Larin Scheme in QCD

• Larin Scheme is a dimensional-regularization prescription that rigorously treats γ5 and axial currents in higher-order QCD calculations.
• It employs a D-dimensional approach using Levi-Civita tensors to maintain consistency when polarized observables and chiral effects are present.
• The scheme underpins key applications in polarized deep-inelastic scattering, heavy flavor matching, and operator renormalization, ensuring all components are defined consistently.

The Larin scheme is a dimensional-regularization prescription for handling $\gamma_5$ and axial-vector structures in perturbative QCD beyond leading order. It is used when polarized observables, axial currents, or chiral projectors are present and naive four-dimensional anticommutation of $\gamma_5$ is not consistent in $D\neq 4$. In contemporary higher-order calculations, the scheme is not merely a local definition of a Dirac matrix: it can function as the operative renormalization and factorization convention for the polarized sector, with Wilson coefficients, anomalous dimensions, operator matrix elements (OMEs), and parton distribution functions (PDFs) all defined consistently within the same framework (Blümlein et al., 2024, Ablinger et al., 2024).

1. Definition and conceptual status

In the literature considered here, the Larin scheme is defined as a consistent dimensional-regularization treatment of $\gamma_5$ and axial currents for higher-order QCD calculations involving polarized quantities, axial-vector couplings, or chiral effects (Blümlein et al., 2024, Blümlein et al., 2022). It is used in polarized deep-inelastic scattering (DIS), axial-vector form factors, heavy-flavor OMEs, variable flavor number scheme (VFNS) matching, semi-inclusive DIS, and unresolved infrared limits of polarized matrix elements (Ablinger et al., 2024, Gehrmann et al., 2021, Behring et al., 2021, Gehrmann et al., 2024, Bonino et al., 30 Sep 2025).

A central feature of the scheme is that it replaces a naive four-dimensional treatment of $\gamma_5$ by a $D$-dimensional prescription based on Levi-Civita tensors and ordinary gamma matrices. In the axial-vector current context, one writes

$$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$

which is the key Larin replacement used explicitly in multiloop form-factor and electroweak coefficient-function calculations (Gehrmann et al., 2021, Bonino et al., 30 Sep 2025). In related discussions of polarized operator insertions, the scheme is described as rooted in the ’t Hooft–Veltman/Breitenlohner–Maison approach, but with the practical advantage that Lorentz algebra can be carried out in $d$ dimensions without splitting the space into four-dimensional and $(d-4)$-dimensional parts (Gehrmann et al., 2021).

In modern polarized QCD applications, the Larin scheme is frequently treated as a full polarized factorization/renormalization scheme rather than a purely technical $\gamma_5$ prescription. This is explicit in three-loop heavy-flavor DIS and NNLO polarized PDF evolution, where the perturbative building blocks and the evolved PDFs are all defined in the same scheme (Ablinger et al., 2024, Blümlein et al., 2024). A plausible implication is that the term “Larin scheme” now denotes both a prescription for axial structures in dimensional regularization and, in polarized phenomenology, a convention for organizing renormalized perturbative quantities.

2. Treatment of $\gamma_5$0, axial currents, and finite renormalization

The need for the scheme arises because $\gamma_5$1 is intrinsically four-dimensional and does not generalize consistently by naive anticommutation to $\gamma_5$2 dimensions in higher-order calculations (Gehrmann et al., 2021). This issue appears in polarized DIS, singlet axial-vector form factors, flavor-singlet axial-current renormalization, and baryonic operators containing $\gamma_5$3 (Ablinger et al., 2024, Gehrmann et al., 2021, Gracey, 2020, Gracey, 2012).

The practical prescription is accompanied by a characteristic renormalization pattern. Because the axial current defined in this way does not automatically preserve chiral Ward identities, one requires finite renormalization in addition to the usual ultraviolet renormalization. In the axial-vector form-factor calculation, the renormalization of the vertex involves

$\gamma_5$4

where $\gamma_5$5 is the divergent part and $\gamma_5$6 is the finite renormalization factor (Gehrmann et al., 2021). The same logic appears in the renormalization of the flavor-singlet axial current, where the Larin method is used to ensure that the chiral anomaly is properly included and the correct anomalous Ward identity is restored after finite renormalization (Gracey, 2020).

