Polarized Massive Operator Matrix Elements

• The paper demonstrates precise three-loop computations of polarized massive OMEs using IBP reduction, nested sums, and matching techniques, crucial for heavy quark corrections in polarized DIS.
• It employs advanced computer algebra, summation theory, and differential equations to derive analytic results for both single-mass and two-mass configurations across all parton channels.
• These results enable accurate determination and evolution of polarized PDFs in the variable flavor number scheme, enhancing predictions for spin-dependent observables.

Polarized massive operator matrix elements (OMEs) are central, universal quantities in perturbative Quantum Chromodynamics (QCD) that encode the effects of heavy quarks on polarized deep-inelastic scattering (DIS) observables, particularly the spin-dependent structure functions. At three-loop order, the theoretical landscape has reached full completion for both single-mass and two-mass scenarios, involving all parton channels (quark and gluon, non-singlet and singlet). These results are obtained in the Larin scheme for the polarized sector, employing advanced techniques from computer algebra, summation theory, and analytic continuation, and are crucial for the precise definition and evolution of polarized parton distribution functions (PDFs) in the variable flavor number scheme (VFNS).

1. Theoretical Structure of Polarized Massive OMEs

Polarized OMEs, typically denoted as ΔA_{ij}(N, m^2/Q^2) where N is the Mellin moment, are process-independent building blocks encoding heavy quark effects in the factorization framework of high–Q^2 spin-dependent observables. In the VFNS, they enter as matching coefficients that connect the PDFs in schemes with different numbers of active flavors. The color and flavor structure—including non-singlet (NS), pure singlet (PS), and gluon–initiated channels—parallels the unpolarized case, but with distinctive operator projections that isolate spin-dependent contributions. Evaluation is performed predominantly in the Larin scheme to ensure a consistent treatment of γ_5 and the axial anomaly in D-dimensional regularization.

The OMEs are expanded perturbatively: $$ΔA_{ij}(N, m^2/Q^2) = δ_{ij} + \sum_{k=1}^\infty a_s^k ΔA_{ij}^{(k)}(N, m^2/Q^2),$$ with $a_s = \alpha_s/(4\pi)$.

2. Computational Methodologies and Analytic Structures

Three-loop calculations of polarized massive OMEs exploit elaborate reduction and summation methods:

• IBP Reduction and Master Integrals: Diagrams are generated (QGRAF) and reduced to master integrals using IBP (e.g., Reduze 2).
• Nested Sums and Differential Equations: Solutions for master integrals are obtained through systems of recurrences (difference equations in N or t-space) and differential equations in auxiliary parameters, producing closed analytic results wherever possible.
• First-order-factorizable vs. Non-first-order-factorizable Parts: The decomposition is based on whether the differential/difference equations of master integrals fully factorize at first order in the variable of interest ("first-order factorizable") (Ablinger et al., 2023). Non-first-order-factorizable parts (notably those involving _2F_1 functions and higher transcendentalities such as elliptic integrals) require semi-analytic expansions and precise series matching across singularities and thresholds (Ablinger et al., 1 Mar 2024).

In Mellin space (N-space), the results are given in terms of rational functions, generalized harmonic sums, finite (inverse) binomial sums, and, for certain topologies, cyclotomic and Kummer–Poincaré sums. After Mellin inversion, the x-space expressions (often required for phenomenology) are written using iterated integrals over alphabets that may involve square root–valued letters and, for two-mass cases, nontrivial algebraic dependencies on mass ratios.

Table 1: Analytic and Algorithmic Components

Structure	N-space	x-space (Bjorken-x)
Basis functions	Nested harmonic sums, binomial sums	Iterated integrals, polylogarithms
Special constants	Zeta values, Kummer-Poincaré blocks	Algebraic constants/values (e.g., at x=1/4, 1/2, 1)
Alphabet (letters)	(variable) 17-letter (t-space)	14-letter (x-space) incl. √ terms

3. Treatment of Single-Mass and Two-Mass Contributions

At three loops, single-mass and two-mass corrections both enter, starting for the first time at O(α_s^3). Single-mass OMEs have been computed completely for all parton channels (Ablinger et al., 2022, Ablinger et al., 1 Mar 2024, Ablinger et al., 2023). The two-mass contributions, where two distinct massive quark flavors (e.g., charm and bottom) run in the loops, are more involved. They are essential in precision phenomenology: the two-mass terms typically account for up to 50% of the full O(T_F^2) and O(T_F^3) contributions in the polarized sector (Ablinger et al., 10 Oct 2025). Their calculation involves:

• Resummation techniques introducing generating functions with auxiliary parameters (t), mapping local operator insertions to effective propagators.
• Reduction to master integrals, followed by the solution of large coupled differential systems in t-space or with respect to the mass ratio variable η.
• Expansion in the mass ratio and matching of high-precision series expansions to construct highly accurate x-space representations.

