Statistical Patterns of Theory Uncertainties

Aishik Ghosh1,2, Benjamin Nachman1,3, Tilman Plehn4, Lily Shire2,

Tim M.P. Tait2, and Daniel Whiteson2

1 Physics Division, Lawrence Berkeley National Laboratory, Berkeley, USA

2 Department of Physics & Astronomy, University of California, Irvine, USA

3 Berkeley Institute for Data Science, University of California, Berkeley, USA

4 Institut für Theoretische Physik, Universität Heidelberg, Germany

August 29, 2024

## Abstract

A comprehensive uncertainty estimation is vital for the precision program of
the LHC. While experimental uncertainties are often described by stochastic
processes and well-defined nuisance parameters, theoretical uncertainties lack
such a description. We study uncertainty estimates for cross-section
predictions based on scale variations across a large set of processes. We find
patterns similar to a stochastic origin, with accurate uncertainties for
processes mediated by the strong force, but a systematic underestimate for
electroweak processes. We propose an improved scheme, based on the scale
variation of reference processes, which reduces outliers in the mapping from
leading order to next-to-leading-order in perturbation theory.

###### Contents

  1. 1 Introduction
  2. 2 Theory uncertainties from scale variations
  3. 3 NLO dataset and results
  4. 4 Reference-process method
  5. 5 Outlook

##  1 Introduction

A core goal of particle physics is to measure the parameters of the Standard
Model (SM) and its extensions. For physics at the Large Hadron Collider (LHC),
the SM is connected to observables via a combination of predictions for
perturbative cross sections, event generation, and detector simulations. To
measure fundamental parameters, particle physics relies on likelihood
functions extracted from experimental data and expressed as functions of the
model parameters for a fixed dataset. With the likelihood functions, we can
determine confidence intervals for the target. In addition to parameters of
interest, such as e.g. the mass of the Higgs boson or a Wilson coefficient in
an effective field theory, models depend on nuisance parameters, which
describe systematic uncertainties, such as the modeling of the experimental
response or auxiliary components of the theoretical predictions. The
dependence on nuisance parameters is typically treated by profiling or
marginalization [1, 2, 3, 4, 5].

With large datasets at next-generation facilities such as the high-luminosity
LHC and the Deep Underground Neutrino Experiment (DUNE), systematic
uncertainties will be a limiting factor for many important measurements.
Therefore, a careful assessment of the statistical treatment of systematic
uncertainties is critical. Despite a unified statistical treatment in
practice, in reality, nuisance parameters describe sources of uncertainty
which are quite disparate in their origin and statistical behavior. In one
category are uncertainties due to auxiliary measurements which have inherent
statistical uncertainty because of finite sample sizes. An example is the
calibrated detector response from dedicated analyses, which is used to model
the expected data under various hypotheses. This uncertainty would vanish for
an infinite-sized calibration sample, and hypothetical similar finite
calibration samples would be expected to yield different, typically Poisson-
distributed, results due to the stochastic nature of the data.

Another category are uncertainties that arise due to our inability to perform
infinitely precise calculations [6], or due to a lack of first principle
theory predictions. These uncertainties are not determined by stochastic
processes, in that hypothetical similar efforts would result in identical
results if one were to repeat them making the same assumptions and
approximations. As such, one does not probe a well-defined abstract space of
theoretical possibilities when comparing different approaches based on
different assumptions. In some cases, as in choice between two ad hoc models,
such a distribution has limited ability to probabilistically encapsulate the
space of possibilities. In other cases, such as uncertainties due to limited-
order calculations of cross-sections, the interpretation of such distributions
is at best unclear. The common practice of treating all of the nuisance
parameters on the same statistical footing as if every case is drawn via a
stochastic process from a well-defined distribution can result in misleadingly
small estimations of the uncertainties in the derived parameters [4].

We explore this second type of uncertainty, focusing on fixed-order
perturbation theory for inclusive cross section calculations at the LHC. Our
discussion begins in Sec. 2 with a review of the standard method of estimating
the theoretical uncertainty due to limited-order calculations of inclusive LHC
cross-sections via scale variations. In Sec. 3, we assess this method from a
global perspective, examining inclusive cross sections at next-to-leading
order (NLO) and leading order (LO) in QCD for many available reactions. We
examine the distribution of those predictions, revealing some surprising
trends and behavior. We identify a class of electroweak processes for which
leading-order uncertainty estimates fail spectacularly, and propose an
alternative scheme with significantly improved performance in Sec. 4. Sec. 5
contains our outlook and conclusions.

##  2 Theory uncertainties from scale variations

The description of proton-proton collisions in the collinear approximation is
based on the separation of hard scattering at the parton level convolved with
universal parton densities encapsulating non-perturbative inputs [7, 8],

$\displaystyle\sigma(\theta)\approx\sum_{a,b}\int
dx_{a}x_{b}\,f_{a}(x_{a};\mu_{F})\,f_{b}(x_{b};\mu_{F})\,\hat{\sigma}_{ab}(\theta;\mu_{F},\mu_{R})\,,$
(1)

where the sum runs over partons inside the protons, $f_{a/b}$ are parton
distribution functions (PDFs), $\hat{\sigma}$ is the partonic hard-scattering
cross section, and $\theta$ represents parameter(s) of interest. A typical LHC
analysis infers information on the parameter of interest $\theta$ through
comparison with data measuring the observable $\sigma(\theta)$. We will be
concerned primarily with inclusive cross sections, which are under better
theoretical control than differential cross sections, including QCD radiation
of jets, or fully exclusive event generation.

The partonic cross section $\hat{\sigma}$ is computed as a perturbative
expansion in the QCD coupling $\alpha_{s}$ [6], but estimating the precision
of a given truncation is challenging. Estimates based on the size of the
expansion parameter $\alpha_{s}$, or $\alpha_{s}/\pi$, are fraught because the
perturbative series formally has a zero radius of convergence, implying that
at a high enough order one does not expect subsequent terms in the expansion
to be smaller than the preceding ones. While most predictions for inclusive
cross sections at the LHC appear to be convergent at least up to next-to-next-
to-leading order (NNLO), naive estimates based on the size of the expansion
parameter are found to be insufficiently conservative [6].

The PDFs appearing in Eq.(1) are extracted from a large orthogonal dataset
(keeping track of the scale dependence) under the assumption that the relevant
processes are described by the SM [9], and include their own comprehensive
uncertainty treatment [10, 11, 12, 13].

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].

