Transverse-momentum-dependent parton distribution functions up to twist 4 for spin-1 hadrons

We show possible transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and twist-4 functions in addition to the leading twist-2 ones by investigating all the possible decomposition of a quark correlation function in the Lorentz-invariant way. The Hermiticity and parity invariance are imposed in the decomposition; however, the time-reversal invariance is not used due to an active role of gauge links in the TMDs. Therefore, there exist time-reversal-odd functions in addition to the time-reversal even ones in the TMDs. We list all the functions up to twist-4 level because there were missing terms associated with the light cone vector $n$ in previous works on the twist-2 part and there was no correlation-function study in the twist-3 and twist-4 parts for spin-1 hadrons. We show that 40 TMDs exist in the tensor-polarized spin-1 hadron in twists 2–4. Some expressions of twist-2 structure functions are modified from previous derivations due to the new terms with $n$, and we find 30 new structure functions in twists 3 and 4 in this work. Since time-reversal-odd terms of the collinear correlation function should vanish after integrals over the partonic transverse momentum, we obtain new sum rules for the time-reversal-odd structure functions, $\int d^2{k}_T{g}_{LT}=\int d^2{k}_T{h}_{LL}=\int d^2{k}_T{h}_{3LL}=0$. In addition, we indicate that new transverse-momentum-dependent fragmentation functions exist in tensor-polarized spin-1 hadrons. The TMDs are rare observables to find explicit color degrees of freedom in terms of color flow, which cannot be usually measured because the color is confined in hadrons. Furthermore, the studies of TMDs enable us not only to find three-dimensional structure of hadrons, namely, hadron tomography including transverse structure, but also to provide unique opportunities for creating interesting interdisciplinary physics fields such as gluon condensates, color Aharonov-Bohm effect, and color entanglement. The tensor structure functions may not be easily measured in experiments. However, high-intensity facility such as the Thomas Jefferson National Accelerator Facility, the Fermilab Main Injector, and future accelerators like electron-ion collider may probe such observables. In addition, since the Nuclotron-based Ion Collider fAcility focuses on spin-1 deuteron structure functions, there is a possibility to study the details of polarized structure functions of the deuteron at this facility.

Figure 1

Three-dimensional structure functions (TMD, GPD, GDA) from the generalized transverse-momentum-dependent parton distribution (GTMD) and the Wigner function, together with the form factor and parton distribution function.

Figure 2

Quark correlation function $\mathrm{\Phi }$ with the quark (hadron) momentum $k$ ($P$) and the hadron vector and tensor polarizations $S$ and $T$, respectively.

Figure 3

A typical semi-inclusive DIS process ${\gamma }^{*}+H\to h+X$ ($\ell \to {\gamma }^{*}{\ell }^{\prime }$, $\ell =e$ or $\mu$) with a gluon interaction in the final state.

Figure 4

Gauge link for (a) semi-inclusive DIS with the spacelike correlation function ${\mathrm{\Phi }}^{[+]}$ and (b) Drell-Yan process with the timelike correlation function ${\mathrm{\Phi }}^{[-]}$.

Figure 5

A typical Drell-Yan process $H+H^{\prime }\to {\gamma }^{*}+X$ (${\gamma }^{*}\to {\mu }^{-}{\mu }^{+}$) with a gluon interaction in the initial state.