Nonperturbative limits of the Drell–Yan picture in QCD: factorization and universality at very small x and very small transverse momentum

Politzer's 1977 calculation of the $O(g^{2})$ gluon corrections to Drell–Yan processes verified, at that order, the factorization structure of the parton-model picture — the cancellation of double infrared logarithms, the separation of single logarithms into the two incoming hadrons, and the equality of the resulting anomalous dimensions with those of electroproduction — and posed the question of whether the parton model can be justified in Drell–Yan production to the same extent as in lepton–hadron scattering. The purely perturbative parts of that program (an all-orders factorization proof for inclusive observables, and the computation of the $O(g^{4})$ and higher corrections with coupled quark–gluon mixing) have since been completed. The surviving open core lies in the two regimes Politzer identified where perturbation theory is not a guide: (i) the very small-$x$ (wee-parton/saturation) regime, where it is not established whether and how QCD factorization and the universality of parton and transverse-momentum-dependent (TMD) distributions persist, and what the complete factorization theorem or controlled breakdown statement is; and (ii) the very small transverse-momentum regime $q_{T}\sim\Lambda_{\mathrm{QCD}}$, where the nonperturbative transverse-momentum sector (intrinsic transverse momentum and the nonperturbative Collins–Soper/Sudakov component) is still fitted to data rather than determined from first principles and matched to the perturbative resummation in a controlled way.

Background

In the Drell–Yan process, named after Drell and Yan's 1971 parton-model analysis, two hadrons collide at center-of-mass energy $\sqrt{S}$ and annihilate into a lepton pair (e.g. a $\mu^{+}\mu^{-}$ pair or a $W/Z$ boson) of invariant mass $M$. The parton-model prediction writes the cross section as a convolution of parton distribution functions (PDFs) — the probabilities $f_{i}(x)$ for finding a quark or gluon carrying momentum fraction $x$ in each hadron — with a free-parton annihilation cross section. For this to be a prediction rather than a parametrization, the same PDFs measured in deep-inelastic lepton–hadron scattering (electroproduction) must be usable in hadron–hadron collisions, which requires QCD factorization: all infrared sensitivity from gluon radiation must either cancel or be absorbed into the universal PDFs, leaving a perturbatively computable hard part. Politzer's 1977 paper computed the $O(g^{2})$ gluon radiative corrections to Drell–Yan processes in QCD and verified three non-trivial necessary conditions for this: the double logarithms $\log^{2}$ cancel between real and virtual gluon diagrams (a Sudakov-type cancellation), the surviving single logarithms $\log(-p_{1}^{2})$ and $\log(-p_{2}^{2})$ attach separately to the two incoming hadrons (here $p_{1}^{2},p_{2}^{2}$ are off-shell virtualities used as infrared regulators), and their coefficients coincide with the electroproduction anomalous dimensions (the coefficient functions of the renormalization-group evolution of the PDFs). The paper explicitly left open: (1) a proof that this structure persists to all orders — its conclusion states that the computational algorithm 'suggests a method of proof of the Drell–Yan picture which deserves further attention'; (2) the $O(g^{4})$ and higher corrections, in which quark and gluon distributions mix through a matrix (coupled) evolution and constants beyond the leading $\log M^{2}$ terms are needed for precision; (3) the behavior at very small $x$, where hadrons contain arbitrarily many soft 'wee' partons whose interactions stay strong — 'asymptotic freedom is no guide to their behavior' — so that whether wee partons spoil the Drell–Yan picture was unknown; (4) the rate of heavy-quark production from nonperturbative string breaking; and (5) the transverse-momentum ($k_{\perp}$) spectrum, where the perturbative formulae hold only for $k_{\perp}^{2}\gtrsim 1\ \mathrm{GeV}^{2}$, while for $k_{\perp}^{2}\lesssim(400\ \mathrm{MeV})^{2}$ primordial intrinsic transverse momentum, hadronic coherence effects, and final-state interactions enter with no controlled matching. The subsequent history: Collins, Soper and Sterman summarized the standard all-orders factorization theorems for hard processes in QCD and their proofs (1989), including inclusive Drell–Yan at leading power, and their 1985 work established transverse-momentum-dependent (TMD) factorization — the factorization of the small-$q_{T}$ spectrum into TMD parton distributions (PDFs differential in intrinsic transverse momentum) evolved in the Collins–Soper–Sudakov soft-radiation kernel (the resummed effect of soft gluon emission, which governs how the transverse-momentum spectrum depends on the hard scale). The higher fixed orders were subsequently computed through next-to-next-to-leading order and next-to-next-to-next-to-leading order ($\alpha_{s}^{3}$), and the small-$x$ problem was transformed into the BFKL (Balitsky–Fadin–Kuraev–Lipatov small-$x$ evolution)/saturation/Color Glass Condensate (CGC) program, an effective-field-theory description of hadrons at high energy in which the relevant small-$x$ distributions are Wilson-line (gauge-link) correlators and their evolution becomes nonlinear when gluon densities saturate. For the Drell-Yan hadronic tensor specifically, the gauge-invariant small-$x$ TMD factorization obtained so far holds only in the Sudakov region and, for the reduction of higher-twist to leading-twist TMDs, only at leading order in the number of colors $N_{c}$ (refs 10 and 11); the saturation-regime theorem or controlled breakdown statement at all orders and finite $N_{c}$, together with the universal-versus-process-dependent classification of the small-$x$ matrix elements, therefore remains open. What remains unresolved is the completion of Politzer's question precisely where asymptotic freedom is not a guide: the saturation regime and the genuinely nonperturbative transverse-momentum region.

