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

TL;DR

The paper tackles the problem that fixed-order perturbative predictions come with theoretical uncertainties (MHOU) that are often estimated with scale variation, which can be unreliable. It introduces theory nuisance parameters (TNPs) with a $k=2$ polynomial basis (Bernstein or Chebyshev) in a rescaled kinematic variable to parameterize the unknown N+1 contributions, enabling MHOU estimation by varying the TNPs. The authors validate the method on several LHC processes at NLO and NNLO (ttbar, gamma+jj, WW, DY), showing that TNP-based uncertainties align with or improve upon scale variation and, in problematic cases, provide significantly more credible uncertainty bands (e.g., WW). They find the TNP coefficients theta_i to be O(1) and approximately Gaussian in fits, and provide public tools (HighTEA notebook) to apply the approach across processes, highlighting its potential to extend to electroweak corrections and correlation-structured uncertainties.

Abstract

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.

Figures (6)

• Figure 1: Comparison between MHOU estimates from TNP and scale variation for differential distributions in $t\bar{t}$ production, with QCD corrections to the production and decay processes treated separately and combined in the narrow width approximation. From left to right, we show the lepton pair invariant mass $m(\ell\bar{\ell})$, the transverse momentum of the hardest $b$-jet $p_T(b_1)$ and the ratio of the lepton transverse momenta $p_T(\ell_1)/p_T(\ell_2)$. We take $\mu=H_T/4$ as our central scale choice and use a Bernstein parameterisation. The solid lines represent the LO (green), NLO (blue) and NNLO (red) central predictions and the corresponding bands of the estimated MHOU from scale variations, the dashed bands those from TNPs. The vertical lines indicate statistical uncertainties. We do not associate a TNP uncertainty with the LO prediction -- see text for details.
• Figure 2: As in fig. \ref{['fig:ttbar-HT4']}, but for $\gamma jj$ production. We take $\mu = H_T$ as our central scale choice and use a Chebyshev parameterisation. From left to right we show the hardest jet transverse momentum $p_T^{\rm jet}$, the photon transverse energy $E_T(\gamma)$, the invariant mass of the system $m(\gamma j_1 j_2)$ and the absolute rapidity difference between photon and hardest jet $|\Delta y^{\gamma- \rm{jet}}|$.
• Figure 3: As in fig. \ref{['fig:ttbar-HT4']}, but for off-shell, fully decayed $WW$ production. We take $\mu = H_T/2$ as our central scale choice and use a Bernstein parameterisation. From left to right we show the pair invariant mass $m(W^+W^-)$, the transverse momentum of the muon $p_T(\mu^-)$ and the rapidity of one boson $y(W^-)$.
• Figure 4: Comparison between scale and TNP uncertainties for charged-current ($W^+$) and neutral-current Drell-Yan ($Z/\gamma$) through next-to-next-to-next-to-leading order. The coloured bands indicate the uncertainties from 7-point scale variations, and dotted lines the quadrature combination of the TNP uncertainties at a 95% CL.
• Figure 5: Fit of the NLO uncertainty estimate using Bernstein polynomials (red) to the true NNLO result (blue) for the process $pp\to ZZ^*\to e^+e^-\mu^+\mu^-$. For each kinematic distribution the corresponding fit values of the nuisance parameters are shown.