Tensor-Polarized Twist-3 Parton Distributions

• Tensor-polarized twist-3 PDFs are quark distributions in spin-1 hadrons that incorporate additional tensor polarization beyond the vector component.
• The twist-3 functions (eLL and fLT) are derived from a light-cone quark correlator and are constrained by WW-like approximations and BC-like sum rules.
• These distributions play a key role in processes like deep-inelastic scattering, SIDIS, and proton–deuteron Drell–Yan experiments, offering insights into quark-gluon dynamics.

Tensor-polarized twist-3 parton distribution functions are subleading-power collinear quark distributions specific to spin-1 hadrons, where the target state is characterized not only by vector polarization but also by a symmetric traceless rank-2 spin tensor $T^{\mu\nu}$. In the collinear sector, the distinctive tensor-polarized PDFs are $f_{1LL}$ at twist 2, $e_{LL}$ and $f_{LT}$ at twist 3, and $f_{3LL}$ at twist 4; among these, $f_{LT}$ has become the central object because it admits a Wandzura-Wilczek-like decomposition in terms of the twist-2 tensor-polarized PDF $f_{1LL}$, together with a genuine twist-3 quark-gluon contribution (Kumano et al., 2021, Kumano et al., 2021). The subject sits at the intersection of collinear factorization, transverse-momentum-dependent factorization, local and nonlocal operator product expansion, and spin-1 phenomenology for the deuteron, with explicit applications to deep-inelastic scattering, semi-inclusive DIS, and proton-deuteron Drell-Yan (Qiao et al., 2024).

1. Spin-1 tensor polarization and the operator definition of twist-3 PDFs

The defining structural feature of a spin-1 hadron is the existence of tensor polarization in addition to ordinary vector polarization. In the light-cone parametrization used in the spin-1 PDF literature, the tensor polarization is decomposed into three sectors,

$S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$

corresponding to longitudinal-longitudinal, longitudinal-transverse, and transverse-transverse tensor polarization, respectively (Kumano et al., 2022). This extra polarization content has no analogue for a spin-$\tfrac12$ target and is the reason tensor-polarized PDFs exist at all.

The basic collinear quark correlator is obtained from the fully unintegrated correlator after integrating over transverse momentum and the light-cone minus component. In the tensor sector, its standard decomposition is

$\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$

or equivalently with $f_{1LL}$0 instead of $f_{1LL}$1 in the light-cone convention (Kumano et al., 1 May 2026, Kumano et al., 2021). In this decomposition, $f_{1LL}$2 is twist-2, $f_{1LL}$3 and $f_{1LL}$4 are twist-3, and $f_{1LL}$5 is twist-4.

The two twist-3 collinear tensor-polarized PDFs have distinct Dirac and polarization content. The function $f_{1LL}$6 is associated with the $f_{1LL}$7 sector and belongs to the chiral-even channel, while $f_{1LL}$8 belongs to the $f_{1LL}$9 sector and is chiral-odd (Kumano et al., 1 May 2026). The former is therefore the direct tensor-polarized analogue of the familiar nucleon twist-3 distribution $e_{LL}$0, whereas the latter is governed by a different set of equation-of-motion constraints (Kumano et al., 2022, Kumano et al., 2021).

A compact summary of the collinear tensor-polarized PDF hierarchy is as follows.

Twist	PDF	Tensor sector
2	$e_{LL}$1	$e_{LL}$2
3	$e_{LL}$3	$e_{LL}$4
3	$e_{LL}$5	$e_{LL}$6
4	$e_{LL}$7	$e_{LL}$8

This hierarchy is not merely notational. It organizes which tensor structures survive in the strict collinear limit and which are instead only meaningful as transverse-momentum-dependent distributions.

2. From the full TMD tower to the collinear twist-3 PDFs

The modern classification of tensor-polarized twist-3 PDFs emerged from the Lorentz-invariant decomposition of the quark correlator for spin-1 hadrons with explicit inclusion of the light-cone vector $e_{LL}$9, together with Hermiticity and parity invariance (Kumano et al., 2021, Kumano et al., 2020). In that framework, the $f_{LT}$0-dependent terms are essential: without them the twist-3 and twist-4 sectors are incomplete, and even some twist-2 expressions are modified.

