9. Quantum Chromodynamics

1

9. Quantum Chromodynamics
Revised August 2023 by J. Huston (Michigan State U.), K. Rabbertz (KIT) and G. Zanderighi(MPP, Garching; München Tech. U.).

9.1

9.2

9.3

9.4

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Running coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.2 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Structure of QCD predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Fully inclusive cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Processes with initial-state hadrons . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 Cross sections with phase-space restrictions . . . . . . . . . . . . . . . . . . .
9.2.4 Accuracy of predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental studies of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Hadronic final-state observables . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 QCD measurements at colliders . . . . . . . . . . . . . . . . . . . . . . . . . .
Determinations of the strong coupling constant . . . . . . . . . . . . . . . . . . . .
9.4.1 Hadronic τ decays and low Q2 continuum: . . . . . . . . . . . . . . . . . . .
9.4.2 Heavy quarkonia decays: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.3 PDF fits: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.4 Hadronic final states of e+ e− annihilations: . . . . . . . . . . . . . . . . . . .
9.4.5 Observables from hadron-induced collisions: . . . . . . . . . . . . . . . . . . .
9.4.6 Electroweak precision fit: . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.7 Lattice QCD: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.8 Determination of the world average value of αs (m2Z ) : . . . . . . . . . . . . .

1
2
3
4
4
5
10
17
19
20
23
30
31
32
32
34
35
37
38
39

9.1 Basics
Quantum Chromodynamics (QCD), the gauge field theory that describes the strong interactionsof colored quarks and gluons, is the SU(3) component of the SU(3)×SU(2)×U(1) Standard Modelof Particle Physics. The Lagrangian of QCD is given byL=

1 A A µνCψ̄q,a (iγ µ ∂µ δab − gs γ µ tC,
ab Aµ − mq δab )ψq,b − Fµν F
4
q

X

(9.1)

where repeated indices are summed over. The γ µ are the Dirac γ-matrices. The ψq,a are quark-fieldspinors for a quark of flavor q and mass mq , with a color-index a that runs from a = 1 to Nc = 3,
i.e. quarks come in three “colors.” Quarks are said to be in the fundamental representation of theSU(3) color group.
2
The ACµ correspond to the gluon fields, with C running from 1 to Nc − 1 = 8, i.e. there are eightkinds of gluon. Gluons transform under the adjoint representation of the SU(3) color group. The
tC
ab correspond to eight 3 × 3 matrices and are the generators of the SU(3) group (cf. the section
C
on “SU(3) isoscalar factors and representation matrices” in this Review, with tC
ab ≡ λab /2). They
encode the fact that a gluon’s interaction with a quark rotates the quark’s color in SU(3) space.
gs2
The quantity gs (or αs = 4π
) is the QCD coupling constant. Besides quark masses, which have
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:18am

9. Quantum Chromodynamics

2

A iselectroweak origin, it is the only fundamental parameter of QCD. Finally, the field tensor Fµν
given by
A
A
B C
Fµν
= ∂µ AAν − ∂ν Aµ − gs fABC Aµ Aν ,[tA , tB ] = ifABC tC ,(9.2)

where the fABC are the structure constants of the SU(3) group.
Neither quarks nor gluons are observed as free particles. Hadrons are color-singlet (i.e. colorneutral) combinations of quarks, anti-quarks, and gluons.
Ab-initio predictive methods for QCD include lattice gauge theory and perturbative expansions
in the coupling. The Feynman rules of QCD involve a quark-antiquark-gluon (q q̄g) vertex, a 3-gluon
vertex (both proportional to gs ), and a 4-gluon vertex (proportional to gs2 ). A full set of Feynman
rules is to be found for example in Refs. [1, 2].
Adopting a standard notation where repeated indices are summed over, useful color-algebraA
2
relations include: tA
ab tbc = CF δac , where CF ≡ (Nc − 1)/(2Nc ) = 4/3 is the color factor (“Casimir”)
associated with gluon emission from a quark; fACD fBCD = CA δAB , where CA ≡ Nc = 3 is theBcolor factor associated with gluon emission from a gluon; tA
ab tab = TR δAB , where TR = 1/2 is thecolor factor for a gluon to split to a q q̄ pair.
There is freedom for an additional CP -violating term to be present in the QCD Lagrangian,A , where θ is an additional free parameter, and F̃ A is the dual of the gluon field tensor,
θ α8πs F A µν F̃ µν
µν
1
A
σρ

F
,
with

being
the
fully
antisymmetricLevi-Civita
symbol. Experimental limits onµνσρ
2 µνσρ
the electric dipole moment (EDM) of ultracold neutrons [3, 4] and atomic mercury [5] constrain
the QCD vacuum angle to satisfy |θ| . 10−10 . For an overview of current and future experiments
measuring EDMs see [6]. Further discussion is to be found in Ref. [7] and in the Axions section in
the Listings of this Review.
This section will concentrate mainly on perturbative aspects of QCD as they relate to collider
physics. Related textbooks and lecture notes include Refs. [1, 2, 8–11]. Aspects specific to Monte
Carlo event generators are reviewed in the dedicated section 43. Lattice QCD is also reviewed ina section of its own, Sec. 17, with further discussion of perturbative and non-perturbative aspects
to be found in the sections on “Quark Masses”, “The CKM quark-mixing matrix”, “Structure
functions”, “Fragmentation Functions”, “High Energy Soft QCD and Diffraction”, “Passage ofParticles Through Matter” and “Heavy-Quark and Soft-Collinear Effective Theory” in this Review.
9.1.1 Running coupling
In the framework of perturbative QCD (pQCD), predictions for observables are expressed interms of the renormalized coupling αs (µ2R ), a function of an (unphysical) renormalization scaleµR . When one takes µR close to the scale of the momentum transfer Q in a given process, thenαs (µ2R ' Q2 ) is indicative of the effective strength of the strong interaction in that process.
The coupling satisfies the following renormalization group equation (RGE):µ2Rdαs= β(αs ) = −(b0 αs2 + b1 αs3 + b2 αs4 + · · · ) ,
dµ2R

(9.3)

where b0 = (11CA − 4nf TR )/(12π) = (33 − 2nf )/(12π) is referred to as the 1-loop β-function
2 −n T (10C +6C ))/(24π 2 ) = (153−19n )/(24π 2 ),coefficient, the 2-loop coefficient is b1 = (17CA
A
F
f R
f
325 2
3 ) for the SU(3) values of C
and the 3-loop coefficient is b2 = (2857 − 5033
n+n)/(128πAf
927 f
and CF . Here nf is the number of quark flavors. The 4-loop coefficient, b3 , is to be found in
Refs. [12, 13], while the 5-loop coefficient, b4 , is in Refs. [14–18]. The coefficients b2 and b3 (andbeyond) are renormalization-scheme-dependent and given here in the modified minimal subtraction
scheme (MS) [19], by far the most widely used scheme in QCD and the one adopted in the following.
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9. Quantum Chromodynamics

3

The minus sign in Eq. (9.3) is the origin of “asymptotic freedom” [20, 21], i.e. the fact that thestrong coupling becomes weak for processes involving large momentum transfers (“hard processes”).
For momentum transfers in the 0.1–1 TeV range, αs ∼ 0.1, while the theory is strongly interacting
for scales around and below 1 GeV.
The β-function coefficients, the bi , are given for the coupling of an effective theory in which nfof the quark flavors are considered light (mq  µR ), and in which the remaining heavier quark
flavors decouple from the theory. One may relate the coupling for the theory with nf + 1 lightflavors to that with nf flavors through an equation of the form
(n +1)(n )αs f (µ2R ) = αs f (µ2R )

∞ X
n
X

µ2
(n )
1+
cn` [αs f (µ2R )]n ln` R2
mh
n=1 `=0

!

,

(9.4)

1
, c10 = 0,
where mh is the mass of the (nf +1)th flavor, and the first few cn` coefficients are c11 = 6π
11
7
11
2
c22 = c11 , c21 = 24π2 , and c20 = − 72π2 when mh is the MS mass at scale mh , while c20 = 24π2when mh is the pole mass (mass definitions are discussed below in Sec. (9.1.2) and in the review
on “Quark Masses”). Terms up to c4` with 0 ≤ ` ≤ 4 are to be found in Refs. [22, 23]. Numerically,
when one chooses µR = mh , the matching is a modest effect, owing to the zero value for the c10
coefficient. Relations between nf and (nf +2) flavors where the two heavy flavors are close in mass
are given to three loops in Ref. [24].
Working in an energy range where the number of flavors is taken constant, a simple exact
analytic solution exists for Eq. (9.3) only if one neglects all but the b0 term, giving αs (µ2R ) =(b0 ln(µ2R /Λ2 ))−1 . Here Λ is a constant of integration, which corresponds to the scale where theperturbatively-defined coupling would diverge. Its value is indicative of the energy range where
non-perturbative dynamics dominates. A convenient approximate analytic solution to the RGE
that includes terms up to b4 is given by solving iteratively Eq. (9.3)αs (µ2R ) '1
b1 ` b2 (`2 − ` − 1) + b0 b2+
1− 2 + 1
b0 t
b0 t
b40 t2b3 −2`3 + 5`2 + 4` − 1 − 6b0 b2 b1 ` + b20 b3+ 1+2b60 t3
18b0 b2 b21 2`2 −`−1 +b41 6`4 −26`3 −9`2 +24`+7+6b80 t4−b2 b3 b1 (12`+1)+2b20 5b22 +b0 b4+ 06b80 t4

!

,

(9.5)

µ2

with t ≡ ln ΛR2 and ` = ln t, again parameterized in terms of a constant Λ. Note that Eq. (9.5) is
one of several possible approximate 4-loop solutions for αs (µ2R ), and that a value for Λ only defines
αs (µ2R ) once one knows which particular approximation is being used. An alternative to the use offormulas such as Eq. (9.5) is to solve the RGE exactly, numerically (including the discontinuities,
Eq. (9.4), at flavor thresholds). In such cases the quantity Λ does not directly arise (though it canbe defined, cf. Eqs. (1–3) of Ref. [25]). For these reasons, in determinations of the coupling, it has
become standard practice to quote the value of αs at a given scale (typically the mass of the Zboson, mZ ) rather than to quote a value for Λ.
A discussion of determinations of the coupling and a graph illustrating its scale dependence(“running”) are to be found in Section 9.4.
9.1.2 Quark masses
Free quarks have never been observed, which is understood as a result of a long-distance,
confining property of the strong QCD force: up, down, strange, charm, and bottom quarks all
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9. Quantum Chromodynamics

4

hadronize, i.e. become part of a meson or baryon, on a timescale ∼ 1/Λ; the top quark instead
decays before it has time to hadronize. This means that the question of what one means by thequark mass is a complex one, which requires one to adopt a specific prescription. A perturbatively
defined prescription is the pole mass, mq , which corresponds to the position of the divergence ofthe quark propagator. This is close to one’s physical picture of mass. However, when relating it toobservable quantities, it suffers from a badly behaved perturbative series which makes it ambiguous
to an amount related to Λ (see e.g. Ref. [26–28]). An alternative is the MS mass, mq (µ2R ), which
depends on the renormalization scale µR .
Results for the masses of heavier quarks are often quoted either as the pole mass or as the MS
mass evaluated at a scale equal to the mass, mq (m2q ); light quark masses are often quoted in the
MS scheme at a scale µR ∼ 2 GeV. The pole and MS masses are related by a series that starts as
mq = mq (m2q )(1 +order by4αs (m2q )+ O(αs2 )), while the scale-dependence of MS masses is given at lowest
3π"dmq (µ2R )
=
µ2R
dµ2R

#

αs (µ2R )+ O(αs2 ) mq (µ2R ) .
−
π

(9.6)

A more detailed discussion is to be found in a dedicated section of the Review, “Quark Masses”,
with detailed formulas also in Ref. [29] and references therein.
In perturbative QCD calculations of scattering processes, it is customary to employ an approximation where the masses of all quarks, whose magnitudes are much smaller than the momentum
transfer involved in the process, are neglected or set to zero.

9.2 Structure of QCD predictions
9.2.1 Fully inclusive cross sections
The simplest observables in perturbative QCD are those that do not depend on the initial-state
hadrons and are fully inclusive, disregarding specific details of the final state. An example of suchan observable is the total cross section for the process e+ e− → hadrons at a center-of-mass energy
Q. It can be expressed as follows:σ(e+ e− → hadrons, Q)≡ R(Q) = REW (Q)(1 + δQCD (Q)) ,σ(e+ e− → µ+ µ− , Q)

(9.7)

where REW (Q) is the purely electroweak prediction for the ratio and δQCD (Q) is the correction due
to QCD effects.
For the sake of simplicity, we can focus on energies where Q  mZ , where the primary contribution to the process comes from photon exchange (while disregarding electroweak and finiteP
quark-mass corrections) denoted as REW = Nc q e2q , with eq representing the electric charges of
the quarks. The QCD correction reads
δQCD (Q) =

∞
X

cn ·

n=1

αs (Q2 )
π

!n

+O

Λ4
Q4

!

.

(9.8)

The first four terms in the αs series expansion are [30]c1 = 1 ,c2 = 1.9857 − 0.1152nf ,c3 = −6.63694 − 1.20013nf − 0.00518n2f − 1.240η ,c4 = −156.61 + 18.775nf − 0.7974n2f + 0.0215n3f − (17.828 − 0.575nf )η ,31st May, 2024

(9.9a)(9.9b)(9.9c)

9. Quantum Chromodynamics

5with η = ( eq )2 /(Nc e2q ) and Nc = 3. For corresponding expressions including also Z exchange
and finite-quark-mass effects, see Refs. [31–33].
A related series holds also for the QCD corrections to the hadronic decay width of the τ lepton,
which essentially involves an integral of R(Q) over the allowed range of invariant masses of thehadronic part of the τ decay (see e.g. Ref. [34]). The series expansions for QCD corrections to
Higgs-boson hadronic (partial) decay widths in the limit of heavy top quark and massless lightflavors at N4 LO are given in Ref. [35].
One characteristic feature of Eqs. (9.8) and (9.9) is that the coefficients of αsn increase order byorder: calculations in perturbative QCD tend to converge more slowly than would be expected based
just on the size of αs . The situation is significantly worse near thresholds or in the presence of tightkinematic cuts. Another feature is the existence of an extra “power-correction” term O(Λ4 /Q4 )in Eq. (9.8), which accounts for contributions that are fundamentally non-perturbative. All highenergy QCD predictions involve power corrections (Λ/Q)p , although typically the suppression ofthese corrections with Q is smaller than given in Eq. (9.8), where p = 4. The exact power pdepends on the observable and, for many processes and observables, it is possible to introduce anoperator product expansion and associate power-suppressed terms with specific higher-dimension(non-perturbative) operators [36].
Scale dependence. In Eq. (9.8) the renormalization scale for αs has been chosen equal to Q. The
result can also be expressed in terms of the coupling at an arbitrary renormalization scale µR ,PPδQCD (Q) =

∞
X
n=1

cn

µ2R
Q2

!

·

αs (µ2R )
π

!n

+O

Λ4
Q4

!

,

(9.10)

where c1 (µ2R /Q2 ) ≡ c1 , c2 (µ2R /Q2 ) = c2 + πb0 c1 ln(µ2R /Q2 ), c3 (µ2R /Q2 ) = c3 + (2b0 c2 π + b1 c1 π 2 )× ln(µ2R /Q2 ) + b20 c1 π 2 ln2 (µ2R /Q2 ), etc. Given an infinite number of terms in the αs expansion, theµR dependence of the cn (µ2R /Q2 ) coefficients will exactly cancel that of αs (µ2R ), and the final result
will be independent of the choice of µR : physical observables do not depend on unphysical scales.1
With just terms up to some finite n = N , a residual µR dependence will remain, which implies anuncertainty on the prediction of R(Q) due to the arbitrariness of the scale choice. This uncertainty
will be O(αsN +1 ), i.e. of the same order as the neglected higher-order terms. For this reason it
is customary to use the scale dependence of QCD predictions as an estimate of the uncertainties
due to neglected terms. One usually takes a central value for µR ∼ Q, in order to avoid the poor
convergence of the perturbative series that results from the large lnn−1 (µ2R /Q2 ) terms in the cn
coefficients when µR  Q or µR  Q. Uncertainties are then commonly determined by varying µR
by a factor of two up and down around the central scale choice. This is not a rigorous prescription
for determining the uncertainty, but is motivated by the requirement that there should not be large
logarithms introduced into a calculation by large ratios of scales. A more detailed discussion on
the accuracy of theoretical predictions and on ways to estimate the theoretical uncertainties canbe found in Section 9.2.4.
9.2.2 Processes with initial-state hadrons
Deep-Inelastic Scattering. To illustrate the key features of QCD cross sections in processes
with initial-state hadrons, let us consider deep-inelastic scattering (DIS), e + p → e + X, where
an electron e with four-momentum k emits a highly off-shell photon (momentum q) that interacts
with the proton (momentum p). For photon virtualities Q2 ≡ −q 2 far above the squared proton
mass (but far below the squared Z mass), the differential cross section in terms of the kinematic
1

With respect to pQCD there is an important caveat to this statement: at sufficiently high orders, perturbative
series generally suffer from “renormalon” divergences αsn n! (reviewed in Ref. [26]). This phenomenon is not usuallyvisible with the limited number of perturbative terms available today. However, it is closely connected with nonperturbative contributions and sets a limit on the possible precision of perturbative predictions. The cancellation ofscale dependence will also ultimately be affected by this renormalon-induced breakdown of perturbation theory.
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69. Quantum Chromodynamics

variables Q2 , x = Q2 /(2p · q) and y = (q · p)/(k · p) isi
4πα2 h
d2 σ
2
2
2
2
=
(1
+
(1
−
y)
)F
(x,
Q
)
−
y
F
(x,
Q
)
,
2
L
dxdQ2
2xQ4

(9.11)

where α is the electromagnetic coupling and F2 (x, Q2 ) and FL (x, Q2 ) are proton structure functions,which encode the interaction between the photon and the proton. In the presence of parity-violatinginteractions (e.g. νp scattering) an additional F3 structure function is present. For an extended
review, including equations for the full electroweak and polarized cases, see Sec. 18 of this Review.
Structure functions are not calculable in perturbative QCD, nor is any other cross section thatinvolves QCD interactions and initial-state hadrons. To zeroth order in αs , the structure functions
are given directly in terms of non-perturbative parton (quark or gluon) distribution functions(PDFs),X
F2 (x, Q2 ) = x
e2q fq/p (x) ,FL (x, Q2 ) = 0 ,(9.12)q

where fq/p (x) is the non-perturbative PDF for quarks of type q inside the proton, i.e. the number
density of quarks of type q inside a fast-moving proton that carry a fraction x of its longitudinal
momentum.
PDFs are primarily determined from data in global fits. PDF sets are available at different
orders in perturbation theory (LO, NLO and NNLO) based on the order at which cross sections
used in the fits are calculated. The evolution equations for the PDFs from one scale to anothermatches the perturbative accuracy of the cross sections. In modern global PDF fits, data are
included from DIS, Drell-Yan (DY), jets and tt̄ processes, and more LHC collider data, with theglobal PDF fits using 3000-4000 data points. There is a large change in the PDFs from LO to
NLO, with a much smaller change from NLO to NNLO. LO PDFs can be unreliable for collider
predictions, especially at low and high x. The uncertainties for the resulting PDFs are determinedprimarily from the experimental uncertainties of the data that serve as input to the global PDF
fits. There is also a component due to the limitations of the parameterizations used, as discussed
e.g. in Ref. [37], although this may be reduced with the use of more flexible forms. The PDF
uncertainties can either be determined through a Hessian approach or through the use of Monte
Carlo replicas. It is now relatively straightforward to convert results from one approach to the other.
There has been a recent combination (PDF4LHC21) of the CT18, MSHT20 and NNPDF3.1 PDFs,
each produced from a global PDF fit, that serves to provide a more comprehensive view of both thecentral PDFs and of their uncertainties [38]. Recently, theoretical uncertainties related to missing
higher orders have been included in global PDF determinations but so far only at NLO [39–42]. Bytying renormalization scales for related processes, and factorization scales for all processes, together
in the global fit, an estimate of the higher order uncertainties can be added to the PDF fit. An
alternative approach [43] is to attempt to deal with cross sections rather than PDFs themselves.
Another approach is to use as much available information as possible from N3 LO to examine
the impact of such information on the resultant PDFs [44]. The result, of course, is sensitive tothe higher order information that is still missing, for example from an incomplete knowledge of the
splitting functions and the N3 LO matrix elements.
Since the original work of Ref. [45], one of the intriguing questions is whether lattice calculations can provide valuable insights for extracting PDFs. Despite initial debates surrounding theunderlying methodologies [46, 47], it is now understood that lattice data in Euclidean space can beconnected to light-cone PDFs through factorization theorems, once lattice observables have beenrenormalized and extrapolated to the continuum limit [48]. Initial determinations based on latticedata have begun to emerge [48–61], and the comparisons with determinations based on conventional data show great promise. However, the uncertainties in lattice-based determinations are
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9. Quantum Chromodynamics

7

still quite large, therefore, for practical applications, PDFs are currently derived from global fits
of experimental data (cf. Sec. 18 of this Review and also Refs. [62, 63]). It is likely that latticesimulations for light-cone PDFs will have an impact in the short-medium term if they focus on flavor components (strange and charm) and on the kinematic regions that are poorly covered by theexperimental data [64]. Similarly, lattice determinations of transverse momentum dependent PDF(TMDs) and generalized parton distributions (GPDs) will be more relevant for phenomenology
since the experimental data for these quantities are much less abundant and precise.
The result in Eq.(9.12), with PDFs fq/p (x) that are independent of the scale Q, corresponds
to the “quark-parton model” picture, in which the photon interacts with point-like free quarks, or
equivalently, one has incoherent elastic scattering between the electron and individual constituentsof the proton. As a consequence, in this picture also F2 and FL are independent of Q [65]. When
including higher orders in pQCD, one has
F2 (x, Q2 ) = xZ∞Xαn (µ2 ) X 1 dz
s

n=0

R
(2π)n

i=q,g xz(n)C2,i (z, Q2 , µ2R , µ2F )fi/p

x

z



, µ2F + O Λ2 

Q2.

