Fixed-Order Organization and Scale Dependence

A fixed-order prediction is a truncation of a coupling expansion after ultraviolet renormalization and, when needed, initial-state factorization. Its order label, central scales, scale-variation prescription, normalization convention, and numerical errors are part of the result. Residual dependence on unphysical scales is a useful diagnostic of omitted terms, but it is not by itself a probability distribution or an error theorem.

Required background. Real–Virtual Cancellation and Subtraction supplies the finite NLO coefficients being organized here.

Perturbative coefficients and order names

For an observable whose Born contribution starts at $\alpha_s^p$, let each $C_k$ below include any required convolution with parton distributions or other factorized inputs. It then inherits their factorization-scale dependence. Write

$$\begin{aligned} \sigma(\mu_R,\mu_F) =\alpha_s^p(\mu_R)\Big[{}&C_0(\mu_F)\\ &+\alpha_s(\mu_R)C_1(\mu_R,\mu_F)\\ &+\alpha_s^2(\mu_R)C_2(\mu_R,\mu_F)+\cdots\Big]. \end{aligned}$$

LO retains $C_0$, NLO retains $C_0$ and $C_1$, and NNLO retains through $C_2$, relative to that process’s first nonzero term. An electroweak, mixed-coupling, or loop-induced Born process needs a multi-index label rather than a single ambiguous “NLO.” State which powers of every coupling are present.

$\mu_R$ is the renormalization scale. $\mu_F$ appears when long-distance initial-state collinear contributions have been factorized into distributions or other nonperturbative functions. A purely final-state process may have no factorization scale. The scheme and input coupling must be stated, but their primary derivation belongs to the renormalization and factorization volumes.

Why a truncated result depends on scale

An all-orders physical observable is independent of $\mu_R$:

$$\frac{\mathrm d\sigma}{\mathrm d\ln\mu_R}=0.$$

The running coupling obeys

$$\frac{\mathrm d\alpha_s}{\mathrm d\ln\mu_R} =-2\beta_0\alpha_s^2+O(\alpha_s^3)$$

in a convention where $\beta_0$ absorbs the chosen powers of $4\pi$. The explicit logarithms in $C_1,C_2,\ldots$ cancel this running order by order. If the expansion is truncated at relative order $k$, differentiation leaves terms beginning one order higher:

$$\frac{\mathrm d\sigma^{[k]}}{\mathrm d\ln\mu_R} =O(\alpha_s^{p+k+1}).$$

Residual scale dependence therefore probes a structured subset of missing higher-order terms. It does not probe new channels, accidental constants, power corrections, or all possible kinematic dependence uniformly.

For two scales, a common diagnostic evaluates a set of points around central values $(\mu_R^0,\mu_F^0)$, often using factors of two and excluding extreme antipodal ratios. That convention is not universal. Publish the exact point set, the envelope rule, whether numerator and denominator scales are correlated, and any bin-to-bin correlation model.

Expanded and unexpanded quantities

Derived ratios require a perturbative prescription. With

$$N=N_0+\alpha_sN_1, \qquad D=D_0+\alpha_sD_1,$$

one may report the consistently expanded ratio

$$R_{\mathrm{exp}} =\frac{N_0}{D_0} +\alpha_s\left( \frac{N_1}{D_0}-\frac{N_0D_1}{D_0^2} \right),$$

or the unexpanded quotient $R_{\mathrm{quot}}=(N_0+\alpha_sN_1)/(D_0+\alpha_sD_1)$. They agree through NLO and differ at NNLO and beyond. Neither is intrinsically wrong, but comparing them without declaring the prescription confuses a higher-order convention with a calculation error.

A $K$ factor,

$$K_{\mathrm{NLO}}=\frac{\sigma_{\mathrm{NLO}}}{\sigma_{\mathrm{LO}}},$$

is likewise conditional on scales, input parameters, cuts, partonic channels, and whether the same-order or order-specific normalization is used. A large $K$ factor may signal a newly opened channel, a poor central scale, a kinematic logarithm, or an accidentally small Born term. It is not a universal property of the process.

Interpreting scale variation

Scale variation is most informative when it is treated as a falsifiable diagnostic:

• the central scale follows the characteristic hard kinematics and is stated as a fixed or dynamic function;
• the same prescription is applied at successive perturbative orders;
• the next known coefficient is compared with the preceding band;
• independent physical scale hierarchies are not collapsed into one variation without justification;
• integration errors are much smaller than the variation being interpreted;
• correlations across bins and between observables are stated.

Studies of known QCD orders find that conventional factor-two scale envelopes do not possess a process-independent confidence-level interpretation Bagnaschi et al. 2015, § 2.1, pp. 4–5; § 3, pp. 8–12; § 5, p. 25. Bayesian models can assign probabilities only after adding explicit assumptions about coefficient distributions Cacciari and Houdeau 2011, §§ 2–3, pp. 3–17. Accordingly, this chapter reports scale variation as a perturbative diagnostic unless a separate statistical model is defined and validated.

A reporting example

Suppose a dimensionless observable has

$$\sigma^{\mathrm{NLO}}(Q) =\alpha_s^2(Q) \left[C_0+\alpha_s(Q)C_1\right].$$

A complete statement includes the coupling definition, $Q$, coefficient normalization, cuts, measurement function, subtraction scheme, integration error, and the values obtained at the declared scale points. It also checks that the $\mu_R$ derivative of the NLO result begins at $O(\alpha_s^4)$ after the explicit logarithm in $C_1(\mu_R)$ is restored.

The reusable validation and uncertainty matrix collects this scale check with pole cancellation, gauge invariance, numerical convergence, benchmark, and physical-limit tests. Scale stability alone is never used as a substitute for those independent checks.

Common pitfalls

Calling the first computed term “tree level.” Some processes begin at one loop. Order labels should follow powers of couplings relative to the first nonzero contribution, with loop origin stated separately.

Varying only the coupling. Coefficients contain explicit scale logarithms. Change the complete prediction consistently, including evolved factorized inputs where applicable.

Reading an envelope as a confidence interval. Without an additional probability model and calibration, the scale band is a diagnostic of perturbative sensitivity.

Comparing mismatched inputs. A change in PDFs, couplings, masses, cuts, or normalization can dominate the apparent order correction. Freeze an input specification before comparing coefficients.

Check your understanding

Differentiate an NLO expression in which $C_1(\mu_R)=C_1(Q)+2p\beta_0C_0\ln(\mu_R/Q)$. Verify explicitly that the $O(\alpha_s^{p+1})$ scale derivative cancels and identify the first uncanceled order.

Where to continue

• Resummation and Fixed-Order Matching separates logarithmic accuracy from fixed-order accuracy and removes double counting.
• Validation and Theory Uncertainties distinguishes scale diagnostics from numerical, parametric, and model uncertainties.
• Unstable-Particle Observables and Controlled Resonance Approximations applies order counting when a width introduces an additional small parameter.

References

• Bagnaschi, Emanuele, Matteo Cacciari, Alberto Guffanti, and Laura Jenniches. “An Extensive Survey of the Estimation of Uncertainties from Missing Higher Orders in Perturbative Calculations.” Journal of High Energy Physics 02 (2015): 133. DOI. Open preprint.
• Cacciari, Matteo, and Nicolas Houdeau. “Meaningful Characterisation of Perturbative Theoretical Uncertainties.” Journal of High Energy Physics 09 (2011): 039. DOI. Open preprint.