The dependence of the theoretical prediction on the choice of scale introduces
an arbitrariness into the prediction, and is an artifact arising from the
truncation of the perturbative series. Formally, at all orders, the scale
dependence would vanish, and thus the degree to which the prediction depends
on the choice of scale at any finite order can be thought of as a measure of
how significant the contribution of the uncomputed remaining terms in the
series is expected to be. Traditionally, the uncertainty on the predicted
hadronic cross section in Eq.(1) is estimated by varying $\mu_{R}$ and
$\mu_{F}$ around a suitable central scale. The logarithmic dependence on the
scale requires $\mu\sim E$ where $E$ characterizes the physical energy scale
appearing in the observable of interest. This choice insures that terms
growing as $\log E/\mu$ do not spoil perturbation theory, but it cannot
distinguish choices of scale differing by order-one factors.

While the scale variation may provide a measure of the impact of missing
higher orders, it is not the ‘true’ uncertainty – which would quantify the
likelihood distribution of the difference between the estimated cross section
and the all-orders prediction. Nonetheless, despite the fact that the theory
uncertainty on the cross section prediction has no rigorous statistical
interpretation, in fits to data, the corresponding nuisance parameters are
often treated as Gaussian-distributed random variables. Trying to be
reasonably conservative, one could instead think of a rate prediction as a
range of expected values [1, 2, 3, 4, 5]

$\displaystyle\sigma\in\left[\sigma_{-},\sigma_{+}\right]\;.$ (2)

The implementation of an allowed range in terms of a nuisance parameter is not
trivial. A flat distribution of the corresponding nuisance parameter is not
invariant under transformations such as changing the prediction from $\sigma$
to $\log\sigma$. It induces volume effects in the marginalization, which one
can only partially avoid by using a profile likelihood.

Based on Eq.(2), a lower bound on the uncertainty can be estimated through the
dependence on the unphysical scales at a given order in perturbation theory.
For pure QCD processes, the dependence on the renormalization scale typically
dominates, and the range of predictions effectively corresponds to a range of
$\mu_{R}$,

$\displaystyle\sigma\in\left[\sigma_{-},\sigma_{+}\right]\equiv\left[\sigma(\mu_{R,+}),\sigma(\mu_{R,-})\right]\;,$
(3)

and a suitable central scale choice $\sigma_{0}=\sigma(\mu_{R,0})$ is often
the transverse mass of the final state,

$\displaystyle\mu_{R,0}=\sum_{i}\sqrt{m_{i}^{2}+p^{2}_{\textrm{T},i}}\;.$ (4)

where the sum is over final state particles of mass $m_{i}$ and transverse
momentum $p_{\textrm{T},i}$. Assuming that the leading dependence on the
renormalization scale enters through the running of the strong coupling, the
scale-based uncertainty can be expressed

$\displaystyle\Delta\sigma_{\text{scale}}=\frac{\partial\sigma}{\partial\alpha_{s}}\;\Delta\alpha_{s}=\frac{\partial\sigma}{\partial\alpha_{s}}\;\frac{\alpha_{s}(\mu_{R,-})-\alpha_{s}(\mu_{R,+})}{2}\;.$
(5)

where the appropriate range of scales defined by $\mu_{R,\pm}$ is discussed
below. An approach based on lower and upper limits defined by a scale
variation ensures that:

  1. 1.

no ${\cal O}(1)$ variation of the unphysical scale around the central choice
is considered more likely than any other;

  2. 2.

there is no long tail of exponentially suppressed probability to obtain a very
large deviation from perturbative QCD; and

  3. 3.

perturbative and process-dependent arguments always leave an order-one
uncertainty on the scale choice.

Two features of using scale dependence as the uncertainty measure are that it
decreases as one considers higher order calculations, and that it increases
when additional particles are added to the final state. We illustrate this for
an $n$-particle production process at leading order in QCD and assuming that
the renormalization scale only occurs implicitly through $\alpha_{s}$,

$\displaystyle\sigma\propto\alpha_{s}^{n}\qquad\Rightarrow\qquad\frac{\Delta\sigma_{\text{scale}}}{\sigma_{0}}$
$\displaystyle=n\times\frac{\alpha_{s}(\mu_{R,-})-\alpha_{s}(\mu_{R,+})}{2\alpha_{s}(\mu_{R,0})}\;.$
(6)

Until now we have not discussed the actual size of the scale variation. One
can gain insight into the appropriate range of scales that one should consider
in forming the uncertainty estimate from the process that is arguably the
best-understood QCD reaction at hadron colliders, $t\bar{t}$ production.
Experience from its perturbative behavior at the Tevatron [14, 15, 16, 17] and
at the LHC [18, 19, 20, 21, 22] motivates the standard choice:

$\displaystyle\mu_{R,0}=m_{t}\qquad\quad\mu_{R,+}=2m_{t}\qquad\quad\mu_{R,-}=\frac{m_{t}}{2}\;,$
(7)

such that the LO and NLO uncertainty estimates cover the known higher-order
corrections. This recipe is simply generalized to QCD production processes of
pairs of other heavy particles at hadron colliders. The LO scale dependence
through the strong coupling is

$\displaystyle\alpha_{s}(p^{2})$
$\displaystyle=\frac{1}{b_{0}\log\dfrac{p^{2}}{\Lambda_{\text{QCD}}^{2}}}\qquad\text{with}\quad
b_{0}=\frac{1}{4\pi}\left(\frac{11}{3}N_{c}-\frac{2}{3}n_{f}\right)$
$\displaystyle\Rightarrow\qquad\frac{\alpha_{s}(p^{2}/4)}{\alpha_{s}(p^{2})}$
$\displaystyle\approx 1+b_{0}\alpha_{s}(p^{2})\log 4\approx 1.04$
$\displaystyle\frac{\alpha_{s}(4p^{2})}{\alpha_{s}(p^{2})}$
$\displaystyle\approx 1-b_{0}\alpha_{s}(p^{2})\log 4\approx 0.96\;.$ (8)

Around $\mu_{R}=m_{t}$, $\alpha_{s}^{\text{(LO)}}\in[0.112,0.120]$, and
similar ranges are obtained at NLO and NNLO. Including higher orders in
perturbation theory, the scale dependence of the $t\bar{t}$ production rate
decreases through explicit appearance of $\log\mu_{R}$ in the rate prediction,
which partially compensates the implicit dependence through $\alpha_{s}$. This
cancellation is sensitively dependent on the details of the process,
preventing one from obtaining a general rule to infer higher-order corrections
($K$-factors) for a wider selection of processes.