(9.13)(n)Just as in Eq. (9.10), we have a series in powers of αs (µ2R ), each term involving a coefficient C2,ithat can be calculated using Feynman diagrams. At variance with the parton model, the PDFs inpQCD depend on an additional scale, the factorization scale µF , whose significance will be discussed
in the following. Another important difference is the additional integral over z. The parton thatcomes from the proton can undergo a splitting before it enters the hard scattering. As a result,(n)the C2,i coefficients are functions that depend on the ratio, z, of the parton’s momentum before
and after radiation, and one must integrate over that ratio. For the electromagnetic component of(0)
DIS with light quarks and gluons, the zeroth order coefficient functions are C2,q = e2q δ(1 − z) and(0)

C2,g = 0. Corrections are known up to O(αs3 ) (next-to-next-to-next-to-leading order, N3 LO) for
both electromagnetic [66] and weak currents [67, 68]. For heavy-quark production they are known
to O(αs2 ) [69, 70] (next-to-leading order, NLO, insofar as the series starts at O(αs )). The current
status of the theoretical description of unpolarized and polarized DIS is summarized in Ref. [71].
For precise comparisons of LHC cross sections with theoretical predictions, the photon PDF of the
proton is also needed. It has been computed precisely in Ref. [72,73] and has now been implemented
in most global PDF fits. More recently, the PDF for leptons has also been computed [74]. The
photon PDF has an elastic and an inelastic component, where the photon is generated from theintact proton or from the splitting of quarks inside the proton, respectively, while the lepton PDF
is generated by photons splitting to collinear leptons.
The majority of the emissions that modify a parton’s momentum are collinear (parallel) tothat parton, and do not depend on whether the parton will interact with a photon. It is natural
to view these emissions as modifying the proton’s structure rather than being part of the coefficient function for the parton’s interaction with the photon. Technically, one uses a procedureknown as collinear factorization to give a well-defined meaning to this distinction, most commonly
through the MS factorization scheme, defined in the context of dimensional regularization. The
MS factorization scheme involves an arbitrary choice of factorization scale, µF , whose meaning
can be understood roughly as follows: emissions with transverse momenta above µF are included(n)in the C2,q (z, Q2 , µ2R , µ2F ); emissions with transverse momenta below µF are accounted for within
the PDFs, fi/p (x, µ2F ). While collinear factorization is generally believed to be valid for suitable(sufficiently inclusive) observables in processes with hard scales, Ref. [75], which reviews the factorization proofs in detail, is cautious in the statements it makes about their exhaustivity, notably
for the hadron-collider processes which we shall discuss below. Interesting considerations on the
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9. Quantum Chromodynamics

8

current status of our understanding of factorization can be found in Refs. [76, 77]. While thetransverse degrees of freedom have been integrated over for collinear PDFs, it is also possible toproduce transverse-momentum dependent (or unintegrated) PDFs where these degrees of freedom
remain. For a recent comprehensive review of transverse-momentum-dependent parton distributionfunctions and fragmentation functions see Ref. [78].
For collinear PDFs, the resulting dependence on µF is described by the Dokshitzer-GribovLipatov-Altarelli-Parisi (DGLAP) equations [79], which to leading order (LO) read22
X αs (µ2 ) Z 1 dz (1)x 2
2 ∂fi/p (x, µF )F
µF=Pi←j (z)fj/p
,µ ,
2π
z F
∂µ2F
x z
j

(9.14)(1)

with, for example, Pq←g (z) = TR (z 2 + (1 − z)2 ). The other LO splitting functions are listed
in Sec. 18 of this Review, while results up to NLO, αs2 , and NNLO, αs3 , are given in Refs. [80]and [81] respectively. At N3 LO accuracy, only partial results are currently available in Refs. [82–86].
Splitting functions for PDFs in the helicity dependent case are given in Ref. [87].
(1)
Beyond LO, the coefficient functions are also µF dependent, for example C2,i (x, Q2 , µ2R , µ2F ) =(1)µ2  P

(0)

(1)

1 dz
x
C2,i (x, Q2 , µ2R , Q2 ) − ln QF2
j x z ×C2,j ( z )Pj←i (z). Progress in higher-order QED and mixed
QED-QCD corrections to the splitting functions can be found in refs. [88, 89].
As with the renormalization scale, the choice of factorization scale is arbitrary, but if one has an
infinite number of terms in the perturbative series, the µF -dependencies of the coefficient functions
and PDFs will compensate each other fully. Given only N terms of the series, a residual O(αsN +1 )uncertainty is associated with the ambiguity in the choice of µF . As with µR , varying µF provides
an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choicefor the scales is usually µR = µF = Q.
As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds
near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered tobe part of the proton’s structure, while above it is considered to be part of the proton’s structureand evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with masslesscoefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near thethreshold, it is instead necessary to appropriately adapt the standard massive coefficient functions
to account for the heavy-quark contribution already included in the PDFs [90–92].
At sufficiently small x and Q2 in inclusive DIS, resummation of small x logarithms may benecessary [93, 94]. This may in fact have been observed in Refs. [95] based on HERA data [96], ina kinematic region where useful information for PDFs for collider predictions is present.
√
In situations, in which the center-of-mass energy s is much larger than all other momentumtransfer scales in the problem (e.g. Q in DIS, mb for bb̄ production in pp collisions, etc.), each powerof αs beyond LO can be accompanied by a power of ln(s/Q2 ) (or ln(s/m2b ), etc.). This is variously
referred to as the high-energy, small-x or Balitsky-Fadin-Kuraev-Lipatov (BFKL) limit [94, 97, 98].
Currently it is possible to account for the dominant and first sub-dominant [99, 100] power of suchlogarithms at each order of αs , and also to estimate further sub-dominant contributions that are
numerically large (see Refs. [101–104] and references therein). Progress towards NNLO is discussed
in Ref. [105].
Physically, the summation of all orders in αs can be understood as leading to a growth with sof the gluon density in the proton. At sufficiently high energies this implies non-linear effects
2

R

LO is generally taken to mean the lowest order at which a quantity is non-zero.

31st May, 2024

99. Quantum Chromodynamics(commonly referred to as parton saturation), whose treatment has been the subject of intense
study (see for example Refs. [106–108] and references thereto).
Hadron-hadron collisions.
The extension to processes with two initial-state hadrons can be illustrated with the example ofthe total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2 , which
can be written asσ(h1 h2 → W + X) =∞X

αsn (µ2R )n=0(n)× σ̂ij→W +X

XZ





i,j





dx1 dx2 fi/h1 x1 , µ2F fj/h2 x2 , µ2F

x1 x2 s, µ2R , µ2F



+O

Λ2
m4W



(9.15)!

,

where s is the squared center-of-mass energy of the collision. At LO, n = 0, the hard (partonic)(0)
cross section σ̂ij→W +X (x1 x2 s, µ2R , µ2F ) is simply proportional to δ(x1 x2 s − m2W ), in the narrow W boson width approximation (see Sec. 51 of this Review for detailed expressions for this and otherhard scattering cross sections). It is non-zero only for choices of i, j that can directly give a W ,¯ At higher orders, n ≥ 1, new partonic channels contribute, such as gq, andsuch as i = u, j = d.
x1 x2 s ≥ m2W in the narrow W -boson width approximation.
Equation (9.15) involves a collinear factorization between the hard cross section and the PDFs,
just like Eq. (9.13). As long as the same factorization scheme is used in DIS and pp or pp̄ (usually
the MS scheme), then PDFs extracted in DIS can be directly used in pp and pp̄ predictions [75,109](with the anti-quark distributions in an anti-proton being the same as the quark distributions in a
proton).
A number of fully inclusive cross-sections have been computed up to N3 LO, i.e. including corrections up to relative order αs3 , in recent years. These include inclusive Higgs production throughgluon fusion in the large mt limit, calculated at N3 LO in Refs. [110, 111]. Higgs boson pair production via gluon fusion in the same approximation was computed at N3 LO in Ref. [112]. Calculations
at this order, differential in the Higgs rapidity [113, 114] and fully differential [115] have been presented recently. Bottom-induced Higgs boson production has also been computed at N3 LO [116].
Vector-boson fusion single- [117] and double-pair [118] production is also known to N3 LO [117] inthe factorized approximation. Neutral [119, 120] and charged [121] Drell-Yan processes, as well as
associated Higgs production [116] have also been computed at N3 LO. Differential rapidity distributions are available, but not yet with fiducial cuts. A number of public codes are now available whichallow for phenomenological studies at N3 LO [116, 117, 122–124]. The uncertainty band derived byvarying the renormalization and factorization scales in the perturbative predictions is not always
contained within the corresponding uncertainty band of the previous order, as might be expectedfrom a well-behaved process. This has been ascribed to cancellations between channels at NNLO
that lead to underestimates of the uncertainty at that order. This effect does not seem to be present
in other processes, such as Higgs boson production through gluon-gluon fusion. Note that there iscurrently a mis-match in N3 LO calculations in that PDFs at the same order are not yet available.
Preliminary approximated N3 LO PDFs [44] suggest that N3 LO effects can be large, especially as
far as heavy-quark distributions and low scales are concerned, as well as the gluon distribution in aregion important for Higgs boson production through gluon-gluon fusion. The fully inclusive hardcross sections for several other processes are known to NNLO,3 for instance, Higgs-boson productionin association with a vector boson [125], Higgs-pair production in the large mt approximation [126],
and with full mt dependence [127], top-antitop production [128, 129], bottom-anti-bottom production [130] and vector-boson pair production [131–133]. For a comprehensive overview on other3
Processes with jets or photons in the final state have divergent cross sections unless one places a cut on the jetor (dressed) photon momentum. Accordingly, they are discussed below in Section 9.2.3.2.

31st May, 2024

109. Quantum Chromodynamics

recent NNLO 2 → 2 calculations see Ref. [134, 135]. Other NNLO calculations with fiducial cuts
will be discussed in Sec. 9.2.3.
Photoproduction.
γp (and γγ) collisions are similar to pp collisions, with the subtlety that the photon can behave in
two ways: there is “direct” photoproduction, in which the photon behaves as a point-like particle
and takes part directly in the hard collision, with hard subprocesses such as γg → q q̄; there isalso resolved photoproduction, in which the photon behaves like a hadron, with non-perturbative
partonic substructure and a corresponding PDF for its quark and gluon content, fi/γ (x, Q2 ). Whileuseful to understand the general structure of γp collisions, the distinction between direct andresolved photoproduction is not well defined beyond leading order, as discussed for example inRef. [136].
9.2.3 Cross sections with phase-space restrictions
QCD final states always consist of hadrons, while perturbative QCD calculations deal with partons. Physically, an energetic parton fragments (“showers”) into many further partons, which then,on later timescales, undergo a transition to hadrons (“hadronization”). Fixed-order perturbation
theory captures only a small part of these dynamics. This does not matter for the fully inclusive
cross sections discussed above: the showering and hadronization stages are approximately unitary,i.e. they do not substantially change the overall probability of hard scattering, because they occur
long after it has taken place (they introduce at most a correction proportional to a power of theratio of timescales involved, i.e. a power of Λ/Q, where Q is the hard scattering scale).
Less inclusive measurements, in contrast, may be affected by the extra dynamics. For thosesensitive just to the main directions of energy flow (jet rates, event shapes, cf. Sec. 9.3.1) fixedorder perturbation theory is often still adequate, because showering and hadronization do notsubstantially change the overall energy flow. This means that one can make a prediction using justa small number of partons, which should correspond well to a measurement of the same observablecarried out on hadrons. For observables that instead depend on distributions of individual hadrons(which, e.g., are the inputs to detector simulations), it is mandatory to account for showeringand hadronization. The range of predictive techniques available for QCD final states reflects thisdiversity of needs of different measurements.
While illustrating the different methods, we shall for simplicity mainly use expressions that
hold for e+ e− scattering. The extension to cases with initial-state partons will be mostly straightforward (space constraints unfortunately prevent us from addressing diffraction and exclusive
hadron-production processes; extensive discussion is to be found in Sec. 20 of this Review andin Refs. [137, 138]).
9.2.3.1 Soft and collinear limits
Before examining specific predictive methods, it is useful to be aware of a general property of
QCD matrix elements in the soft and collinear limits. Consider a squared tree-level matrix element|Mn2 (p1 , . . . , pn )| for the process e+ e− → n partons with momenta p1 , . . . , pn , and a corresponding
phase-space integration measure dΦn . If particle n is a gluon, which becomes collinear (parallel)
to another particle i and additionally its momentum tends to zero (is “soft”), the matrix element
simplifies as follows,limθin →0, En →0dΦn |Mn2 (p1 , . . . , pn )|2= dΦn−1 |Mn−1(p1 , . . . , pn−1 )|31st May, 2024

2 dE
αs Ci dθin
n
,
2
π θin
En

(9.16)

11

9. Quantum Chromodynamics

where Ci = CF (CA ) if i is a quark (gluon). This formula has non-integrable divergences both forthe inter-parton angle θin → 0 and for the gluon energy En → 0, which are mirrored also in the
structure of divergences in loop diagrams. These divergences are important for at least two reasons:firstly, they govern the typical structure of events (inducing many emissions either with low energy
or at small angle with respect to hard partons); secondly, they will determine which observables
can be calculated within fixed-order perturbative QCD.
9.2.3.2 Fixed-order predictions
Let us consider an observable O that is a function On (p1 , . . . , pn ) of the four-momenta of the nfinal-state particles in an event (either partons or hadrons). In what follows, we shall consider the
cross section for events weighted with the value of the observable, σO . As examples, if On ≡ 1 for
all n, then σO is just the total cross section; if On ≡ τ̂ (p1 , . . . , pn ) where τ̂ is the value of the thrust
for that event (see Eq. (9.22) in Sec. 9.3.1.2), then the average value of the thrust is hτ i = σO /σtot ;if On ≡ δ(τ − τ̂ (p1 , . . . , pn )) then one gets the differential cross section as a function of the thrust,σO ≡ dσ/dτ .
In the expressions below, we shall omit to write the non-perturbative power correction term,
which for most common observables is proportional to a single power of Λ/Q.
Leading Order. If the observable O is non-zero only for events with at least n final-state particles,
then the LO QCD prediction for the weighted cross section in e+ e− annihilation isLOσO= αsn−2 (µ2R )Z

dΦn |Mn2 (p1 , . . . , pn )| On (p1 , . . . , pn ) ,(9.17)

where the squared tree-level matrix element, |Mn2 (p1 , . . . , pn )|, including relevant symmetry factors,
has been summed over all subprocesses (e.g. e+ e− → q q̄q q̄, e+ e− → q q̄gg) and has had all factorsof αs extracted in front. In processes other than e+ e− collisions, the center-of-mass energy of the
LO process is generally not fixed, and so the powers of the coupling are often brought inside theintegrals, with the scale µR chosen event by event, as a function of the event kinematics.
Other than in the simplest cases (see the review on Cross Sections in this Review), the matrix
elements in Eq. (9.17) are usually calculated automatically with programs such as CompHEP [139],
MadGraph [140], Alpgen [141], Comix/Sherpa [142], and Helac/Phegas [143]. Some of these(CompHEP, MadGraph) use formulas obtained from direct evaluations of Feynman diagrams.
Others (Alpgen, Helac/Phegas and Comix/Sherpa) use methods designed to be particularly
efficient at high multiplicities, such as Berends-Giele recursion [144], which builds up amplitudes
for complex processes from simpler ones (see also Refs. [145–147] for other tree-level calculational
methods).
The phase-space integration is usually carried out by Monte Carlo sampling, because of the highdimensionality of the integration and in order to deal with the possibly involved kinematic cuts thatare used in the corresponding experimental measurements. Perturbatively calculable observables
should be insensitive to the emission of soft and collinear radiation. Because of the divergences in
the matrix element, Eq. (9.16), at lowest order the integral converges only if the observable vanishes
for kinematic configurations in which one of the n particles is arbitrarily soft or it is collinear toanother particle. As an example, the cross section for producing any configuration of n partons
will lead to an infinite integral, whereas a finite result will be obtained for the cross section for
producing n deposits of energy (or jets, see Sec. 9.3.1.1), each above some energy threshold andwell separated from each other in angle.
At a practical level, LO calculations can be carried out for 2 → n processes with n . 6 − 10.
The exact upper limit depends on the process, the method used to evaluate the matrix elements(recursive methods are more efficient), and the extent to which the phase-space integration can be
optimized to work around the large variations in the values of the matrix elements.
31st May, 2024

9. Quantum Chromodynamics

12

NLO. Given an observable that is non-zero starting from n final-state particles, its prediction
at NLO involves supplementing the LO result, Eq. (9.17), with the 2 → (n + 1)-particle squared2 |), and the interference of a 2 → n tree-level and 2 → n 1-looptree-level matrix element (|Mn+1∗
amplitude (2Re(Mn Mn,1−loop )),
Z

NLO
LO
2
σO
= σO
+ αsn−1 (µ2R ) dΦn+1 |Mn+1(p1 , . . . , pn+1 )|On+1 (p1 , . . . , pn+1 )+αsn−1 (µ2R )Z(9.18)∗
dΦn 2Re [ Mn (p1 , . . . , pn ) Mn,1−loop(p1 , . . . , pn ) ] On (p1 , . . . , pn ) .

Relative to LO calculations, two important issues appear in the NLO calculations. Firstly, the extra
complexity of loop-calculations relative to tree-level calculations means that automated calculations
started to appear only about fifteen years ago (see below). Secondly, loop amplitudes are infinite
in four dimensions, while tree-level amplitudes are finite, but their integrals over the phase space
are infinite, due to the divergences of Eq. (9.16). These two sources of infinities have the same softand collinear origins and cancel after the integration only if the observable O satisfies the property
of infrared and collinear safety, which means that the observable is non-sensitive to soft emissions
or to collinear splittings, i.e.
On+1 (p1 , . . . , ps , . . . , pn )→ On (p1 , . . . , ps−1 , ps+1 , . . . , pn )On+1 (p1 , . . . , pa , pb , . . . , pn ) → On (p1 , . . . , pa + pb , . . . , pn )if ps → 0

if pa || pb .

(9.19)

Examples of infrared-safe quantities include event-shape distributions and jet cross sections (withappropriate jet algorithms, see below). Unsafe quantities include the distribution of the momentumof the hardest QCD particle (which is not conserved under collinear splitting), observables that
require the complete absence of radiation in some region of phase space (e.g. rapidity gaps or100% isolation cuts, which are affected by soft emissions), or the particle multiplicity (affected byboth soft and collinear emissions). The non-cancellation of divergences at NLO due to infrared or
collinear unsafety compromises the usefulness not only of the NLO calculation, but also that of a LO
calculation, since LO is only an acceptable approximation if one can prove that higher-order terms
are smaller. Infrared and collinear unsafety usually also implies large non-perturbative effects.
As with LO calculations, the phase-space integrals in Eq. (9.18) are usually carried out by Monte
Carlo integration, so as to facilitate the study of arbitrary observables. Various methods exist
to obtain numerically efficient cancellation among the different infinities. These include notably
dipole [148], FKS [149] and antenna [150] subtraction.
Thanks to new ideas like the OPP method [151], generalized [152] and D-dimensional [153]unitarity, on-shell methods [154], and on-the-fly reduction algorithms [155], recent years have seen abreakthrough in the calculation of one-loop matrix elements (for reviews on unitarity based method
see Ref. [156,157]). Thanks to these innovative methods, automated tools for NLO calculations havebeen developed and a number of programs are available publicly: MadGraph5_aMC@NLO [140]and Helac-NLO [158] provide full frameworks for NLO calculations; GoSam [159], Njet [160],
OpenLoops [161] and Recola [162] calculate just the 1-loop part and are typically interfaced withan external tool for a combination with the appropriate tree-level amplitudes. Other tools suchas NLOJet++ [163], MCFM [164], VBFNLO [165], the Phox family [166] or BlackHat [167]
implement analytic calculations and provide full frameworks to compute NLO cross sections forselected classes of processes. Recently, a lot of attention has also been paid to the calculationof NLO electroweak corrections. Electroweak corrections are especially important for transverse
momenta significantly above the W and Z masses, because they are enhanced by two powers ofln pt /mW for each power of the electroweak coupling, and close to Sudakov peaks4 , where most of4

For the definition of Sudakov form factors or Sudakov peaks see e.g. Refs. [1, 2, 11].

