Completing the optimized perturbation theory program for factorization- and mass-scheme dependence in finite-order perturbative QCD

H. D. Politzer's 1982 paper applied Stevenson's optimized perturbation theory (OPT, based on the principle of minimal sensitivity) to the factorization-scheme and mass-scheme dependence of finite-order perturbative QCD predictions. The paper addressed the optimization of factorization-scheme dependence in inclusive hadron reactions and of problems with mass dependence, and discussed 'a serious shortcoming of the procedure, common to all applications': the non-uniqueness of optimized finite-order results owing to implicit assumptions about uncomputed higher-order terms. An ambiguity in the 1982 optimization equations for factorized quantities was later resolved (a 1986 correction of an algebraic error in the 1982 analysis, refined to a firm footing by Stevenson in 2019 for the non-singlet, moment-space case), the then-incomplete factorization proofs for two-hadron processes were closed by the Collins-Soper-Sterman factorization proofs, and the truncation shortcoming has been transformed into the problem of quantifying missing-higher-order uncertainties, without an accepted solution. The surviving open problem is to complete the optimization program as a practical, principled procedure for perturbative QCD: extend it from the non-singlet, moment-space case to the flavour-singlet sector and to DGLAP parton evolution; develop the optimization of mass-scheme dependence, including how mass-dependent schemes and the decoupling of heavy fields (threshold matching) should be organized together with optimization; and quantify the residual dependence of optimized answers on uncomputed higher orders. A recent challenge to the framework's assumptions — 2026 results forcing the factorization and renormalization scales to be equal in generalized pole-subtraction schemes — is relevant context a solution may have to confront.

Background

In quantum chromodynamics (QCD), predictions for hard processes such as deep-inelastic scattering (DIS) — where a lepton scatters off a hadron by exchanging a photon of large virtuality $Q^2$ — are computed as truncated power series in the running coupling $a(\mu)\equiv\alpha_s(\mu)/\pi$. Here $\mu$ is the renormalization scale, an arbitrary mass parameter introduced when ultraviolet divergences are absorbed into a redefinition of the coupling; the coupling's scale evolution is governed by the $\beta$ function, $\mu\,\partial a/\partial\mu=-ba^2(1+ca+c_2a^2+\cdots)$, whose first two coefficients $b$ and $c$ are scheme-independent while $c_2,c_3,\ldots$ depend on the renormalization scheme (RS) — the convention for the finite part of the coupling's definition, with $\mu$ and $c_2,c_3,\ldots$ serving as RS labels. Beyond leading order, a truncated ('finite-order') prediction depends on these arbitrary choices; the exact all-orders result does not. The dependence is also encoded in the integration constant $\Lambda$ of the renormalization-group (RG) equation, which sets the momentum scale at which the coupling grows large.

Stevenson (1981) proposed to remove this unphysical arbitrariness by optimization: since the residual scheme dependence of a truncated series is spurious, choose the scheme parameters so that the prediction is locally insensitive to them, i.e. impose stationarity with respect to each artificial parameter — the principle of minimal sensitivity (PMS), also called optimized perturbation theory (OPT). For purely perturbative quantities, such as the ratio $R$ for electron-positron annihilation into hadrons, this eliminates ad hoc guesses of the 'right' scale or scheme.

For quantities involving non-perturbative physics an extra layer of arbitrariness appears. Moments (Mellin transforms in the momentum-fraction variable $x$) of DIS structure functions factorize as

$$F_n(Q)=\langle O_n(M)\rangle\,C_n(Q,M),$$

where $\langle O_n(M)\rangle$ is a non-perturbative hadronic matrix element (the ancestor of the parton distribution functions, PDFs, which describe the momentum distribution of partons in a hadron), $C_n$ is a perturbatively calculable coefficient function, and $M$ is the factorization scale. The split between the two factors is itself a convention: finite redefinitions of the operators $O_n$ (equivalently, of the PDFs) move finite terms between $\langle O\rangle$ and $C$. A factorization scheme (FS) is fixed by $M$ together with the scheme-dependent coefficients $d_1,d_2,\ldots$ of the anomalous dimension $\gamma(a)=d\,a+d_1a^2+\cdots$ that governs the $M$-evolution of $\langle O\rangle$. Some combinations of labels are redundant: to first nontrivial order only the combination $d_1+b\,d\log M$ is physical. Evolution of PDFs in $x$-space is conventionally described by the DGLAP equations (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi), the standard parton-evolution equations used in phenomenology. When renormalized masses enter, a further scheme choice appears: in mass-dependent schemes the RG functions depend nontrivially on $m/\mu$, while mass-independent schemes such as modified minimal subtraction ($\overline{\mathrm{MS}}$) must be supplemented by explicit decoupling — the effective disappearance of heavy fields from amplitudes at scales far below their mass — implemented by matching conditions that join the theories with and without the heavy field at a threshold scale.