The same method with the same range of scales has been demonstrated to work
for multi-jet production [23, 24, 25, 26, 27, 28, 29]. However, it fails e.g.
in the case of Higgs production, where notably the NLO $K$–factor is much
larger than expected [30, 31, 32, 33, 34]. In such cases one can often
identify the reason for the larger-than-expected higher-order contributions,
such as a loop-induced ($2\to 1$)-scattering at LO, higher order contributions
from new partonic channels, and/or large logarithms. For processes which are
governed by more than one numerically relevant scale, for example the
$\mu_{R}$ and $\mu_{F}$, one estimates the allowed range of the cross section
by using the extremal values in Eq.(3),

$\displaystyle\sigma_{-}=\min_{\mu_{R},\mu_{F}}\sigma(\mu_{R},\mu_{F})\qquad\text{and}\qquad\sigma_{+}=\max_{\mu_{R},\mu_{F}}\sigma(\mu_{R},\mu_{F})\;.$
(9)

An alternative to including the full multi-scale dependence is to exclude the
largest scale ratios [35].

Bayesian approaches [36] attempt to parse the confidence interval on a
particular inclusive rate in a different way, for processes in which higher
order QCD corrections are known. The Bayesian approach estimates the size of
the higher order corrections based on the inferred patterns based on the
available terms, under the assumption that the perturbative series should
converge upon a stable result at any fixed order (which is true in practice
for known LHC predictions). By including process-specific information [37],
they are able to infer a non-trivial shape for the distribution of the
nuisance parameter, related to missing higher-order corrections from known
higher orders in perturbation theory.

Such analyses typically settle on a very similar interval for the prediction
as the naive scale-based method described above, though it remains difficult
to meaningfully predict an optimal scale choice, even at the process-by-
process level [35]. Ref [38] finds that they generally work well provided one
corrects for generic features from QFT such as the factorial growth of the
perturbative coefficients and expands the estimated size of the expansion
parameter from $\alpha_{s}\to\alpha_{s}/\lambda$ with $\lambda\simeq 0.6$ for
hadronic processes. It further discerned that the interval corresponding to
scale variation by a factor of two is not always sufficiently conservative at
the 95% confidence level and reported an increased uncertainty for Higgs
production, $\sigma(pp\to H)$, where, as mentioned above, specific features
are known to spoil the scale variation estimate.

Consequently, while the flat distribution of Eq.(2) with range determined by
scale variation by a factor of two, as suggested by $t\bar{t}$ production,
does not encapsulate a nuanced shape for the corresponding nuisance parameter
distribution, it remains a reasonably well-justified approximation that holds
for a variety of hard processes. As we consider trends for a plethora of
different processes, we adopt it below.

##  3 NLO dataset and results

To assess the standard scale variation method of assigning a theory
uncertainty to predictions of inclusive cross sections at the LHC, we analyze
the comprehensive set of LO and NLO calculations presented in Ref. [39]. For
each case, we investigate to what degree the uncertainty derived from scale
variation of the LO prediction is a reasonable estimate of the difference
between it and the known NLO calculation. While in many cases these NLO
predictions are not the state of the art for the particular process at hand,
and it would be very interesting to perform the same study using higher-order
calculations, the number of NNLO calculations is comparably much smaller,
which would limit the scope of the available processes. Furthermore, many
searches at the LHC still use LO for generating signal samples.

We extract the central value and the scale uncertainties for 136
processes***All $pp$-initiated processes other than $pp\rightarrow\gamma
W^{+}W^{-}j$, for which the NLO cross section has a typo, confirmed by the
authors of Ref. [39] via private communication., and neglect the statistical
uncertainties from Monte Carlo integration, which are typically small in
comparison, and PDF uncertainties, which are due to measurements in auxiliary
datasets and so are in a different category and with complicated correlations
between processes. All cross sections are calculated by MadGraph5_aMC@NLO
[40]. The Higgs boson is assumed to have $m_{H}=125$ GeV, and the top quark
$m_{t}=173.2$ GeV. The parton distribution functions MSTWnlo2008[41] are used
with $n_{f}=4,5$. The choice of a central scale is

$\displaystyle\mu_{0}=\frac{H_{\text{T}}}{2}=\frac{1}{2}\sum_{\text{final
state}}\sqrt{p_{\text{T}}^{2}+m^{2}}\;.$ (10)

The LO uncertainty is estimated by varying the factorization and
renormalization scales by factors of two relative to the central scale, as
defined in Eq.(9).

Given an estimator $x$ for the parameter $\mu$ with an estimated uncertainty
$\Delta x$, we can analyze that uncertainty estimate using the pull

$\displaystyle t=\frac{x-\mu}{\Delta x}\;.$ (11)

If $x$ is a Gaussian random variable around $\mu$, then the pull will follow a
Gaussian distribution with zero mean and unit variance. We treat
$\sigma_{\text{LO}}$ as an estimator for $\sigma_{\text{NLO}}$ and analyze the
behavior of its uncertainty $\Delta\sigma_{\text{LO scale}}$ by examining

$\displaystyle
t_{\text{scale}}=\frac{\sigma_{\text{NLO}}-\sigma_{\text{LO}}}{\Delta\sigma_{\text{LO
scale}}}\;,$ (12)

using the positive $\Delta\sigma_{\text{LO scale}}$ if
$\sigma_{\text{NLO}}>\sigma_{\text{LO}}$, and the negative otherwise. Note
that in Eq. 12 we reorder the terms in the numerator relative to Eq. 11 in
anticipation of $\sigma_{\text{NLO}}$ being larger than $\sigma_{\text{LO}}$;
the statistical behavior is unchanged.

Figure 1: Performance of the uncertainty estimation in LO cross section
calculations. Shown is the scale-based pull $t_{\text{scale}}$, defined in
Eq.(12). Pulls greater than 25 are shown at 25. Blue entries with
$|t_{\text{scale}}|<4$ are included in the Gaussian fit (red).

Figure 1 shows the distribution of the pull for our dataset. The distribution
is clearly not Gaussian with zero mean and unit variance, though it displays
some expected behavior. For example, the pull is almost always greater than
zero, which aligns with our expectation that cross-section estimates tend to
grow as additional partonic channels are included beyond LO. There also
appears to be a core distribution which is locally approximately normally
distributed with unit variance, indicating that for many processes, the
central scale and the scale variation are accurate indicators for the NLO
result.