Within the tensor-polarized sectors $f_{LT}$1, the complete analysis yields 40 TMDs through twist 4, of which 30 are new and first appear at twist 3 or 4 (Kumano et al., 2021). At twist 3, the tensor-polarized TMDs include

$f_{LT}$2

$f_{LT}$3

while the collinear limit retains only a small subset (Kumano et al., 2021).

After transverse-momentum integration,

$f_{LT}$4

many tensor-polarized TMDs vanish. The surviving genuinely new collinear tensor-polarized PDFs are

$f_{LT}$5

(Kumano et al., 2021). This sharply distinguishes the collinear twist-3 problem from the much larger TMD classification problem.

Time-reversal symmetry is decisive in this reduction. Because the collinear correlator is time-reversal invariant once transverse momentum is integrated out, T-odd collinear PDFs must vanish. Accordingly,

$f_{LT}$6

are zero as collinear PDFs (Kumano et al., 2021). The same reasoning produces transverse-momentum sum rules such as

$f_{LT}$7

and related relations were presented for higher-twist tensor-polarized functions as well (Kumano et al., 2020, Kumano et al., 2021). By contrast, the corresponding fragmentation functions need not vanish, because time-reversal constraints do not apply there in the same way; the analogues $f_{LT}$8, $f_{LT}$9, $f_{3LL}$0, and $f_{3LL}$1 can therefore exist (Kumano et al., 2021).

A common misconception is that every tensor-polarized twist-3 object identified at the TMD level survives as a collinear PDF. The classification program shows the opposite: the collinear tensor-polarized twist-3 sector is much smaller and is essentially centered on $f_{3LL}$2 and $f_{3LL}$3.

3. The $f_{3LL}$4 relation to $f_{3LL}$5: WW-like decomposition and BC-like sum rule

The most developed piece of twist-3 spin-1 PDF theory concerns $f_{3LL}$6. Using a nonlocal operator analysis, one obtains the differential relation

$f_{3LL}$7

where the higher-twist term is expressed through twist-3 quark-gluon distributions (Kumano et al., 2022). Integration yields

$f_{3LL}$8

and for the charge-conjugation-even combination,

$f_{3LL}$9

(Kumano et al., 2021, Kumano et al., 2022).

Neglecting genuine twist-3 quark-gluon effects gives the WW-like approximation,

$f_{LT}$0

which is the tensor-polarized analogue of the usual $f_{LT}$1 relation in the nucleon (Kumano et al., 2022). The corresponding $f_{LT}$2-like combination is defined as

$f_{LT}$3

so that, in the WW-like approximation,

$f_{LT}$4

Its first moment then obeys the BC-like sum rule

$f_{LT}$5

(Kumano et al., 2021, Kumano et al., 2022).

The relation is not only heuristic. A later local-OPE derivation reproduced the same WW-like relation and BC-like sum rule through the moment expansion of local gauge-invariant operators, with twist-2 reduced matrix elements $f_{LT}$6 and genuine twist-3 reduced matrix elements $f_{LT}$7 (Kumano et al., 1 May 2026). In that formulation, the twist-2 part of $f_{LT}$8 is

$f_{LT}$9

while the higher-twist remainder is determined by $f_{1LL}$0. The same work notes an important caveat: because the local operators are defined for $f_{1LL}$1, the OPE does not rigorously prove the BC-like sum rule in the strict mathematical sense, although the result is consistent with the formal derivation and with the earlier nonlocal-operator analysis (Kumano et al., 1 May 2026).

A second sum rule is more conditional. If one assumes the parton-model sum rule

$f_{1LL}$2

together with vanishing tensor-polarized antiquark distributions, then

$f_{1LL}$3

follows (Kumano et al., 2021). This should not be conflated with the BC-like sum rule for $f_{1LL}$4: the former needs extra assumptions, whereas the latter follows from the WW-like structure itself.

