Statistical Patterns of Theory Uncertainties

2025-05-03 0 0 336.72KB 18 页 10玖币

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SciPost Physics Submission

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

Tim M.P. Tait2, and Daniel Whiteson2

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

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

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

4Institut für Theoretische Physik, Universität Heidelberg, Germany

May 8, 2023

Abstract

A comprehensive uncertainty estimation is vital for the precision program of the LHC. While ex-

perimental 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 Introduction 2

2 Theory uncertainties from scale variations 3

3 NLO dataset and results 6

4 Reference-process method 8

5 Outlook 9

A Comparison to simple correction of uncertainties 11

References 12

arXiv:2210.15167v4 [hep-ph] 5 May 2023

SciPost Physics Submission

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 ob-

servables via a combination of predictions for perturbative cross-sections, event generation,

and detector simulations. To measure fundamental parameters, particle physics relies on like-

lihood 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 pa-

rameters, 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 [1–5].

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 prac-

tice, 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 mea-

surements which have inherent statistical uncertainty because of ﬁnite sample sizes. An ex-

ample 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 pre-

cise 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 approxima-

tions. 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. 2with a review

of the standard method of estimating the theoretical uncertainty due to limited-order calcu-

lations 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. 5con-

tains our outlook and conclusions.

SciPost Physics Submission

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],

σ(θ)≈X

a,bZd xaxbfa(xa;µF)fb(xb;µF)ˆ

σab(θ;µF,µR), (1)

where the sum runs over partons inside the protons, fa/bare parton distribution functions

(PDFs), ˆ

σis the partonic hard-scattering cross-section, and θrepresents parameter(s) of in-

terest. A typical LHC analysis infers information on the parameter of interest θthrough com-

parison with data measuring the observable σ(θ). We will be concerned primarily with inclu-

sive 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 ˆ

σis computed as a perturbative expansion in the QCD coupling

αs[6], but estimating the precision of a given truncation is challenging. Estimates based on

the size of the expansion parameter αs, or αs/π, 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 [10–13].

At each perturbative order, ultraviolet (UV) divergences in cross-section predictions are

removed through renormalization, introducing a logarithmic dependence on an unphysical

renormalization scale µRin 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 µF. 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 arbi-

trariness 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 se-

ries is expected to be. Traditionally, the uncertainty on the predicted hadronic cross-section

in Eq.(1) is estimated by varying µRand µFaround a suitable central scale. The choice of the

central scale can vary depending on the physics process, it could be for example the scalar sum

of transverse mass of all ﬁnal state particles,the invariant mass of the system being produced,

the average transverse energy of jets produced, or centre-of-mass energy of the collider. The

logarithmic dependence on the scale requires µ∼Ewhere Echaracterizes the physical en-

ergy scale appearing in the observable of interest. This choice insures that terms growing as

log E/µ 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 differ-

SciPost Physics Submission

ence 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 statisti-

cal 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 [1–5]

σ∈[σ−,σ+]. (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 σto log σ. It induces volume effects in the marginaliza-

tion, 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 depen-

dence 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 µR,

σ∈[σ−,σ+]≡σ(µR,+),σ(µR,−), (3)

and a suitable central scale choice σ0=σ(µR,0)is often the transverse mass of the ﬁnal state,

µR,0 =X

iÇm2

i+p2

T,i. (4)

where the sum is over ﬁnal state particles of mass miand transverse momentum pT,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

∆σscale =∂ σ

∂ αs

∆αs=∂ σ

∂ αs

αs(µR,−)−αs(µR,+)

2. (5)

where the appropriate range of scales deﬁned by µR,±is discussed below. An approach based

on lower and upper limits deﬁned by a scale variation ensures that:

1. no O(1)variation of the unphysical scale around the central choice is considered more

likely than any other;

2. there is no long tail of exponentially suppressed probability to obtain a very large deviation

from perturbative QCD; and

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 αs,

σ∝αn

s⇒∆σscale

σ0

=n×αs(µR,−)−αs(µR,+)

2αs(µR,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¯

production. Experience from its perturbative behavior at the Tevatron [14–17]and at the

LHC [18–22]motivates the standard choice:

µR,0 =mtµR,+=2mtµR,−=mt

2, (7)

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SciPostPhysicsSubmissionStatisticalPatternsofTheoryUncertaintiesAishikGhosh1,2,BenjaminNachman1,3,TilmanPlehn4,LilyShire2,TimM.P.Tait2,andDanielWhiteson21PhysicsDivision,LawrenceBerkeleyNationalLaboratory,Berkeley,USA2DepartmentofPhysics&Astronomy,UniversityofCalifornia,Irvine,USA3BerkeleyInstitutef...

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Statistical Patterns of Theory Uncertainties

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