Validity and scope of generalized factorization for power-suppressed corrections to hard scattering in QCD

Politzer's 1980 paper 'Power corrections at short distances' proposed a generalized factorization theorem: inclusive hard-scattering cross sections at a single large scale $Q$ expand in powers of $1/Q^{2}$ as convolutions of process-independent, non-perturbative multi-constituent distribution functions (the $\Gamma$ hierarchy) with perturbatively calculable short-distance parts, with the equations of motion eliminating off-shellness and transverse momentum in favor of additional on-shell collinear constituents. The known limitations of this algorithm are that it is established only perturbatively and only outside exceptional classes - non-perturbative phenomena could spoil it at some high power of $1/Q^{2}$; disconnected amplitudes (e.g. in double Drell-Yan) produce plane-wave volume factors $(2\pi)^{4}\delta^{(4)}(0)$ for which the only early treatment was the transverse-size rule $(2\pi)^{4}\delta^{(4)}(0)\to 1/(m^{2}S)$; processes with unavoidable color-Coulomb (Glauber) exchanges evade soft-gluon Ward-identity factorization; and operator mixing beyond perturbation theory is uncontrolled. The subsequent literature sharpened each point: leading-power factorization with Glauber cancellation was proven for Drell-Yan-type processes (Collins-Soper-Sterman); the leading $1/Q^{2}$ terms were factorized by Qiu and Sterman; the double-parton-scattering framework replaced the transverse-size estimate by position-space double parton distributions; renormalons clarified the operator-mixing question; and soft-collinear effective theory built case-by-case subleading-power factorizations that encounter a generic endpoint-divergence obstruction. The surviving open core is the general question: whether the generalized factorization of power corrections holds beyond the perturbatively proven cases - at high powers and for the exceptional disconnected and Glauber classes - together with a first-principles classification of which processes violate it.

Background

Quantum chromodynamics (QCD) predicts 'hard' scattering processes - reactions dominated by a single large momentum transfer $Q$, such as production of a lepton pair of invariant mass $Q$ (the Drell-Yan process), deep-inelastic scattering (DIS), or high-transverse-momentum jet production - through a separation of scales. Because QCD is asymptotically free, physics at distances of order $1/Q$ is computable in perturbation theory, while long-distance physics of order $1/\Lambda_{\mathrm{QCD}}$ (the intrinsic QCD scale, a few hundred MeV) is not. A factorization theorem states that an inclusive cross section equals a perturbatively calculable short-distance 'hard part' convoluted with universal, process-independent non-perturbative functions - parton distribution functions (PDFs) at leading power. Collins, Soper and Sterman proved all-order factorization at leading power in $1/Q$ for Drell-Yan-type inclusive cross sections, including the cancellation of soft gluons and of gluons exchanged in the soft, nearly instantaneous Glauber (color-Coulomb) region between the two colliding hadrons' spectator systems.

Corrections suppressed by powers $\Lambda_{\mathrm{QCD}}^{2}/Q^{2}$ - called 'higher twist', the twist of an operator being its mass dimension minus its spin, so that twist-2 PDFs give the leading term and twist-4 operators generate $1/Q^{2}$ corrections - are numerically important and set the ultimate precision limit of perturbative QCD. In 'Power corrections at short distances' (1980), Politzer proposed a generalized factorization for them: every hadron channel is characterized by an infinite hierarchy of process-independent longitudinal multi-constituent distribution functions $\Gamma_{H,i}(\{x_{\alpha}\};M^{2})$, one for each set of on-shell, collinear constituents carrying longitudinal momentum fractions $x_{\alpha}$; the inclusive cross section is a sum over constituent sets in each channel of convolutions of these $\Gamma$ functions with perturbatively calculable short-distance cross sections,

$$\mathrm{d}\sigma^{\mathrm{hadron}} = \sum_{i,j,k}\int \Gamma_{A,i}\,\Gamma_{B,j}\,\Gamma_{C,k}\;\mathrm{d}\sigma^{\mathrm{hard}}_{ijk}\;+\;(\text{non-factorizable residuals});$$

