Robust estimates of theoretical uncertainties at fixed-order in perturbation theory

Calculations truncated at a fixed order in perturbation theory are accompanied by an associated theoretical uncertainty, which encodes the missing higher orders (MHOU). This is typically estimated by a scale variation procedure, which has well-known shortcomings. In this work, we propose a simple prescription to directly encode the missing higher order terms using theory nuisance parameters (TNPs) and estimate the uncertainty by their variation. We study multiple processes relevant for Large Hadron Collider physics at next-to-leading and next-to-next-to-leading order in perturbation theory, obtaining MHOU estimates for differential observables in each case. In cases where scale variations are well-behaved we are able to replicate their effects using TNPs, while we find significant improvement in cases where scale variation typically underestimates the uncertainty.

💡 Research Summary

The paper addresses a long‑standing problem in perturbative quantum field theory: how to reliably quantify the uncertainty due to missing higher‑order terms (MHOU) when a calculation is truncated at a fixed order. The standard practice is to vary the unphysical renormalisation and factorisation scales (μ) around a central value, typically by a factor of two, and to take the envelope of the resulting predictions as an estimate of the theoretical uncertainty. While this method is simple and widely used, it suffers from several well‑known drawbacks. The choice of the central scale is arbitrary, the size of the variation is ad‑hoc, and the procedure can dramatically underestimate the true uncertainty when new partonic channels open at higher orders, when large cancellations make lower‑order corrections artificially small, or when “giant K‑factors’’ appear in certain kinematic regions. Moreover, scale variations do not provide a proper correlation of uncertainties across bins of a differential distribution or between different observables.

To overcome these issues, the authors propose a new, complementary approach based on Theory Nuisance Parameters (TNPs). The idea is to model the unknown (N + 1)‑th order contribution using information from the already computed lower‑order terms. Concretely, they write the (N + 1)‑th order differential cross‑section as a linear combination of the known lower‑order pieces multiplied by a set of polynomial functions of the kinematic variable x, whose coefficients are the TNPs, denoted θ_i. Two families of orthogonal polynomials are considered: Bernstein polynomials and Chebyshev polynomials of the first kind. In all phenomenological studies the polynomial degree is fixed to k = 2, which provides enough shape flexibility without over‑parameterising the model. The (N + 1)‑th order term is then expressed as

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