31st May, 2024

13

9. Quantum Chromodynamics

the data lie and the best experimental precision can be achieved. In some cases the above programs
can be used to calculate also NLO electroweak or beyond-standard-model corrections [168–174].
Given the progress in QCD and EW fixed-order computations, the largest unknown from fixedorder corrections is often given by the mixed QCD-electroweak corrections of O(αs α). These
mixed two-loop corrections are often available in an approximate form [175–188]. The first complete
computation of the mixed QCD–EW corrections to the neutral-current Drell–Yan process appeared
recently [189–191]. For a review on EW corrections to collider processes see Ref. [192].
NNLO. NNLO is considerably more complicated than NLO as it involves a further order in αs ,consisting of: the squared (n + 2)-parton tree-level amplitude, the interference of the (n + 1)-parton
tree-level and 1-loop amplitudes, the interference of the n-parton tree-level and 2-loop amplitudes,
and the squared n-parton 1-loop amplitude.
Each of these elements involves large numbers of soft and collinear divergences, satisfying relations analogous to Eq. (9.16) which now involve multiple collinear or soft particles and higher loop
orders (see e.g. Refs. [193–195]). Arranging for the cancellation of the divergences after numericalMonte Carlo integration has been one of the significant challenges of NNLO calculations, as has
been the determination of the relevant 2-loop amplitudes. For the cancellations of divergences awide range of methods has been developed. Some of them [196–205] retain the approach, inherent in
NLO methods, of directly combining the separate loop and tree-level amplitudes. Others combine asuitably chosen, partially inclusive 2 → n NNLO calculation with a fully differential 2 → n+1 NLO
calculation [206–212]. The qT -subtraction method was extended to deal with mixed QCD–QED
corrections at NNLO [213]. For an overview of NNLO subtraction methods see Ref [214].
Quite a number of processes have been calculated differentially at NNLO so far. The state of
the art for e+ e− collisions is e+ e− → 3 jets [215–218].
For DIS, dijet production is known at NNLO [219, 220] and the description jet production has
been recently pushed even to N3 LO using the Projection-to-Born method [221, 222]. For hadroncolliders, all 2 → 1 processes are known, specifically vector boson [223, 224] and Higgs boson
production in the large mt limit [206, 225]. The finite top-mass corrections at this order have also
been computed [226]. This calculation eliminates one important source of theoretical uncertainty
to inclusive Higgs production at the LHC. Substantial progress has been made in the years forhadron-collider 2 → 2 processes, with calculations having been performed for nearly all relevant
processes: ZZ [132] W W [131] and W Z [227], γγ [228, 229], Zγ [230] and W γ [231] (many ofthese color singlet processes are available also in MCFM [232, 233] or MATRIX [133]), inclusive
photon [234, 235], γ+ jet [235, 236], W + jet [208, 237], Z+ jet [236, 238, 239] H+ jet [240–243], W H[244] and ZH [245], s-channel [246] and t-channel single-top [247–250], tt̄ production [251, 252],
dijet production [253, 254], W production in association with a c-jet [255, 256] and HH [126] (inlarge-top-mass approximation, see also the exact (two-loop) NLO result [127]). Recently, also the
NNLO corrections to identified B-hadron hadro-production have been computed [257]. The frontierof NNLO calculation has now reached the complexity of 2 → 3 processes. The first 2 → 3 process
known at NNLO has been Higgs production through vector-boson fusion, using an approximationin which the two underlying DIS-like q → qV scatterings are factorized, the so-called structure
function approximation [211, 258]. Corrections beyond the structure function approximation are
expected to be small, on the order of a percent or less [259]. More recently, first genuine 2 → 3 LHC
processes have been described at NNLO accuracy, including three photon [260, 261], two photons
and one jet [262], two jets and one photon [263], three-jets [264] and W bb production [265]. The
calculation of three-jet production might be relevant for future extractions of αs (m2Z ) from three-jet
observables at the LHC [266].
Cross sections at the LHC are most often measured with fiducial cuts, for example on thetransverse momenta and rapidities of the measured objects, restricting the measurement to regions
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14

9. Quantum Chromodynamics

where the objects have a good efficiency to be detected and are well-reconstructed. Ideally, thetheoretical predictions should also be constructed at the same fiducial level; the other possibility
is to extrapolate the experimental fiducial results to the full phase space. Such an extrapolation,
however, requires the extrapolation (typically using a parton shower Monte Carlo) to be accurate
over the full phase space.
Comparisons of fixed order predictions to fiducial measurements can sometimes result in thepresence of large logarithms which degrade the accuracy of the prediction. Such is the case, forexample, in the calculation of the Higgs rapidity distribution, in the diphoton final state. The
imposition of transverse momenta and rapidity cuts on the two photons leads to an uncertainty
notably greater at N3 LO then at NNLO, due to the presence of these logs. One possible solution is
to change the form of the kinematic cut on the photons, to a product of the two photon transverse
momenta [267], reducing the impact of the logs; another solution is to perform a resummation of
the logs, restoring the expected precision [268].
The Les Houches precision wishlist compiles predictions needed to fully exploit the data thatwill be taken at the High Luminosity LHC [269]. Most of the needed calculations require accuracyof at least NNLO QCD and NLO EW, and many require the prediction of 2 → 3 processes, such as
W/Z+ ≥ 2 jets, H+ ≥ 2 jets, and ttH to NNLO. The latter process has been computed at NNLO
in Ref. [270] in the approximation where the Higgs boson in the double-virtual contribution is soft.
This calculation considerably reduces the theory uncertainty on this process.
As discussed in this section, calculations at NLO can now be relatively easily generated by nonexperts using the programs mentioned above. By now there are also a number of publicly available
tools to compute a range of processes at NNLO accuracy in QCD, such as Eerad3 [216] (fore+ e− collisions), [206], Dynnlo [224], Dyturbo [271], Fehip [225], Fewz [272], Hturbo [273],
Matrix [133], MCFM [233] or Sushi [124, 274]. However, in some cases NNLO calculations canbe too complex and CPU-intensive to allow such an approach. In these cases, the relevant matrix
element information for a specific observable can be stored by means of interpolation grids developed originally for cross sections at NLO [275–277] and recently extended to include electroweak
corrections [278]. The application to NNLO has been demonstrated for tt̄ and for DIS jet crosssections in Refs. [279, 280]. Each such interpolation grid corresponds to one fixed observable withspecific selection criteria and binning, but is flexible with respect to the renormalization or factorization scale, the PDF, or the αs evolution chosen for the cross-section computation. An even more
flexible method, at the cost of requiring large amounts of storage space, saves huge numbers ofpartonic events in the form of ROOT n-tuples, which allow predictions to be generated on-the-fly
for many observables of a particular process [281,282]. Recent developments of both techniques forpp collisions at NNLO are described in Ref. [269].
9.2.3.3 Resummation
Many experimental measurements place tight constraints on emissions in the final state. For
example, in e+ e− events, that (one minus) the thrust should be less than some value τ  1,or, in pp → Z, events that the Z-boson transverse momentum or the transverse momentum of
the accompanying jet should be much smaller than the Z-boson mass. A further example isthe production of heavy particles or jets near threshold (so that little energy is left over for realemissions) in DIS and pp collisions.
In such cases, the constraint vetoes a significant part of the integral over the soft and collinear
divergence of Eq. (9.16). As a result, there is only a partial cancellation between real emission
terms (subject to the constraint) and loop (virtual) contributions (not subject to the constraint),
causing each order of αs to be accompanied by a large coefficient ∼ L2 , where e.g. L = ln τ or
2 n
L = ln(mZ /pZt ). One ends up with a perturbative series, whose terms go as ∼ (αs L ) . It is
31st May, 2024

9. Quantum Chromodynamics

15

not uncommon that αs L2  1, so that the perturbative series converges very poorly if at all.5 In
such cases one may carry out a “resummation”, which accounts for the dominant logarithmicallyenhanced terms to all orders in αs , by making use of known properties of matrix elements for
multiple soft and collinear emissions, and of the all-orders properties of the divergent parts ofvirtual corrections, following original works such as Refs. [283–292] and also through soft-collinear
effective theory [293,294] (cf. also the section on “Heavy-Quark and Soft-Collinear Effective Theory”
in this Review, as well as Ref. [295]).
For cases with double logarithmic enhancements (two powers of logarithm per power of αs ),there are two classification schemes for resummation accuracy. Writing the cross section including
the constraint as σ(L) and the unconstrained (total) cross section as σtot , the series expansion takes
the form∞ X
2nX

σ(L) ' σtot

Rnk αsn (µ2R )Lk ,L  1,(9.20)

n=0 k=0

and leading log (LL) resummation means that one accounts for all terms with k = 2n, next-toleading-log (NLL) includes additionally all terms with k = 2n − 1, etc. Often σ(L) (or its Fourier
or Mellin transform) exponentiates,6σ(L) ' σtot exp" ∞ n+1XX#

Gnk αsn (µ2R )Lk ,L  1,(9.21)

n=1 k=0

where one notes the different upper limit on k (≤ n + 1) compared to Eq. (9.20). This is a more
powerful form of resummation: the G12 term alone reproduces the full LL series in Eq. (9.20). With
the form Eq. (9.21) one still uses the nomenclature LL, but this now means that all terms withk = n + 1 are included, and NLL implies all terms with k = n, etc.
For a large number of observables, NLL resummations are available in the sense of Eq. (9.21)(see Refs. [299–301] and references therein). NNLL has been achieved for the DY and Higgs-boson
pt distributions [302–305] (also available in the CuTe [306], HRes [307] and ResBos [308] families ofprograms and also differentially in vector-boson decay products [309]) and related variables [310],
for the pt of vector-boson pairs [311, 312], for the back-to-back energy-energy correlation in e+ e−[313], the jet broadening in e+ e− collisions [314], the jet-veto survival probability in Higgs and Z
boson production in pp collisions [315–317],7 an event-shape type observable known as the beam
thrust [318], hadron-collider jet masses in specific limits [319] (see also Ref. [320]), the productionof top anti-top pairs near threshold [321–323] (and references therein), and high-pt W and Z
production [324]. Automation of NNLL jet-veto resummations for different processes has beenachieved in Ref. [325] (cf. also the NLL automation in Ref. [326]), while automation for a certainclass of e+ e− observables has been achieved in Ref. [327]. N3 LL resummations are available for thethrust variable, C-parameter and heavy-jet mass in e+ e− annihilations [328–330], for pt distributionof the Higgs boson [331, 332] and weak gauge bosons [333–335] and for Higgs- and vector-boson
production near threshold [336]. In order to make better contact with experimental measurements,
recent years have seen an increasing interest in resummations in exclusive phase-space regions
and joint resummations [337–346]. Finally, there has also been considerable progress in resummed
5

To be precise one should be aware of two causes of the divergence of perturbative series. That which interests us
here is associated with the presence of a new large parameter (e.g. ratio of scales). It is distinct from the “renormalon”
induced factorial divergences of perturbation theory discussed above.
6
Whether or not this happens depends on the quantity being resummed. A classic example involves two-jet rate
in e+ e− collisions as a function of a jet-resolution parameter ycut . The logarithms of 1/ycut exponentiate for the kt(Durham) jet algorithm [296], but not [297] for the JADE algorithm [298] (both are discussed below in Sec. 9.3.1.1).
7
A veto on the jet phase space can be severe, for example by requiring exactly zero jets above a given transverse
momentum cut accompanying a Higgs boson, or relatively mild, for example by placing a transverse momentum cutof 30 GeV on the measurement of the production of a Higgs boson with one or more jets. In general, inclusive crosssections are preferable, as uncertainties on both the theoretical and experimental sides are smaller.
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16

calculations for jet substructure, whose observables involve more complicated definitions than is thecase for standard resummations, see e.g. Refs. [347–351] and references therein (see also Sec. 9.3.1.3).
9.2.3.4 Fragmentation functions
Since the parton-hadron transition is non-perturbative, it is not possible to perturbatively calculate quantities such as the energy-spectra of specific hadrons in high-energy collisions. However,
one can factorize perturbative and non-perturbative contributions via the concept of fragmentation
functions. These are the final-state analogue of the parton distribution functions which are used
for initial-state hadrons. Like parton distribution functions, they depend on a (fragmentation)
factorization scale and satisfy a DGLAP evolution equation.
It should be added that if one ignores the non-perturbative difficulties and just calculates theenergy and angular spectrum of partons in perturbative QCD with some low cutoff scale ∼ Λ√(using resummation to sum large logarithms of s/Λ), then this reproduces many features of thecorresponding hadron spectra [352]. This is often taken to suggest that hadronization is “local”, inthis sense it mainly involves partons that are close both in position and in momentum.
Section 19 of this Review provides further information and references on these topics, including
also the question of heavy-quark fragmentation.
9.2.3.5 Parton-shower Monte Carlo event generators
Parton-shower Monte Carlo (MC) event generators like Pythia [353–355], Herwig [356–358],
and Sherpa [359]8 provide fully exclusive simulations of QCD events at the level of measurable
particles, the so-called “particle level” or “hadron level”. Here, “measurable” refers to color-neutral
particles with mean lifetimes τ long enough to be associated with tracks or decay vertices in particle
detectors. Usually, this requires mean decay lengths cτ of around 10 mm. As such MC event
generators are a crucial tool for all applications that involve simulating the response of detectors
to QCD events. Here we give only a brief outline of how they work and refer the reader to Sec. 43
and Ref. [361] for a full overview.
In general, we expect parton-shower matched predictions to differ from the underlying fixedorder results in regions where (1) there is a large sensitivity to jet shapes (for instance small R jets),(2) there is a restriction in phase space such that soft gluon resummation effects become important,(3) the observable contains multiple disparate scales, (4) there are perturbative instabilities at fixedorder, e.g. related to kinematical cuts, and (5) the observable is sensitive to higher multiplicity states
than those described by the fixed-order calculation (for an explicit study of some of these effects
see e.g. [362]).
The MC generation of an event involves several stages. It starts with the random generationof the kinematics and partonic channels of whatever hard scattering process the user has requested
at some high scale Q0 (for complex processes, this may be carried out by an external program).
This is followed by a parton shower, usually based on the successive random generation of gluon
emissions (and g → q q̄ splittings). Leading contributions to the shower have emissions that areordered according to some ordering variable. Common choices of scale for the ordering of emissions
are virtuality, transverse momentum or angle. Each emission is generated at a scale lower thanthe previous emission, following a (soft and collinear resummed) perturbative QCD distribution,
which depends on the momenta of all previous emissions. Parton showering stops at a scale of order
1 GeV, at which point a hadronization model is used to convert the resulting partons into hadrons.
One widely-used model involves stretching a color “string” across quarks and gluons, and breaking
it up into hadrons [363,364]. Another breaks each gluon into a q q̄ pair and then groups quarks andanti-quarks into colorless “clusters”, which then give the hadrons [356]. As both models are tuned
primarily to LEP data, the cluster and string models provide similar results for most observables
8

The program ARIADNE [360] has also been widely used for simulating e+ e− and DIS collisions.
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sensitive to hadronization [362]. For pp and γp processes, modeling is also needed to treat thecollision between the two hadron remnants, which generates an underlying event (UE), usually
implemented via additional 2 → 2 scatterings (“multiple parton interactions”, MPI) at a scale of afew GeV, following Ref. [365]. The parameter values for the MPI models must be determined fromfits to minimum-bias and/or underlying-event observables of LHC collision data. As the different
MC event generators are adapted to essentially the same measurements, ideally the respective MPI
implementations should lead to similar predictions for each program. One complication, however,is some non-universality of the underlying event among different physics processes.
A deficiency of the soft and collinear approximations that underlie parton showers is that they
may fail to reproduce the full pattern of hard wide-angle emissions, important, for example, inmany new physics searches. It is therefore common to use LO multi-parton matrix elements togenerate hard high-multiplicity partonic configurations as additional starting points for the showering, supplemented with some prescription (CKKW [366], MLM [367]) for consistently merging
samples with different initial multiplicities. Monte Carlo generators, as described above, computecross sections for the requested hard process that are correct at LO.
A wide variety of processes are available in MC implementations that are correct to NLO, using
the MC@NLO [368] or Powheg [369] prescriptions, through the MadGraph5_aMC@NLO [140],
Powheg-Box [370] and Sherpa [142] programs. Techniques have also been developed to combine NLO plus parton shower accuracy for different multiplicities of final-state jets [371]. While
NLO+PS accurate predictions can now be implemented in the above tools rather easily and areavailable for a range of processes, given the advances in NNLO calculations it is natural to seek for
NNLO+PS accurate Monte Carlos as well. The two main approaches today are the MiNNLO [372]and Geneva [373] ones. NNLO plus shower accuracy was achieved first for Drell-Yan and Higgs
production [374, 375] and is now available for several 2 → 2 color singlet processes [376–390], as
well as for heavy-quark pair-production [391–393].
It is important to understand/verify the accuracy of the parton shower predictions in the Monte
Carlo programs. A general framework for assessing the logarithmic accuracy of parton-showeralgorithms has been formulated, based on their ability to reproduce the singularity structure ofmulti-parton matrix elements, and their ability to reproduce logarithmic resummation results [394].
The dominant contributions relevant for the extension of parton showers to higher logarithmic
accuracy have been computed [395–399] and included in some algorithms.
There exist ways to improve on current parton-shower algorithms [400, 401] and to demonstrate the parton shower accuracy through a comparison to analytic next-to-leading logarithmic
calculations for a range of observables [401]. Considerable progress has been achieved in recent
years [400–412]. References [401, 404, 405, 408, 409] introduced the first shower implementations
towards NLL, but these implementations did not include spin-correlation effects and lacked fullcolor dependence at LL level. Ref. [402] incorporated full color effects at the NLL level, while
spin correlations at NLL were included in Refs. [406, 407], resulting in an NLL accurate parton
shower for proton-proton collisions [410, 411]. The modifications necessary for processes such as
Deep Inelastic Scattering (DIS) and Vector Boson Fusion (VBF) are addressed in [412]. The next
steps involve incorporating non-perturbative effects, accounting for quark masses, and tuning NLL
parton shower on experimental data.
9.2.4 Accuracy of predictions
Estimating the accuracy of perturbative QCD predictions is not an exact science. It is often
said that LO calculations are accurate to within a factor of two. This is based on experience
with NLO corrections in the cases where these are available. In processes involving new partonic
scattering channels at NLO and/or large ratios of scales (such as jet observables in processes with
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vector bosons, or the production of high-pt jets containing B-hadrons), the ratio of the NLO to LO
predictions, commonly called the “K factor”, can be substantially larger than two. NLO corrections
tend to be large for processes for which there is a great deal of color annihilation in the interaction.
In addition, NLO corrections tend to decrease as more final state legs are added.
For calculations beyond LO, one approach to estimate the perturbative uncertainty is to base iton the last known perturbative correction; this may lead to misleading results if new sub-processes
are present at the next-higher order. A more widely used method is to estimate it from the change
in the prediction when varying the renormalization and factorization scales around a central value
Q that is taken close to the physical scale of the process. A conventional range of variation is Q/2 <µR , µF < 2Q, varying the two scales independently with the restriction 12 µR < µF < 2µR [413].
This constraint limits the risk of misleadingly small uncertainties due to fortuitous cancellations
between the µF and µR dependence when both are varied together, while avoiding the appearanceof large logarithms of µ2R /µ2F when both are varied completely independently. Where possible,
it can be instructive to examine the two-dimensional scale distributions (µR vs. µF ) to obtain abetter understanding of the interplay between µR and µF . This procedure should not be assumed
to always estimate the full uncertainty from missing higher orders, but it does indicate the size of
one important known source of higher-order ambiguity.9
Most 2 → 2 processes at the LHC are now known to NNLO. This typically results in a large
reduction in the uncertainty from that obtained at NLO. However, care must be taken in theestimate of the uncertainty in processes containing jets, as accidental cancellations of the scale
uncertainties may result an artificial reduction of the scale uncertainty; for some jet R values, theuncertainty may even be zero. There are several possibilities for providing a more realistic estimateof the scale uncertainty for such processes [362], but none have been widely adopted at the LHC.
Calculations that involve resummations usually have an additional source of uncertainty assopZ
ciated with the choice of argument of the logarithms being resummed, e.g. ln(2 mtZ ) as opposedpZ

to ln( 12 mtZ ), as well as a prescription to switch off resummation effects when the logarithm is notlarge. In addition to varying renormalization and factorization scales, it is common practice to vary
the argument of the logarithm by a suitable factor in either direction with respect to the default
argument.
The accuracy of QCD predictions is limited also by non-perturbative (or hadronization) corrections, which typically scale as a power of Λ/Q.10 For measurements that are directly sensitive
to the structure of the hadronic final state, the corrections are usually linear in Λ/Q. The nonperturbative corrections are further enhanced in processes with a significant underlying event (i.e.
in pp and pp̄ collisions) and the impact of non-perturbative corrections is larger in cases where theperturbative cross sections fall steeply as a function of pt or some other kinematic variable, forexample in inclusive jet pt or dijet mass spectra. Under high-luminosity running conditions, suchas 13 TeV at the LHC, there can be on the order of 50 minimum-bias interactions occurring at
each beam-beam crossing. This additional energy needs to be subtracted, and is typically removed
by means of a rapidity-dependent transverse energy density determined on an event-by-event basis [419]. This subtraction, of necessity, also removes the underlying event, which must be added
back in by means of MC event generator modelling if one wants to restore the measured event to
the particle level.
Non-perturbative corrections are commonly estimated from the difference between Monte Carloevents at the “parton level” and at particle level. Parton level refers to the stage of the parton
shower, where the evolution is stopped at an energy scale of typical hadron masses of a few GeV.
9
Various studies have been carried out on how to estimate uncertainties from missing higher orders that go beyond
scale variations [414–418].
10
In some circumstances, the scale in the denominator could be a smaller kinematic or physical scale that depends
on the observable.

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An issue to be aware of is that “parton level” is not a uniquely defined concept. For example, in a
MC event generator such a procedure depends on an arbitrary and tunable internal cutoff scale that
separates the parton showering from the hadronization. In contrast, no such cutoff scale exists in
an NLO or NNLO partonic calculation. The uncertainties in these corrections are often estimatedby comparing different tunes of the various MC event generators. It should be noted that such
estimates are not guaranteed to fully cover the true uncertainties.
Alternative methods exist for estimating hadronization corrections, that attempt to analyticallydeduce non-perturbative effects in one observable based on measurements of other observables (seethe reviews [26, 420]). While they directly address the problem of different possible definitions ofparton level, it should also be said that they are far less flexible than Monte Carlo programs andnot always able to provide equally good descriptions of the data.
One of the main issues is whether the fixed-order partonic final state of a NLO or NNLO prediction can match the parton shower in its ability to describe the experimental jet shape (minus any
underlying event). Calculations at NNLO provide a better match to the parton shower predictions
than do NLO ones, as might be expected from the additional gluon available to describe the jet
shape. The hadronization predictions appear to work for both orders, but at an unknown accuracy.