the equations of motion (EOM) of QCD eliminate a constituent's off-shellness (its deviation from the on-shell energy-momentum relation) and transverse momentum in favor of amplitudes with additional on-shell constituents; and for energy-weighted jet observables the hierarchy collapses to a single number $\gamma_{i}$ per constituent set. The limitations of this algorithm are that it is established only perturbatively and only outside exceptional classes: the proof uses Ward identities (exact relations among gauge-field amplitudes that follow from gauge invariance) to resum soft gluon emissions eikonally (into Wilson-line factors), and it can fail for color-Coulomb exchanges; 'disconnected' amplitudes - two independent hard subamplitudes, as in double Drell-Yan, where one hadron pair produces two lepton pairs through two short-distance interactions - generate plane-wave volume factors $(2\pi)^{4}\delta^{(4)}(0)$ for which the only early treatment was the transverse-size rule $(2\pi)^{4}\delta^{(4)}(0)\to 1/(m^{2}S)$, with $m^{-2}$ a hadronic transverse area and $S$ the hadron-hadron center-of-mass energy squared; operators of different mass dimension do not mix to all orders in massless perturbation theory, but nothing was known beyond perturbation theory; and purely non-perturbative phenomena (for example instanton-like vacuum structure - tunneling events between topologically distinct QCD vacuum sectors) could spoil the factorization at some high power of $1/Q^{2}$.

Later work sharpened parts of this program. The asymptotic expansion of Feynman integrals near threshold (Beneke and Smirnov, 1998) underlies soft-collinear effective theory (SCET), an effective field theory that organizes fluctuations by off-shellness and supports systematic power expansions. Double parton scattering (DPS) is now described by double parton distribution functions (dPDFs) in transverse position space, whose overlap profile is measured by the effective cross section $\sigma_{\mathrm{eff}}$ of order 15 mb. The factorial divergence of perturbative series ('renormalons') is understood to be tied to power corrections and to the ambiguity of separating an operator product expansion (OPE) into perturbative coefficient functions and non-perturbative matrix elements.

Problem Statement

Does the generalized factorization of power corrections proposed by Politzer hold in QCD beyond the cases that can be proven perturbatively, and what is its complete scope? Specifically:

1. Non-perturbative validity. To what extent - at which power of $1/Q^{2}$, if any, and by which mechanisms - do purely non-perturbative phenomena (for example instanton-like vacuum structure) spoil the factorization of inclusive hard-scattering cross sections into process-independent multi-constituent distribution functions and perturbatively calculable short-distance parts? The perturbative arguments are necessary but not sufficient, since the existence of hadrons with finite matrix elements is itself non-perturbative.

2. Exceptional classes. Which processes violate this factorization algorithm through disconnected multi-constituent amplitudes or unavoidable color-Coulomb (Glauber) exchanges? Can the disconnected contributions - such as the $(2\pi)^{4}\delta^{(4)}(0)$ factor in the double Drell-Yan disconnected term, for which the only early treatment was the transverse-area rule $1/(m^{2}S)$, with $m^{-2}$ a hadronic transverse area and $S$ the hadron-hadron center-of-mass energy squared - be derived from first principles, so that these classes are either incorporated into the factorization algorithm or definitively excluded from it?

3. Operator mixing beyond perturbation theory. Do operators of different mass dimension, which do not mix to all orders in perturbation theory in the massless theory, mix once logarithms are resummed or non-perturbative mass scales are present, and with what consequences for the separation between perturbative hard parts and non-perturbative functions?

The general question spans all powers of $1/Q^{2}$ and the general class of inclusive hard-scattering processes, without restriction to particular observables, geometries, or fixed perturbative order, with the three numbered aspects above as its components.