Politzer (1982) extended OPT to precisely these problems: the simultaneous optimization of RS and FS choices for factorized quantities and the optimization of mass-scheme dependence. The paper also discussed a serious shortcoming common to all applications of optimization: imposing stationarity on a truncated series implicitly fixes particular combinations of the uncomputed higher-order coefficients, so optimized finite-order answers are not unique — different, equally consistent choices differ by terms of the next order in $a$.

The subsequent history is essential to the present problem. Stevenson and Politzer (1986) found that the main difficulty of the 1982 analysis — in which the optimization equations for factorized quantities appeared to have no solution — came from a trivial algebraic error, and corrected it. Nakkagawa and Niégawa (1982-1984) developed a simultaneous treatment of renormalization- and factorization-scheme dependence whose optimization equations are equivalent to the modern ones, though Stevenson (2019) judges their starting point unnecessarily complicated. Stevenson (2019), elaborated in his 2022 monograph, placed the non-singlet factorization-scheme optimization on a 'firm footing': with the proper scheme variables, RG equations, and invariants identified, all optimized perturbative coefficients of the coefficient function vanish ($C^{\mathrm{opt}}=1$), making the procedure as simple as for purely perturbative quantities — but only for non-singlet quantities (structure-function combinations without quark-gluon mixing) and in moment space. The factorization gap for processes with two incoming hadrons, such as Drell-Yan lepton-pair production, was closed by the proofs of Collins, Soper and Sterman (1989), now textbook material (Collins, 2011). The truncation ambiguity, by contrast, was never eliminated: it is handled today as an uncertainty-estimation problem for missing higher orders, via Bayesian models (Cacciari and Houdeau, 2011; Duhr, Huss, Mazeliauskas and Szafron, 2021) and conventional scale variation, while competing scale-setting schools — effective charges (the coupling redefined as a physical observable; Chýla, 1989), complete renormalization-group improvement (Maxwell and Mirjalili, 2000), and the principle of maximal conformality (Yan, Brodsky, Di Giustino, Ratcliffe, Wang and Wu, 2025, claiming that it 'correctly resolves' the scheme-dependence problem) — continue to dispute the right prescription. Stevenson (1984) analyzed whether optimization improves the ultimate convergence of QCD perturbation theory. Heavy-quark decoupling is now implemented to four loops in fixed conventions (Chetyrkin, Kühn and Sturm, 2006), but by convention rather than optimization. Most recently, Rogers and Whitehill (2026) showed, in explicit examples in Yukawa theory (a toy field theory in which a scalar boson couples to a fermion, used because all relevant quantities remain explicitly calculable in perturbation theory), that in generalized pole-subtraction schemes using dimensional regularization, simultaneously preserving RG invariance, Ward identities (the gauge-invariance relations constraining counterterms), and parton sum rules (exact conservation laws, such as quark-number conservation, encoded in PDFs) forces the factorization and renormalization scales to be equal — contesting the independent variation of $M$ and $\mu$ on which the optimization parametrization relies.

Problem Statement

The problem is to complete the optimization (PMS) program for the renormalization-, factorization-, and mass-scheme dependence of finite-order perturbative QCD predictions — the program initiated by Stevenson (1981) and extended to factorized and massive quantities by Politzer (1982) — by resolving the components that the source and its follow-up literature identify as unresolved:

(1) Singlet and DGLAP extension. Extend the (now well-founded) non-singlet, moment-space optimization equations to the flavour-singlet sector, where quark and gluon operators mix and anomalous dimensions are matrices, and determine whether the moments-based approach can be reformulated in the language of DGLAP parton evolution and put into practice for the parton distribution functions used in phenomenology (Stevenson 2019, Sec. 9).

(2) Mass-scheme dependence and decoupling. Develop the optimization of problems with mass-scheme dependence announced in Politzer (1982), including how optimization should be organized when a mass-dependent renormalization scheme is used together with the decoupling of heavy fields in the $m/\mu\gg1$ regime: how threshold matching conditions and stationarity conditions combine, and whether the resulting procedure is stable and practical.

(3) Residual truncation ambiguity. Provide a principled treatment and quantification of the residual dependence of optimized finite-order answers on the uncomputed higher-order terms — the 'serious shortcoming' the source identified as common to all applications of optimization — going beyond the observation of non-uniqueness.

These components are individually identified open items of the program, not a single conjunctive acceptance contract. Recent challenges to the framework's assumptions — notably the result of Rogers and Whitehill (2026) that RG invariance, Ward identities, and parton sum rules force the factorization and renormalization scales to coincide in generalized pole-subtraction schemes — are relevant context a solution may have to confront, not standalone required components.

References

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