In addition, there is a very long positive tail, indicating many processes,
where the uncertainty is dramatically underestimated. The complete list of
processes and the associated pulls are given in Tables 2-4. Many of the
processes with underestimated uncertainties are those with many particles and
without QCD vertices. Indeed, unlike pure QCD processes, QCD corrections to
electroweak processes do not appear to be covered by the scale-based
uncertainty estimate. Numerically, the leading scale dependence in QCD
processes is the renormalization scale, and this scale dependence is absent in
electroweak processes at LO, and is small at NLO. For instance, di-lepton
production at the LHC is a purely electroweak process and only has a small
factorization scale dependence at leading order, which does not cover the NLO
corrections. This problem extends beyond Drell-Yan [42], to di-boson [43], and
tri-boson production [44].

In addition to, generally predicting uncertainty estimates which are too
small, electroweak processes encounter large higher-order corrections for
identifiable reasons [45, 46], e.g. due to flavor symmetries and constrained
topologies in Feynman diagrams. An example is the process $q\bar{q}\to
Zb\bar{b}$, where the leading topology is $q\bar{q}\to Zg^{*}\to Zb\bar{b}$,
and a $t$-channel contribution favoring large $m_{bb}$ only appears when an
additional jet is added to the final state. This challenges the scale
variation scheme and leads to large QCD corrections [47]. Enhanced
uncertainties also appear in kinematic tails of electroweak processes, for
example as electroweak Sudakov logarithms and at times as large threshold
corrections. Generally, for any observable strongly sensitive to more than one
relevant scale, large logarithms of their ratio tend to enhance perturbative
corrections, challenging the standard estimation of their uncertainties.

All these considerations depend on the details of the process and the phase
space region under consideration. While there is sufficient understanding to
understand and post-dict processes for which there are large deviations at NLO
not covered by varying the scale of the LO prediction, it is clear that the
treatment of the uncertainty in Eq.(2) with the implementation of Eq.(7) must
be generalized to cover such processes. For this purpose we will define a set
of reference processes, specifically those processes populating the core of
Fig. 1.

##  4 Reference-process method

Process | $\dfrac{\Delta\sigma}{\sigma_{0}}$ |  $n$ | $\dfrac{\Delta\sigma}{n\;\sigma_{0}}$
---|---|---|---
p p > j j |  +$2.49\times 10^{-1}$ | $-1.88\times 10^{-1}$ | 2 |  +$1.24\times 10^{-1}$ | $-9.40\times 10^{-2}$
p p > b b |  +$2.52\times 10^{-1}$ | $-1.89\times 10^{-1}$ | 2 |  +$1.26\times 10^{-1}$ | $-9.45\times 10^{-2}$
p p > t t |  +$2.90\times 10^{-1}$ | $-2.11\times 10^{-1}$ | 2 |  +$1.45\times 10^{-1}$ | $-1.06\times 10^{-1}$
p p > j j j |  +$4.38\times 10^{-1}$ | $-2.84\times 10^{-1}$ | 3 |  +$1.46\times 10^{-1}$ | $-9.47\times 10^{-2}$
p p > b b j |  +$4.41\times 10^{-1}$ | $-2.85\times 10^{-1}$ | 3 |  +$1.47\times 10^{-1}$ | $-9.50\times 10^{-2}$
p p > t t j |  +$4.51\times 10^{-1}$ | $-2.90\times 10^{-1}$ | 3 |  +$1.50\times 10^{-1}$ | $-9.67\times 10^{-2}$
p p > b b j j |  +$6.18\times 10^{-1}$ | $-3.56\times 10^{-1}$ | 4 |  +$1.54\times 10^{-1}$ | $-8.90\times 10^{-2}$
p p > b b b b |  +$6.17\times 10^{-1}$ | $-3.56\times 10^{-1}$ | 4 |  +$1.54\times 10^{-1}$ | $-8.90\times 10^{-2}$
p p > t t j j |  +$6.14\times 10^{-1}$ | $-3.56\times 10^{-1}$ | 4 |  +$1.53\times 10^{-1}$ | $-8.90\times 10^{-2}$
p p > t t t t |  +$6.38\times 10^{-1}$ | $-3.65\times 10^{-1}$ | 4 |  +$1.60\times 10^{-1}$ | $-9.12\times 10^{-2}$
p p > t t b b |  +$6.21\times 10^{-1}$ | $-3.57\times 10^{-1}$ | 4 |  +$1.55\times 10^{-1}$ | $-8.93\times 10^{-2}$
average |  |  |  | $+1.47\times 10^{-1}$ | $-9.34\times 10^{-2}$
Table 1: Scale dependence for LHC processes with only QCD particles in the
final state. For each process, we report the relative scale uncertainty, the
number of final state particles, and the per-particle relative scale
uncertainty.

To define a conservative way of estimating theory uncertainties and the
corresponding nuisance parameters, we look at some of the processes which form
the controlled core in Fig. 1. The processes where a scale variation by a
factor two provides a quantitatively correct estimate of the size of the NLO-
corrections are shown in Tab. 1, and include the QCD processes: top pair
production, bottom pair production, di-jet production, and those same
processes with up to two additional jets. We observe that the $K$-factors for
massive and massless quarks are very similar, despite the fact that the
central scale choices and the total or fiducial rates vary extensively. In
addition, the relative uncertainty per final state particle only has a small
variation across these processes, suggesting that the the scale uncertainty
indeed simply reflects the implicit renormalization scale dependence through
the corresponding power of $\alpha_{s}$. It also suggests that one could
estimate the uncertainty for a new arbitrary process with $n$ particles in the
final state from the approximately universal values of the re-scaled relative
uncertainty $(\Delta\sigma/\sigma_{0})/n$.

It suggests that a reasonable treatment for electroweak processes could be to
use the set of QCD processes in Tab. 1 as a reference class by generalizing
the QCD-driven uncertainty estimate from Eq.(6). In practice, one replaces
their process-specific LO scale uncertainty with

$\displaystyle\frac{\Delta\sigma_{\text{ref}}}{\sigma_{0}}=n\times\Bigg{\langle}\frac{\Delta\sigma}{n\sigma_{0}}\bigg{\rangle}_{\text{QCD}}\;.$
(13)

Of course, there will always be certain processes for which our simple
generalization does not describe the underlying physics and therefore will not
apply. For example, $pp\to H+n$ jets production, as discussed above, is a case
where the NLO corrections are enhanced by a combination of the fact that the
LO process is loop-induced and has ($2\to 1$)-kinematics, and where the rate
scales like $\sigma\propto\alpha_{s}^{n+2}$. To accommodate such processes we
modify Eq.(13) to our final proposal

$\displaystyle\frac{\Delta\sigma_{\text{ref}}}{\sigma_{0}}=\max\left(\frac{\Delta\sigma_{\text{scale}}}{\sigma_{0}},n\times\left\langle\frac{\Delta\sigma}{n\sigma_{0}}\right\rangle_{\text{QCD}}\right)\;,$
(14)

where the QCD reference value is computed from Eq.(9), making use of the
typical behavior shown in Tab. 1. This generalization allows for a unified
treatment in which the corresponding range is reliably implemented for each
corresponding nuisance parameter relevant for a given (global) analysis, be it
QCD or electroweak.

