Short-Distance QCD Parameters, Mass Schemes, and Thresholds

A short-distance QCD parameter is not just a number: it is a renormalized quantity, a scheme, a scale, an active-flavor theory, a perturbative order, and—when inferred jointly—a covariance. This page gives a reproducible procedure for evolving and converting $\alpha_s$ and quark masses without using a pole mass or a current world summary as an unqualified input.

Required background. QCD fields, scales, and the perturbative domain fixes the coupling and color conventions. Decoupling theorems and threshold corrections supplies the matching logic at a heavy mass.

Helpful background. Scheme transformations and RG invariants supplies the rule that scheme changes must be truncated consistently with the observable.

Renormalized coupling and masses

In the $\overline{\mathrm{MS}}$ scheme, record the coupling as $\alpha_s^{(n_f)}(\mu)$ and a quark mass as $\overline m_q^{(n_f)}(\mu)$. To one loop,

$$\mu\frac{d\alpha_s}{d\mu} =-\frac{\beta_0}{2\pi}\alpha_s^2+\cdots, \qquad \mu\frac{d\overline m_q}{d\mu} =-\overline m_q\frac{\gamma_0}{4\pi}\alpha_s+\cdots,$$

where $\beta_0=11-2n_f/3$ and $\gamma_0=6C_F=8$. Dividing the equations and integrating within one fixed-$n_f$ interval gives

$$\frac{\overline m_q(\mu)}{\overline m_q(\mu_0)} =\left[\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\right]^{\gamma_0/(2\beta_0)} \left[1+O(\alpha_s)\right] =\left[\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\right]^{12/(33-2n_f)} \left[1+O(\alpha_s)\right].$$

This relation is a derivation checkpoint: increasing $\mu$ decreases both $\alpha_s$ and the running mass. Precision evolution uses beta and anomalous-dimension coefficients at a common stated loop order, rather than splicing a high-order coupling into a one-loop mass formula.

The coupling evolution and the role of renormalized quark masses in QCD are introduced in Schwartz 2014, §§26.3–26.6, pp. 513–28.

Matching an active-flavor threshold

Suppose the scale crosses a heavy quark $h$. The theories above and below the threshold have different parameters and different operator bases. At a matching scale $\mu_h$ of order a declared short-distance mass,

$$\alpha_s^{(n_f-1)}(\mu_h) =\zeta_\alpha\!\left(\alpha_s^{(n_f)}(\mu_h), \ln\frac{\mu_h^2}{\overline m_h^2(\mu_h)}\right) \alpha_s^{(n_f)}(\mu_h),$$

and analogous decoupling factors relate light-quark masses and composite operators. The leading term is $\zeta_\alpha=1$; nontrivial matching enters at higher orders. Multi-loop decoupling coefficients and their scale dependence are derived in Chetyrkin, Kniehl, and Steinhauser 1998, §§2–4, pp. 61–87.

A controlled threshold workflow is:

1. evolve in the $n_f$-flavor theory to $\mu_h$;
2. apply every required matching coefficient at one declared order;
3. evolve in the $(n_f-1)$-flavor theory;
4. vary $\mu_h$ within a justified neighborhood of the mass; and
5. treat the residual change as perturbative sensitivity correlated across all predictions using that match.

Running through a threshold with one beta function, or changing $n_f$ without matching PDFs and hard coefficients in a factorized observable, leaves an uncancelled scheme dependence.

Why the pole mass is not a precision input

For a heavy quark, perturbation theory begins with a relation of the form

$$m_q^{\mathrm{pole}} =\overline m_q(\overline m_q) \left[1+\frac{4}{3}\frac{\alpha_s(\overline m_q)}{\pi} +O(\alpha_s^2)\right].$$

At high order the pole-mass series has an infrared renormalon, so the all-order separation between a pole mass and long-distance physics is ambiguous by $O(\!\left(\Lambda_{\mathrm{QCD}}\right))$ Beneke 1999, §§3.1 and 5.4, pp. 20–28 and 67–75. This is a structural ambiguity, not an uncertainty that more pole-scheme coefficients remove.

Choose a mass adapted to the scale and observable:

Use case	Suitable short-distance choice	Required extra datum
hard process with virtuality of order $m_q$ or larger	$\overline{\mathrm{MS}}$ mass	reference scale and active flavors
nonrelativistic threshold	threshold mass such as PS, 1S, kinetic, or MSR	subtraction scale and its evolution
conversion between two schemes	explicit finite relation	conversion order and common $\alpha_s$ input

Low-scale short-distance masses cancel the leading pole renormalon while retaining sensitivity appropriate to threshold dynamics; the MSR construction provides an explicit scale-evolution example Hoang et al. 2018, §§2–4.

Propagating correlated inputs

Let the input vector be $p=(\alpha_s,m_1,\ldots,\eta)$ with covariance $C_p$, where $\eta$ denotes shared fit or calibration parameters. For a scheme conversion $p'=f(p)$, the linearized covariance is

$$C_{p'}=J C_p J^{\mathsf T}, \qquad J_{ij}=\frac{\partial f_i}{\partial p_j}.$$

For nonlinear or non-Gaussian inputs, propagate replicas or nuisance parameters through the complete conversion. Do not convert central values while retaining a covariance defined in the old scheme, and do not add an $\alpha_s$ variation to a PDF ensemble that already varies the same fitted coupling unless that correlation has been separated.

A portable parameter record contains:

Field	What must be stated
definition	$\overline{\mathrm{MS}}$, MSR, PS, 1S, kinetic, or another explicit scheme
scale and flavors	$\mu$, $n_f$, and threshold convention
evolution	beta/anomalous-dimension order and implementation version
matching	matching scales, coefficients, and orders
uncertainty	covariance, replicas, or named nuisance variations
provenance	source release or calculation, date, and any correction history

Checks and failure modes

Round trip. Evolve and match from $\mu_0$ to $\mu_1$ and back at the same truncation. The residual should be of omitted order and should decrease when the calculation is consistently improved.

Threshold continuity. A physical observable computed on either side of a threshold must agree through the claimed order after parameter, PDF, and coefficient matching. The coupling alone need not be naively identical beyond leading matching.

Scheme cancellation. Convert both the parameter and every coefficient that depends on it. A large residual scheme shift can diagnose a poor scale choice, an asymptotic series, or an inconsistent truncation.

Circular inference. Do not use a value extracted under a particular QCD theory setup as if it were independent evidence for validating the same setup. Preserve the fit assumptions and correlations.

Handoff

Subsequent calculations should receive

$$\left\{p(\mu_0),C_p,\ \mathcal S,\ n_f,\ U(\mu,\mu_0),\ M_{n_f\to n_f-1}(\mu_h),\ \text{orders and versions}\right\},$$

where $U$ is the evolution map and $M$ the complete threshold match. No current numerical parameter value is required to define this method.

References

• Beneke, Martin. “Renormalons.” Physics Reports 317, nos. 1–2 (1999): 1–142. DOI. Open PDF.
• Chetyrkin, Konstantin G., Bernd A. Kniehl, and Matthias Steinhauser. “Decoupling Relations to $O(\alpha_s^3)$ and Their Connection to Low-Energy Theorems.” Nuclear Physics B 510, nos. 1–2 (1998): 61–87. DOI. Open PDF.
• Hoang, André H., Ambar Jain, Ignazio Scimemi, and Iain W. Stewart. “The MSR Mass and the $O(\Lambda_{\mathrm{QCD}})$ Renormalon Sum Rule.” Journal of High Energy Physics 2018, no. 4 (2018): 003. DOI. Open PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §§26.3–26.6, pp. 513–28. DOI.