Statistical Patterns of Theory Uncertainties

1. An exploration of NLO/LO corrections in a large number of collider inclusive processes.
2. A potentially useful rule-of-thumb for quick estimations of such corrections in experimental analyses.

1. Weak conceptual support for the proposed rule-of-thumb.
2. Overselling the generality of the approach.
3. Limitation to a specific choice of the factorization scale.

I am not ready to recommend the current version of the manuscript for the SciPost publication, as in my view it does not appear to meet either of four mandatory acceptance criteria listed at https://scipost.org/SciPostPhys/about#criteria . My assessment of the significance of the manuscript largely sides with the concerns of referee #2 about the conceptual foundations and generality of the proposed approach. At the same time, the authors raise a practically relevant question about simple approximations of higher-order QCD contributions for various collider processes using the LO computations that continue to be widely used in LHC analyses. Finding such an approximation can benefit the experimental analyses in many ways, but it has been difficult to do it given the versatility of contributing QCD dynamics. I thus think that the proposed formula for estimating the theoretical uncertainty has some limited, non-zero value and could be published with the appropriate disclaimers and warnings, although not necessarily in a journal.

The key weakness of the manuscript, as I see it, is that it limits itself to exploring the "how", instead of the "why", of the observed behavior of the NLO/LO corrections. From the list of references, it is clear that the authors are aware of the large body of literature dedicated to the estimates of missing higher-order uncertainties (MHOUs) from the available lower orders and experimental measurements, for example in the Cacciari-Houdeau's approach. Not mentioned here are the articles on "improved" PDFs for LO parton showers, such as arXiv:0711.2473, 0910.4183, hep-ph/020412, which had studied in-depth the underlying issues on the example of well-understood QCD processes. The manuscript itself seems to base the estimation formula on vague definitions and a rather primitive picture of the actual QCD scattering, as e.g. reflected by the following paragraph:

"At each perturbative order, ultraviolet (UV) divergences in cross-section predictions are removed through renormalization, introducing a logarithmic dependence on an unphysical renormalization scale mu_R in the prediction. Similarly, infrared (IR) and collinear divergences are absorbed into the definition of the parton densities, introducing logarithms of an equally unphysical factorization scale mu_F . Both scales can be related through the resummation of large collinear logarithms, but generally are independent scales with different infrared and ultraviolet origins and can be chosen independently [7]."

This might pass as a sloppy description of a fixed-order calculation of a hard cross section, but it is not an adequate summary to relate radiative contributions of different orders in an arbitrary hadronic cross section. Parton distributions (not densities) do not absorb infrared and collinear divergences, while mu_R and mu_F are separate scales that are not related by resummation of collinear logarithms. The behavior of the full radiative contribution reflects the diagram topologies, color factors, flavor composition, kinematics, and leading radiation configurations. Many modern textbooks on QCD elaborate on these factors.

Furthermore, the scale HT/2 advocated in Eq. (10) is not special. Other scales are commonly used, resulting in different NLO and especially LO values, and often with as good or better description of data.

I agree with referee #2 that Eqs. (6) and (13) estimate the uncertainty due to the running of the strong coupling. This can certainly be useful, but a large part of the full radiative contribution does not have much to do with the scale dependence.

In the historic example of Drell-Yan pair production in 1970's, before the QCD theory was developed, the practitioners first realized that the fixed-target DY inclusive data at pair virtuality Q can be described by the parton-model (LO) prediction multiplied by a factor that is very close to K = 1 + 3\alpha_s(Q) in a large range of sqrt{s} and Q. The easy rule-of-thumb formula for the DY K factor has been used for many years; the manuscript follows a similar logic by proposing an empirical formula to approximate many NLO cross sections using LO cross sections. The NLO QCD computation for DY reveals the limitations of this approach. The NLO/LO K factor is so close to 1+3 *alpha_s because the fixed-target inclusive DY cross section is dominated by the _hard_ virtual correction whose color and pi factors combine to a net constant of about 3. This NLO hard correction is not related to alpha_s or PDF running. It gives a good approximation for Q and y distributions at x>0.01 (at fixed targets and the Tevatron), and it fails at the LHC or FCC-hh, where rapidly varying PDFs introduce large x-dependent terms, as well as for pT distributions dominated by soft and collinear dynamics. The 3*alpha_s term is proportional to the pi^2 term that arises in timelike processes like DY, gg->Higgs, s-channel single-top or jet production. It is absent in spacelike processes like DIS, t-channel single-top or jet production. The proposed formula does not capture the pi^2 contribution or [integrated-over] hard real-emission contributions at NLO.