Problem Statement

The source paper (Politzer, 'Gluon Corrections to Drell-Yan Processes', 1977) asked whether the parton model can be justified in Drell-Yan production within QCD to the same extent as in lepton-hadron scattering. Its perturbative components — the all-orders factorization proof for inclusive observables and the computation of the $O(g^{4})$ and higher corrections with coupled quark-gluon mixing — have been settled by later work, so the open problem is the surviving core in the regimes the paper itself identified as beyond the reach of perturbation theory:

1. Small-$x$ (wee-parton/saturation) regime: determine, within QCD, whether and how Drell-Yan factorization and the universality of the parton and transverse-momentum-dependent distributions persist as $x\to0$, where the density of soft partons grows, interactions remain strong, and nonlinear saturation dynamics sets in — the source's question of whether the wee partons spoil the Drell-Yan picture. This requires a factorization theorem for the Drell-Yan hadronic tensor in the saturation/Color Glass Condensate regime, or a controlled statement of breakdown with the modified leading structure, together with the identification of which small-$x$ matrix elements are universal and which are process-dependent.

2. Small-transverse-momentum regime: determine from first principles (rather than by phenomenological fit) the nonperturbative transverse-momentum sector that controls the Drell-Yan spectrum at $q_{T}\sim\Lambda_{\mathrm{QCD}}$ (the intrinsic nonperturbative scale of the strong interaction) — the intrinsic transverse momentum of partons and the nonperturbative component of the Collins-Soper/Sudakov evolution — and construct a controlled matching between this sector and the perturbative transverse-momentum resummation across the region where the perturbative formulae cease to be a guide, $q_{T}^{2}\lesssim1\ \mathrm{GeV}^{2}$, the matching reaching down to and through the region $q_{T}^{2}\lesssim(400\ \mathrm{MeV})^{2}$ in which, per the source, primordial intrinsic transverse momentum, hadronic coherence effects, and final-state interactions enter.

These two threads share one objective: to complete the justification of the Drell-Yan picture in QCD precisely where asymptotic freedom is no guide, by establishing the domain of validity of the factorized, universal description and the controlled perturbative/nonperturbative matching at its boundaries. The question is resolved by settling, in each regime, whether the factorized, universal description persists and what the corresponding resolution is — a factorization theorem, a controlled breakdown with the modified leading structure, or a first-principles determination of the relevant sector — together with, in the small-$x$ regime, the identification of which small-$x$ matrix elements are universal and which are process-dependent, and, in the small-transverse-momentum regime, the controlled matching between the nonperturbative sector and the perturbative transverse-momentum resummation. A fit of a particular observable does not by itself resolve the question. The two regimes are coupled aspects of the same question. Throughout this statement, 'controlled' means derived within a systematic expansion with a stated domain of validity and a quantified or estimable uncertainty, rather than fitted to data or merely asserted.

References

1. H. David Politzer, 'Gluon Corrections to Drell-Yan Processes', Nuclear Physics B 129 (1977) 301-318, DOI 10.1016/0550-3213(77)90197-3, https://doi.org/10.1016/0550-3213(77)90197-3 (source paper; open-access copy: https://authors.library.caltech.edu/records/gf7xd-0e153)
2. S. D. Drell and T.-M. Yan, 'Partons and their Applications at High-Energies', Annals of Physics 66 (1971) 578-623, DOI 10.1016/0003-4916(71)90071-6, https://doi.org/10.1016/0003-4916(71)90071-6
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