Boer Mulders FunctionEdit
The Boer-Mulders function is a fundamental object in the study of how quarks move and spin inside hadrons. Located within the broader framework of transverse momentum dependent distributions (Transverse momentum dependent distributions), it encodes a correlation between the intrinsic transverse momentum of a quark and the quark’s transverse spin inside an unpolarized hadron. Denoted often as h_1^{⊥ q}(x, k_T^2), this distribution is chiral-odd and time-reversal odd in the naive sense, which means its effects do not appear in simple inclusive measurements but reveal themselves in specific angular modulations of produced leptons in processes like Drell–Yan and SIDIS. The function is named after Daniël Boer and Pieter Mulders, who helped formalize its role in the TMD picture of nucleon structure.
Physically, a nonzero Boer-Mulders function signals that quarks carry orbital angular momentum that is correlated with their spin orientation in a way that survives even when the parent hadron is unpolarized. This is a subtle but important piece of the spin puzzle for the proton and other hadrons, linking microscopic spin-orbit dynamics to observable asymmetries. The study of h_1^{⊥ q} thus complements other TMDs such as the Sivers function and the transversity distribution, and it plays a role in interpreting azimuthal asymmetries measured in experiments across different reactions. See, for example, the broader discussions of hadron structure and orbital angular momentum in the nucleon, as well as related TMD concepts like Sivers function.
Boer-Mulders function
Definition and physical interpretation
The Boer-Mulders function h_1^{⊥ q}(x, k_T^2) describes the distribution of transversely polarized quarks q with longitudinal momentum fraction x and intrinsic transverse momentum k_T inside an unpolarized hadron. It is a chiral-odd quantity, meaning it flips quark chirality, and it is naively time-reversal odd because it requires initial- or final-state interactions to generate the necessary phase. In equations, one can think of h_1^{⊥ q} as a function that ties the transverse spin of a quark to its transverse momentum, even when the parent hadron carries no net spin. In practice, the Boer-Mulders function contributes to angular modulations in processes like unpolarized Drell–Yan Drell–Yan and SIDIS Semi-inclusive deep inelastic scattering, and it can be accessed when it couples with other chiral-odd or fragmentation effects.
Theoretical framework and process dependence
The Boer-Mulders function is part of the TMD factorization approach, where cross sections are written in terms of TMDs that depend on x and k_T and on a hard scale Q. A key feature in this framework is the role of gauge links, which encode color interactions with the spectator system and give rise to process dependence. Consequently, many TMDs, including the Boer-Mulders function, are predicted to change sign between SIDIS and Drell–Yan under the standard TMD factorization picture. This sign change is a robust, testable prediction that has driven much of the experimental program and lattice studies of TMDs. For the broader formalism, see TMD factorization and Gauge link as well as Chiral-odd distributions that must appear in pairs to be observable.
Experimental status
Empirical evidence for the Boer-Mulders effect comes from multiple channels. Unpolarized SIDIS experiments have observed azimuthal asymmetries that can be interpreted, in part, as arising from h_1^{⊥ q} when combined with fragmentation functions. Major experiments at facilities such as HERMES and COMPASS (experiment) have contributed to our understanding of these asymmetries and their flavor structure, while fixed-target Drell–Yan measurements have provided complementary access to the same physics in a different process. Classical fixed-target studies, including the NA10 experiment at CERN and the E615 experiment at Fermilab, reported significant cos(2φ) modulations in unpolarized Drell–Yan that are consistent with Boer-Mulders-type correlations between quark spin and transverse momentum. Ongoing global fits and lattice-QCD-inspired analyses continue to refine the x- and k_T-dependence of h_1^{⊥ q} and its flavor decomposition, connecting experimental data to the underlying orbital motion of quarks inside hadrons. See also discussions of hadron structure and orbital angular momentum in the nucleon for broader context.
Universality and controversies
Within the modern QCD framework, the universality properties of TMDs are subtle. The presence of gauge links means that certain TMDs are not strictly universal in the same way as collinear parton distributions, but they obey predictable process-dependent modifications (for example, the predicted sign change between SIDIS and Drell–Yan for the Boer-Mulders function). This has sparked debates about the precise interpretation of extractions from different experiments and the limits of TMD factorization, especially at large transverse momenta or in kinematic regimes where factorization breaks down. Proponents emphasize that TMDs provide a coherent and predictive language for spin-momentum correlations across processes, while critics sometimes point to large theoretical uncertainties or model dependencies in phenomenological extractions. See the related discussions in TMD factorization and Gauge link for the formal foundations, and in Lattice QCD for first-principles attempts to constrain moments of h_1^{⊥ q}.
Controversies and debates (from a results-focused perspective)
The magnitude and flavor structure of the Boer-Mulders function remain subject to interpretation because different processes probe different combinations of quark flavors and fragmentation effects. This has led to a range of fits and models that are sometimes difficult to reconcile across SIDIS and Drell–Yan data.
Critics who push for simpler global pictures sometimes discount TMDs as overcomplicating the analysis or overfitting data. Supporters argue that the observed angular modulations are systematically connected to the same underlying spin-orbit physics that generates orbital angular momentum in the nucleon, and that a disciplined TMD approach yields coherent, testable predictions across multiple experiments.
Some criticisms emphasize broader social or political narratives about scientific funding or research directions. From a pragmatic, results-oriented standpoint, the case for continuing to study spin-momentum correlations rests on the consistent experimental signatures, the cross-process testable predictions (such as sign changes), and the potential to illuminate the nucleon’s internal dynamics in a way that complements lattice QCD and other nonperturbative approaches.
Proponents of the TMD program stress that the science is driven by data and phenomenology that map onto QCD dynamics, rather than ideology. Dismissing the field on ideological grounds would be misreading the strength of the empirical program and the predictive power of its framework. In this view, skeptical scrutiny of methods and assumptions is healthy, but it should be anchored in reproducible results and falsifiable theories rather than extrapolated judgments about the people doing the science.