4. Genuine twist-3 structure: multiparton correlators, EOM relations, and Lorentz-invariance relations

The WW-like approximation isolates only the kinematical twist-2 part of $f_{1LL}$5. The genuine twist-3 content is encoded in quark-gluon-quark correlators. In the tensor-polarized spin-1 case, four twist-3 multiparton distributions were identified: $f_{1LL}$6 appearing in the detailed operator parametrization as $f_{1LL}$7, $f_{1LL}$8, $f_{1LL}$9, and $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$0 (Kumano et al., 2021). They probe tensor-polarized multiparton correlations beyond the leading-parton picture.

The first exact equation-of-motion relation ties the collinear twist-3 PDF $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$1 to the $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$2-moment of a twist-2 TMD and to the genuine three-parton correlators: $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$3 (Kumano et al., 2021). This makes explicit that $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$4 is not an unconstrained independent function: its dynamical content is coupled to both transverse-momentum moments and quark-gluon correlations.

The second equation-of-motion relation governs the other collinear twist-3 tensor-polarized PDF,

$S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$5

Since $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$6, the quark-mass term is numerically small and $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$7 is directly governed by the twist-3 quark-gluon-quark correlator $S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$8 (Kumano et al., 2021).

Combining the EOM analysis with the previously derived twist-2/twist-3 relation yields a Lorentz-invariance relation,

$S_{LL},\qquad S_{LT}^{\mu},\qquad S_{TT}^{\mu\nu},$9

which links the transverse-momentum moment $\tfrac12$0, the twist-3 PDF $\tfrac12$1, the twist-2 PDF $\tfrac12$2, and the genuine quark-gluon-quark correlator $\tfrac12$3 (Kumano et al., 2021).

These relations also delimit the scope of the collinear sector. The functions $\tfrac12$4 and $\tfrac12$5 are explicitly related to multiparton correlators, whereas $\tfrac12$6 or $\tfrac12$7 has no corresponding twist-3 collinear PDF because there is no twist-3 collinear PDF associated with the $\tfrac12$8 tensor-polarization structure (Kumano et al., 2021). A plausible implication is that the $\tfrac12$9 and $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$0 sectors dominate any strictly collinear twist-3 phenomenology, while the $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$1 sector is intrinsically more TMD-like.

5. Phenomenology: deuteron estimates, Drell-Yan access, and SIDIS observables

The deuteron is the principal phenomenological target for tensor-polarized twist-3 studies. A dedicated numerical analysis computed $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$2 for the spin-1 deuteron using the WW-like relation and tensor-polarized twist-2 PDFs $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$3 at

$\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$4

(Kumano et al., 5 Sep 2025). In that study, the resulting $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$5 has a shape very similar to that of $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$6, and its magnitude is roughly of the same order as $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$7. The BC-like sum rule,

$\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$8

was also confirmed numerically, with the residual of order $\Phi(x,P,T) = \frac12\left[ S_{LL}\slashed{\bar n}\, f_{1LL}(x) +\frac{M}{P\cdot n}S_{LL}e_{LL}(x) +\frac{M}{P\cdot n}\slashed S_{LT}\, f_{LT}(x) +\frac{M^2}{(P\cdot n)^2}S_{LL}\slashed n\, f_{3LL}(x) \right],$9 when integrated from $f_{1LL}$00 (Kumano et al., 5 Sep 2025). Because Jefferson Lab operates at moderate $f_{1LL}$01, the same work argues that higher-twist effects could be sizable and that $f_{1LL}$02 may be experimentally relevant there.

In proton-deuteron Drell-Yan, twist-3 tensor-polarized PDFs enter directly into the hadronic tensor and observable angular modulations. For Fermilab kinematics, where the dilepton invariant mass is

$f_{1LL}$03

the twist-3 contribution is not negligible compared to the twist-2 contribution (Qiao et al., 2024). The differential cross section contains the tensor-polarized twist-3 combination

$f_{1LL}$04

modulated by $f_{1LL}$05, while a weighted cross section with

$f_{1LL}$06

isolates $f_{1LL}$07 (Qiao et al., 2024). Together with the angle-integrated cross section for $f_{1LL}$08, these observables provide an explicit strategy for disentangling $f_{1LL}$09, $f_{1LL}$10, and $f_{1LL}$11.