9.3 Experimental studies of QCD
Since we are not able to directly measure partons (quarks or gluons), but only hadrons and theirdecay products, a central issue for every experimental study of perturbative QCD is establishinga correspondence between observables obtained at the parton and the hadron level. The only
theoretically sound correspondence is achieved by means of infrared and collinear safe quantities,
which allow one to obtain finite predictions at any order of perturbative QCD.
As stated above, the simplest case of infrared- and collinear-safe observables are inclusive crosssections. More generally, when measuring fully inclusive observables, the final state is not analyzed
at all regarding its (topological, kinematical) structure or its composition. Basically the relevant
information consists in the rate of a process ending up in a partonic or hadronic final state. In
e+ e− annihilation, widely used examples are the ratios of partial widths or branching ratios for theelectroweak decay of particles into hadrons or leptons, such as Z or τ decays, (cf. Sec. 9.2.1). Suchratios are often favored over absolute cross sections or partial widths because of large cancellations ofexperimental and theoretical systematic uncertainties. The strong suppression of non-perturbative
effects, O(Λ4 /Q4 ), is one of the attractive features of such observables, however, at the same time,
the sensitivity to radiative QCD corrections is small, which for example leads to a larger statisticaluncertainty when using them for the determination of the strong coupling constant. In the caseof τ decays not only the hadronic branching ratio is of interest, but also moments of the spectral
functions of hadronic τ decays, which sample different parts of the decay spectrum and thus provide
additional information. Other examples of fully inclusive observables are structure functions (andrelated sum rules) in DIS. These are extensively discussed in Sec. 18 of this Review.
On the other hand, often the structure or composition of the final state are analyzed, andcross sections differential in one or more variables characterizing this structure are of interest.
Examples are jet rates, jet substructure, event shapes or transverse momentum distributions of jets
or vector bosons in hadron collisions. The case of fragmentation functions, i.e. the measurement of
hadron production as a function of the hadron momentum relative to some hard scattering scale,is discussed in Sec. 19 of this Review.
It is worth mentioning that, besides the correspondence between the parton and hadron level,also a correspondence between the hadron level and the actually measured quantities in the detector
has to be established. The simplest examples are corrections for finite experimental acceptance andefficiencies. Whereas acceptance corrections essentially are of theoretical nature, since they involve
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extrapolations from the measurable (partial) to the full phase space, other corrections such as
for efficiency, resolution and response are of experimental nature. For example, measurements ofdifferential cross sections such as jet rates require corrections in order to relate, e.g., the energy
deposits in a calorimeter to the jets at the hadron level. Typically detector simulations and/or
data-driven methods are used in order to obtain these corrections. Care should be taken here inorder to have a clear separation between the parton-to-hadron level and hadron-to-detector level
corrections. Finally, for the sake of an easy comparison to the results of other experiments and/or
theoretical calculations, it is suggested to provide, whenever possible, measurements corrected for
detector effects and/or all necessary information related to the detector response (e.g., the detector
response matrix). Any fiducial phase space for measurements should be defined as close as possible
to the detector-level selection in order to minimize model-dependent extrapolations. A versatile
repository for storing such information is the use of Rivet routines [421].
9.3.1 Hadronic final-state observables
9.3.1.1 Jets
In hard interactions, final-state partons and hadrons appear predominantly in collimated bunches, which are generically called jets. To a first approximation, a jet can be thought of as a hardparton that has undergone soft and collinear showering and then hadronization. Jets are used bothfor testing our understanding and predictions of high-energy QCD processes, and also for identifying the hard partonic structure of decays of massive particles such as top quarks and W, Z and
Higgs bosons.
In order to map observed hadrons onto a set of jets, one uses a jet definition. The mapping
involves explicit choices: for example when a gluon is radiated from a quark, for what range ofkinematics should the gluon be part of the quark jet, or instead form a separate jet? Good jet
definitions are infrared and collinear safe, simple to use in theoretical and experimental contexts,
applicable to any type of inputs (parton or hadron momenta, charged particle tracks, and/or energy
deposits in the detectors) and lead to jets that are not too sensitive to non-perturbative effects.
An extensive treatment of the topic of jet definitions is given in Ref. [422] (for e+ e− collisions)and Refs. [423–425]. Here we briefly review the two main classes: cone algorithms, extensively usedat hadron colliders before the LHC, and sequential recombination algorithms, more widespread ine+ e− and ep colliders and at the LHC.
Very generically, most (iterative) cone algorithms start with some seed particle i, sum themomenta of all particles j within a cone of opening-angle R, typically defined in terms of differencesin rapidity and azimuthal angle. They then take the direction of this sum as a new seed and repeat
until the direction of the cone is stable, and call the contents of the resulting stable cone a jet if
its transverse momentum is above some threshold pt,min . The parameters R and pt,min should bechosen according to the needs of a given analysis.
There are many variants of the cone algorithm, and they differ in the set of seeds they use andthe manner in which they ensure a one-to-one mapping of particles to jets, given that two stable
cones may share particles (“overlap”). The use of seed particles is a problem w.r.t. infrared andcollinear safety. Seeded algorithms are generally not compatible with higher-order (or sometimes
even leading-order) QCD calculations, especially in multi-jet contexts, as well as potentially subject
to large non-perturbative corrections and instabilities. Seeded algorithms (JetCLU, MidPoint, andvarious other experiment-specific iterative cone algorithms) are therefore to be deprecated. Suchalgorithms are not used at the LHC, but were at the Fermilab Tevatron, where data still provide
useful information, for example for global PDF fits.11 A modern alternative is to use a seedlessvariant, SISCone [427].
11

In the data, the difference between the use of the Midpoint algorithm and the use of the SISCone algorithm issmall [426], allowing the use of the SISCone algorithm for any theory comparisons at higher order in QCD.
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Sequential recombination algorithms at hadron colliders (and in DIS) are characterized by a
2p 2p
distance dij = min(kt,i, kt,j )∆2ij /R2 between all pairs of particles i, j, where ∆ij is their separationin the rapidity-azimuthal plane, kt,i is the transverse momentum w.r.t. the incoming beams, and R
is a free parameter. At the LHC, R is typically in the range from 0.4 to 0.8, although analyses can
2palso use jet sizes up to 1.0-1.2. They also involve a “beam” distance diB = kt,i. One identifies thesmallest of all the dij and diB , and if it is a dij , then i and j are merged into a new pseudo-particle(with some prescription, a recombination scheme, for the definition of the merged four-momentum).
If the smallest distance is a diB , then i is removed from the list of particles and called a jet. As
with cone algorithms, one usually considers only jets above some transverse-momentum threshold
pt,min . The parameter p determines the kind of algorithm: p = 1 corresponds to the (inclusive-)kt
algorithm [296, 428, 429], p = 0 defines the Cambridge-Aachen algorithm [430, 431], while for theanti-kt algorithm p = −1 [432]. All these variants are infrared and collinear safe. Whereas theformer two lead to irregularly shaped jet boundaries, the latter results in cone-like boundaries,
except in situations where there are nearby jets. The anti-kt algorithm has become the de-facto
standard for the LHC experiments.
In e+ e− annihilation the kt algorithm [296] uses yij = 2 min(Ei2 , Ej2 )(1 − cos θij )/Q2 as distancemeasure between two particles/partons i and j and repeatedly merges the pair with smallest yij ,until all yij distances are above some threshold ycut , the jet resolution parameter. Q is a measureof the overall hardness of the event. The (pseudo)-particles that remain at this point are called
the jets. Here it is ycut (rather than R and pt,min ) that should be chosen according to the needsof the analysis. The two-jet rate in the kt algorithm has the property that logarithms ln(1/ycut )exponentiate. This is one reason why it is preferred over the earlier JADE algorithm [298], whichuses the distance measure yij = 2 Ei Ej (1 − cos θij )/Q2 . Note that other variants of sequential recombination algorithms for e+ e− annihilations, using different definitions of the resolution measure
yij , exhibit much larger sensitivities to fragmentation and hadronization effects than the kt and
JADE algorithms [433]. Efficient implementations of the above algorithms are available throughthe FastJet package [434].
While building infrared (IR) safe jets is generally considered a solved problem, this is not quite
the case when one also wishes to assign a flavor to a jet. In fact, the general experimental definition
considers a flavored-jet to be a jet that contains at least one flavor tag (such as a B or D meson)
above a given transverse momentum threshold. Because of collinear or soft wide-angle g → q q̄
splittings, it is easy to see that such a definition is neither collinear nor infrared safe. This problem
was addressed in Ref. [435,436] in the context of heavy-flavor production at the Tevatron. However,
the jet-algorithm proposed in that work was impractical to implement experimentally because it
was based on the kt -algorithm. Furthermore, it required tagging two nearby flavored hadrons andinvolved a rather complex beam-distance measure. Since most experimental studies instead rely on
the anti-kt algorithm, recent investigations have focused on developing algorithms that preserve theanti-kt kinematics of the jets while assigning jet flavors in an infrared safe way [437–439]. It turns
out that the problem is more involved than anticipated, and a formulation of infrared-safe anti-kt like jets could only be achieved by introducing an interleaved flavor neutralization procedure [440].
An unfolding procedure will be necessary to convert experimental measurements of flavor-kt jets
to a form that can be directly compared to theoretical predictions.
9.3.1.2 Event Shapes
Event-shape variables are functions of the four momenta of the particles in the final state andcharacterize the topology of an event’s energy flow. They are sensitive to QCD radiation (andcorrespondingly to the strong coupling) insofar as gluon emission changes the shape of the energy
flow.
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The classic example of an event shape is the thrust [441, 442] in e+ e− annihilations, defined asP
pi · ~nτ |i |~τ̂ = max P
,
~
nτ

pi |
i |~

(9.22)

where p~i are the momenta of the particles or the jets in the final-state and the maximum is obtained
for the thrust axis ~nτ . In the Born limit of the production of a perfect back-to-back q q̄ pair, thelimit τ̂ → 1 is obtained, whereas a perfectly spherical many-particle configuration leads to τ̂ → 1/2.
Further event shapes of similar nature have been extensively measured at LEP and at HERA, andfor their definitions and reviews we refer to Refs. [1, 8, 420, 443, 444]. The energy-energy correlation
function [445], namely the energy-weighted angular distribution of produced hadron pairs, andits associated asymmetry are further shape variables which have been studied in detail at e+ e−
colliders. For hadron colliders the appropriate modification consists in only taking the transverse
momentum component [446]. The event shape variable N-jettiness has been proposed [447], thatmeasures the degree to which the hadrons in the final state are aligned along N jet axes or the
beam direction. It vanishes in the limit of exactly N infinitely narrow jets.
Phenomenological discussions of event shapes at hadron colliders can be found in Refs. [447–451].
Measurements of hadronic event-shape distributions have been published by CDF [452], ATLAS[453–459] and CMS [460–463].
Event shapes are used for many purposes. These include measuring the strong coupling (seee.g. Ref. [459]), tuning the parameters of Monte Carlo programs, investigating analytical modelsof hadronization and distinguishing QCD events from events that might involve decays of newparticles (giving event-shape values closer to the spherical limit).
9.3.1.3 Jet substructure, quark vs. gluon jets
Jet substructure, which can be resolved by finding subjets or by measuring jet shapes, is sensitive
to the details of QCD radiation in the shower development inside a jet and has been extensively
used to study differences in the properties of quark and gluon induced jets, strongly related totheir different color charges. There is clear experimental evidence that gluon jets have a softer
particle spectrum and are “broader” than (light-) quark jets (as expected from perturbative QCD)
when looking at observables such as the jet shape Ψ (r/R) (see e.g. Ref. [464]). This is the fractionof transverse momentum contained within a sub-cone of cone-size r for jets of cone-size R. It issensitive to the relative fractions of quark and gluon jets in an inclusive jet sample and receives
contributions from soft-gluon initial-state radiation and the underlying event. Therefore, it has been
widely employed for validation and tuning of Monte Carlo parton-shower models. Furthermore,this quantity turns out to be sensitive to the modification of the gluon radiation pattern in heavy
ion collisions (see e.g. Ref. [465]).
Jet shape measurements using proton-proton collision data have been presented for inclusive
jet samples [466–468] and for top-quark production [469]. Further discussions, references andsummaries can be found in Refs. [444, 470–472] and Sec. 4 of Ref. [473].
The use of jet substructure has also been investigated in order to distinguish QCD jets from jetsthat originate from hadronic decays of boosted massive particles (high-pt electroweak bosons, top
quarks and hypothesized new particles). A considerable number of experimental studies have been
carried out with Tevatron and LHC data, in order to investigate the performance of the proposed
algorithms for resolving jet substructure and to apply them to searches for new physics, as well as
to the reconstruction of boosted top quarks, vector bosons and the Higgs boson. For reviews of this
rapidly growing field, see Sec. 5.3 of Ref. [423], Refs. [350,351,473–479] and references thereto. One
convenient representation for visualizing jet substructure is through the Lund plane [480]. From atheoretical perspective, the Lund plane would be constructed using quarks and gluons, but similar
observables can be constructed through the use of jets, as described in detail in Ref. [481].
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Neural network techniques and deep learning methods have also been applied to jet and top
physics and jet substructure, see e.g. Refs. [482–485]. Perhaps no other sub-field has benefited asmuch from machine learning techniques as the study of jet substructure. As a jet can have O(100)constituents each with kinematic and other information, jet substructure analysis is naturally ahighly multivariate problem. Deep learning techniques can use all of the available information to
study jets in their natural high dimensionality. Such techniques have not only improved discrimination between different final states/types of jets, but have also improved our understanding of
perturbative QCD. See for example the review in Ref. [349].
9.3.2 QCD measurements at colliders
There exists a wealth of data on QCD-related measurements in e+ e− , ep, pp, and pp̄ collisions,to which a short overview like this would not be able to do any justice. Reviews of the subject havebeen published in Refs. [443, 444] for e+ e− colliders and in Ref. [486] for ep scattering, whereas forhadron colliders overviews are given in, e.g., Refs. [424, 471] and Refs. [2, 11, 487–489].
Below we concentrate our discussion on measurements that are most sensitive to hard QCD
processes with focus on jet production.
9.3.2.1 e+ e− colliders
Analyses of jet production in e+ e− collisions are mostly based on data from the JADE experiment at center-of-mass energies between 14 and 44 GeV, as well as on LEP collider data at the Z
resonance and up to 209 GeV. The analyses cover the measurements of (differential or exclusive)
jet rates (with multiplicities typically up to 4, 5 or 6 jets), the study of three-jet events and particle
production between the jets, as well as four-jet production and angular correlations in four-jet
events.
Event-shape distributions from e+ e− data have been an important input to the tuning of parton
shower MC models, typically matched to matrix elements for three-jet production. In general thesemodels provide good descriptions of the available, highly precise data. Especially for the large LEP
data sample at the Z peak, the statistical uncertainties are mostly negligible and the experimental
systematic uncertainties are at the percent level or even below. These are usually dominated by
the uncertainties related to the MC model dependence of the efficiency and acceptance corrections(often referred to as “detector corrections”).
Observables measured in e+ e− collisions have been used for determinations of the strong coupling constant (cf. Section 9.4 below) and for putting constraints on the QCD color factors (cf.
Sec. 9.1 for their definitions), thus probing the non-Abelian nature of QCD. Angular correlations in four-jet events are sensitive at leading order. Some sensitivity to these color factors,
although only at NLO, is also obtained from event-shape distributions. Scaling violations offragmentation functions and the different subjet structure in quark and gluon induced jets also
give access to these color factors. A compilation of results [444] quotes world average values of
CA = 2.89 ± 0.03 (stat) ± 0.21 (syst) and CF = 1.30 ± 0.01 (stat) ± 0.09 (syst), with a correlation
coefficient of 82%. These results are in perfect agreement with the expectations from SU(3) of
CA = 3 and CF = 4/3.
9.3.2.2 DIS and photoproduction
Jet measurements in ep collisions, both in the DIS and photoproduction regimes, allow for testsof QCD factorization (as they involve only one initial state proton and thus one PDF function), andprovide sensitivity to both the gluon PDF and to the strong coupling constant. Calculations areavailable at NNLO in both regimes [219, 220]. An N3 LO calculation using the Projection-to-Born
method was also presented in Ref. [221,222]. Experimental uncertainties of the order of 5–10% havebeen achieved, whereas statistical uncertainties are negligible to a large extent. For comparison to
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theoretical predictions, at large jet pt the PDF uncertainty dominates the theoretical uncertainty(typically of order 5–10%, in some regions of phase space up to 20%), therefore jet observables
become useful inputs for PDF fits.
In general, the data are well described by the NLO and NNLO matrix-element calculations,combined with DGLAP evolution equations, in particular at large Q2 and central values of jet
pseudo-rapidity. At low values of Q2 and x, in particular for large jet pseudo-rapidities, certain
features of the data have been interpreted as requiring BFKL-type evolution, though the predictions for such schemes are still limited. It is worth noting that there is no indication that the
BKFL approximation is needed within the currently probed phase space in the x, Q2 plane, andan alternative approach [490], which implements the merging of LO matrix-element based event
generation with a parton shower (using the Sherpa framework), successfully describes the datain all kinematical regions, including the low Q2 , low x domain. At moderately small x values, it
should perhaps not be surprising that the BFKL approach and fixed-order matrix-element merging
with parton showers may both provide adequate descriptions of the data, because some part of themulti-parton phase space that they model is common to both approaches.
In the case of photoproduction, a wealth of measurements with low pt jets were performed inorder to constrain the photon content of the proton (which is by now determined with percent
accuracy thanks to the LUX approach [72, 73]). A few examples of measurements can be found in
Refs. [491–495] for photoproduction and in Refs. [496–505] for DIS.
9.3.2.3 Hadron-hadron colliders
The spectrum of observables and the number of measurements performed at hadron colliders
is enormous, probing many regions of phase space and covering a huge range of cross sections, as
exemplified in Fig. 9.1 for a wide class of processes by the ATLAS experiment, and specialised to
top-quark related processes by the CMS experiment at the LHC. In general, the theory agreement with data is excellent for a wide variety of processes, indicating the success of perturbative
QCD with the PDF and the strong coupling as inputs. For the sake of brevity, in the following
only certain classes of those measurements will be discussed, which permit various aspects of the
QCD studies to be addressed. Most of our discussion will focus on LHC results, which are available for center-of-mass energies of 2.76, 5, 7, 8 and 13 TeV with integrated luminosities of up to
140 fb−1 . As of writing of this update, new results at 13.6 TeV are starting to appear. Generally
speaking, besides representing precision tests of the Standard Model and QCD in particular, thesemeasurements serve several purposes, such as: (i) probing pQCD and its various approximations
and implementations in MC models to quantify the order of magnitude of not yet calculated contributions and to gauge their precision when used as background predictions to searches for new
physics, or (ii) extracting/constraining model parameters such as the strong coupling constant or
PDFs.
The final states measured at the LHC include single, double and triple gauge boson production,
top production (single top, top pair and four top production), Higgs boson production, alone andin conjunction with a W or Z boson, and with a top quark pair. Many/most of these events areaccompanied by additional jets. So far only limits have been placed on double Higgs production.
The volume of LHC results prohibits a comprehensive description in this Review; hence, only a few
highlights will be presented.
Among the most important cross sections measured, and the one with the largest dynamic
range, is the inclusive jet spectrum as a function of the jet transverse momentum (pt ), for several
rapidity regions and for pt up to 700 GeV at the Tevatron and ∼ 3.5 TeV at the LHC. It is worth
noting that this latter upper limit in pt corresponds to a distance scale of ∼ 10−19 m: no otherexperiment so far is able to directly probe smaller distance scales of nature than this measurement.
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25

CMS Preliminary

Production Cross Section, σ [pb]Aug 2023

5.02 TeV CMS measurement (L ≤ 302 pb-1)
7 TeV CMS measurement (L ≤ 5.0 fb-1)
8 TeV CMS measurement (L ≤ 19.6 fb-1)
13 TeV CMS measurement (L ≤ 138 fb-1)
13.6 TeV CMS measurement (L ≤ 1 fb-1)
Theory prediction
CMS 95%CL limits at 7, 8 and 13 TeV

103
102

=n jet(s)

10
1
10−1

10−2
10−3
tt

0j

1j

2j

3j

4j

2c

2b

t t-chtWt s-ch

ttγ

tZq

ttZ

tγ

ttW

tWZ

tttt

ttH

tH

All results at: http://cern.ch/go/pNj7

Figure 9.1: Overview of cross section measurements for a wide class of processes by the ATLAS[506] experiment, and specialised to top-quark related processes by the CMS [507] experiment at
the LHC, for center-of-mass energies of 5, 7, 8, 13, and 13.6 TeV. Also shown are the theoreticalpredictions and their uncertainties.
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The Tevatron inclusive jet measurements in Run 2 (Refs. [508–511]) were carried out with the
MidPoint jet clustering algorithm (or its equivalent) and, in a few cases, with the kt jet clustering
algorithm. Most of the LHC measurements use the anti-kt algorithm, with a variety of jet radii. The
use of multiple jet radii in the same analysis allows a better understanding of the underlying QCD
dynamics. Measurements by ALICE, ATLAS and CMS have been published in Refs. [512–522].
In general, we observe a good description of the data by the NLO and NNLO QCD predictions
over about 11 orders of magnitude in cross section, as long as care is taken for the functional formof the central scale choice [523]. The experimental systematic uncertainties are dominated by thejet energy scale uncertainty, quoted to be in the range of a few percent (see for instance the review
in Ref. [524]), leading to uncertainties of ∼ 5 − 30% on the cross section, increasing with pt andrapidity. The PDF uncertainties dominate the theoretical uncertainty at large pt and rapidity. In
fact, inclusive jet data are one of the most important inputs to global PDF fits, in particular forconstraining the high-x gluon PDF [525–527]. Constraints on the PDFs can also be obtained fromratios of inclusive cross sections at different center-of-mass energies [513, 518]. Ratios of jet crosssections are a means to (at least partially) cancel the jet energy scale uncertainties and thus provide
jet observables with significantly improved precision.
Dijet events are typically analyzed in terms of their invariant mass or average dijet pt and angular
distributions, which allows for tests of NLO and NNLO QCD predictions (see e.g. Refs. [517,528,529]
for recent LHC results), and for setting stringent limits on deviations from the Standard Model, suchas quark compositeness or contact interactions (some examples can be found in Refs. [520,530–536]).
Furthermore, dijet azimuthal correlations between the two leading jets, normalized to the total dijetcross section, are an extremely valuable tool for studying the spectrum of gluon radiation in theevent. The azimuthal separation of the two leading jets is sensitive to multi-jet production, avoiding
at the same time large systematic uncertainties from the jet energy calibration. For example, resultsfrom the Tevatron [537, 538] and the LHC [458, 539–543] show that the LO (non-trivial) prediction
for this observable, with at most three partons in the final state, is not able to describe the data
for an azimuthal separation below 2π/3, where NLO contributions (with 4 partons) restore theagreement with data. In addition, this observable can be employed to tune Monte Carlo predictionsof soft gluon radiation. Further examples of dijet observables that probe special corners of phasespace are those that involve forward (large rapidity) jets and where a large rapidity separation,
possibly also a rapidity gap, is required between the two jets. Reviews of such measurements canbe found in Ref. [471], showing that no single prediction is capable of describing the data in allphase-space regions. In particular, no conclusive evidence for BFKL effects in these observables
has been established so far.
Beyond dijet final states, measurements of the production of three or more jets, including crosssection ratios, have been performed (see Refs. [471, 544] for recent reviews), as a means of testing
perturbative QCD predictions, determining the strong coupling constant, and probing/tuning MC
models, in particular those combining multi-parton matrix elements with parton showers [459,545].
The calculation of three-jet production to NNLO [264] will allow more precise predictions of thethree-jet to two-jet ratio, and thus the extraction of αs (m2Z ) from this observable. This calculation
also allows for the use of transverse energy-energy correlations (TEEC) for the determination ofαs (m2Z ) [459].
W and Z production serve as benchmark cross sections at the LHC. The large boson mass stabilizes perturbative predictions, which results in better theoretical accuracy. In terms of experimental
precision, measurements of inclusive vector boson (W, Z) production provide the most precisely determined observables at hadron colliders so far. This is because the experimental signatures are
based on charged leptons which are measured much more accurately than jets or photons. At the
LHC [546–553], the dominant uncertainty stems from the luminosity determination (≤2–4%), while
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27

other uncertainties (e.g. statistical or from lepton efficiencies) are controlled at the ∼0.5–3% level.
The uncertainty from the acceptance correction of about ∼1–2% can be reduced by measuringso-called fiducial cross sections, i.e. by applying kinematic cuts also to the particle level of thetheoretical predictions. Measurements combining the electron and muon final states were able toachieve a precision a few per mille level in the dilepton final state for the normalized cross section
for pllT less than 30 GeV [554, 555]. This level of precision can also be achieved by measuring crosssection ratios (W/Z or W + /W − ). On the theory side, as discussed earlier in this Review, theproduction of these color-singlet states has been calculated to N3 LO [119–121]. Since the dominant
theoretical uncertainty is related to the choice of PDFs, these high-precision data provide useful
handles for PDF determinations.
Further insights are obtained from measurements of differential vector boson production, as afunction of the invariant dilepton mass, the boson’s rapidity or its transverse momentum. For example, the dilepton invariant mass distribution has been measured [556–561] for masses between 15
and 3000 GeV, covering more than 8 orders of magnitude in cross section. NNLO QCD predictions,
together with modern PDF sets and including higher-order electroweak and QED final-state radiation corrections, describe the data to within 5–10% over this large range, whereas NLO predictions
show larger deviations, unless matched to a parton shower.
Similar conclusions can be drawn from the observed rapidity distribution of the dilepton system(see e.g. Refs. [546, 557, 562]) or, in the case of W production, from the observed charged lepton
rapidity distribution and its charge asymmetry. The latter is particularly sensitive to differencesamong PDF sets [546,563–565], also thanks to the high precision achieved by the ATLAS and CMS
experiments for central rapidity ranges. These measurements are extended to the very forward
region, up to 4.5 in lepton rapidity, by the LHCb experiment [552, 553, 566].
An overview of these kinds of measurement can be found in Ref. [471]. There one can alsofind a discussion of and references to LHC results from studies of the vector boson’s transverse
momentum distribution, pVt (see also Refs. [567–569]). This observable covers a wide kinematic
range and probes different aspects of higher-order QCD effects. It is sensitive to jet production inassociation with the vector boson, without suffering from the large jet energy scale uncertainties.
In the pVt region of several tens of GeV to over 1 TeV, the NNLO predictions for V+jet can beused to predict the high-pt boson production cross section.12 The NNLO predictions agree with thedata to within about 10%, and agree somewhat better at high transverse momentum than do the
NLO predictions [570]. At transverse momenta below ∼20 GeV, the fixed-order predictions fail andsoft-gluon resummation is needed to restore the agreement with data. The soft gluon resummation
can either be performed analytically, or effectively using parton showering implemented in Monte
Carlo programs. While analytic approaches reach a higher perturbative precision, they typicallyrefer to inclusive measurements without fiducial cuts.
The addition of further jets to the final state extends the kinematic range as well as increasing thecomplexity of the calculation/measurements. The number of results obtained both at the Tevatron
and at the LHC is extensive, summaries can be found in Refs. [471, 571], and more recent results
can be found in Refs. [570, 572–575]. The measurements cover a very large phase space, e.g. with
jet transverse momenta between 30 GeV and ∼ 1.5 TeV and jet rapidities up to |y| < 4.4 [570, 576].
Jet multiplicities as high as seven jets accompanying the vector boson have already been probed
at the LHC, together with a substantial number of other kinematical observables, such as angularcorrelations among the various jets or among the jets and the vector boson, or the sum of jet
transverse momenta, HT . Whereas the jet pt and HT distributions are dominated by jet energy
scale uncertainties at levels similar to those discussed above for inclusive jet production, angular12