These results are compared with independent large-moment calculations (N=2000–3000) to confirm precision.

4. Analytic Continuation, Special Constants, and Endpoint Behaviors

For practical applications (Mellin inversion, convolutions with PDFs), analytic continuation of the N-space expressions to complex N is required. The presence of fast-growing intermediate contributions (e.g., factors of 2^N or 4^N) in individual summands necessitates elaborate cancellation mechanisms for physical results, especially as N→∞. Iterated integrals in x-space use extended alphabets, including up to 17 letters in t-space and 14 in x-space, many involving square roots and roots of quadratic expressions.

Special constants (up to weight 5) arise when evaluating these integrals at particular arguments (e.g., x=1/4, 1/2, 1), and are rationalized to Kummer–Poincaré iterated integrals. These constants must be handled carefully for consistency and to ensure all endpoint limits are correct.

The endpoint behaviors are:

• Small–x (x → 0) limit: In the unpolarized case, the leading behavior is

$$a_{Qg}^{(3), x \to 0}(x) = \frac{64}{243} C_A^2 T_F [1312 + 135\,\zeta_2 - 189\,\zeta_3] \frac{\ln x}{x},$$

while in the polarized sector, the leading behavior may be of the form

$$\Delta a_{Qg}^{(3)}(x) \propto \frac{4}{3} C_F T_F^2 N_F \ln^5 x,$$

with subleading terms contributing significantly and requiring precise cancellation between singular contributions from various master integrals.

• Large–x (x → 1) limit: The behavior is dominated by soft-gluon logarithms, typically of the form

$$a_{Qg}^{(3), x\to 1}(x) \simeq \frac{8}{3}(C_A - C_F)^2 \ln^5(1-x) + \mathcal{O}(\ln^4(1-x)).$$

The unpolarized and polarized OMEs become numerically very close at large x.

5. Role in the Variable Flavor Number Scheme and Phenomenological Impact

The polarized massive OMEs are fundamental for implementing matching in the VFNS, allowing heavy-quark PDFs to be defined process-independently and consistently across flavor thresholds (Ablinger et al., 2 Oct 2025). The explicit matching relations in Mellin-N space involve all single-mass and two-mass OMEs, ensuring that heavy flavors are properly included as active partons above their respective matching scales. These matching coefficients, being process–independent, enable the reabsorption of all logarithmic and constant heavy mass dependence into PDFs, allowing for precision analyses of polarized DIS data and other hard scattering processes.

Fast numerical implementations and libraries (‘libome’, OMEPOL3) are now available, providing high-precision evaluations of these OMEs across the kinematic range in x and Q^2.

6. Mathematical and Algorithmic Innovations

The final three-loop results for the polarized OMEs integrate multiple algorithmic advances:

• Automated summation and recurrence solving (Sigma, HarmonicSums, difference–ring theory).
• Series solution and matching methodologies for non-first-order-factorizable master integrals, achieving up to 250-digit precision for analytic continuation and threshold matching (Ablinger et al., 1 Mar 2024).
• Iterated integral technology over extended alphabets and the rationalization of constants (Kummer–Poincaré blocks) enable fully analytic x-space representations, critical for Mellin inversion and convolution with experimental and theory PDFs.

These innovations address the unique challenges posed by new transcendental structures beyond multiple zeta values (MZVs) at high weights, especially for V-topology and two-mass master integrals.

7. Current Status and Outlook

With recent results (Ablinger et al., 10 Oct 2025), the computation of all unpolarized and polarized massive operator matrix elements at three loops is complete, covering both single-mass and two-mass contributions, and all relevant partonic channels. This comprehensive understanding:

• Enables model-independent, high-precision evolution and matching of polarized PDFs,
• Underpins O(α_s^3) predictions for spin-dependent observables in DIS and related processes,
• Is essential for global fits and the extraction of QCD parameters (α_s, heavy quark masses, polarized PDFs) at next-to-next-to-leading order (NNLO).

Future directions include extending these results to even higher loop orders, incorporating potential elliptic and modular structures, and their implementation in global PDF fitting frameworks and Monte Carlo generators.