For singlet quantities, the renormalization is more intricate than for non-singlet ones. In the three-loop singlet axial-vector quark form factor, separate renormalization constants are required for singlet and non-singlet axial currents, and the pure-singlet contribution is organized through differences of singlet and non-singlet renormalization factors (Gehrmann et al., 2021). For the flavor-singlet axial current at the symmetric subtraction point, the singlet finite renormalization factor is given as

$\gamma_5$7

while the non-singlet factor differs in the $\gamma_5$8 term (Gracey, 2020).

The same structural principle extends beyond current renormalization. In the three-loop renormalization of proton-like three-quark operators, the Larin prescription is generalized from a single finite renormalization constant to a finite renormalization matrix, because the operators mix under renormalization (Gracey, 2012). There, the physical anomalous dimension is obtained from a “naive” $\gamma_5$9-dimensional result plus an additional anomalous-dimension matrix generated by the finite renormalization. This shows that the scheme is compatible with operator mixing and evanescent-operator sectors, not only with simple current insertions (Gracey, 2012).

A recurrent technical consequence is that intermediate finite terms can be scheme dependent even when physical remainders are not. In the singlet axial-vector form factor, the finite parts of bare and renormalized amplitudes contain scheme-dependent $D\neq 4$0-terms, while the infrared-finite remainder after universal infrared subtraction is scheme independent (Gehrmann et al., 2021). In the unresolved limits of polarized tree-level matrix elements, the polarized splitting amplitudes contain evanescent $D\neq 4$1-dependent terms tied to the $D\neq 4$2 scheme, and finite transformation to a helicity-conserving scheme restores the axial Ward identity and helicity conservation on the polarized quark line (Gehrmann et al., 2024).

3. Polarized DIS, heavy flavor, and the emergence of a polarized scheme

The most extensive present-day use of the Larin scheme is in polarized DIS and heavy-flavor factorization. In the three-loop heavy-flavor DIS program, the polarized Wilson coefficients, anomalous dimensions, OMEs, and PDFs are all kept in the Larin scheme so that the polarized observable $D\neq 4$3 can be computed directly without requiring a three-loop conversion to $D\neq 4$4 (Ablinger et al., 2024). The same perspective is explicit in NNLO evolution of polarized parton densities, where the evolved distributions are provided in Bjorken $D\neq 4$5 space in the Larin scheme for the first time (Blümlein et al., 2024).

This usage is driven by the fact that polarized heavy-flavor calculations combine multiple ingredients that all depend on the axial prescription. In three-loop heavy-flavor DIS these include three-loop polarized massive OMEs, three-loop polarized massless anomalous dimensions, and polarized Wilson coefficients (Ablinger et al., 2024). In the asymptotic heavy-flavor program, all relations are expressed in the Larin scheme, which is described as a consistent scheme also in the massive case (Blümlein et al., 2021). In the massless three-loop Wilson-coefficient calculation, the polarized coefficient functions for $D\neq 4$6 are computed in the Larin scheme, whereas the unpolarized coefficients for $D\neq 4$7, $D\neq 4$8, and the unpolarized part of $D\neq 4$9 are given in $\gamma_5$0 (Blümlein et al., 2022).

The scheme also organizes the polarized sector of the VFNS. The polarized transition matrix element $\gamma_5$1 is calculated to three loops in the Larin scheme and enters the matching from $\gamma_5$2 to $\gamma_5$3 active flavors (Behring et al., 2021). The same is true for the single-mass and two-mass polarized OMEs $\gamma_5$4, $\gamma_5$5, and $\gamma_5$6, which are all computed in the Larin scheme and interpreted as polarized matching functions in the heavy-flavor sector (Ablinger et al., 2022, Ablinger et al., 10 Oct 2025, Ablinger et al., 2019). In the three-loop single-mass VFNS, the polarized case generally works in the Larin scheme, while two-loop polarized massive OMEs are also presented in $\gamma_5$7 (Ablinger et al., 2 Oct 2025).

The practical rule stated repeatedly is that physical observables remain scheme independent only when all ingredients are used consistently in the same scheme. In polarized DIS and VFNS matching, this means Larin-scheme PDFs must be combined with massless and massive Wilson coefficients and OMEs computed in the Larin scheme (Blümlein et al., 2024, Behring et al., 2021, Blümlein et al., 2021). The same principle is reiterated for heavy-flavor corrections to $\gamma_5$8, where the scaling violations of the polarized massless parton densities are described in this scheme by the corresponding anomalous dimensions (Ablinger et al., 2024).