For these calculations, there is a requirement of the presence of a jet, but the pt cut is typically small (30 GeV)
compared to the high pt region being discussed here.
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correlations and jet multiplicity ratios have been measured with a precision of ∼ 10%, see e.g.
Refs. [461, 577].
NLO calculations for up to five jets [578] in addition to the vector boson are in good agreementwith the data over that phase space, where the calculations are applicable. Predictions for V+jet
at NNLO improve the description of the data for distributions involving the vector boson or theleading jet. MC models that implement parton shower matching to matrix elements (either at LO
or NLO) have mixed results.
The challenges get even more severe in the case of vector boson plus heavy quark (b, c) production: on the theory side because an additional scale is introduced by the heavy quark mass,and different schemes exist for the handling of heavy quarks and their mass effects in the initial
and/or final state; and on the experimental side because additional uncertainties related to theheavy-flavor tagging must be considered [579] (see also the discussion regarding flavor-jet definitionin Sec. 9.3.1.1). A review of heavy quark production at the LHC is presented in Ref. [580], where
the di-b-jet pt and mass spectra are found to be well modeled within uncertainties by most generators for b-jet production with or without associated W and Z bosons. However, sizable differences
between data and predictions are seen in the modeling of events with single b jets, particularly atlarge b-jet pt , where gluon splitting processes become dominant, as also confirmed by studies ofb-hadron and b-jet angular correlations.
The precision reached in photon measurements is in between that for lepton and jet measurements. The photon energy and angles can be measured at about the same precision as the lepton
energy and angles in Drell-Yan production, but there are greater challenges encountered in photon
reconstruction (for example isolation) and in purity determination. Note, though, that the photon
purity approaches unity as the photon pt increases. At high pt , it becomes increasingly difficult for a
jet to fragment into an isolated neutral electromagnetic cluster which mimics the photon signature.
The inclusive photon cross section can be measured [528, 581–584], as well as the production of aphoton accompanied by one or more jets [584–586, 586–590]. The kinematic range for photon production is less than that for jet production because of the presence of the electromagnetic coupling,
but still reaches about 2 TeV. Better agreement is obtained with NNLO predictions for photon
production than for NLO predictions, except when the latter are matched to matrix element plus
parton shower predictions. Photon production in association with a heavy-flavor jet is a useful
input for the determination of the b and c quark PDFs [591].
Electroweak corrections are expected to become more and more relevant now that the TeV energy range starts to be explored, and EW corrections can now also be computed automatically [171].
For a comprehensive review on electroweak corrections see Ref. [192]. For example, such corrections
were found [592, 593] to be sizable (tens of percent) when studying the ratio (dσ γ /dpt )/(dσ Z /dpt )in γ (Z)+jet production, pt being the boson’s transverse momentum, and might account for (someof) the differences observed in a CMS measurement [594] of this quantity.
A number of interesting developments, in terms of probing higher-order QCD effects, haveoccurred in the sector of diboson production, in particular for the W W and γγ cases. Regarding
the former, an early disagreement of about 10% between the LHC measurements and the NLO
predictions had led to a number of speculations of possible new physics effects in this channel.
However, more recent ATLAS and CMS measurements [595–598] are in agreement with the NNLO
prediction [131]. The statistical reach of the LHC has resulted in the discovery of triple massive
gauge boson production [599–601].
In the case of diphoton production, ATLAS [602–604] and CMS [605] have provided accurate
measurements, in particular for phase-space regions that are sensitive to radiative QCD corrections(multi-jet production), such as small azimuthal photon separation. While there are large deviations
between data and NLO predictions in this region, a calculation [228] at NNLO accuracy manages to
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mostly fill this gap. This is an interesting example where scale variations can not provide a reliable
estimate of missing contributions beyond NLO, since at NNLO new channels appear in the initialstate (gluon fusion in this case). These missing channels can be included in a matrix element plus
parton shower calculation in which two additional jets are included at NLO. The result exhibits
a similar level of agreement as that obtained at NNLO. Three-photon production has also beenmeasured [606] and is in good agreement with NNLO theory predictions [260, 261].
In terms of heaviest particle involved, top-quark production at the LHC has become an important tool for probing higher-order QCD calculations, thanks to very impressive achievements bothon the experimental and theoretical side, as extensively summarized in Ref. [607]. Regarding tt̄
production, the most precise inclusive cross section measurements are achieved using the dilepton(e µ) final state, with a total uncertainty of 4% [608–612]. This is of about the same size as theuncertainty on the most advanced theoretical predictions [128, 129, 613, 614], obtained at NNLO
with additional soft-gluon resummation at NNLL accuracy [615]. There is excellent agreement
between data and the QCD predictions.
The tt̄ final state allows multiple observables to be measured. A large number of differential
cross section measurements have been performed at 7, 8 and 13 TeV center-of-mass energy, studying distributions such as the top-quark pt and rapidity, the transverse momentum and invariant
mass of the tt̄ system (probing the TeV range), or the number of additional jets. These measurements have been compared to a wide range of predictions, at fixed order up to NNLO as well as
using LO or NLO matrix elements matched to parton showers. Each of the observables provides
information on the high x gluon. However, there are tensions among the multiple observables thatcan lead to difficulties in PDF fits. Four top production has been measured in multi-lepton final
states [616] by the ATLAS and CMS collaboration, with results consistent with the standard model
prediction [617].
Thanks to both the precise measurements of, and predictions for, the inclusive top-pair crosssection, which is sensitive to the strong coupling constant and the top-quark mass, this observablehas been used to measure the strong coupling constant at NNLO accuracy from hadron collider
data [618, 619] (cf. Section 9.4 below), as well as to obtain a measurement of the top-quark’s polemass without employing direct reconstruction methods [618, 620, 621].
The Higgs boson provides a tool for QCD studies, especially as the dominant production mechanism is gg fusion, which is subject to very large QCD corrections. Higgs boson production has
been measured in the ZZ, γγ, bb̄, W W and τ τ decay channels. A measurement of the cross sectionin the ZZ and γγ channels is one of the first measurements at 13.6 TeV to be published [622].
The experimental cross section is now known with a precision approaching 10% [623, 624], similar
to the size of the theoretical uncertainty [625], of which the PDF+αs uncertainty is the largest
component. Part of this systematic is the mis-match in orders between the PDF determination(NNLO) and the cross section evaluation (N3 LO), as discussed earlier. The experimental precision
has allowed detailed fiducial and differential cross section measurements. For example, with thediphoton final state, the transverse momentum of the Higgs boson can be measured out to the orderof 650 GeV [626–629], where top quark mass effects become important. The production of a Higgs
boson with up to 4 jets has been measured [624, 626]. The experimental cross sections have been
compared to NNLO predictions (for H+ ≥ 1 jet), NLO for 2 and 3 jets, and NNLO+NNLL for thetransverse momentum distribution. In addition, finite top quark mass effects have been taken into
account at NLO. The use of the boosted H → bb̄ topology allows probes of Higgs boson transverse
momenta on the order of 600 GeV and greater [624]. So far the agreement with the perturbative
QCD corrections is good.
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9.4 Determinations of the strong coupling constant
Beside the quark masses, the only free parameter in the QCD Lagrangian is the strong coupling
constant αs . The coupling constant in itself is not a physical observable, but rather a quantity defined in the context of perturbation theory, which enters predictions for experimentally measurable
observables, such as R in Eq. (9.7). The value of the strong coupling constant must be inferred fromsuch measurements and is subject to experimental and theoretical uncertainties. The incomplete
knowledge of αs propagates into uncertainties in numerous precision tests of the Standard Model.
Here, we present an update of the 2022 PDG average value of αs (m2Z ) and its uncertainty [630].
Many experimental observables are used to determine αs . A number of recent determinations
are collected in Refs. [631,632]. Further discussions and considerations on determinations of αs canalso be found in Refs. [633–635]. Such considerations include:• The observable’s sensitivity to αs as compared to the experimental precision. For example, forthe e+ e− cross section to hadrons (cf. R in Sec. 9.2.1), QCD effects are only a small correction,
since the perturbative series starts at order αs0 ; three-jet production or event shapes in e+ e−annihilations are directly sensitive to αs since they start at order αs ; the hadronic decay
width of heavy quarkonia, Γ (Υ → hadrons), is very sensitive to αs since its leading orderterm is ∝ αs3 .
• The accuracy of the perturbative prediction, or equivalently of the relation between αs and thevalue of the observable. Several observables have been known to NNLO for quite some time.
These include, for instance, inclusive observables, as well as three-jet rates and event shapes in
e+ e− collisions, inclusive jet and dijet production in DIS, and inclusive jet, dijet, tt̄, W /Z+jet
and three-jet production cross sections in pp or pp̄ collisions. The e+ e− hadronic cross sectionand τ , W and Z branching fractions to hadrons are even known to N3 LO, if one denotes the
LO as the first non-trivial term. In certain cases, fixed-order predictions are supplemented
with resummation. The precise magnitude of the associated theory uncertainties usually isestimated as discussed in Sec. 9.2.4.
• The size of non-perturbative effects. Sufficiently inclusive quantities, like the e+ e− cross section
to hadrons, have small non-perturbative contributions ∼ Λ4 /Q4 . Others, such as event-shape
distributions, have typically contributions ∼ Λ/Q.
• The scale at which the measurement is performed. An uncertainty δ on a measurement of αs (Q2 ),at a scale Q, translates to an uncertainty δ 0 = (αs2 (m2Z )/αs2 (Q2 )) · δ on αs (m2Z ). For example,
this enhances the already important impact of precise low-Q measurements, such as from τ
decays, in combinations performed at the mZ scale.
The selection of results from which to determine the world average value of αs (m2Z ) is restricted tothose that are- published in a peer-reviewed journal at the time of writing this report,- based on the most complete perturbative QCD predictions of at least NNLO accuracy,- accompanied by reliable estimates of all experimental and theoretical uncertainties.
Numerous measurements from jet production in DIS and at hadron colliders are still excluded
from the average presented here, because the determination of αs (m2Z ) from those data sets has
not yet been upgraded to NNLO. A few new results did appear comparing to theory at NNLO andare included in the corresponding section. NLO analyses will still be discussed in this Review, as
they are important ingredients for the experimental evidence of the energy dependence of αs , i.e.
for asymptotic freedom, one of the key features of QCD.
In order to calculate the world average value of αs (m2Z ), we apply, as in earlier editions, anintermediate step of pre-averaging results within the sub-fields now labeled “Hadronic τ decays and31st May, 2024

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low Q2 continuum” (τ decays and low Q2 ), “Heavy quarkonia decays” (QQ̄ bound states), “PDF
fits” (PDF fits), “Hadronic final states of e+ e− annihilations” (e+ e− jets & shapes), “Observablesfrom hadron-induced collisions” (hadron colliders), and “Electroweak precision fit” (electroweak)
as explained in the following sections. For each sub-field, the unweighted average of all selected
results is taken as the pre-average value of αs (m2Z ), and the unweighted average of the quoted
total uncertainties is assigned to be the respective overall error of this pre-average. Asymmetrictotal uncertainties are symmetrised beforehand by adopting the larger of the two values as the (±)
uncertainty. For the “Lattice QCD” (lattice) sub-field we do not perform a pre-averaging; instead,
we adopt for this sub-field the αs (m2Z ) estimate and uncertainty derived by the Flavour Lattice
Averaging Group (FLAG) in Ref. [636].
Assuming that the six sub-fields (excluding lattice) are largely independent of each other, we
determine a non-lattice world average value using a ‘χ2 averaging’ method. In a last step we perform
an unweighted average of the values and uncertainties of αs (m2Z ) from our non-lattice result andthe lattice result presented in the FLAG 2021 report [636].
9.4.1 Hadronic τ decays and low Q2 continuum:
Based on complete N3 LO predictions [34], analyses of the τ hadronic decay width and spectral
functions have been performed, e.g. in Refs. [34, 637–642], and lead to precise determinations of
αs at the energy scale of m2τ . They are based on different approaches to treat perturbative andnon-perturbative contributions, the impacts of which have been a matter of intense discussions for along time, see e.g. Refs. [641–644]. In particular, in τ decays there is a significant difference between
results obtained using fixed-order (FOPT) or contour-improved perturbation theory (CIPT), suchthat analyses based on CIPT generally arrive at larger values of αs (m2τ ) than those based on FOPT.
In addition, some results show discrepancies in αs (m2τ ) among groups of authors using thesame data sets and perturbative calculations, most likely due to different treatments of the nonperturbative contributions, cf. Ref. [642] with Refs. [641, 645]. References [646, 647] question thevalidity of using a truncated operator product expansion (OPE) at Q2 = m2τ based on the disagreement found between experimental values of the spectral moments and the theory representations
based on the truncated OPE fits at Q2 > m2τ .
Recent developments now have shed light on the disagreement between FOPT- and CIPT-based
analyses. References [648–650] argue that CIPT-based calculations require a dedicated estimation of
non-perturbative effects instead of “standard” ones used so far. Otherwise an asymptotic separation
between results using the two perturbative approaches would remain. Potential ways forward aresuggested in Refs. [651, 652]. As a consequence we remove for the time being determinations of thestrong coupling constant based on CIPT from the derivation of the central value and the uncertaintyof αs (m2Z ).
We determine the pre-average value of αs (m2Z ) for this sub-field only from studies that employ
FOPT expansions and remove any eventual averaging with CIPT central values or increased uncertainties due to the differences in CIPT vs. FOPT. As the results from Refs. [34,644,645] are nottotally independent, we pre-averaged as a first step these three results in the previous edition of thisReview. Lacking, however, an estimate of the theory uncertainty for the FOPT method, Ref. [645]had to be left out, such that the first entry to this category of αs determinations is pre-averaged fromαs (m2Z ) = 0.1183 ± 0.0026 [34] and αs (m2Z ) = 0.1181 ± 0.0015 [644] to αs (m2Z ) = 0.1182 ± 0.0021(summarized as BP 2008-16 FO in Fig. 9.2).
Subsequently, this is combined with αs (m2Z ) = 0.1158 ± 0.0022 [653] and αs (m2Z ) = 0.1171 ±0.0010 [654], which replaces the previous result from Ref. [642]. We also include the result fromτ decay and lifetime measurements, obtained in Sec. Electroweak Model and constraints on New
Physics of the 2022 edition of this Review [630], αs (m2Z ) = 0.1171 ± 0.0018. The latter result,31st May, 2024

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being a global fit of τ data, involves some correlations with the other extractions of this category.
However, since we perform an unweighted average of the central value and uncertainty, the effectsof the potential correlations are reduced. Finally, a new determination reported in Ref. [655] isadded to this category: αs (m2Z ) = 0.1183+0.0009−0.0012 . A recent publication evaluating data on Rhad in
the continuum below the charm threshold exhibits huge experimental uncertainties and therefore
has not been considered [656].
All these results are summarized in Fig. 9.2. Determining the unweighted average of the central
values and their overall uncertainties, we arrive at αs (m2Z ) = 0.1173 ± 0.0017, which we will use as
the first input for determining the world average value of αs (m2Z ). This corresponds to αs (m2τ ) =0.314 ± 0.014.
9.4.2 Heavy quarkonia decays:
Two determinations of αs (m2Z ) have been performed [657, 658] that are based on N3 LO accurate
predictions. Reference [657] performs a simultaneous fit of the strong coupling and the bottom mass
mb , including states with principal quantum number up to n ≤ 2 in order to break the degeneracy
between αs and mb , finding αs (m2Z ) = 0.1178 ± 0.0051. Reference [658] instead uses as input of
the fit the renormalon-free combination of masses of the meson Bc , the bottomonium ηb and thecharmonium ηc , MBc − Mηb /2 − Mηc /2, which is weakly dependent on the heavy quark masses, butshows a good dependence on αs . Using this observable, they obtain αs (m2Z ) = 0.1195 ± 0.0053.
Two further values are derived at NNLO in Ref. [659, 660] from mass splittings and sum rules
giving αs (m2Z ) = 0.1183 ± 0.0019 and αs (m2Z ) = 0.1175 ± 0.0032 when evolved from the relevant
charmonium respectively bottomonium mass scales to m2Z . Finally, by means of quarkonium sum
rules, Refs. [661,662] quote αs (m2Z ) = 0.1168±0.0019 and αs (m2Z ) = 0.1186±0.0048 for charmonium
and bottomonium respectively. These six determinations satisfy our criteria to be included in theheavy-quarkonia category of the world average. Their unweighted combination leads to the preaverage for this category of αs (m2Z ) = 0.1181 ± 0.0037.
9.4.3 PDF fits:
Another class of studies, analyzing structure functions at NNLO QCD (and partly beyond), provide
results that serve as relevant inputs for the world average of αs . Some of these studies do not,however, explicitly include estimates of theoretical uncertainties when quoting fit results of αs . In
such cases we add, in quadrature, half of the difference between the results obtained in NNLO andNLO to the quoted errors.
A combined analysis of non-singlet structure functions from DIS [663], based on QCD predictions up to N3 LO in some of its parts, results in αs (m2Z ) = 0.1141 ± 0.0022 (BBG). Studies
of singlet and non-singlet structure functions, based on NNLO predictions, result in αs (m2Z ) =0.1162 ± 0.0017 [664] (JR14). The AMBP group [665, 666] determined a set of parton distributionfunctions using data from HERA, NOMAD, CHORUS, from the Tevatron and the LHC using
the Drell-Yan process and the hadro-production of single-top and top-quark pairs, and determined
αs (m2Z ) = 0.1147 ± 0.0024 [665].
The MSHT group [667], also including hadron collider data, determined a set of parton density
functions (MSHT20) together with αs (m2Z ) = 0.1174 ± 0.0013. Similarly, the CT group [37] determined the CT18 parton density set together with αs (m2Z ) = 0.1164 ± 0.0026. The NNPDF group[668] presented NNPDF3.1 parton distribution functions together with αs (m2Z ) = 0.1185 ± 0.0012.
New in this category is the result of αs (m2Z ) = 0.1156 ± 0.0031 reported as HERAPDF2.0Jets in
Ref. [669]. In addition to DIS data from HERA also jet cross sections are considered in the fit totheory at NNLO.
We note that criticism has been expressed on some of the above extractions. Among the issues
raised, we mention the neglect of singlet contributions at x ≥ 0.3 in pure non-singlet fits [670], the31st May, 2024

9. Quantum Chromodynamics

33

BP 2008-16 FO

τ decays&
low Q 2

Boito 2018 FO
Boito 2021 FO
PDG 2022 tau
Ayala 2023
Mateu 2018
Peset 2018
Narison 2018 (cc)̄
Narison 2018 (bb)̄

QQ
bound
statesBM19 (cc)̄
BM20 (bb)̄
BBG06
JR14
ABMP16

PDF fits

NNPDF31
CT18
MSHT20HERAPDF20jets
ALEPH (j&s)
OPAL (j&s)

e+e−
jets
&
shapesJADE (j&s)
Dissertori (3j)
JADE (3j)
Verbytskyi (2j)Kardos (EEC)
Klijnsma (tt)̄
CMS (tt)̄
H1 (jets)*

hadroncolliderd'Enterria (W/Z)
HERA (jets)
CMS (incl. jets)*
ATLAS ([A]TEEC)
Gfitter 2018

electroweak

HEPfit 2022
PDG 2022 EW

lattice

FLAG 2021

0.110

0.115

0.120

0.125

0.130

αs(m2Z)