There is no reason to expect that these issues will be simpler in the other processes, besides DY, or at higher orders, when new color and kinematic configurations contribute.

1. Rewrite the text to make it very clear that the estimation formula (13) applies to the QCD coupling dependence only for a finite list of QCD observables that the authors explored.

2. List specifically what colliders, center-of-mass energies, and QCD observables (inclusive distributions only?) can be safely described by this prescription. Note that the distinction between "QCD" and "electroweak" processes is spurious, as both QCD and EW radiation is present in the actual processes. The proposed formula makes better sense when the all-order QCD observable is dominated by t-channel Born kinematics.

3. Elaborate on the other scale choices besides HT/2.

4. The theoretical motivation for the estimation formula could be sharpened throughout the text to avoid venturing into insufficiently understood areas or overselling the prescription for the situations when it will clearly fail. With these revisions in place, the article may satisfy the acceptance criterion #4, "Provide a novel and synergetic link between different research areas."

The authors do not address the main criticism in the revised manuscript and I remain highly sceptical on the usefulness of the proposed method.

1. I have not doubted the use of LO predictions in experimental analyses but questioned whether the interpretation in these cases is actually limited by the robustness of the uncertainties that are assigned to those LO predictions. I have not seen convincing evidence that improving LO uncertainty estimates is a critical issue that needs to be addressed. The fact that LO predictions, which are known to only provide order-of-magnitude estimates, are used is already an indicator that these analyses do not rely crucially on this aspect of the theory predictions.

2. I have explained in detail in my initial report that
- by considering QCD processes at LO only, the "universality" property observed is simply the renormalisation group evolution.
- the proposed "reference-process method" is just a convoluted way of assigning $\mu_R$-variation uncertainties in $\alpha_s$ to the EW coupling $\alpha$, i.e. is as ad-hoc as dressing $\alpha$ with a $\pm10\%$ uncertainty.

I expected the authors to explicitly test this claim but instead they chose to consider a very naive approach of inflating all uncertainties by a constant factor. Obviously, that will perform rather poorly given that it will not take into account different powers of $\alpha$ in the varying EW component of the processes.
I therefore did this exercise for them and in the attachment to this report ("hist.pdf") a comparison is shown for the pull distribution. We simply take the number of EW bosons (W, Z, $\gamma$, ...) as a proxy of the number of $\alpha$ powers ($n_\alpha$) in the process and add in quadrature to the scale uncertainty an additional $\pm10\%$ uncertainty from $\alpha$
\[
\frac{\Delta\sigma_{\alpha_{\pm10\%}}}{\sigma_0}
\equiv
\sqrt{
\left(\frac{\Delta\sigma}{\sigma_0}\right)^2
+
(n_\alpha \cdot 0.1)^2
}
\]
There is no need for a special treatment as in Eq.(14).
The comparison shows that the two methods are virtually the same; if anything, the completely ad-hoc $\alpha$ error inflation is performing slightly better. This clearly illustrates what the "reference-process method" is effectively doing and how arbitrary it is without any physics justification.

3. The authors comment in their reply that a generalisation to differential distributions is straightforward. I strongly disagree. How do the authors envision transferring uncertainties from reference processes to e.g. observables associated with colour-neutral particles such as the $p_T$ spectrum of a Z boson? All their reference processes are pure QCD ones containing no colourless particles. Not to mention differences in fiducial cuts, ...

1- Proposes novel way of determining theoretical uncertainties associated with scale variations, trying to identify the weakness of usual determinations and proposing to rely on some quasi-universal properties of these uncertainties to correct this weakness.
2- Interesting proposal, backed by some interesting statistical considerations from the pulls between LO and NLO predictions.
3- Article well written

The authors answered my questions in an appropriate manner.