$O(\alpha_s^2)$ and $O(\alpha_s^3)$ Heavy Flavor Contributions to Transversity at $Q^2 \gg m^2$

Motivation

• Transversity cannot be measured in inclusive deep-inelastic scattering because its corresponding contribution is proportional to m^2, motivating semi-inclusive and polarized Drell–Yan measurements.The paper emphasizes charm tagging in semi-inclusive scattering as a way to access heavy-flavor contributions.Source: §1 · pp. 2–3
• Heavy-flavor corrections remain relevant for transversity and factorize at Q^2 ≫ m^2 into process-independent massive OMEs and light-flavor Wilson coefficients, up to power corrections.The analysis applies this factorization to twist-2 flavor non-singlet transversity.Source: §1 · p. 2

Contributions

• The paper computes the flavor non-singlet transversity OME at O(α_s^2) for general Mellin N and at O(α_s^3) for moments N = 1 to 13.The three-loop calculation includes the complete two-loop anomalous dimension in the double pole and extracts T_F terms from the single pole.Source: §1 · pp. 2–3
• The two-loop calculation verifies the T_F terms of earlier NLO transversity anomalous dimensions, while three-loop moments N = 1 to 8 enable comparison with existing results and N = 9 to 13 extend them.The new moments are presented alongside confirmations of results in the literature.Source: §1 · p. 3

Implications

• The results are connected to future lattice simulations with (2+1+1)- and (2+1)-dynamical fermions and to the Soffer bound for transversity.The paper discusses these applications after calculating the massive OME.Source: §1 · p. 3

Scope

• The calculation restricts transversity to twist 2 and the collinear parton model, excluding k_⊥ effects or considering them only after final-state integration.This restriction controls the twist expansion and renormalization of the calculated processes.Source: §2 · p. 3

Semi-inclusive scattering

• Semi-inclusive scattering can determine transversity by measuring the polarization of a transversely polarized final-state hadron, with fragmentation functions entering through Mellin convolutions.The cross section depends on Bjorken variables, the fragmentation variable z, and transverse spin vectors.Source: §2 · pp. 3–4

Process choice

• SIDIS is emphasized because high-luminosity experiments can tag heavy-flavor contributions, unlike polarized Drell–Yan where they enter as inclusive radiative corrections.The paper therefore mainly considers SIDIS.Source: §2 · p. 4

Factorization

• For Q^2 ≫ m^2, non-power heavy-quark Wilson-coefficient contributions factorize into light-flavor Wilson coefficients and massive operator matrix elements.The OMEs contain the heavy-quark degrees of freedom and are process independent.Source: §2 · pp. 4–5

OME construction

• The massive OME is obtained from a projected Green’s function for the local tensor operator with on-shell light-quark states and is represented in Mellin space.The calculation uses a light-quark matrix element with the heavy-quark contribution selected explicitly.Source: §2 · pp. 5–6

Scope

• The light-flavor SIDIS Wilson coefficients were not yet calculated even at O(α_s), limiting a complete phenomenological heavy-flavor prediction for that process.The corresponding calculation and soft exponentiation were described as straightforward but unavailable.Source: §2 · p. 5

3 The O(a2 s) Massive Operator Matrix Element

• The calculation uses dimensional regularization in the MS scheme, expands the OME through O(ε), and evaluates finite terms with harmonic sums S_k(N).The spherical factor Sε is set to one in the MS-scheme treatment, while the un-renormalized OME supplies finite and O(ε) contributions.Source: §3 · p. 6
• The two-loop OME calculation confirms the T_F terms of earlier transversity anomalous-dimension results [15] [16] [17].The agreement is stated for the corresponding anomalous-dimension contribution appearing in the calculation.Source: §3 · p. 6
• The renormalized two-loop massive transversity OME is constructed in Mellin space from the un-renormalized OME, anomalous dimensions, and heavy-flavor beta-function contributions.The renormalization setup includes the heavy-flavor contribution β0,Q and the relevant non-singlet anomalous dimensions.Source: §3 · pp. 6–7
• At large Mellin moment N, the leading anomalous dimensions in the vector and transversity cases approach one another.The same large-N behavior was also observed for the two-loop transversity anomalous dimension in Ref. [15].Source: §3 · p. 7

