetries affected by additional color factors are double Sivers and double Boer–Mulders asymmetries.Double Boer–Mulders is accessible with unpolarized hadron beams, including at the LHC.Source: DISCUSSION AND CONCLUSIONS · p. 4

Origin of entanglement

  • Multiple transverse directions prevent the gauge-link entanglement from being undone as straightforwardly as in the collinear case.A single light-like direction can be handled by a gauge transformation, but transverse directions associated with different correlators complicate this.Source: DISCUSSION AND CONCLUSIONS · p. 4

Scope and limitations

  • The results do not prove factorization because next-to-leading-order effects, factorization breaking, and the required treatment of soft factors are not included.The authors note that soft factors may require modifications or SCET techniques.Source: DISCUSSION AND CONCLUSIONS · p. 4

Conclusion

  • The analysis concludes that gauge links already produce color entanglement at tree level and that factorized expressions require additional color factors, causing suppression and sign changes in Drell–Yan asymmetries.The factorized terms retain characteristic azimuthal structure but deviate from naive expectations.Source: DISCUSSION AND CONCLUSIONS · p. 4
Loading 1309.4681v2…