August 2023

Figure 9.2: Summary of determinations of αs (m2Z ) with uncertainty in the seven sub-fields as discussed in the text. The yellow (light shaded) bands and dotted lines indicate the pre-average valuesof each sub-field. The dashed line and blue (dark shaded) band represent the final world averagevalue of αs (m2Z ). The “*” symbol within the “hadron colliders” sub-field indicates a determination
including a simultaneous fit of PDFs.
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9. Quantum Chromodynamics

impact and detailed treatment of particular classes of data in the fits [670, 671], possible biases due
to insufficiently flexible parameterizations of the PDFs [672] and the use of a fixed-flavor number
scheme [673, 674]. For recent more extensive discussions see e.g. Ref. [38].
Summarizing the results from world data on structure functions, taking the unweighted averageof the central values and errors of all selected results, leads to a pre-average value of αs (m2Z ) =0.1161 ± 0.0022, see also Fig. 9.2.
9.4.4 Hadronic final states of e+ e− annihilations:Re-analyses of jets and event shapes in e+ e− annihilation (j&s), measured around the Z peak
and at LEP2 center-of-mass energies up to 209 GeV, using NNLO predictions matched to NLL
resummation and Monte Carlo models to correct for hadronization effects, resulted in αs (m2Z ) =0.1224 ± 0.0039 (ALEPH) [675], and in αs (m2Z ) = 0.1189 ± 0.0043 (OPAL) [676]. Similarly, ananalysis of JADE data [677] at center-of-mass energies between 14 and 46 GeV gives αs (m2Z ) =0.1172 ± 0.0051, with contributions from the hadronization model and from perturbative QCD
uncertainties of 0.0035 and 0.0030, respectively. Precise determinations of αs from three-jet production alone (3j), at NNLO, resulted in αs (m2Z ) = 0.1175 ± 0.0025 [678] from ALEPH data and in
αs (m2Z ) = 0.1199 ± 0.0059 [679] from JADE. A recent determination is based on an NNLO+NNLL
accurate calculation that allows to fit the region of lower three-jet rate (2j) using data collected
at LEP and PETRA at different energies. This fit gives αs (m2Z ) = 0.1188 ± 0.0013 [680], where
the dominant uncertainty is the hadronization uncertainty, which is estimated from Monte Carlosimulations. A fit of energy-energy-correlation (EEC), also based on an NNLO+NNLL calculation, together with a Monte Carlo based modeling of hadronization corrections gives αs (m2Z ) =0.1175 ± 0.0029 [681]. These results are summarized in the e+ e− sector of Fig. 9.2.
Another class of αs determinations is based on analytic modeling of non-perturbative andhadronization effects, rather than on Monte Carlo models [682–685], using methods like power
corrections, factorization of soft-collinear effective field theory, dispersive models and low scale
QCD effective couplings. In these studies, the world data on thrust distributions (T), or the Cparameter distributions (C), are analyzed and fitted to perturbative QCD predictions at NNLO
matched with resummation of leading logs up to N3 LL accuracy, see Sec. 9.2.3.3. The results are
2
αs (m2Z ) = 0.1135 ± 0.0011 [683] and αs (m2Z ) = 0.1134+0.0031−0.0025 [684] from thrust, and αs (mZ ) =0.1123 ± 0.0015 [685] from C-parameter.
A long-standing question was why this latter class of determinations led systematically to rather
small values of αs (m2Z ) as compared to the former one based on Monte Carlo models to correct for
hadronization effects. New insights have now been gained in this regard. In a recent calculationof the leading non-perturbative contribution to the C-parameter in the three-jet symmetric limit
based on an effective coupling approach it was found that it differs by a factor of two from the one
in the two-jet limit [686]. Subsequently, using the same effective coupling approach, the leading
1/Q power correction was computed for thrust and the C-parameter in the full three-jet region
under the assumption of a large (negative) nf limit [687]. Expanding on this approach, Ref.
[688] extended the calculation of non-perturbative corrections to include the heavy-jet mass, thedifference of jet masses, wide broadening, and, with some caveat, the three-jet resolution variable in
the Durham algorithm. The key finding in Ref. [688] is that non-perturbative corrections computedin the three-jet region significantly deviate from those computed in the two-jet limit and hence theaforementioned fits based on power corrections in the two-jet limit result in smaller values ofαs (m2Z ). Another important observation is that the inclusion of resummation effects introduces arelatively substantial ambiguity outside the two-jet limit. Additionally, other factors such as the
choice of mass-scheme used to extend the definition of event shapes to massive hadrons can havesignificant effects.
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9. Quantum Chromodynamics

These findings are inconsistent with the very small experimental, hadronization, and theoreticaluncertainties of only 2, 5, and 9 per-mille, respectively, as reported in Refs. [683, 685]. For thesereasons, we exclude the results of Refs. [683–685] from the average. Determinations based oncorrections for non-perturbative hadronization effects using QCD-inspired Monte Carlo generators
have also faced criticism due to the differing nature of parton-level simulations compared to fixedorder calculations. However, these determinations typically exhibit a more conservative theoreticaluncertainty.
Not included in the computation of the world average but worth mentioning are a computation of
the NLO corrections to 5-jet production and comparison to the measured 5-jet rates at LEP [689],
giving αs (m2Z ) = 0.1156+0.0041−0.0034 , and a computation of non-perturbative and perturbative QCD
contributions to the scale evolution of quark and gluon jet multiplicities, including resummation,
resulting in αs (m2Z ) = 0.1199 ± 0.0026 [690].
The unweighted average of the considered determinations as shown in the e+ e− sector of Fig. 9.2
yields αs (m2Z ) = 0.1189 ± 0.0037.
9.4.5 Observables from hadron-induced collisions:
Until recently, determinations of αs using hadron collider data, mostly from jet or tt̄ production
processes, could be performed at NLO only. NNLO calculations have now become available fortt̄ [128, 613, 614] and for inclusive jet, dijet, and three-jet production [253, 264, 266, 691, 692]. Fortt̄ production, in addition, logarithms to NNLL have been resummed [615]. Both should be supplemented by electroweak corrections [693–696], which become important for high-pT collisions at
the LHC. Z+jet production, studied with respect to an αs determination at NLO from multi-jet
events in Ref. [697], is also known at NNLO for the 1-jet case [239, 698].
Determinations of αs from production cross sections at hadron colliders also require a knowledgeof the relevant PDFs for those αs values. Two strategies are pursued for the extraction of αs , one
using pre-determined PDFs as input and a second strategy fitting the proton PDFs together withthe strong coupling constant. Each global PDF group produces PDF sets not only for a value of0.118 that is close to the world average, but also for a wide range above and below in increments of
0.001, which can be used in such determinations. The latter technique of simultaneously fitting αsand the PDFs is technically more accurate, given that the new data used in the determination of
αs may modify the PDFs in a manner not taken into account by the αs -variation PDFs provided by
the fitting collaborations. The former technique may result in a bias of unknown magnitude [699].
As the LHC experiments have the ability to combine a PDF fit with the αs determination, forexample with tools like described in Ref. [700], we expect more experimental joint determinationsof PDFs and of αs for future iterations of this review.
The first determination of αs at NNLO accuracy in QCD has been reported by CMS from the tt̄
√production cross section at s = 7 TeV [618]. In former Reviews this opened up a new sub-field on√
its own. In the meantime, multiple datasets on tt̄ production from Tevatron at s = 1.96 TeV and√from LHC at s = 7, 8, and 13 TeV have been analyzed simultaneously to determine αs [619] toαs (m2Z ) = 0.1177+0.0034−0.0036 , where the largest uncertainties are associated with missing higher orders
and with PDFs, and where an additional complication arises from the top-mass dependence. Since
this combined analysis contains among other things an updated measurement as compared to thedataset used by CMS, the latter is replaced in the averaging by this combined result. A second√
entry into this sub-field is given by an analysis of tt̄ production data at s = 13 TeV from the CMS
Collaboration [610]. From the four values derived for different PDF sets, the unweighted average
is taken: αs (m2Z ) = 0.1145+0.0036−0.0031 . A second analysis by CMS using differential distributions of tt̄
production [611] has been performed at NLO only and is not further considered.
From collisions at HERA the αs determination at NNLO using inclusive jet and dijet measure31st May, 2024

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9. Quantum Chromodynamics

ments in addition to DIS data of the H1 Collaboration [505] has been corrected for an issue withthe theory prediction reported in Ref. [220]. We choose the result αs (m2Z ) = 0.1147 ± 0.0025 thatis derived from a simultaneous fit of the strong coupling constant together with proton PDFs. We
note that results of this section derived from such a simultaneous fit will be marked with a “*” inthe corresponding figures. A second determination [280] combines multiple datasets on inclusive jet
production of the H1 and ZEUS collaborations into one fit using interpolation grids at NNLO. The
result of αs (m2Z ) = 0.1178±0.0026 has been updated for the same issue reported above [220] and isincluded in the unweighted average, although some of the inclusive jet data have already been usedin the previous analysis. We note that the ZEUS Collaboration has presented preliminary resultsfrom a new inclusive jet measurement [701].
A first new determination of αs (m2Z ) at NNLO from inclusive jet production at the LHC has beenpresented by the CMS Collaboration [522]. A simultaneous fit of the strong coupling constant andthe proton PDFs to the HERA DIS and the new LHC jet data gives: αs (m2Z ) = 0.1166±0.0016. The
ATLAS Collaboration has published an extraction of αs (m2Z ) at NNLO from the transverse energyenergy correlation (TEEC) and its asymmetry (ATEEC) [459]. The results of this first derivationfrom an event shape observable requiring three-jet predictions at NNLO are αs (m2Z ) = 0.1175+0.0035−0.0018
for the TEEC and αs (m2Z ) = 0.1185+0.0027−0.0015 for the ATEEC, respectively. We include the unweighted+0.0031
average of the two numbers αs (m2Z ) = 0.1180−0.0017
as a new result in this category. Very recently,
two preliminary determinations of the strong coupling constant, albeit at NLO, have been reported
by CMS from energy correlators inside jets and from azimuthal correlations among jets [702, 703].
Finally, Ref. [704] extracts the strong coupling constant from a fit at NNLO to measurements
of inclusive Z and W boson production by experiments at the LHC and Tevatron colliders. From
the four values quoted for different PDF sets we include the unweighted average for the CT14 andMMHT14 PDFs αs (m2Z ) = 0.1180+0.0017−0.0015 . The other results suffer from either a bad fit quality or
αs (m2Z ) values outside the range of validity for the PDF set. Furthermore, two determinations
using measurements of the Z boson’s recoil at low pT have been submitted using data of the CDF
respectively ATLAS experiments [705, 706]. The very small size of the estimated uncertainties has
stimulated some discussion.
As unweighted pre-average for this sub-field we now obtain αs (m2Z ) = 0.1168 ± 0.0027. If thestricter requirement of simultaneous fits with PDFs is imposed, then only the H1 and CMS results
are left giving αs (m2Z ) = 0.1157 ± 0.0021 for this sub-field. In sect. 9.4.8 below, we will also report
the outcome for this choice.
Many further αs determinations from jet measurements have not yet been advanced to NNLO
accuracy. A selection of results from inclusive jet [504, 518, 707–712], dijet [529], and multi-jet
measurements [455, 457, 458, 504, 713–717] is presented in Fig. 9.3, where the uncertainty in mostcases is dominated by the impact of missing higher orders estimated through scale variations. From√
the CMS Collaboration we quote for the inclusive jet production at s = 7 and 8 TeV, and for
dijet production at 8 TeV the values that have been derived in a simultaneous fit with the PDFs
and marked with “*” in the figure. The last point of the inclusive jet sub-field from Ref. [712] isderived from a simultaneous fit to six datasets from different experiments and partially includes
data used already for the other data points, e.g. the CMS result at 7 TeV.
The multi-jet αs determinations are based on three-jet cross sections (m3j), three- to twojet cross-section ratios (R32), dijet angular decorrelations (RdR, RdPhi), and the aforementioned√
transverse energy-energy-correlations, but at s = 7 and 8 TeV only. The H1 result is extracted
from a fit to inclusive one-, two-, and three-jet cross sections (nj) simultaneously.
All NLO results are within their large uncertainties in agreement with the world average andthe associated analyses provide valuable new values for the scale dependence of αs at energy scales
31st May, 2024

9. Quantum Chromodynamics

37

CDF 1.96 TeV (1j)inclusive jets

ZEUS 320 GeV (1j)D0 1.96 TeV (1j)Mal.&Star. 7 TeV (1j)H1 319 GeV (1j)
CMS 7 TeV (1j)*
CMS 8 TeV (1j)*Britzger (1j)dijetsCMS 8 TeV (2j)*
ZEUS 318 GeV (R32)
D0 1.96 TeV (RdR)
CMS 7 TeV (R32)

multi-jets

CMS 7 TeV (m3j)
ATLAS 7 TeV (TEEC)ATLAS 7 TeV (ATEEC)
H1 319 GeV (nj)
ATLAS 8 TeV (TEEC)ATLAS 8 TeV (ATEEC)ATLAS 8 TeV (RdPhi)
August 2023 (NLO)

0.110 0.115 0.120 0.125 0.130αs(m2Z)

Figure 9.3: Summary of determinations of αs (m2Z ) at NLO from inclusive jet, dijet, and multi-jet
measurements at hadron colliders. The uncertainty is dominated by estimates of the impact of
missing higher orders. The yellow (light shaded) bands and dotted lines indicate average values
for each sub-field. The dashed line and blue (dark shaded) band represent the final world averagevalue of αs (m2Z ). The “*” symbol indicates determinations including a simultaneous fit of PDFs.

now extending beyond 2.0 TeV as shown in Fig. 9.5.
9.4.6 Electroweak precision fit:
For this category, we take the global electroweak fit of Ref. [718], which includes kinematic top
quark and W boson mass measurements from the LHC, determinations of the effective leptonic
electroweak mixing angles from the Tevatron, a Higgs mass measurement from ATLAS and CMS,
and an evaluation of the hadronic contribution to the running of the electromagnetic coupling at theZ-boson mass. We also consider the fit of αs (m2Z ) from the global fit to electroweak data presented
in the supplemental material of Ref. [719]. We choose to include the result of their “conservative
scenario”13 in accounting for the uncertainties of mt and mW , which avoids a potential bias from13
We note that this result agrees with the previous fit in the “conservative scenario” of Ref. [720], which precedes
the latest CDF W mass determination.

31st May, 2024

9. Quantum Chromodynamics

38

inconsistencies between the world average of the W boson mass and the new measurement reportedby the CDF Collaboration in Ref. [721]. In addition, we use the newer results of the electroweak fit
at the Z mass pole from LEP and SLC data presented in Sec. Electroweak Model and constraints on
New Physics of the 2022 edition of this Review. All three determinations, αs (m2Z ) = 0.1194±0.0029[718], αs (m2Z ) = 0.1193 ± 0.0028 [719], and αs (m2Z ) = 0.1221 ± 0.0027 [630], are also in perfectagreement with the original result obtained from LEP and SLD data [722]. Our pre-averaging gives
αs (m2Z ) = 0.1203 ± 0.0028.
We note, however, that results from electroweak precision data strongly depend on the strict
validity of Standard Model predictions and the existence of the minimal Higgs mechanism to implement electroweak symmetry breaking. Any - even small - deviation of nature from this model
could strongly influence this extraction of αs .
9.4.7 Lattice QCD:
Several methods exist to extract the strong coupling constant from lattice QCD, as reviewed alsoin Sec. Lattice QCD of this Review and in Ref. [723]. The Flavour Lattice Averaging Grouphas considered the most up-to-date determinations and combined them to produce an updated
αs (m2Z ) estimate [636]. Their final result is obtained by considering a multitude of possible input
calculations and by retaining in their final estimate only those that fulfill their predefined quality
criteria, detailed in the Sec. 9.2.1 of Ref. [636]. In summary, a determination of αs needs to satisfy
the following requirements:• The determination of αs should be based on a comparison of a short-distance quantity Q at
scale µ with a well-defined continuum limit, which does not involve UV or IR divergences,when the quantity is expressed by using an expansion in terms of a short distance definitionof the strong coupling (e.g. in the M S scheme);• The scale µ, at which the determination is carried out, has to be sufficiently large such thatthe error associated with the perturbative truncation remains small enough;• If Q is defined by physical quantities in infinite volume, one needs to satisfy the constraints
L/a  µ/ΛQCD , where L is the lattice size and a the lattice spacing;• Only results for nf ≥ 3 are considered.
• As is the case for the PDG average, a calculation must be published in a peer-reviewed journal
to be eligible for inclusion into the FLAG estimate.
We note that the FLAG criteria applied now are unchanged compared to the FLAG 2019 Review [724] and are considered to be relatively loose. More stringent criteria have already beenformulated by FLAG, and it is likely that in future their estimate includes only those results thatsatisfy these stricter criteria.
Altogether, from a large number of old and new results six from the 2019 Review also enter the
final FLAG 2021 estimate [725–729]14 and three new ones qualify for inclusion [730–732]. These
determinations, together with their uncertainties, are displayed in Fig. 9.4. The yellow (lightshaded) band and dotted line indicate the FLAG 2021 estimate, while the dashed line and blue
(dark shaded) band represent the world average (see later). The level of agreement of individual
results to the world average, or to the non-lattice world average is very similar.
Similarly to what is done here, the FLAG Collaboration built pre-averages of results within thevarious classes. The five categories that contribute to their final estimate are: step-scaling methods(αs (m2Z ) = 0.11848 ± 0.00081), the potential at short distances (αs (m2Z ) = 0.11782 ± 0.00165),
Wilson loops (αs (m2Z ) = 0.11871 ± 0.00128), heavy-quark current two-point functions (αs (m2Z ) =0.11826 ± 0.00200) and, for the first time, light quark vacuum polarization (αs (m2Z ) = 0.11863 ±14

Reference [727] contains two results.

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9. Quantum Chromodynamics

0.00360). Other categories such as the calculation of QCD vertices, or of the eigenvalue spectrumof the Dirac operator have not yet published results that fulfill all requirements.
We note that, in addition to presenting new results and improvements on previous works, the
ALPHA Collaboration [733] has introduced a novel approach to non-perturbative renormalization
called decoupling. This strategy shifts the perspective on results involving unphysical flavor numbers, particularly nf = 0. By performing a non-perturbative matching calculation, these results
can be non-perturbatively related to results with nf > 0. Consequently, obtaining precise and controlled nf = 0 results becomes of great importance, with significant implications for future FLAG
reports.
The final value is obtained by performing a weighted average of the pre-averages. The final
uncertainty however is not the combined uncertainty of the pre-averages (which is 0.0006), since
the errors on almost all determinations are dominated by the perturbative truncation error. Instead,the error on the pre-range for αs from the step-scaling method is taken, since perturbative truncation
errors are sub-dominant in this method. The final FLAG 2021 estimate (rounded to four digits) is
αs (m2Z ) = 0.1184 ± 0.0008(FLAG 2021 estimate) ,(9.23)

which is fully compatible with the FLAG 2019 result of αs (m2Z ) = 0.1182 ± 0.0008.
We believe that this result expresses to a large extent the consensus of the lattice community andthat the imposed criteria and the rigorous assessment of systematic uncertainties qualify for a direct
inclusion of this FLAG estimate here. As in the previous review, we therefore adopt the FLAG
estimate with its uncertainty as our value of αs for the lattice category. Moreover, this lattice result
will not be directly combined with any other sub-field average, but with our non-lattice average to
give our final world average value for αs .
9.4.8 Determination of the world average value of αs (m2Z ) :Obtaining a world average value for αs (m2Z ) is a non-trivial exercise. A certain arbitrariness andsubjective component is inevitable because of the choice of measurements to be included in the
average, the treatment of (non-Gaussian) systematic uncertainties of mostly theoretical nature, as
well as the treatment of correlations among the various inputs, of theoretical as well as experimental
origin.
We have chosen to determine pre-averages for sub-fields of measurements that are considered
to exhibit a maximum degree of independence among each other, considering experimental as wellas theoretical issues. The six pre-averages, illustrated also in Fig. 9.2, are listed in column two of
Table 9.1. We recall that these are exclusively obtained from extractions that are based on (at least)
NNLO QCD predictions, and are published in peer-reviewed journals at the time of completing thisReview. To obtain our final world average, we first combine these six pre-averages, excluding thelattice result, using a χ2 averaging method. This gives
αs (m2Z ) = 0.1175 ± 0.0010(PDG 2023 without lattice) .

(9.24)

This result is fully compatible with the lattice estimate Eq. (9.23) and has a comparable error.
To avoid a possible over-reduction, we combine these two numbers using an unweighted averageand take as an uncertainty the average between these two uncertainties. This gives our final world
average value
αs (m2Z ) = 0.1180 ± 0.0009(PDG 2023 average) .
(9.25)
If for the sub-field of hadron colliders we are more restrictive and instead only accept results froma simultaneous fit of PDFs, we arrive at 0.1157 ± 0.0021 for this sub-field leading to 0.1172 ± 0.0010(without lattice) and αs (m2Z ) = 0.1178 ± 0.0009 for the final average. Both the new world average
31st May, 2024

9. Quantum Chromodynamics

40

ALPHA 17
PACS-CS 09A
Ayala 20
TUMQCD 19
Cali 20
HPQCD 10 (Wl)Maltman 08
HPQCD 14A
HPQCD 10 (2p)

August 2023

0.115

0.120

αs(m2Z)

Figure 9.4: Lattice determinations that enter the FLAG 2021 estimate. The yellow (light shaded)band and dotted line indicate the unweighted average value for this sub-field. The dashed line andblue (dark shaded) band represent our final world average value of αs (m2Z ).

value and the restricted result are compatible with each other and changed only marginally ascompared to the values reported in the last edition of this Review.
It also stands to question whether the sub-fields of PDF fits and hadron colliders still are as
independent as originally assumed. To test the potential impact, the fit has been repeated whilegrouping all αs (m2Z ) determinations of both sub-fields into a common one. We obtain 0.1164±0.0024
for the new larger sub-field and 0.1178 ± 0.0011 for the combination with all other sub-fields except
lattice, which is well within the estimated uncertainties. Moreover, we present in column fourof Table 9.1 the combined result for αs (m2Z ) when the respective sub-field is omitted from the
combination. The variation in values obtained for αs (m2Z ) is ±0.0004, which is less than half ourestimated uncertainty.
Since two long-standing issues causing offsets among determinations of αs (m2Z ) in the subfields of τ decays and low Q2 and e+ e− jets & shapes have been resolved, it may be argued thata weighted fit among the non-lattice sub-fields may be warranted. We compare the outcome ofweighted fits with our standard procedure in columns two and three of Table 9.1. We observe
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9. Quantum Chromodynamics

41

Table 9.1: Unweighted and weighted pre-averages of αs (m2Z ) for each subfield in columns two and three. The bottom line corresponds to the combined result (without lattice gauge theory) using the χ2 averaging method.
The same χ2 averaging is used for column four combining all unweighted
averages except for the sub-field of column one. See text for more details.
averages per sub-fieldτ decays & low Q2
QQ̄ bound states
PDF fits
e+ e− jets & shapes
hadron colliders
electroweak
PDG 2023 (without lattice)unweighted
0.1173 ± 0.0017
0.1181 ± 0.0037
0.1161 ± 0.0022
0.1189 ± 0.0037
0.1168 ± 0.0027
0.1203 ± 0.0028
0.1175 ± 0.0010

0.35

unweighted without subfield
0.1177 ± 0.0013
0.1175 ± 0.0011
0.1179 ± 0.0011
0.1174 ± 0.0011
0.1177 ± 0.0011
0.1171 ± 0.0011
n/aτ decay (N3LO)
low Q2 cont. (N3LO)
Heavy Quarkonia (NNLO)
HERA jets (NNLO)
e+e- jets/shapes (NNLO+NLLA)
e+e- Z0 pole fit (N3LO)
pp/p-p jets (NLO)
pp top (NNLO)
pp TEEC (NNLO)

0.3
0.25
αs(Q2)

weighted
0.1174 ± 0.0009
0.1177 ± 0.0011
0.1168 ± 0.0014
0.1187 ± 0.0017
0.1169 ± 0.0014
0.1203 ± 0.0016
0.1178 ± 0.0005

0.2
0.15
0.1

αs(mZ2) = 0.1180 ± 0.0009
0.05

1August 2023

10100
Q [GeV]

1000

Figure 9.5: Summary of determinations of αs as a function of the energy scale Q compared to
the running of the coupling computed at five loops taking as an input the current PDG average,
αs (m2Z ) = 0.1180 ± 0.0009. Compared to the previous edition, numerous points have been updated
or added.