For our new method of computing theoretical uncertainties we again calculate
the pull as

$\displaystyle
t_{\text{ref}}=\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}\;.$
(15)

The pull distribution based on the reference process method is shown in Fig.
2, and given for specific processes in Tables 2\- 4. Some processes still have
large pulls, indicating significant underestimate of the uncertainty but the
occurrence is much rarer and far less significant. While the reference process
method is conservative in that it leads to a smaller width of the Gaussian fit
in Fig. 2, it clearly does much better at the tails compared to Fig. 1 or what
one would get by simply inflating the uncertainties by some arbitrary factor.
The processes with pulls $t>2$ are dominated by reactions with photons in the
final state (which suffer from the known issues of soft and collinear
divergences), triple and quartic gauge boson production (with numerically
significant gauge cancellations), associated $b$-pair production with flavor-
locked channels and massless $b$-quarks (leading to large logarithms), and
Higgs production (with large corrections to the loop-induced $(2\to
1)$-process). For those processes the large NLO $K$-factors are understood as
arising from specific physics effects, which cannot be easily extracted from
lower orders in perturbation theory. For those processes, it would be
interesting to investigate how our proposal fares in describing how the
contribution of NNLO compares to the variation of the NLO rate.

Figure 2: Performance of the uncertainty estimation in LO cross section
calculations. Shown is the new pull $t_{\text{ref}}$, defined in Eq.(15).
Pulls greater than 2.5 are shown at 2.5. Blue entries with
$|t_{\text{ref}}|<1.2$ are included in the Gaussian fit.

##  5 Outlook

Figure 3: Comparison of the original and modified relative uncertainty via
distribution of $\log t$, both for the scale-based and the reference-process
definition.

We have analyzed the behavior of the estimation of theory uncertainties at LO
and the known NLO-QCD corrections for a large number of processes [39]. It
turns out that the classic recipe of varying the renormalization and
factorization scales provides a reliable uncertainty estimate for a sizeable
set of QCD processes, but fails for electroweak processes, sometimes by orders
of magnitude. This problem is known and related to the dominance of the
implicit effects of the renormalization scale through the running of the
strong coupling. We have proposed a new method of estimating theory
uncertainties, based not on the scale dependence of a specific process, but on
the scale dependence of the set of QCD reference processes for which the scale
variation estimate is well founded and quantitatively reliable. The allowed
range of cross sections can be described by a nuisance parameter. While it
does not predict a nuanced distribution for that parameter, this detail is not
expected to matter if one profiles over it, as opposed to marginalizing.

We illustrate the benefit of our proposed method using the original and
modified pull distributions for our process dataset in Fig. 3. The pull based
on our new model shows a significantly improved behavior in the tail of
underestimated uncertainties and reflects a few aspects of perturbative cross
section predictions:

  1. 1.

the central peak of the curve indicates that there exist universal patterns in
this set of cross sections, as observed in Tab. 1;

  2. 2.

the shifted core of the pull distribution reflects, for instance, additional
partonic channels;

  3. 3.

the seemingly stochastic drop away from the peak indicates that for instance
the central factorization and renormalization scales are indeed described as
random choices of the unphysical parameters;

  4. 4.

the sizable tails indicate the existence of physics effects which are not
accounted for by our assumption of a homogeneous reference sample.

At this stage one might be tempted to turn the argument around and speculate
what can be learned from the pull distributions about the expected size of NLO
predictions. The first problem in applying Fig. 2 to completely new processes
is that they might have a physics reason for large higher-order corrections.
Even if that is not the case, our method is based on a sample of independent
inclusive processes. Even if we assume a stochastic pattern, it does not
provide predictions for individual new processes. The only way we could
justify such a prediction would be through the same equivalence class, i.e. a
physics argument based on the known properties of a given process.

While we do not advocate to use the pull distribution as a theory nuisance
parameter, an estimation of theory uncertainties based on the scale variation
of reference processes shows a very significant improvement over the current
scheme for the processes that we have considered. The implementation of a
range of predictions in terms of nuisance parameters is fairly straightforward
for a profile likelihood, but would need to be better understood for Bayesian
marginalization. Moreover, our reference process method should be further
tested with regard to higher orders in perturbation theory. A similar study at
higher orders in perturbation theory may inform us about methods to find more
such patterns.

### Acknowledgments

This work has been partly financed by the program Internationale
Spitzenforschung of the Baden-Württemberg-Stiftung, project Uncertainties —
Teaching AI its Limits (BWST_IF2020-010). The research of TP is supported by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
grant 396021762 – TRR 257 Particle Physics Phenomenology after the Higgs
Discovery. AG/DW and BN are supported by the U.S. Department of Energy, Office
of Science under grant DE-SC0009920 and contract DE-AC02-05CH11231,
respectively. The work of TMPT is supported in part by the US National Science
Foundation through grant PHY-2210283.

