Transverse-momentum-dependent parton distribution functions for spin-1 hadrons

TL;DR

This paper develops a comprehensive classification of transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons, revealing 30 new TMDs at higher twists and establishing sum rules and relations that deepen understanding of hadronic structure.

Contribution

It introduces 30 new TMDs for spin-1 hadrons at twists 3 and 4, along with sum rules and relations, expanding the theoretical framework of hadronic parton distributions.

Findings

Identified 30 new TMDs at twists 3 and 4 for spin-1 hadrons.

Derived sum rules indicating certain T-odd TMDs vanish after transverse momentum integration.

Established new collinear PDFs and relations analogous to known sum rules.

Abstract

We show transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions by taking the decomposition of a quark correlation function in the Lorentz-invariant way with the conditions of Hermiticity and parity invariance. We found 30 new TMDs in the tensor-polarized spin-1 hadron at twists 3 and 4 in addition to 10 TMDs at twist 2. Since time-reversal-odd terms of the collinear correlation function should vanish after integrals over the partonic transverse momentum, we obtained new sum rules for the time-reversal-odd structure functions, $\bm{\int d^2 k_T g_{LT} = \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL}}$$\bm{=0}$, at twists 3 and 4. We also indicated that transverse-momentum-dependent fragmentation functions exist in tensor-polarized spin-1 hadrons. The TMDs can probe color degrees of freedom, so that they are valuable in providing unique opportunities for creating interdisciplinary physics fields such as gluon condensate, color Aharonov-Bohm effect, and color entanglement. We also found three new collinear PDFs at twists 3 and 4, and a twist-2 relation and a sum rule were derived in analogy to the Wandzura-Wilczek relation and the Burkhardt-Cottingham sum rule on the structure function $\bm{g_2}$.