that the weighted averages are rather close to the unweighted ones. However, the uncertainties
become significantly smaller. This approach may be too aggressive as it ignores the correlations
among the data, methods, and theory ingredients of the various determinations. We feel that theuncertainty of ±0.0005 is an underestimation of the true error. We also note that in the unweighted
combination the estimated uncertainty for each sub-field is larger than the spread of the results asgiven by the standard deviation. In the weighted fit this crosscheck fails in four out of six cases.
The last several years have seen clarification of some persistent concerns and a wealth of new
results at NNLO, providing not only a rather precise and reasonably stable world average valueof αs (m2Z ), but also a clear signature and proof of the energy dependence of αs in full agreement
31st May, 2024

9. Quantum Chromodynamics

42with the QCD prediction of asymptotic freedom. This is demonstrated in Fig. 9.5, where resultsof αs (Q2 ) obtained at discrete energy scales Q are summarized, which now mostly include those
based on NNLO QCD.15 Thanks to the results from the LHC, the energy scales, at which αs isdetermined, now extend even beyond 2 TeV.16 The points in this plot are extracted from Refs. [280,458, 459, 518, 619, 630, 653, 660, 675–677, 679, 709, 715, 716].
In this combination, as in past combinations, we have considered lattice QCD calculationsof αs independently of experimental/phenomenological determinations. In the future, when thelattice continuum extrapolations are under better control, it may be useful to group lattice QCD
determinations of αs with experimental determinations of αs that have systematics of similar origin,in a similar manner as we currently group, for example, hadron collider results together [723].
References[1] R. K. Ellis, W. J. Stirling and B. R. Webber, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.
8, 1 (1996).
[2] J. Campbell, J. Huston, F. Krauss “The Black Book of Quantum Chromodynamics, a Primer
for the QCD Era,”Oxford University Press, UK (2017).
[3] C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006), [hep-ex/0602020].
[4] J. M. Pendlebury et al., Phys. Rev. D92, 9, 092003 (2015), [arXiv:1509.04411].
[5] B. Graner et al., Phys. Rev. Lett. 116, 16, 161601 (2016), [Erratum: Phys. Rev.
Lett.119,no.11,119901(2017)], [arXiv:1601.04339].
[6] https://www.psi.ch/en/nedm/edms-world-wide.
[7] J. E. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010), [arXiv:0807.3125].
[8] G. Dissertori, I. G. Knowles and M. Schmelling, High energy experiments and theory, Oxford,
UK: Clarendon (2003).
[9] R. Brock et al. (CTEQ), Rev. Mod. Phys. 67, 157 (1995).
[10] K. Melnikov, CERN Yellow Rep. School Proc. 3, 37 (2018).
[11] F. Gross et al. (2022), [arXiv:2212.11107].
[12] T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B400, 379 (1997),[hep-ph/9701390].
[13] M. Czakon, Nucl. Phys. B710, 485 (2005), [hep-ph/0411261].
[14] P. A. Baikov, K. G. Chetyrkin and J. H. Kühn, Phys. Rev. Lett. 118, 8, 082002 (2017),[arXiv:1606.08659].
[15] T. Luthe et al., JHEP 07, 127 (2016), [arXiv:1606.08662].
[16] F. Herzog et al., JHEP 02, 090 (2017), [arXiv:1701.01404].
[17] T. Luthe et al., JHEP 10, 166 (2017), [arXiv:1709.07718].
[18] K. G. Chetyrkin et al., JHEP 10, 179 (2017), [Addendum: JHEP12,006(2017)],[arXiv:1709.08541].
[19] W. A. Bardeen et al., Phys. Rev. D18, 3998 (1978).
[20] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973), [,271(1973)].
[21] H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973), [,274(1973)].
[22] Y. Schroder and M. Steinhauser, JHEP 01, 051 (2006), [hep-ph/0512058].
15

The uncertainties of the HERA jets points were, by mistake, shown at only half their size. The uncertainties of
the ALEPH points of the e+ e− jets/shapes sub-field now correspond to the total uncertainty.
16
We note, however, that the relevant energy scale of a measurement is not uniquely defined. In addition to being
multiplied by factors of e.g. 1/2 or 2, for instance in studies of the ratio of three- to two-jet cross sections at the
LHC, the relevant scale can be taken to be the average of the transverse momenta of the two leading jets [714], or
alternatively might be chosen to be the transverse momentum of the 3rd jet.
31st May, 2024

43

9. Quantum Chromodynamics[23] K. G. Chetyrkin, J. H. Kuhn and C. Sturm, Nucl. Phys. B744, 121 (2006), [hep-ph/0512060].
[24] A. G. Grozin et al., JHEP 09, 066 (2011), [arXiv:1107.5970].
[25] M. Dalla Brida et al. (ALPHA), Phys. Rev. Lett. 117, 18, 182001 (2016), [arXiv:1604.06193].
[26] M. Beneke, Phys. Rept. 317, 1 (1999), [hep-ph/9807443].
[27] M. Beneke et al., Phys. Lett. B775, 63 (2017), [arXiv:1605.03609].
[28] A. H. Hoang, C. Lepenik and M. Preisser, JHEP 09, 099 (2017), [arXiv:1706.08526].
[29] P. Marquard et al., Phys. Rev. Lett. 114, 14, 142002 (2015), [arXiv:1502.01030].
[30] P. A. Baikov et al., Phys. Lett. B714, 62 (2012), [arXiv:1206.1288].
[31] K. G. Chetyrkin, J. H. Kuhn and A. Kwiatkowski (1996), [Phys. Rept.277,189(1996)], [hepph/9503396].
[32] Y. Kiyo et al., Nucl. Phys. B823, 269 (2009), [arXiv:0907.2120].
[33] P. A. Baikov et al., Phys. Rev. Lett. 108, 222003 (2012), [arXiv:1201.5804].
[34] P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett. 101, 012002 (2008),[arXiv:0801.1821].
[35] F. Herzog et al., JHEP 08, 113 (2017), [arXiv:1707.01044].
[36] V. A. Novikov et al., Nucl. Phys. B174, 378 (1980).
[37] T.-J. Hou et al., Phys. Rev. D 103, 1, 014013 (2021), [arXiv:1912.10053].
[38] R. D. Ball et al. (PDF4LHC Working Group), J. Phys. G 49, 8, 080501 (2022),[arXiv:2203.05506].
[39] R. Abdul Khalek et al. (NNPDF), Eur. Phys. J. C, 79:838 (2019), [arXiv:1905.04311].
[40] R. Abdul Khalek et al. (NNPDF), Eur. Phys. J. C 79, 11, 931 (2019), [arXiv:1906.10698].
[41] R. D. Ball and R. L. Pearson, Eur. Phys. J. C 81, 9, 830 (2021), [arXiv:2105.05114].
[42] Z. Kassabov, M. Ubiali and C. Voisey, JHEP 03, 148 (2023), [arXiv:2207.07616].
[43] L. A. Harland-Lang and R. S. Thorne, Eur. Phys. J. C 79, 3, 225 (2019), [arXiv:1811.08434].
[44] J. McGowan et al., Eur. Phys. J. C 83, 3, 185 (2023), [Erratum: Eur.Phys.J.C 83, 302 (2023)],[arXiv:2207.04739].
[45] X. Ji, Phys. Rev. Lett. 110, 262002 (2013), [arXiv:1305.1539].
[46] G. C. Rossi and M. Testa, Phys. Rev. D96, 1, 014507 (2017), [arXiv:1706.04428].
[47] G. Rossi and M. Testa, Phys. Rev. D 98, 5, 054028 (2018), [arXiv:1806.00808].
[48] L. Del Debbio, T. Giani and C. J. Monahan, JHEP 09, 021 (2020), [arXiv:2007.02131].
[49] H.-W. Lin et al., Phys. Rev. D91, 054510 (2015), [arXiv:1402.1462]; C. Alexandrou et al.,Phys. Rev. D92, 014502 (2015), [arXiv:1504.07455].
[50] H.-W. Lin et al., Prog. Part. Nucl. Phys. 100, 107 (2018), [arXiv:1711.07916].
[51] C. Alexandrou et al., Phys. Rev. Lett. 121, 11, 112001 (2018), [arXiv:1803.02685].
[52] J.-W. Chen et al. (2018), [arXiv:1803.04393].
[53] K. Cichy, L. Del Debbio and T. Giani, JHEP 10, 137 (2019), [arXiv:1907.06037].
[54] K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019),[arXiv:1811.07248].
[55] M. A. Ebert, I. W. Stewart and Y. Zhao, JHEP 09, 037 (2019), [arXiv:1901.03685].
[56] M. Constantinou, Eur. Phys. J. A 57, 2, 77 (2021), [arXiv:2010.02445].
31st May, 2024

44

9. Quantum Chromodynamics[57] M. Constantinou et al., Prog. Part. Nucl. Phys. 121, 103908 (2021), [arXiv:2006.08636].
[58] L. Del Debbio et al., JHEP 02, 138 (2021), [arXiv:2010.03996].
[59] Z. Fan, R. Zhang and H.-W. Lin, Int. J. Mod. Phys. A 36, 13, 2150080 (2021),[arXiv:2007.16113].
[60] T. Khan et al. (HadStruc), Phys. Rev. D 104, 9, 094516 (2021), [arXiv:2107.08960].
[61] Z. Fan, W. Good and H.-W. Lin (2022), [arXiv:2210.09985].
[62] J. Gao, L. Harland-Lang and J. Rojo, Phys. Rept. 742, 1 (2018), [arXiv:1709.04922].
[63] K. Kovařík, P. M. Nadolsky and D. E. Soper, Rev. Mod. Phys. 92, 4, 045003 (2020),[arXiv:1905.06957].
[64] T.-J. Hou et al., Phys. Rev. D 107, 7, 076018 (2023), [arXiv:2211.11064].
[65] J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969).
[66] J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B724, 3 (2005), [hep-ph/0504242].
[67] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B813, 220 (2009), [arXiv:0812.4168].
[68] J. Davies et al., PoS DIS2016, 059 (2016), [arXiv:1606.08907].
[69] E. Laenen et al., Nucl. Phys. B392, 162 (1993); S. Riemersma, J. Smith and W. L. van
Neerven, Phys. Lett. B347, 143 (1995), [hep-ph/9411431].
[70] J. Blümlein et al. (2019), [arXiv:1903.06155].
[71] J. Blümlein (2023), [arXiv:2306.01362].
[72] A. Manohar et al., Phys. Rev. Lett. 117, 24, 242002 (2016), [arXiv:1607.04266].
[73] A. V. Manohar et al., JHEP 12, 046 (2017), [arXiv:1708.01256].
[74] L. Buonocore et al., JHEP 08, 08, 019 (2020), [arXiv:2005.06477].
[75] J. C. Collins, D. E. Soper and G. F. Sterman, Adv. Ser. Direct. High Energy Phys. 5, 1
(1989), [hep-ph/0409313].
[76] J. Collins and J.-W. Qiu, Phys. Rev. D 75, 114014 (2007), [arXiv:0705.2141].
[77] G. Sterman, in “Snowmass 2021,” (2022), [arXiv:2207.06507].
[78] R. Boussarie et al. (2023), [arXiv:2304.03302].
[79] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972), [Yad. Fiz.15,781(1972)];
L. N. Lipatov, Sov. J. Nucl. Phys. 20, 94 (1975), [Yad. Fiz.20,181(1974)]; G. Altarelli and
G. Parisi, Nucl. Phys. B126, 298 (1977); Y. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977),[Zh. Eksp. Teor. Fiz.73,1216(1977)].
[80] G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B175, 27 (1980); W. Furmanski and
R. Petronzio, Phys. Lett. 97B, 437 (1980).
[81] A. Vogt, S. Moch and J. A. M. Vermaseren, Nucl. Phys. B691, 129 (2004), [hep-ph/0404111];
S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B688, 101 (2004), [hep-ph/0403192].
[82] J. Davies et al., Nucl. Phys. B 915, 335 (2017), [arXiv:1610.07477].
[83] S. Moch et al., JHEP 10, 041 (2017), [arXiv:1707.08315].
[84] A. Vogt et al., PoS RADCOR2017, 046 (2018), [arXiv:1801.06085].
[85] A. Vogt et al., PoS LL2018, 050 (2018), [arXiv:1808.08981].
[86] G. Falcioni et al., Phys. Lett. B 842, 137944 (2023), [arXiv:2302.07593].
[87] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B889, 351 (2014), [arXiv:1409.5131].
31st May, 2024

45

9. Quantum Chromodynamics[88] D. de Florian, G. F. R. Sborlini and G. Rodrigo, Eur. Phys. J. C76, 5, 282 (2016),[arXiv:1512.00612]; D. de Florian, G. F. R. Sborlini and G. Rodrigo, JHEP 10, 056 (2016),[arXiv:1606.02887].
[89] D. de Florian and L. P. Conte (2023), [arXiv:2305.14144].
[90] R. S. Thorne, Phys. Rev. D73, 054019 (2006), [hep-ph/0601245].
[91] S. Forte et al., Nucl. Phys. B834, 116 (2010), [arXiv:1001.2312].
[92] M. Guzzi et al., Phys. Rev. D86, 053005 (2012), [arXiv:1108.5112].
[93] V. S. Fadin, E. A. Kuraev and L. N. Lipatov, Phys. Lett. 60B, 50 (1975).
[94] I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978), [Yad. Fiz.28,1597(1978)].
[95] R. D. Ball et al., Eur. Phys. J. C78, 4, 321 (2018), [arXiv:1710.05935].
[96] H. Abramowicz et al. (H1, ZEUS), Eur. Phys. J. C75, 12, 580 (2015), [arXiv:1506.06042].
[97] L. N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976), [Yad. Fiz.23,642(1976)].
[98] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys. JETP 45, 199 (1977), [Zh. Eksp.
Teor. Fiz.72,377(1977)].
[99] V. S. Fadin and L. N. Lipatov, Phys. Lett. B429, 127 (1998), [hep-ph/9802290].
[100] M. Ciafaloni and G. Camici, Phys. Lett. B430, 349 (1998), [hep-ph/9803389].
[101] G. Altarelli, R. D. Ball and S. Forte, Nucl. Phys. B799, 199 (2008), [arXiv:0802.0032].
[102] M. Ciafaloni et al., JHEP 08, 046 (2007), [arXiv:0707.1453].
[103] C. D. White and R. S. Thorne, Phys. Rev. D75, 034005 (2007), [hep-ph/0611204].
[104] E. Iancu et al., Phys. Lett. B744, 293 (2015), [arXiv:1502.05642].
[105] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Phys. Rev. Lett. 115, 25, 251601 (2015),[arXiv:1507.04010]; V. N. Velizhanin (2015), [arXiv:1508.02857]; S. Caron-Huot and M. Herranen, JHEP 02, 058 (2018), [arXiv:1604.07417]; V. Fadin, Chapter 4: BFKL — Past andFuture, 63–90 (2021), [arXiv:2012.11931].
[106] I. Balitsky, Nucl. Phys. B463, 99 (1996), [hep-ph/9509348].
[107] Y. V. Kovchegov, Phys. Rev. D60, 034008 (1999), [hep-ph/9901281].
[108] M. Davier et al., Eur. Phys. J. C 56, 305 (2008), [arXiv:0803.0979].
[109] J. C. Collins, D. E. Soper and G. F. Sterman, Nucl. Phys. B261, 104 (1985).
[110] C. Anastasiou et al., Phys. Rev. Lett. 114, 212001 (2015), [arXiv:1503.06056]; C. Anastasiou
et al., JHEP 05, 058 (2016), [arXiv:1602.00695].
[111] B. Mistlberger, JHEP 05, 028 (2018), [arXiv:1802.00833].
[112] L.-B. Chen et al., Phys. Lett. B 803, 135292 (2020), [arXiv:1909.06808].
[113] F. Dulat, B. Mistlberger and A. Pelloni,[arXiv:1810.09462].

Phys. Rev. D99,

3,034004 (2019),[114] L. Cieri et al., JHEP 02, 096 (2019), [arXiv:1807.11501].
[115] X. Chen et al. (2021), [arXiv:2102.07607].
[116] J. Baglio et al., JHEP 12, 066 (2022), [arXiv:2209.06138].
[117] F. A. Dreyer and A. Karlberg, Phys. Rev. Lett. 117, 7, 072001 (2016), [arXiv:1606.00840].
[118] F. A. Dreyer and A. Karlberg, Phys. Rev. D 98, 11, 114016 (2018), [arXiv:1811.07906].
[119] C. Duhr, F. Dulat and B. Mistlberger, Phys. Rev. Lett. 125, 17, 172001 (2020),[arXiv:2001.07717].
31st May, 2024

46

9. Quantum Chromodynamics[120] X. Chen et al. (2021), [arXiv:2107.09085].
[121] C. Duhr, F. Dulat and B. Mistlberger, JHEP 11, 143 (2020), [arXiv:2007.13313].
[122] M. Bonvini et al., JHEP 08, 105 (2016), [arXiv:1603.08000].
[123] F. Dulat, A. Lazopoulos and B. Mistlberger, Comput. Phys. Commun. 233, 243 (2018),[arXiv:1802.00827].
[124] R. V. Harlander, S. Liebler and H. Mantler, Comput. Phys. Commun. 212, 239 (2017),[arXiv:1605.03190].
[125] O. Brein, A. Djouadi and R. Harlander, Phys. Lett. B579, 149 (2004), [hep-ph/0307206].
[126] D. de Florian and J. Mazzitelli, Phys. Rev. Lett. 111, 201801 (2013), [arXiv:1309.6594].
[127] S. Borowka et al., Phys. Rev. Lett. 117, 1, 012001 (2016), [Erratum:
Lett.117,no.7,079901(2016)], [arXiv:1604.06447].

Phys. Rev.

[128] M. Czakon, P. Fiedler and A. Mitov, Phys. Rev. Lett. 110, 252004 (2013), [arXiv:1303.6254].
[129] S. Catani et al., Phys. Rev. D99, 5, 051501 (2019), [arXiv:1901.04005].
[130] S. Catani et al., JHEP 03, 029 (2021), [arXiv:2010.11906].
[131] T. Gehrmann et al., Phys. Rev. Lett. 113, 21, 212001 (2014), [arXiv:1408.5243].
[132] F. Cascioli et al., Phys. Lett. B735, 311 (2014), [arXiv:1405.2219].
[133] M. Grazzini, S. Kallweit and M. Wiesemann, Eur. Phys. J. C78, 7, 537 (2018),[arXiv:1711.06631].
[134] G. Heinrich, Phys. Rept. 922, 1 (2021), [arXiv:2009.00516].
[135] A. Huss et al., J. Phys. G 50, 4, 043001 (2023), [arXiv:2207.02122].
[136] M. Greco and A. Vicini, Nucl. Phys. B415, 386 (1994).
[137] A. Hebecker, Phys. Rept. 331, 1 (2000), [hep-ph/9905226].
[138] A. V. Belitsky and A. V. Radyushkin, Phys. Rept. 418, 1 (2005), [hep-ph/0504030].
[139] E. Boos et al. (CompHEP), Nucl. Instrum. Meth. A534, 250 (2004), [hep-ph/0403113].
[140] J. Alwall et al., JHEP 07, 079 (2014), [arXiv:1405.0301]; https://launchpad.net/
mg5amcnlo.
[141] M. L. Mangano et al., JHEP 07, 001 (2003), [hep-ph/0206293]; http://cern.ch/mlm/
alpgen/.
[142] T. Gleisberg and S. Hoeche, JHEP 12, 039 (2008), [arXiv:0808.3674]; https://sherpa-team.
gitlab.io/.
[143] A. Cafarella, C. G. Papadopoulos and M. Worek, Comput. Phys. Commun. 180, 1941 (2009),[arXiv:0710.2427]; http://cern.ch/helac-phegas/.
[144] F. A. Berends and W. T. Giele, Nucl. Phys. B306, 759 (1988).
[145] L. J. Dixon, in “QCD and beyond. Proceedings, Theoretical Advanced Study Institute in
Elementary Particle Physics, TASI-95, Boulder, USA, June 4-30, 1995,” 539–584 (1996),[hep-ph/9601359], URL http://www-public.slac.stanford.edu/sciDoc/docMeta.aspx?
slacPubNumber=SLAC-PUB-7106.
[146] R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B715, 499 (2005), [hep-th/0412308].
[147] F. Cachazo, P. Svrcek and E. Witten, JHEP 09, 006 (2004), [hep-th/0403047].
[148] S. Catani and M. H. Seymour, Nucl. Phys. B485, 291 (1997), [Erratum:
Phys.B510,503(1998)], [hep-ph/9605323].
31st May, 2024

Nucl.

47

9. Quantum Chromodynamics[149] S. Frixione, Z. Kunszt and A. Signer, Nucl. Phys. B467, 399 (1996), [hep-ph/9512328].
[150] D. A. Kosower, Phys. Rev. D57, 5410 (1998), [hep-ph/9710213]; J. M. Campbell, M. A.
Cullen and E. W. N. Glover, Eur. Phys. J. C9, 245 (1999), [hep-ph/9809429]; D. A. Kosower,
Phys. Rev. D71, 045016 (2005), [hep-ph/0311272].
[151] G. Ossola, C. G. Papadopoulos and R. Pittau, Nucl. Phys. B763, 147 (2007), [hepph/0609007].
[152] R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B725, 275 (2005), [hep-th/0412103].
[153] R. K. Ellis et al., Nucl. Phys. B822, 270 (2009), [arXiv:0806.3467].
[154] C. F. Berger and D. Forde, Ann. Rev. Nucl. Part. Sci. 60, 181 (2010), [arXiv:0912.3534].
[155] F. Cascioli, P. Maierhofer and S. Pozzorini, Phys. Rev. Lett. 108, 111601 (2012),[arXiv:1111.5206].
[156] Z. Bern, L. J. Dixon and D. A. Kosower, Annals Phys. 322, 1587 (2007), [arXiv:0704.2798].
[157] R. K. Ellis et al., Phys. Rept. 518, 141 (2012), [arXiv:1105.4319].
[158] G. Bevilacqua et al., Comput. Phys. Commun. 184, 986 (2013), [arXiv:1110.1499]; http://cern.ch/helac-phegas/.
[159] G. Cullen et al., Eur. Phys. J. C74, 8, 3001 (2014), [arXiv:1404.7096]; http://gosam.
hepforge.org/.
[160] S. Badger et al., Comput. Phys. Commun. 184, 1981 (2013), [arXiv:1209.0100]; https://
bitbucket.org/njet/wiki/Home/.
[161] F. Buccioni, S. Pozzorini and M. Zoller, Eur. Phys. J. C78, 1, 70 (2018), [arXiv:1710.11452];
https://openloops.hepforge.org/.
[162] S. Actis et al., Comput. Phys. Commun. 214, 140 (2017), [arXiv:1605.01090].
[163] Z. Nagy, Phys. Rev. D68, 094002 (2003), [hep-ph/0307268]; http://www.desy.de/~znagy/
Site/NLOJet++.html.
[164] J. M. Campbell and R. K. Ellis, Phys. Rev. D62, 114012 (2000), [hep-ph/0006304].
[165] J. Baglio et al. (2011), [arXiv:1107.4038]; http://www-itp.particle.uni-karlsruhe.de/~vbfnloweb.
[166] T. Binoth et al., Eur. Phys. J. C16, 311 (2000), [hep-ph/9911340]; http://lapth.in2p3.
fr/PHOX_FAMILY/.
[167] Z. Bern et al., PoS LL2012, 018 (2012), [arXiv:1210.6684].
[168] G. Cullen, N. Greiner and G. Heinrich, Eur. Phys. J. C73, 4, 2388 (2013), [arXiv:1212.5154].
[169] S. Kallweit et al., JHEP 04, 012 (2015), [arXiv:1412.5157].
[170] A. Denner et al., JHEP 04, 018 (2015), [arXiv:1412.7421].
[171] M. Chiesa, N. Greiner and F. Tramontano,[arXiv:1507.08579].