Semi-inclusive DIS off a tensor-polarized spin-1 target provides a broader TMD environment in which twist-3 tensor structure appears across many azimuthal harmonics. In a tree-level TMD-factorization treatment up to twist-3, the complete differential cross section involves 23 structure functions, with 21 nonvanishing at leading and subleading twist (Zhao et al., 8 Aug 2025). Several of the tensor-polarized SIDIS structure functions depend explicitly on twist-3 tensor PDFs. For example,

$f_{1LL}$12

contains $f_{1LL}$13, together with other twist-3 and fragmentation contributions (Zhao et al., 8 Aug 2025). The same framework shows that, after integrating over $f_{1LL}$14, only five tensor-polarized structure functions survive in SIDIS, and after integrating over $f_{1LL}$15, the inclusive tensor-polarized cross section depends only on $f_{1LL}$16 and $f_{1LL}$17 up to twist-3 (Zhao et al., 8 Aug 2025).

These phenomenological results place $f_{1LL}$18 in a role analogous to nucleon $f_{1LL}$19: formally suppressed by $f_{1LL}$20, but potentially numerically important in the few-GeV regime.

6. Conceptual significance, limitations, and current outlook

Tensor-polarized twist-3 PDFs enlarge the QCD description of hadronic structure beyond the spin-$f_{1LL}$21 paradigm. They encode subleading but structured information on quark transverse motion, quark-gluon-quark correlations, and tensor-specific spin-momentum couplings that are absent in the proton (Kumano et al., 2021, Zhao et al., 8 Aug 2025). In practical terms, $f_{1LL}$22 is the best-developed example because it is simultaneously constrained by WW-like and BC-like relations, local and nonlocal OPE, equation-of-motion identities, and concrete Drell-Yan and SIDIS observables.

Several limitations are integral to the current formalism. The nonlocal-operator derivation of the WW-like relation neglects twist-4 effects and total derivatives in the operator identity used to isolate the twist-3 sector (Kumano et al., 2022). The local-OPE confirmation is restricted to moments with $f_{1LL}$23, which is why the BC-like sum rule is not established in the fully rigorous moment-by-moment sense (Kumano et al., 1 May 2026). The additional sum rule for $f_{1LL}$24 itself requires the parton-model $f_{1LL}$25 sum rule and vanishing tensor-polarized antiquark distributions (Kumano et al., 2021). More broadly, the WW-like approximation neglects genuine twist-3 quark-gluon correlations, so any experimentally observed deviation from

$f_{1LL}$26

would directly signal dynamical higher twist rather than kinematical twist alone (Kumano et al., 2022).

The broader spin-1 program also retains a TMD motivation. The tensor-polarized TMDs from which the collinear twist-3 PDFs descend were described as valuable because TMDs can probe color degrees of freedom, with proposed relevance to the gluon condensate, color Aharonov-Bohm effect, color entanglement, and the color glass condensate (Kumano et al., 2021). This does not mean that the collinear twist-3 PDFs directly measure those phenomena; rather, it situates the tensor-polarized twist-3 sector within a wider QCD program in which spin-1 observables may illuminate nontrivial color dynamics.

Experimentally, the literature repeatedly points to Jefferson Lab, Fermilab, NICA, future electron-ion colliders, and related hadron facilities as the main settings where tensor-polarized twist-3 structure could be tested (Kumano et al., 2022, Kumano et al., 5 Sep 2025). The central theoretical expectation is stable across these studies: $f_{1LL}$27 provides the cleanest twist-3 tensor-polarized benchmark, $f_{1LL}$28 remains an essential but less explored companion distribution, and the decisive physics lies in separating WW-like kinematics from genuine quark-gluon correlations.