Matching between Collinear Twist-3 and TMD Fragmentation Function Contributions to Polarized Hyperon Production in SIDIS

We investigate the consistency between the collinear twist-3 factorization and the transverse-momentum-dependent (TMD) factorization for the transversely polarized hyperon production in semi-inclusive deep inelastic scattering, $ep\to eΛ^\uparrow X$. In particular, we focus on the contributions from the twist-3 fragmentation functions (FFs) and the TMD polarizing FF in the region of the hyperon’s intermediate transverse momentum $P_T$, $Λ_{\rm QCD}\ll P_T\ll Q$, where both frameworks are valid. In this region the polarizing FF can be expressed in terms of the twist-3 FFs including the purely gluonic ones, and the resulting TMD factorization formula for $ep\to eΛ^\uparrow X$ agrees with the small-$P_T$ limit of the corresponding twist-3 cross section. This matching of the two calculations indicates that the two frameworks describe the same effect in QCD and provide complementary frameworks for the process in different kinematic regions.

💡 Research Summary

The paper investigates the theoretical consistency between two QCD factorization approaches—collinear twist‑3 and transverse‑momentum‑dependent (TMD) factorization—in the semi‑inclusive deep‑inelastic scattering (SIDIS) process ep → e Λ↑ X, where a transversely polarized Λ hyperon is produced from an unpolarized electron‑proton collision. The focus is on the intermediate transverse‑momentum region, defined by Λ_QCD ≪ P_T ≪ Q, where both frameworks are applicable. In this regime the polarizing TMD fragmentation function (FF), D_⊥^1T(z,k_T²), can be expressed in terms of collinear twist‑3 FFs, including both quark‑ and purely gluonic contributions. The authors demonstrate that the TMD factorization formula derived from this expression exactly reproduces the small‑P_T limit of the corresponding collinear twist‑3 cross section, thereby establishing a rigorous matching between the two descriptions.

The paper is organized as follows. Section 1 introduces the longstanding puzzle of large transverse single‑spin asymmetries (SSAs) observed in hyperon production (e.g., pp → Λ↑ X) and explains why the conventional twist‑2 collinear factorization fails to account for them. It motivates the need for higher‑twist (twist‑3) collinear and TMD frameworks, which incorporate multi‑parton correlations and intrinsic transverse momentum, respectively. While previous studies have shown matching for distribution‑function contributions (e.g., Boer‑Mulders versus twist‑3 PDFs), the fragmentation‑function side remained largely unexplored, especially for hyperons where gluonic twist‑3 FFs play a role.

Section 2 sets up the kinematics of SIDIS, defining the usual variables: the virtual photon momentum q, Bjorken x, the fragmentation variable z, and the transverse momentum q_T (with P_T = z q_T). The authors adopt the hadron frame, introduce orthogonal basis vectors (T, X, Y, Z), and express the differential cross section in terms of six Lorentz tensors V_k^{μν} with azimuthal dependence encoded in coefficients A_k(φ). This formalism makes explicit that only the relative azimuthal angle φ = ϕ − χ enters the observable.

Section 3 reviews the relevant fragmentation functions. The collinear twist‑3 FFs are classified into intrinsic, kinematical, and dynamical types for both quarks and gluons. For quarks, intrinsic functions include the T‑odd D_T(z) and the T‑even G_T(z); kinematical functions are D_⊥^{(1)}(z) and G_⊥^{(1)}(z); dynamical functions involve three‑point correlators b D_FT(z,z₁) and b G_FT(z,z₁). For gluons, analogous sets appear: intrinsic Δb G₃¯^T(z), kinematical b G^{(1)}(z), Δb H^{(1)}(z), and dynamical functions built from three‑field‑strength correlators (b N_i, b O_i) and mixed quark‑gluon correlators e D_FT(z). The paper lists the QCD equation‑of‑motion (EOM) relations and Lorentz‑invariance relations (LIRs) that connect these functions, emphasizing that the T‑odd pieces (imaginary parts) are the sources of SSA.

Section 4 summarizes the previously derived collinear twist‑3 cross section for the hyperon polarization. Two distinct mechanisms contribute: (i) a twist‑3 unpolarized distribution in the proton convoluted with the transversity FF H₁(z), and (ii) the twist‑3 fragmentation sector, which itself splits into quark‑type (a) and gluon‑type (b) contributions. The complete leading‑order (LO) expression for (ii) is assembled from earlier works