4 The O(a3 s) Massive Operator Matrix Element

• The three-loop calculation determines the massive transversity OME and associated anomalous-dimension contributions for Mellin moments N = 1 to 13.The computation uses fixed Mellin moments because of the complexity of the transversity projector and available resources.Source: §4 · pp. 7–8
• For larger Mellin N, the transversity OME approaches the corresponding vector-current quantity at O(a3_s), extending the behavior already observed at O(a2_s).The paper reports that the transversity and vector quantities become increasingly similar as N grows.Source: §4 · pp. 7–8
• The T_F parts of the three-loop transversity anomalous dimension are confirmed for N = 1 to 8 and newly provided for N = 9 to 13.The complete NLO anomalous dimension is also confirmed for moments N = 1 to 13.Source: §4 · p. 8
• The first transversity moment does not vanish, unlike the vector-current case, because no conservation law enforces its vanishing.This distinguishes the transversity OME from the corresponding vector-current OME at N = 1.Source: §4 · p. 9
• Gauge-parameter dependence cancels in checks through N = 4, and the massive OME has no ζ2 dependence.The OME structure is reported to resemble the unpolarized flavor-nonsinglet result.Source: §4 · p. 9
• Phenomenological studies of heavy-flavor effects cannot yet be performed because the required light-flavor Wilson coefficients are unavailable at two- and three-loop order.The results can instead be compared with lattice simulations including (2+1+1)-dynamical fermions and charm.Source: §4 · p. 9

5 Remarks on the Soffer Bound

• The two- and three-loop corrections preserve the Soffer bound for characteristic values of α_s.The comparison is performed using the available finite set of Mellin moments for the three-loop transversity anomalous dimension.Source: §5 · p. 10
• The heavy-flavor analysis is restricted to massive OMEs because the corresponding light-flavor Wilson coefficients have not been calculated.Consequently, the OME results alone cannot determine Soffer-bound validity at the level of structure functions.Source: §5 · p. 11
• At lower scales, the heavy-flavor OME becomes negative in an intermediate-x region, while remaining positive at small x and descending toward x ≃ 1.At larger scales, the corresponding behavior is examined through the displayed scale-dependent quantities.Source: §5 · p. 11
• Maintaining the Soffer bound requires the light-flavor Wilson coefficients to compensate the negative heavy-flavor contributions.This compensation is identified from the combined heavy-flavor and light-flavor coefficient analysis.Source: §5 · p. 11
• The Mellin-space heavy-flavor contribution also exhibits a sign change, with three-loop behavior expected to resemble the two-loop result.The paper compares the three-loop behavior with the corresponding O(a2_s) quantity.Source: §5 · p. 11

• The flavor non-singlet transversity massive OME is calculated at two loops and for N = 1 to 13 at three loops, enabling asymptotic heavy-flavor Wilson coefficients for Q^2 ≫ m^2.Source: §6 · p. 11
• The T_F parts of the three-loop transversity anomalous dimension are verified for N = 1 to 8 and extended through N = 13.Source: §6 · p. 11
• The evolution operator obeys the Soffer bound through three-loop order.Source: §6 · p. 11
• The massive OME at O(a_s^2) alone cannot establish Soffer-bound preservation across the full phase space because A^NS,MS_qq,Q changes sign at lower Q^2 and medium x.Source: §6 · p. 11
• A firm conclusion for the massive Wilson coefficients requires the still-unknown massless Wilson coefficients.Source: §6 · p. 11

• The appendix lists the T_F contributions to the three-loop anomalous dimensions for N = 1 to 13.Source: §7 · p. 12
• It gives the constant parts a^(3),TR_qq,Q(N) of the massive three-loop OME for N = 1 to 13.Source: §7 · p. 14
• The tabulated three-loop OME constants likewise contain B_4 and detailed color-factor-dependent expressions.Source: §7 · pp. 16–17
• The tabulated anomalous-dimension expressions include color factors such as C_F and constants including B_4.Source: §7 · pp. 13, 17

• Figure 2 compares the ratio of transversity and vector evolution operators at two and three loops as a function of N for α_s,0 = 0.3.Source: Figure 2 · p. 22
• Figures 3 and 4 compare differences of the two-loop massive OMEs for different Q^2/m^2 ratios.Source: Figure 3 · p. 23, 3-loop Finite Parts · pp. 23–24, Figure 4 · p. 24
• Figure 5 presents the three-loop massive-OME difference, whose behavior is described as quite similar to the two-loop result.Source: Figure 5 · p. 25