Process | $n_{\textrm{part}}$ | $\Delta\sigma/\sigma_{0}$ | $\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma}$ | $\Delta\sigma_{\text{ref}}/\sigma_{0}$ | $\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}$
---|---|---|---|---|---
p p > wpm | 1 | $1.54\times 10^{-1}$ | 1.84 | $1.47\times 10^{-1}$ | 1.92
p p > wpm j | 2 | $1.97\times 10^{-1}$ | 1.96 | $2.94\times 10^{-1}$ | 1.31
p p > wpm j j | 3 | $2.45\times 10^{-1}$ | 0.59 | $4.41\times 10^{-1}$ | 0.33
p p > wpm j j j | 4 | $4.10\times 10^{-1}$ | 0.25 | $5.88\times 10^{-1}$ | 0.18
p p > z | 1 | $1.46\times 10^{-1}$ | 1.87 | $1.47\times 10^{-1}$ | 1.86
p p > z j | 2 | $1.93\times 10^{-1}$ | 1.82 | $2.94\times 10^{-1}$ | 1.19
p p > z j j | 3 | $2.43\times 10^{-1}$ | 0.56 | $4.41\times 10^{-1}$ | 0.31
p p > z j j j | 4 | $4.08\times 10^{-1}$ | 0.27 | $5.88\times 10^{-1}$ | 0.19
p p > a j | 2 | $3.12\times 10^{-1}$ | 5.33 | $2.94\times 10^{-1}$ | 5.66
p p > a j j | 3 | $3.28\times 10^{-1}$ | 0.85 | $4.41\times 10^{-1}$ | 0.63
p p > w+ w- | 2 | $5.00\times 10^{-2}$ | 7.99 | $2.94\times 10^{-1}$ | 1.36
p p > z z | 2 | $4.50\times 10^{-2}$ | 6.46 | $2.94\times 10^{-1}$ | 0.99
p p > z wpm | 2 | $3.60\times 10^{-2}$ | 17.09 | $2.94\times 10^{-1}$ | 2.09
p p > a a | 2 | $2.21\times 10^{-1}$ | 7.36 | $2.94\times 10^{-1}$ | 5.53
p p > a z | 2 | $9.90\times 10^{-2}$ | 4.73 | $2.94\times 10^{-1}$ | 1.59
p p > a wmp | 2 | $9.50\times 10^{-2}$ | 14.88 | $2.94\times 10^{-1}$ | 4.81
p p > w+ w- j | 3 | $1.16\times 10^{-1}$ | 2.58 | $4.41\times 10^{-1}$ | 0.68
p p > z z j | 3 | $1.09\times 10^{-1}$ | 2.93 | $4.41\times 10^{-1}$ | 0.73
p p > z wmp j | 3 | $1.16\times 10^{-1}$ | 2.57 | $4.41\times 10^{-1}$ | 0.68
p p > a a j | 3 | $2.03\times 10^{-1}$ | 6.13 | $4.41\times 10^{-1}$ | 2.83
p p > a z j | 3 | $1.45\times 10^{-1}$ | 3.23 | $4.41\times 10^{-1}$ | 1.06
p p > a wpm j | 3 | $1.37\times 10^{-1}$ | 3.32 | $4.41\times 10^{-1}$ | 1.03
p p > w+ w+ j j | 4 | $2.54\times 10^{-1}$ | 2.05 | $5.88\times 10^{-1}$ | 0.89
p p > w- w- j j | 4 | $2.54\times 10^{-1}$ | 1.90 | $5.88\times 10^{-1}$ | 0.82
p p > w+ w- j j | 4 | $2.72\times 10^{-1}$ | 0.84 | $5.88\times 10^{-1}$ | 0.39
p p > z z j j | 4 | $2.66\times 10^{-1}$ | 1.04 | $5.88\times 10^{-1}$ | 0.47
p p > z wpm j j | 4 | $2.67\times 10^{-1}$ | 0.51 | $5.88\times 10^{-1}$ | 0.23
p p > a a j j | 4 | $2.62\times 10^{-1}$ | 1.50 | $5.88\times 10^{-1}$ | 0.67
p p > a z j j | 4 | $2.43\times 10^{-1}$ | 1.24 | $5.88\times 10^{-1}$ | 0.51
p p > a wpm j j | 4 | $2.47\times 10^{-1}$ | 0.72 | $5.88\times 10^{-1}$ | 0.30
p p > w+ w- wpm | 3 | $1.00\times 10^{-3}$ | 610.69 | $4.41\times 10^{-1}$ | 1.39
p p > z w+ w- | 3 | $8.00\times 10^{-3}$ | 92.39 | $4.41\times 10^{-1}$ | 1.68
p p > z z wpm | 3 | $1.00\times 10^{-2}$ | 85.00 | $4.41\times 10^{-1}$ | 1.93
p p > z z z | 3 | $1.00\times 10^{-3}$ | 302.75 | $4.41\times 10^{-1}$ | 0.69
p p > a w+ w- | 3 | $1.90\times 10^{-2}$ | 42.33 | $4.41\times 10^{-1}$ | 1.82
p p > a a wpm | 3 | $4.40\times 10^{-2}$ | 47.24 | $4.41\times 10^{-1}$ | 4.72
p p > a z wpm | 3 | $1.00\times 10^{-3}$ | 1244.49 | $4.41\times 10^{-1}$ | 2.82
p p > a z z | 3 | $2.00\times 10^{-2}$ | 17.24 | $4.41\times 10^{-1}$ | 0.78
p p > a a z | 3 | $5.60\times 10^{-2}$ | 8.99 | $4.41\times 10^{-1}$ | 1.14
p p > a a a | 3 | $9.80\times 10^{-2}$ | 17.44 | $4.41\times 10^{-1}$ | 3.88
p p > w+ w- wpm j | 4 | $1.50\times 10^{-1}$ | 2.06 | $5.88\times 10^{-1}$ | 0.53
p p > z w+ w- j | 4 | $1.56\times 10^{-1}$ | 1.81 | $5.88\times 10^{-1}$ | 0.48
p p > z z wmp j | 4 | $1.61\times 10^{-1}$ | 1.88 | $5.88\times 10^{-1}$ | 0.51
p p > z z z j | 4 | $1.43\times 10^{-1}$ | 2.21 | $5.88\times 10^{-1}$ | 0.54
p p > a a wpm j | 4 | $1.18\times 10^{-1}$ | 3.51 | $5.88\times 10^{-1}$ | 0.70
p p > a z wpm j | 4 | $1.44\times 10^{-1}$ | 2.30 | $5.88\times 10^{-1}$ | 0.56
p p > a z z j | 4 | $1.25\times 10^{-1}$ | 2.96 | $5.88\times 10^{-1}$ | 0.63
p p > a a z j | 4 | $1.09\times 10^{-1}$ | 7.57 | $5.88\times 10^{-1}$ | 1.40
p p > a a a j | 4 | $1.43\times 10^{-1}$ | 6.72 | $5.88\times 10^{-1}$ | 1.64
