Scale setting for asymptotic-freedom predictions: nonperturbative inputs, singlet mixing, and the spacelike-timelike continuation

The 1974 analysis "Setting the scale for predictions of asymptotic freedom" (Politzer) reduced the predictions of asymptotically free QCD for observable scaling at accessible energies to two nonperturbative inputs - the coupling $g$, equivalently the scale $\Lambda_{\rm QCD}$ marking where the logarithmic approach to the ultraviolet fixed point begins, and the hadronic matrix elements $A_n^i$ of the twist-2 quark and gluon operators, including the unknown relative weight of the two components that mix in the flavor-singlet channel - and flagged that the continuation of the spacelike results to timelike observables such as $\sigma(e^+e^-\to{\rm hadrons})$ is uncontrolled without independent information on the hadronic spectral function. The operational requests of 1974 - dedicated moment fits with controlled singlet/nonsinglet decomposition, and nonperturbative or empirical determination of the onset scale - have been achieved and far exceeded by the DGLAP/global-PDF program and by lattice determinations of $\Lambda_{\rm QCD}$ and $\alpha_s$ at the percent level. What remains open is the deeper layer of the same core: determining the hadronic matrix elements (and the singlet mixing ratio) from first principles at phenomenologically competitive precision, and establishing a controlled, model-independent continuation to timelike observables with quantified quark-hadron duality violations.

Background

In 1973 it was discovered that non-Abelian gauge theories are asymptotically free: the renormalization-group beta function $\beta(g)$ - which governs how the coupling $g$ changes with the arbitrary normalization scale $\mu$ through $\partial g/\partial\ln\mu=\beta(g)$ - is negative near $g=0$, so the effective interaction weakens logarithmically at short distances. Applied to the strong interactions, this gave the first dynamical explanation of the approximate Bjorken scaling observed in deep-inelastic scattering (DIS): inelastic lepton-nucleon scattering is described by structure functions $F(x,q^2)$ of the Bjorken variable $x$ (the fraction of the virtual photon's momentum carried by the struck quark or gluon; the constituents of hadrons are collectively called partons) and the photon virtuality $q^2<0$, which at large $|q^2|$ depend on $q^2$ only through logarithms rather than powers.

The quantitative instrument is Wilson's operator product expansion (OPE): at short distances the product of two electromagnetic currents expands into a series of local operators with coefficient functions computable in perturbation theory. Taking the $n$-th moment (the $x$-integral weighted by $x^{n-1}$) of a structure function projects onto the hadron matrix element $A_n^i$ of a twist-2 operator: a composite local quark or gluon operator whose twist (dimension minus spin) takes its minimal value 2, with spin $n$ matching the moment index, while the $q^2$ dependence is governed by the operator's anomalous dimension $\gamma_n(g)$, the deviation of its renormalization from that of the elementary fields. In an asymptotically free theory each moment therefore behaves as a calculable power of $\ln(-q^2)$.

Politzer's 1974 paper "Setting the scale for predictions of asymptotic freedom" packages these results into a form confrontable with then-available data: each even moment of the structure functions $xW_1$ or $\nu W_2$ (the two standard deep-inelastic structure-function combinations, with $\nu$ the energy transferred to the target) is the product of the constant $A_n^i$, a calculable coupling prefactor, and a predicted power of $[1+b\,g^2\ln(-q^2/M^2)]$, where $b$ is the positive, purely group-theoretic first coefficient of $\beta(g)$ and $M^2$ is any scale marking the onset of the approximate-scaling region. For a fixed gauge group and fermion content everything is predicted except two nonperturbative inputs: the coupling $g$ - equivalently the single mass scale $\Lambda_{\rm QCD}$, since the choice of $M$ is a convention and the theory depends only on the combination - and the constants $A_n^i$. Moments of flavor-nonsinglet combinations - differences of structure functions in which the flavor-singlet content cancels, for example the proton-minus-neutron difference - evolve with a single exponent; the flavor-singlet moments, which sum over quark flavors, mix the quark and gluon twist-2 operators, so two independent $q^2$-dependent eigencomponents appear in an unknown ratio (for the first singlet moment one eigenvalue vanishes: this is the momentum sum rule, the statement that the momentum fractions carried by all partons add to one, and the one number predicted without fitting). The companion computations of Georgi and Politzer and of Gross and Wilczek supply the group-theory ingredients.

The paper flags three needs it cannot itself meet. First, the observed rapid onset of scaling is decided by $\beta(g)$ at moderately strong coupling, where the small-$g$ expansion - which controls only the tail approaching the ultraviolet fixed point (the coupling-free point $g=0$ that the running approaches at short distances) - says nothing. Second, the matrix elements $A_n^i$ (including the singlet ratio) must be fitted from data, with the higher moments experimentally the least certain. Third, predictions for timelike observables - $R(s)=\sigma(e^+e^-\to{\rm hadrons})/\sigma(e^+e^-\to\mu^+\mu^-)$, proportional to the imaginary part of the hadronic vacuum polarization at timelike momentum $s>0$ - obtained by continuing the spacelike results are, in the paper's words, suspect: the dispersion relation ties the vacuum polarization to its spectral function, small Euclidean deviations can produce large timelike ones, and the resonance and threshold structure of the hadronic spectrum - the information that makes the analogous continuation safe in QED - is not known in an interacting non-Abelian gauge theory.

The subsequent history fixes the present status of the question. The moment framework became the DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) evolution equations (the coupled integro-differential equations for the parton distribution functions, whose moments are precisely the $A_n^i$), the beta function is perturbatively known to five loops, and the $A_n^i$ are now fitted in global parton-distribution analyses at next-to-next-to-leading order. $\Lambda_{\rm QCD}$ and the strong coupling $\alpha_s=g^2/4\pi$ are determined nonperturbatively from lattice QCD - the first-principles numerical evaluation of the QCD path integral on a discretized spacetime - via step-scaling studies, in which the running coupling is tracked nonperturbatively across a ladder of energy scales before matching to perturbation theory. The timelike caveat survives as the problem of quark-hadron duality (the assumption that suitably averaged hadronic observables equal their OPE/perturbative descriptions) and of quantifying its violations.

Problem Statement

The problem is the completion of the scale-setting program that Politzer's 1974 analysis left open. The onset of scaling and the determination of $\Lambda_{\rm QCD}$ (equivalently $\alpha_s$ at a reference scale) at percent-level precision are established background. Two parts remain open, posed with no restriction to a particular method, observable class, or theory regime:

(a) Nonperturbative matrix elements: determine the hadronic matrix elements $A_n^i$ of the twist-2 quark and gluon operators entering the moments of the deep-inelastic structure functions - including the relative weight of the two $q^2$-dependent components that mix in the flavor-singlet channel, which the evolution equations leave as an initial condition - from first principles, with quantified uncertainties, notably for the higher moments where the predicted $q^2$ dependence is strongest. Determination of these matrix elements by controlled experimental fits (global parton-distribution analyses) has been achieved; citing existing fits does not by itself close this part.

(b) Timelike continuation: establish a controlled procedure for predicting timelike observables such as $\sigma(e^+e^-\to{\rm hadrons})$ and, in modern form, the spectral functions measured in $\tau$ decay, from the spacelike perturbative results, with quantified deviations from quark-hadron duality rather than uncontrolled modeling.

A complete answer closes both parts at a level of precision and control that makes the asymptotic-freedom predictions quantitatively trustworthy at accessible energies. Solving a single part - fitted matrix elements alone, or a controlled timelike continuation alone - is progress on, not a solution of, the stated problem.

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