4. Relation to $\gamma_5$9 and the structure of scheme dependence

The relation between the Larin scheme and $\gamma_5$0 is central but uneven across observables. At leading order the evolution of parton distributions is scheme invariant, while differences in anomalous dimensions between the Larin and $\gamma_5$1 schemes start at next-to-leading order (Blümlein et al., 2024). The corresponding finite renormalization terms are identified as

$\gamma_5$2

in the polarized evolution analysis (Blümlein et al., 2024).

Where sufficient higher-order information is available, explicit conversion can be carried out. In the non-singlet polarized DIS coefficient functions, the one-loop relation is

$\gamma_5$3

after which the coefficient matches the corresponding $\gamma_5$4 non-singlet coefficient (Blümlein et al., 2022). In polarized SIDIS at NNLO, the Larin-scheme coefficient functions are converted to $\gamma_5$5 by a finite transformation reformulated in an explicit quark-flavor basis, because assignment of finite renormalization terms to partonic channels is nontrivial in semi-inclusive kinematics (Bonino et al., 30 Sep 2025).

In many three-loop singlet applications, however, the full conversion is not yet available. The heavy-flavor DIS analysis states explicitly that “Currently it is not possible to construct the transformation into the $\gamma_5$6 scheme at three-loop order for them,” so the polarized three-loop Wilson coefficients and OMEs are retained in the Larin scheme (Ablinger et al., 2024). The massless three-loop Wilson-coefficient calculation likewise states that converting the singlet polarized $\gamma_5$7 result would require the polarized four-loop anomalous dimensions in the singlet case, so only the non-singlet coefficient is switched to $\gamma_5$8 (Blümlein et al., 2022). The single-mass polarized pure-singlet OME is similarly left in the Larin scheme because the full finite renormalization of the constant term to the $\gamma_5$9-scheme is not known (Ablinger et al., 2019).

The resulting scheme dependence is quantitatively non-negligible in polarized evolution. The comparison ratio

$D$0

is used to compare PDFs in the two schemes (Blümlein et al., 2024). In the heavy-flavor DIS discussion, quark distributions are said to differ by about $D$1 at $D$2 and gluon distributions by about $D$3, implying that high-precision polarized fits must use PDFs evolved in the Larin scheme when the perturbative ingredients are available only there (Ablinger et al., 2024). The NNLO evolution study further states that corrections can reach up to $D$4 in the small-$D$5 range, $D$6, and that the deviations of the evolution of different parton densities are larger than the projected accuracy for future polarized deep-inelastic scattering experiments of $D$7 (Blümlein et al., 2024).

Several structural properties of the scheme difference are explicitly known. The evolution is different in the Larin and $D$8 schemes starting at NLO; for the gluon, the difference is induced by quark–gluon mixing; $D$9 is the same in both schemes; and in the large-$$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$0 limit the two schemes approach each other (Ablinger et al., 2024). In the NNLO evolution analysis, the high-$$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$1 suppression is connected to large-$$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$2 behavior of the splitting-function differences, which vanish sufficiently rapidly that $$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$3 for $$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$4 (Blümlein et al., 2024).

5. Extensions beyond inclusive polarized DIS

The scope of the Larin scheme extends beyond inclusive DIS into form factors, infrared limits, semi-inclusive observables, and operator renormalization.

In the three-loop singlet axial-vector quark form factor, the scheme governs the definition of the axial current, the ultraviolet renormalization of singlet and non-singlet contributions, and the extraction of a pure-singlet finite remainder after infrared subtraction (Gehrmann et al., 2021). The pure-singlet contribution is defined by

$$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$5

and its renormalized three-loop finite remainder is obtained after subtracting the universal infrared singular structure (Gehrmann et al., 2021). This is a clear instance where the Larin prescription shapes the intermediate renormalized amplitude but not the final infrared-finite remainder.

In unresolved infrared limits, the scheme has a more differential role. The full set of splitting amplitudes arising in longitudinally polarized tree-level QCD matrix elements at NNLO has been derived in the Larin $$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$6 scheme (Gehrmann et al., 2024). There the polarized single-collinear splitting functions are the real-radiation parts of the polarized Altarelli–Parisi kernels and contain scheme-dependent $$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$7-dependent evanescent terms (Gehrmann et al., 2024). The same work states that a polarized gluon does not produce a soft singularity, and that if a polarized particle is inside the unresolved cluster, the corresponding splitting amplitude is polarized and the clustered reduced leg is polarized iff the splitting itself is polarized (Gehrmann et al., 2024).