Table 2: A list of processes considered in Ref. [39] with their final state multiplicity ($n_{part}$), their relative uncertainty at leading order ($\Delta\sigma/\sigma_{0}$ ), and the resulting pull ($\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma}$). Also included is the modified relative uncertainty at leading order ($\Delta\sigma_{\text{ref}}/\sigma_{0}$), and the corresponding resulting modified pull ($\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}$). Process | $n_{\textrm{part}}$ | $\Delta\sigma/\sigma_{0}$ | $\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma}$ | $\Delta\sigma_{\text{ref}}/\sigma_{0}$ | $\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}$
---|---|---|---|---|---
p p > w+ w- w+ w- | 4 | $3.70\times 10^{-2}$ | 20.03 | $5.88\times 10^{-1}$ | 1.26
p p > w+ w- wpm z | 4 | $4.40\times 10^{-2}$ | 19.60 | $5.88\times 10^{-1}$ | 1.47
p p > w+ w- wpm a | 4 | $2.50\times 10^{-2}$ | 36.35 | $5.88\times 10^{-1}$ | 1.55
p p > w+ w- z z | 4 | $4.40\times 10^{-2}$ | 14.68 | $5.88\times 10^{-1}$ | 1.10
p p > w+ w- z a | 4 | $3.00\times 10^{-2}$ | 25.40 | $5.88\times 10^{-1}$ | 1.30
p p > w+ w- a a | 4 | $6.00\times 10^{-3}$ | 133.97 | $5.88\times 10^{-1}$ | 1.37
p p > wpm z z z | 4 | $5.10\times 10^{-2}$ | 21.88 | $5.88\times 10^{-1}$ | 1.90
p p > wpm z z a | 4 | $3.60\times 10^{-2}$ | 43.48 | $5.88\times 10^{-1}$ | 2.66
p p > wpm z a a | 4 | $1.70\times 10^{-2}$ | 110.92 | $5.88\times 10^{-1}$ | 3.21
p p > wmp a a a | 4 | $4.00\times 10^{-3}$ | 618.06 | $5.88\times 10^{-1}$ | 4.21
p p > z z z z | 4 | $3.80\times 10^{-2}$ | 8.46 | $5.88\times 10^{-1}$ | 0.55
p p > z z z a | 4 | $1.90\times 10^{-2}$ | 16.92 | $5.88\times 10^{-1}$ | 0.55
p p > z z a a | 4 | $1.00\times 10^{-3}$ | 364.79 | $5.88\times 10^{-1}$ | 0.62
p p > z a a a | 4 | $2.30\times 10^{-2}$ | 20.97 | $5.88\times 10^{-1}$ | 0.82
p p > a a a a | 4 | $4.70\times 10^{-2}$ | 24.09 | $5.88\times 10^{-1}$ | 1.93
p p > j j | 2 | $2.49\times 10^{-1}$ | 1.45 | $2.94\times 10^{-1}$ | 1.23
p p > j j j | 3 | $2.84\times 10^{-1}$ | -0.45 | $2.80\times 10^{-1}$ | -0.46
p p > b b | 2 | $2.52\times 10^{-1}$ | 2.86 | $2.94\times 10^{-1}$ | 2.46
p p > b b j | 3 | $4.41\times 10^{-1}$ | 0.60 | $4.41\times 10^{-1}$ | 0.61
p p > b b j j | 4 | $6.18\times 10^{-1}$ | 0.54 | $5.88\times 10^{-1}$ | 0.57
p p > b b b b | 4 | $6.17\times 10^{-1}$ | 1.18 | $5.88\times 10^{-1}$ | 1.24
p p > t t | 2 | $2.90\times 10^{-1}$ | 1.63 | $2.94\times 10^{-1}$ | 1.61
p p > t t j | 3 | $4.51\times 10^{-1}$ | 0.68 | $4.41\times 10^{-1}$ | 0.70
p p > t t j j | 4 | $6.14\times 10^{-1}$ | 0.53 | $5.88\times 10^{-1}$ | 0.55
p p > t t t t | 4 | $6.38\times 10^{-1}$ | 1.63 | $5.88\times 10^{-1}$ | 1.77
p p > t t b b | 4 | $6.21\times 10^{-1}$ | 2.20 | $5.88\times 10^{-1}$ | 2.33
p p > wpm b b | 3 | $4.23\times 10^{-1}$ | 3.92 | $4.41\times 10^{-1}$ | 3.76
p p > z b b | 3 | $3.35\times 10^{-1}$ | 2.31 | $4.41\times 10^{-1}$ | 1.76
p p > a b b | 3 | $5.19\times 10^{-1}$ | 2.72 | $4.41\times 10^{-1}$ | 3.20
p p > wpm b b j | 4 | $4.25\times 10^{-1}$ | 2.66 | $5.88\times 10^{-1}$ | 1.92
p p > z b b j | 4 | $4.24\times 10^{-1}$ | 1.78 | $5.88\times 10^{-1}$ | 1.29
p p > a b b j | 4 | $5.12\times 10^{-1}$ | 1.12 | $5.88\times 10^{-1}$ | 0.98
p p > t t wpm | 3 | $2.39\times 10^{-1}$ | 2.08 | $4.41\times 10^{-1}$ | 1.13
p p > t t z | 3 | $3.05\times 10^{-1}$ | 1.45 | $4.41\times 10^{-1}$ | 1.00
p p > t t a | 3 | $2.96\times 10^{-1}$ | 1.52 | $4.41\times 10^{-1}$ | 1.02
p p > t t wpm j | 4 | $4.09\times 10^{-1}$ | 1.09 | $5.88\times 10^{-1}$ | 0.76
p p > t t z j | 4 | $4.62\times 10^{-1}$ | 0.61 | $5.88\times 10^{-1}$ | 0.48
p p > t t a j | 4 | $4.54\times 10^{-1}$ | 0.67 | $5.88\times 10^{-1}$ | 0.52
p p > t t w+ w- | 4 | $3.09\times 10^{-1}$ | 1.56 | $5.88\times 10^{-1}$ | 0.82
p p > t t wpm z | 4 | $2.66\times 10^{-1}$ | 1.77 | $5.88\times 10^{-1}$ | 0.80
p p > t t wpm a | 4 | $2.54\times 10^{-1}$ | 1.75 | $5.88\times 10^{-1}$ | 0.76
p p > t t z z | 4 | $2.93\times 10^{-1}$ | 1.24 | $5.88\times 10^{-1}$ | 0.62
p p > t t z a | 4 | $3.01\times 10^{-1}$ | 1.45 | $5.88\times 10^{-1}$ | 0.74
p p > t t a a | 4 | $2.84\times 10^{-1}$ | 1.22 | $5.88\times 10^{-1}$ | 0.59
p p > tt j $$ w+ w- | 3 | $9.40\times 10^{-2}$ | 0.28 | $4.41\times 10^{-1}$ | 0.06
p p > tt a j $$ w+ w- | 4 | $6.40\times 10^{-2}$ | 0.38 | $5.88\times 10^{-1}$ | 0.04
p p > tt z j $$ w+ w- | 4 | $3.50\times 10^{-2}$ | 0.08 | $5.88\times 10^{-1}$ | 0.00
p p > tt bb j $$ w+ w- | 5 | $1.38\times 10^{-1}$ | 2.32 | $7.35\times 10^{-1}$ | 0.44
p p > tt bb j a $$ w+ w- | 6 | $1.68\times 10^{-1}$ | 2.20 | $8.81\times 10^{-1}$ | 0.42