References

1. H. David Politzer, 'Power corrections at short distances', Nuclear Physics B 172 (1980) 349-382. DOI: 10.1016/0550-3213(80)90172-8. URL: https://doi.org/10.1016/0550-3213(80)90172-8
2. John C. Collins, Davison E. Soper, George F. Sterman, 'All-order factorization for Drell-Yan cross sections', Physics Letters B 134 (1984) 263-268. DOI: 10.1016/0370-2693(84)90684-1. URL: https://doi.org/10.1016/0370-2693(84)90684-1
3. John C. Collins, Davison E. Soper, George F. Sterman, 'Soft Gluons and Factorization', Nuclear Physics B 308 (1988) 833-856. DOI: 10.1016/0550-3213(88)90130-7. URL: https://doi.org/10.1016/0550-3213(88)90130-7
4. Jianwei Qiu, George Sterman, 'Power corrections in hadronic scattering (I). Leading $1/Q^{2}$ corrections to the Drell-Yan cross section', Nuclear Physics B 353 (1991) 105-136. DOI: 10.1016/0550-3213(91)90503-P. URL: https://doi.org/10.1016/0550-3213(91)90503-P
5. Jianwei Qiu, George Sterman, 'Power corrections in hadronic scattering (II). Factorization', Nuclear Physics B 353 (1991) 137-164. DOI: 10.1016/0550-3213(91)90504-Q. URL: https://doi.org/10.1016/0550-3213(91)90504-Q
6. M. Beneke, Vladimir A. Smirnov, 'Asymptotic expansion of Feynman integrals near threshold', Nuclear Physics B 522 (1998) 321-344. DOI: 10.1016/S0550-3213(98)00138-2. URL: https://doi.org/10.1016/S0550-3213(98)00138-2
7. M. Beneke, 'Renormalons', Physics Reports 317 (1999) 1-142. DOI: 10.1016/S0370-1573(98)00130-6. URL: https://doi.org/10.1016/S0370-1573(98)00130-6
8. Aneesh V. Manohar, Wouter J. Waalewijn, 'QCD analysis of double parton scattering: Spin and color correlations, interference effects, and evolution', Physical Review D 85 (2012) 114009. DOI: 10.1103/PhysRevD.85.114009. URL: https://doi.org/10.1103/PhysRevD.85.114009 (arXiv:1202.3794)
9. Jonathan R. Gaunt, 'Glauber Gluons and Multiple Parton Interactions', JHEP 07 (2014) 110. DOI: 10.1007/JHEP07(2014)110. URL: https://doi.org/10.1007/JHEP07(2014)110
10. Markus Diehl, Jonathan R. Gaunt, Daniel Ostermeier, Peter Plößl, Andreas Schäfer, 'Cancellation of Glauber gluon exchange in the double Drell-Yan process', JHEP 01 (2016) 076. DOI: 10.1007/JHEP01(2016)076. URL: https://doi.org/10.1007/JHEP01(2016)076 (arXiv:1510.08696)
11. Ian Moult, Iain W. Stewart, Gherardo Vita, 'Subleading Power Factorization with Radiative Functions', JHEP 11 (2019) 153. DOI: 10.1007/JHEP11(2019)153. URL: https://doi.org/10.1007/JHEP11(2019)153
12. Ze Long Liu, Matthias Neubert, 'Factorization at subleading power and endpoint-divergent convolutions in $h\to\gamma\gamma$ decay', JHEP 04 (2020) 033. DOI: 10.1007/JHEP04(2020)033. URL: https://doi.org/10.1007/JHEP04(2020)033
13. Ze Long Liu, Bianka Mecaj, Matthias Neubert, Xing Wang, 'Factorization at subleading power, Sudakov resummation, and endpoint divergences in soft-collinear effective theory', Physical Review D 104 (2021) 014004. DOI: 10.1103/PhysRevD.104.014004. URL: https://doi.org/10.1103/PhysRevD.104.014004 (arXiv:2009.04456)
14. Gunnar S. Bali, Markus Diehl, Benjamin Gläßle, Andreas Schäfer, Christian Zimmermann, 'Double parton distributions in the nucleon from lattice QCD', JHEP 09 (2021) 106. DOI: 10.1007/JHEP09(2021)106. URL: https://doi.org/10.1007/JHEP09(2021)106
15. Vladimir M. Braun, 'Higher Twists', EPJ Web of Conferences 274 (2022) 01012. DOI: 10.1051/epjconf/202227401012. URL: https://doi.org/10.1051/epjconf/202227401012
16. Aurore Courtoy, Angel S. Miramontes, Harut Avakian, Marco Mirazita, Silvia Pisano, 'Extraction of the higher-twist parton distribution $e(x)$ from CLAS data', Physical Review D 106 (2022) 014027. DOI: 10.1103/PhysRevD.106.014027. URL: https://doi.org/10.1103/PhysRevD.106.014027
17. Kyle Lee, Jian-Wei Qiu, George Sterman, Kazuhiro Watanabe, 'Subleading power corrections to heavy quarkonium production in QCD factorization approach', EPJ Web of Conferences 274 (2022) 04005. DOI: 10.1051/epjconf/202227404005. URL: https://doi.org/10.1051/epjconf/202227404005
18. Thomas Becher, Patrick Hager, Sebastian Jaskiewicz, Matthias Neubert, Dominik Schwienbacher, 'Factorization Restoration through Glauber Gluons', Physical Review Letters 134 (2025) 061901. DOI: 10.1103/PhysRevLett.134.061901. URL: https://doi.org/10.1103/PhysRevLett.134.061901 (arXiv:2408.10308)
19. Saad Nabeebaccus, Jakob Schönleber, Lech Szymanowski, Samuel Wallon, 'Demonstration of collinear factorization breaking due to collinear-to-soft Glauber exchanges for a $2\to 3$ exclusive process at leading twist', Physical Review D 111 (2025) L091502. DOI: 10.1103/PhysRevD.111.L091502. URL: https://doi.org/10.1103/PhysRevD.111.L091502 (arXiv:2409.16067)