In polarized SIDIS at NNLO, the Larin scheme is used as the intermediate scheme for all polarized coefficient-function calculations, after which results are converted to $$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$8 by finite renormalization (Bonino et al., 30 Sep 2025). The work computes polarized neutral-current and charged-current coefficient functions for

$$\gamma^\mu\gamma_5 \;\to\; \frac{i}{3!}\,\epsilon^{\mu\nu\rho\sigma}\,\gamma_\nu\gamma_\rho\gamma_\sigma,$$9

and identifies a projector issue specific to polarized quarks in $d$0 dimensions (Bonino et al., 30 Sep 2025). The corrected projector is written as

$d$1

which is presented as the Larin-specific remedy for obtaining the correct NLO and NNLO non-singlet splitting-function structure in SIDIS (Bonino et al., 30 Sep 2025).

Operator renormalization provides further examples. In three-quark operators relevant to the proton, the Larin prescription is extended to a mixing matrix of finite renormalization constants chosen so that chiral symmetry is manifest in four dimensions (Gracey, 2012). In one-loop-corrected Fierz identities, the Larin scheme appears as a target Dirac scheme to which one may transform from a simpler computation in naive dimensional regularization, with renormalization-scheme dependence factorizing from basis change (Aebischer et al., 2023). This suggests that the Larin scheme also serves as a reference scheme in broader operator-basis technology, not solely in polarized phenomenology.

6. Mathematical representations, kinematic patterns, and phenomenological implications

Calculations performed in the Larin scheme commonly use Mellin-$d$2 and momentum-fraction representations that parallel unpolarized analyses but inherit polarized-specific selection rules and analytic structures. In polarized heavy-flavor OMEs, Mellin-space results are expressed through harmonic sums, zeta values, and sometimes generalized sums or binomial structures, while $d$3-space results are written in terms of harmonic polylogarithms (Behring et al., 2021, Blümlein et al., 2021, Ablinger et al., 2022). The scheme does not alter the functional basis itself; it alters the renormalization and factorization interpretation of the polarized quantities (Behring et al., 2021).

Odd–even moment selection is a recurring feature. For the polarized three-loop gluonic OME,

$d$4

so only odd Mellin moments contribute (Ablinger et al., 2022). In the two-mass polarized gluonic OME, the physical values are stated to be $d$5 in the polarized case, and the first polarized moment vanishes: $d$6 (Ablinger et al., 10 Oct 2025). In the heavy-quark–antiquark asymmetry, the polarized asymmetry first appears at three loops through $d$7 terms in $d$8 for even moments in the polarized case, and the calculation is performed in the Larin scheme (Behring et al., 15 Dec 2025).

Small-$d$9 and large-$(d-4)$0 behavior are often analyzed explicitly. For the single-mass polarized pure-singlet OME, the leading small-$(d-4)$1 term is

$(d-4)$2

but the complete result is required because subleading terms are numerically important (Ablinger et al., 2019). For the polarized gluonic OME, the rightmost singularity relevant for small-$(d-4)$3 behavior sits at $(d-4)$4, unlike the unpolarized case where it is at $(d-4)$5, and the polarized small-$(d-4)$6 expansion begins with a $(d-4)$7 term multiplied by a definite color-factor combination (Ablinger et al., 2022). These structural facts do not define the scheme, but they characterize the perturbative objects as they are computed within it.

Phenomenologically, the scheme matters because the perturbative inputs available for polarized analyses are often only known in the Larin scheme at the required order. This is explicit for three-loop heavy-flavor DIS, asymptotic polarized Wilson coefficients, and three-loop VFNS matching (Ablinger et al., 2024, Blümlein et al., 2021, Ablinger et al., 2 Oct 2025). The resulting polarized predictions are scheme-consistent and physically usable, but their PDF representation is scheme specific. Physical observables such as $(d-4)$8 remain scheme independent when all ingredients are used consistently, whereas direct comparison of polarized PDFs across schemes requires the corresponding finite transformation, which may be unavailable at the relevant perturbative order (Ablinger et al., 2024, Blümlein et al., 2024).

In current practice, the Larin scheme is therefore best understood as a technically precise and phenomenologically operative convention for polarized QCD in dimensional regularization: it resolves the $(d-4)$9 problem at higher orders, supplies a consistent framework for renormalization and factorization in the polarized sector, and remains indispensable wherever higher-loop polarized coefficient functions, anomalous dimensions, and heavy-flavor matching relations are known only in that scheme (Blümlein et al., 2022, Ablinger et al., 2024).