p p > tt bb j z $$ w+ w- | 6 | $1.87\times 10^{-1}$ | 2.35 | $8.81\times 10^{-1}$ | 0.50
p p > w+ > t b , p p > w- > t b | 2 | $3.50\times 10^{-2}$ | 9.57 | $2.94\times 10^{-1}$ | 1.14
p p > w+ t b a, p p > w- > t b a | 3 | $1.20\times 10^{-2}$ | 25.73 | $4.41\times 10^{-1}$ | 0.70
p p > w+ > t b z, p p > w- > t b z | 3 | $1.30\times 10^{-2}$ | 33.79 | $4.41\times 10^{-1}$ | 1.00
Table 3: A list of processes considered in Ref. [39] with their final state multiplicity ($n_{part}$), their relative uncertainty at leading order ($\Delta\sigma/\sigma_{0}$ ), and the resulting pull ($\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma}$). Also included is the modified relative uncertainty at leading order ($\Delta\sigma_{\text{ref}}/\sigma_{0}$), and the corresponding resulting modified pull ($\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}$). Process | $n_{\textrm{part}}$ | $\Delta\sigma/\sigma_{0}$ | $\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma}$ | $\Delta\sigma_{\text{ref}}/\sigma_{0}$ | $\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}$
---|---|---|---|---|---
p p > h | 1 | $3.48\times 10^{-1}$ | 3.02 | $1.47\times 10^{-1}$ | 7.15
p p > h j | 2 | $3.94\times 10^{-1}$ | 1.77 | $2.94\times 10^{-1}$ | 2.37
p p > h j j | 3 | $5.91\times 10^{-1}$ | 1.18 | $4.41\times 10^{-1}$ | 1.58
p p > h j j $$ w+ w- z | 3 | $2.00\times 10^{-2}$ | -2.26 | $2.80\times 10^{-1}$ | -0.16
p p > h j j j $$ w+ w- z | 4 | $1.57\times 10^{-1}$ | 0.61 | $5.88\times 10^{-1}$ | 0.16
p p > h wpm | 2 | $3.50\times 10^{-2}$ | 5.24 | $2.94\times 10^{-1}$ | 0.62
p p > h wpm j | 3 | $1.07\times 10^{-1}$ | 1.91 | $4.41\times 10^{-1}$ | 0.46
p p > h wpm j j | 4 | $2.61\times 10^{-1}$ | 1.18 | $5.88\times 10^{-1}$ | 0.52
p p > h z | 2 | $3.50\times 10^{-2}$ | 5.30 | $2.94\times 10^{-1}$ | 0.63
p p > h z j | 3 | $1.06\times 10^{-1}$ | 1.86 | $4.41\times 10^{-1}$ | 0.45
p p > h z j j | 4 | $2.62\times 10^{-1}$ | 0.78 | $5.88\times 10^{-1}$ | 0.35
p p > h w+ w- | 3 | $1.00\times 10^{-3}$ | 284.51 | $4.41\times 10^{-1}$ | 0.65
p p > h wpm a | 3 | $7.00\times 10^{-3}$ | 44.78 | $4.41\times 10^{-1}$ | 0.71
p p > h z wpm | 3 | $1.10\times 10^{-2}$ | 36.99 | $4.41\times 10^{-1}$ | 0.92
p p > h z z | 3 | $1.00\times 10^{-3}$ | 215.31 | $4.41\times 10^{-1}$ | 0.49
p p > h t t | 3 | $3.00\times 10^{-1}$ | 0.96 | $4.41\times 10^{-1}$ | 0.65
p p > h tt j | 4 | $2.40\times 10^{-2}$ | 11.19 | $5.88\times 10^{-1}$ | 0.46
p p > h b b | 3 | $2.81\times 10^{-1}$ | 0.79 | $4.41\times 10^{-1}$ | 0.51
p p > h t t j | 4 | $4.56\times 10^{-1}$ | 0.47 | $5.88\times 10^{-1}$ | 0.36
p p > h b b j | 4 | $4.56\times 10^{-1}$ | 0.49 | $5.88\times 10^{-1}$ | 0.38
p p > h h | 2 | $2.95\times 10^{-1}$ | 1.90 | $2.94\times 10^{-1}$ | 1.90
p p > h h j j $$ w+ w- z | 4 | $7.20\times 10^{-2}$ | 0.68 | $5.88\times 10^{-1}$ | 0.08
p p > h h wpm | 3 | $9.00\times 10^{-3}$ | 18.09 | $4.41\times 10^{-1}$ | 0.37
p p > h h wpm j | 4 | $1.42\times 10^{-1}$ | 1.10 | $5.88\times 10^{-1}$ | 0.27
p p > h h wpm a | 4 | $3.00\times 10^{-2}$ | 6.84 | $5.88\times 10^{-1}$ | 0.35
p p > h h z | 3 | $9.00\times 10^{-3}$ | 17.70 | $4.41\times 10^{-1}$ | 0.36
p p > h h z j | 4 | $1.41\times 10^{-1}$ | 1.06 | $5.88\times 10^{-1}$ | 0.25
p p > h h z a | 4 | $2.40\times 10^{-2}$ | 5.95 | $5.88\times 10^{-1}$ | 0.24
p p > h h z z | 4 | $3.90\times 10^{-2}$ | 4.88 | $5.88\times 10^{-1}$ | 0.32
p p > h h z wpm | 4 | $4.80\times 10^{-2}$ | 6.68 | $5.88\times 10^{-1}$ | 0.55
p p > h h w+ w- | 4 | $3.50\times 10^{-2}$ | 6.65 | $5.88\times 10^{-1}$ | 0.40
p p > h h t t | 4 | $3.02\times 10^{-1}$ | 0.26 | $5.88\times 10^{-1}$ | 0.14
p p > h h tt j | 5 | $1.00\times 10^{-3}$ | 326.09 | $7.35\times 10^{-1}$ | 0.44
p p > h h b b | 4 | $3.43\times 10^{-1}$ | 1.10 | $5.88\times 10^{-1}$ | 0.64
Table 4: A list of processes considered in Ref. [39] with their final state
multiplicity ($n_{part}$), their relative uncertainty at leading order
($\Delta\sigma/\sigma_{0}$ ), and the resulting pull
($\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma}$). Also included is the
modified relative uncertainty at leading order
($\Delta\sigma_{\text{ref}}/\sigma_{0}$), and the corresponding resulting
modified pull
($\frac{\sigma_{\text{NLO}}-\sigma_{0}}{\Delta\sigma_{\text{ref}}}$).

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