J. Phys. G43,

1,013002 (2016),[172] S. Frixione et al., JHEP 06, 184 (2015), [arXiv:1504.03446].
[173] B. Biedermann et al., Eur. Phys. J. C77, 492 (2017), [arXiv:1704.05783].
[174] R. Frederix et al., JHEP 07, 185 (2018), [Erratum: JHEP 11, 085 (2021)], [arXiv:1804.10017].
[175] C. Anastasiou, R. Boughezal and F. Petriello, JHEP 04, 003 (2009), [arXiv:0811.3458].
[176] S. Dittmaier, A. Huss and C. Schwinn, Nucl. Phys. B885, 318 (2014), [arXiv:1403.3216].
[177] S. Dittmaier, A. Huss and C. Schwinn, Nucl. Phys. B904, 216 (2016), [arXiv:1511.08016].
31st May, 2024

48

9. Quantum Chromodynamics[178] D. de Florian, M. Der and I. Fabre, Phys. Rev. D98, 9, 094008 (2018), [arXiv:1805.12214].
[179] M. Bonetti, K. Melnikov and L. Tancredi, Phys. Rev. D97, 3, 034004 (2018),[arXiv:1711.11113].
[180] M. Bonetti, K. Melnikov and L. Tancredi, Phys. Rev. D97, 5, 056017 (2018), [Erratum: Phys.
Rev.D97,no.9,099906(2018)], [arXiv:1801.10403].
[181] C. Anastasiou et al., JHEP 03, 162 (2019), [arXiv:1811.11211].
[182] V. Hirschi, S. Lionetti and A. Schweitzer, JHEP 05, 002 (2019), [arXiv:1902.10167].
[183] M. Becchetti et al., Phys. Rev. D 103, 5, 054037 (2021), [arXiv:2010.09451].
[184] M. Bonetti et al., JHEP 11, 045 (2020), [arXiv:2007.09813].
[185] A. Behring et al., Phys. Rev. D 103, 1, 013008 (2021), [arXiv:2009.10386].
[186] S. Dittmaier, T. Schmidt and J. Schwarz, JHEP 12, 201 (2020), [arXiv:2009.02229].
[187] F. Buccioni et al., Phys. Lett. B 811, 135969 (2020), [arXiv:2005.10221].
[188] L. Buonocore et al., Phys. Rev. D 103, 114012 (2021), [arXiv:2102.12539].
[189] R. Bonciani et al., Phys. Rev. Lett. 128, 1, 012002 (2022), [arXiv:2106.11953].
[190] T. Armadillo et al., JHEP 05, 072 (2022), [arXiv:2201.01754].
[191] F. Buccioni et al., JHEP 06, 022 (2022), [arXiv:2203.11237].
[192] A. Denner and S. Dittmaier, Phys. Rept. 864, 1 (2020), [arXiv:1912.06823].
[193] Z. Bern et al., Nucl. Phys. B425, 217 (1994), [hep-ph/9403226].
[194] J. M. Campbell and E. W. N. Glover, Nucl. Phys. B527, 264 (1998), [hep-ph/9710255].
[195] S. Catani and M. Grazzini, Phys. Lett. B446, 143 (1999), [hep-ph/9810389].
[196] T. Binoth and G. Heinrich, Nucl. Phys. B585, 741 (2000), [hep-ph/0004013].
[197] C. Anastasiou, K. Melnikov and F. Petriello, Phys. Rev. D69, 076010 (2004), [hepph/0311311].
[198] A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, JHEP 09, 056 (2005), [hepph/0505111].
[199] G. Somogyi, Z. Trocsanyi and V. Del Duca, JHEP 01, 070 (2007), [hep-ph/0609042].
[200] M. Czakon, Phys. Lett. B693, 259 (2010), [arXiv:1005.0274].
[201] F. Caola, K. Melnikov and R. Röntsch, Eur. Phys. J. C 77, 4, 248 (2017), [arXiv:1702.01352].
[202] L. Magnea et al., JHEP 12, 107 (2018), [Erratum: JHEP 06, 013 (2019)], [arXiv:1806.09570].
[203] R. M. Prisco and F. Tramontano, JHEP 06, 089 (2021), [arXiv:2012.05012].
[204] G. Bertolotti et al. (2022), [arXiv:2212.11190].
[205] C. Anastasiou and G. Sterman, JHEP 05, 242 (2023), [arXiv:2212.12162].
[206] S. Catani and M. Grazzini, Phys. Rev. Lett. 98, 222002 (2007), [hep-ph/0703012]; http://theory.fi.infn.it/grazzini/codes.html.
[207] R. Bonciani et al., Eur. Phys. J. C 75, 12, 581 (2015), [arXiv:1508.03585].
[208] R. Boughezal et al., Phys. Rev. Lett. 115, 6, 062002 (2015), [arXiv:1504.02131].
[209] R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91, 9, 094035 (2015), [arXiv:1504.02540].
[210] J. Gaunt et al., JHEP 09, 058 (2015), [arXiv:1505.04794].
[211] M. Cacciari et al., Phys. Rev. Lett. 115, 8, 082002 (2015), [Erratum:
Lett.120,no.13,139901(2018)], [arXiv:1506.02660].
31st May, 2024

Phys. Rev.

49

9. Quantum Chromodynamics[212] Z. Capatti, V. Hirschi and B. Ruijl, JHEP 10, 120 (2022), [arXiv:2203.11038].
[213] L. Cieri et al., JHEP 09, 155 (2020), [arXiv:2005.01315].
[214] W. J. Torres Bobadilla et al., Eur. Phys. J. C 81, 3, 250 (2021), [arXiv:2012.02567].
[215] A. Gehrmann-De Ridder et al., Phys. Rev. Lett. 99, 132002 (2007), [arXiv:0707.1285];
A. Gehrmann-De Ridder et al., JHEP 12, 094 (2007), [arXiv:0711.4711]; A. GehrmannDe Ridder et al., Phys. Rev. Lett. 100, 172001 (2008), [arXiv:0802.0813].
[216] A. Gehrmann-De Ridder et al., Comput. Phys. Commun. 185, 3331 (2014), [arXiv:1402.4140];
https://eerad3.hepforge.org/.
[217] S. Weinzierl, Phys. Rev. Lett. 101, 162001 (2008), [arXiv:0807.3241]; S. Weinzierl, JHEP 06,
041 (2009), [arXiv:0904.1077].
[218] V. Del Duca et al., Phys. Rev. D 94, 7, 074019 (2016), [arXiv:1606.03453]; V. Del Duca et al.,Phys. Rev. Lett. 117, 15, 152004 (2016), [arXiv:1603.08927].
[219] J. Currie, T. Gehrmann and J. Niehues, Phys. Rev. Lett. 117, 4, 042001 (2016),[arXiv:1606.03991].
[220] J. Currie et al., JHEP 07, 018 (2017), [Erratum: JHEP 12, 042 (2020)], [arXiv:1703.05977].
[221] J. Currie et al., JHEP 05, 209 (2018), [arXiv:1803.09973].
[222] T. Gehrmann et al., Phys. Lett. B792, 182 (2019), [arXiv:1812.06104].
[223] K. Melnikov and F. Petriello, Phys. Rev. D74, 114017 (2006), [hep-ph/0609070]; http://gate.hep.anl.gov/fpetriello/FEWZ.html.
[224] S. Catani et al., Phys. Rev. Lett. 103, 082001 (2009), [arXiv:0903.2120]; http://theory.
fi.infn.it/gazzani/dy.html.
[225] C. Anastasiou, K. Melnikov and F. Petriello, Nucl. Phys. B724, 197 (2005), [hep-ph/0501130];
http://www.phys.ethz.ch/~pheno/fehipro/.
[226] M. Czakon et al. (2021), [arXiv:2105.04436].
[227] M. Grazzini et al., JHEP 05, 139 (2017), [arXiv:1703.09065].
[228] S. Catani et al., Phys. Rev. Lett. 108, 072001 (2012), [Erratum:
Lett.117,no.8,089901(2016)], [arXiv:1110.2375].

Phys. Rev.

[229] J. M. Campbell et al., JHEP 07, 148 (2016), [arXiv:1603.02663].
[230] M. Grazzini et al., Phys. Lett. B731, 204 (2014), [arXiv:1309.7000].
[231] M. Grazzini, S. Kallweit and D. Rathlev, JHEP 07, 085 (2015), [arXiv:1504.01330].
[232] J. M. Campbell and R. K. Ellis, Phys. Rev. D 60, 113006 (1999), [hep-ph/9905386].
[233] R. Boughezal et al., Eur. Phys. J. C77, 1, 7 (2017), [arXiv:1605.08011].
[234] J. M. Campbell, R. K. Ellis and C. Williams, Phys. Rev. Lett. 118, 22, 222001 (2017),[arXiv:1612.04333].
[235] X. Chen et al., Submitted to: J. High Energy Phys. (2019), [arXiv:1904.01044].
[236] J. M. Campbell, R. K. Ellis and C. Williams, Phys. Rev. D96, 1, 014037 (2017),[arXiv:1703.10109].
[237] A. Gehrmann-De Ridder et al., Phys. Rev. Lett. 120, 12, 122001 (2018), [arXiv:1712.07543].
[238] A. Gehrmann-De Ridder et al., Phys. Rev. Lett. 117, 2, 022001 (2016), [arXiv:1507.02850].
[239] R. Boughezal et al., Phys. Rev. Lett. 116, 15, 152001 (2016), [arXiv:1512.01291].
[240] R. Boughezal et al., Phys. Rev. Lett. 115, 8, 082003 (2015), [arXiv:1504.07922].
31st May, 2024

50

9. Quantum Chromodynamics[241] R. Boughezal et al., Phys. Lett. B748, 5 (2015), [arXiv:1505.03893].
[242] F. Caola, K. Melnikov and M. Schulze, Phys. Rev. D92, 7, 074032 (2015), [arXiv:1508.02684].
[243] X. Chen et al., JHEP 10, 066 (2016), [arXiv:1607.08817].
[244] G. Ferrera, M. Grazzini and F. Tramontano, Phys. Rev. Lett. 107, 152003 (2011),[arXiv:1107.1164].
[245] G. Ferrera, M. Grazzini and F. Tramontano, Phys. Lett. B740, 51 (2015), [arXiv:1407.4747].
[246] Z. L. Liu and J. Gao, Phys. Rev. D 98, 7, 071501 (2018), [arXiv:1807.03835].
[247] M. Brucherseifer, F. Caola and K. Melnikov, Phys. Lett. B736, 58 (2014), [arXiv:1404.7116].
[248] E. L. Berger et al., Phys. Rev. D94, 7, 071501 (2016), [arXiv:1606.08463].
[249] E. L. Berger, J. Gao and H. X. Zhu, JHEP 11, 158 (2017), [arXiv:1708.09405].
[250] J. Campbell, T. Neumann and Z. Sullivan, JHEP 02, 040 (2021), [arXiv:2012.01574].
[251] M. Czakon, P. Fiedler and A. Mitov, Phys. Rev. Lett. 115, 5, 052001 (2015),[arXiv:1411.3007].
[252] M. Czakon et al., JHEP 05, 034 (2016), [arXiv:1601.05375].
[253] J. Currie, E. W. N. Glover and J. Pires, Phys. Rev. Lett. 118, 7, 072002 (2017),[arXiv:1611.01460].
[254] X. Chen et al., JHEP 09, 025 (2022), [arXiv:2204.10173].
[255] M. Czakon et al. (2020), [arXiv:2011.01011].
[256] M. Czakon et al., JHEP 02, 241 (2023), [arXiv:2212.00467].
[257] M. L. Czakon et al. (2021), [arXiv:2102.08267].
[258] J. Cruz-Martinez et al., Phys. Lett. B781, 672 (2018), [arXiv:1802.02445].
[259] T. Liu, K. Melnikov and A. A. Penin (2019), [arXiv:1906.10899].
[260] H. A. Chawdhry et al., JHEP 02, 057 (2020), [arXiv:1911.00479].
[261] S. Kallweit, V. Sotnikov and M. Wiesemann, Phys. Lett. B 812, 136013 (2021),[arXiv:2010.04681].
[262] H. A. Chawdhry et al., JHEP 09, 093 (2021), [arXiv:2105.06940].
[263] S. Badger et al. (2023), [arXiv:2304.06682].
[264] M. Czakon, A. Mitov and R. Poncelet, Phys. Rev. Lett. 127, 15, 152001 (2021), [Erratum:
Phys.Rev.Lett. 129, 119901 (2022)], [arXiv:2106.05331].
[265] H. B. Hartanto et al., Phys. Rev. D 106, 7, 074016 (2022), [arXiv:2205.01687].
[266] M. Alvarez et al., JHEP 03, 129 (2023), [arXiv:2301.01086].
[267] G. P. Salam and E. Slade, JHEP 11, 220 (2021), [arXiv:2106.08329].
[268] M. A. Ebert et al., JHEP 04, 102 (2021), [arXiv:2006.11382].
[269] S. Amoroso et al., in “11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV
Les Houches,” (2020), [arXiv:2003.01700].
[270] S. Catani et al., Phys. Rev. Lett. 130, 11, 111902 (2023), [arXiv:2210.07846].
[271] S. Camarda et al., Eur. Phys. J. C 80, 3, 251 (2020), [Erratum: Eur.Phys.J.C 80, 440 (2020)],[arXiv:1910.07049].
[272] R. Gavin et al., Comput. Phys. Commun. 182, 2388 (2011), [arXiv:1011.3540].
[273] S. Camarda et al., Eur. Phys. J. C 82, 5, 492 (2022), [arXiv:2202.10343].
31st May, 2024

51

9. Quantum Chromodynamics[274] R. V. Harlander, S. Liebler and H. Mantler, Comput. Phys. Commun. 184, 1605 (2013),[arXiv:1212.3249].
[275] T. Carli et al., Eur. Phys. J. C 66, 503 (2010), [arXiv:0911.2985].
[276] T. Kluge, K. Rabbertz and M. Wobisch, in “14th International Workshop on Deep Inelastic
Scattering (DIS 2006),” 483, Tsukuba, Japan, April 20-24 (2006), [hep-ph/0609285].
[277] D. Britzger et al., in “Proceedings, XX. International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS 2012),” 217, Bonn, Germany, March 26-30 (2012),[arXiv:1208.3641].
[278] S. Carrazza et al., JHEP 12, 108 (2020), [arXiv:2008.12789].
[279] M. Czakon, D. Heymes and A. Mitov (2017), [arXiv:1704.08551].
[280] D. Britzger et al., Eur. Phys. J. C 79, 10, 845 (2019), [Erratum: Eur.Phys.J.C 81, 957 (2021)],[arXiv:1906.05303].
[281] D. Maître, J. Phys. Conf. Ser. 1525, 1, 012014 (2020).
[282] D. Maître, G. Heinrich and M. Johnson, PoS LL2016, 016 (2016), [arXiv:1607.06259].
[283] Y. L. Dokshitzer, D. Diakonov and S. I. Troian, Phys. Rept. 58, 269 (1980).
[284] G. Parisi and R. Petronzio, Nucl. Phys. B154, 427 (1979).
[285] G. Curci, M. Greco and Y. Srivastava, Nucl. Phys. B159, 451 (1979).
[286] A. Bassetto, M. Ciafaloni and G. Marchesini, Nucl. Phys. B163, 477 (1980).
[287] J. C. Collins and D. E. Soper, Nucl. Phys. B193, 381 (1981), [Erratum:
Phys.B213,545(1983)].

Nucl.

[288] J. C. Collins and D. E. Soper, Nucl. Phys. B197, 446 (1982).
[289] J. Kodaira and L. Trentadue, Phys. Lett. 112B, 66 (1982).
[290] J. Kodaira and L. Trentadue, Phys. Lett. 123B, 335 (1983).
[291] J. C. Collins, D. E. Soper and G. F. Sterman, Nucl. Phys. B250, 199 (1985).
[292] S. Catani et al., Nucl. Phys. B407, 3 (1993).
[293] C. W. Bauer et al., Phys. Rev. D63, 114020 (2001), [hep-ph/0011336].
[294] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D65, 054022 (2002), [hep-ph/0109045].
[295] T. Becher, A. Broggio and A. Ferroglia, Lect. Notes Phys. 896, pp.1 (2015), [arXiv:1410.1892].
[296] S. Catani et al., Phys. Lett. B269, 432 (1991).
[297] N. Brown and W. J. Stirling, Phys. Lett. B252, 657 (1990).
[298] W. Bartel et al. (JADE), Z. Phys. C33, 23 (1986), [,53(1986)].
[299] N. Kidonakis, G. Oderda and G. F. Sterman, Nucl. Phys. B531, 365 (1998), [hepph/9803241].
[300] R. Bonciani et al., Phys. Lett. B575, 268 (2003), [hep-ph/0307035].
[301] A. Banfi, G. P. Salam and G. Zanderighi, JHEP 03, 073 (2005), [hep-ph/0407286].
[302] D. de Florian and M. Grazzini, Phys. Rev. Lett. 85, 4678 (2000), [hep-ph/0008152].
[303] G. Bozzi et al., Nucl. Phys. B737, 73 (2006), [hep-ph/0508068]; http://theory.fi.infn.
it/grazzini/codes.html.
[304] G. Bozzi et al., Phys. Lett. B696, 207 (2011), [arXiv:1007.2351].
[305] T. Becher and M. Neubert, Eur. Phys. J. C71, 1665 (2011), [arXiv:1007.4005].
[306] http://cute.hepforge.org/.
31st May, 2024

52

9. Quantum Chromodynamics[307] D. de Florian et al., JHEP 06, 132 (2012), [arXiv:1203.6321]; http://theory.fi.infn.it/
grazzini/codes.html.
[308] C. Balazs and C. P. Yuan, Phys. Rev. D56, 5558 (1997), [hep-ph/9704258].
[309] S. Catani et al., JHEP 12, 047 (2015), [arXiv:1507.06937].
[310] A. Banfi et al., Phys. Lett. B715, 152 (2012), [arXiv:1205.4760].
[311] M. Grazzini et al., JHEP 08, 154 (2015), [arXiv:1507.02565].
[312] J. M. Campbell et al., JHEP 03, 080 (2023), [arXiv:2210.10724].
[313] D. de Florian and M. Grazzini, Nucl. Phys. B704, 387 (2005), [hep-ph/0407241].
[314] T. Becher and G. Bell, JHEP 11, 126 (2012), [arXiv:1210.0580].
[315] A. Banfi et al., Phys. Rev. Lett. 109, 202001 (2012), [arXiv:1206.4998]; T. Becher, M. Neubertand L. Rothen, JHEP 10, 125 (2013), [arXiv:1307.0025].
[316] I. W. Stewart et al., Phys. Rev. D 89, 5, 054001 (2014), [arXiv:1307.1808].
[317] J. M. Campbell et al., JHEP 04, 106 (2023), [arXiv:2301.11768].
[318] I. W. Stewart, F. J. Tackmann and W. J. Waalewijn, Phys. Rev. Lett. 106, 032001 (2011),[arXiv:1005.4060].
[319] Y.-T. Chien et al., Phys. Rev. D87, 1, 014010 (2013), [arXiv:1208.0010]; T. T. Jouttenus
et al., Phys. Rev. D88, 5, 054031 (2013), [arXiv:1302.0846].
[320] M. Dasgupta et al., JHEP 10, 126 (2012), [arXiv:1207.1640].
[321] V. Ahrens et al., JHEP 09, 097 (2010), [arXiv:1003.5827].
[322] M. Aliev et al., Comput. Phys. Commun. 182, 1034 (2011), [arXiv:1007.1327].
[323] N. Kidonakis, Phys. Rev. D82, 114030 (2010), [arXiv:1009.4935].
[324] T. Becher, C. Lorentzen and M. D. Schwartz, Phys. Rev. Lett. 108, 012001 (2012),[arXiv:1106.4310].
[325] T. Becher et al., Eur. Phys. J. C75, 4, 154 (2015), [arXiv:1412.8408].
[326] E. Gerwick et al., JHEP 02, 106 (2015), [arXiv:1411.7325].
[327] A. Banfi et al., JHEP 05, 102 (2015), [arXiv:1412.2126].
[328] T. Becher and M. D. Schwartz, JHEP 07, 034 (2008), [arXiv:0803.0342].
[329] A. H. Hoang et al., Phys. Rev. D91, 9, 094017 (2015), [arXiv:1411.6633].
[330] Y.-T. Chien and M. D. Schwartz, JHEP 08, 058 (2010), [arXiv:1005.1644].
[331] W. Bizon et al., JHEP 02, 108 (2018), [arXiv:1705.09127].
[332] X. Chen et al., Phys. Lett. B 788, 425 (2019), [arXiv:1805.00736].
[333] W. Bizon et al., Eur. Phys. J. C 79, 10, 868 (2019), [arXiv:1905.05171].
[334] T. Neumann and J. Campbell, Phys. Rev. D 107, 1, L011506 (2023), [arXiv:2207.07056].
[335] E. Re, L. Rottoli and P. Torrielli (2021), [arXiv:2104.07509].
[336] S. Catani et al., Nucl. Phys. B888, 75 (2014), [arXiv:1405.4827].
[337] A. Banfi et al., JHEP 04, 049 (2016), [arXiv:1511.02886].
[338] A. J. Larkoski and I. Moult, Phys. Rev. D93, 014017 (2016), [arXiv:1510.08459].
[339] G. Lustermans, W. J. Waalewijn and L. Zeune, Phys. Lett. B762, 447 (2016),[arXiv:1605.02740].
[340] C. Muselli, S. Forte and G. Ridolfi, JHEP 03, 106 (2017), [arXiv:1701.01464].
31st May, 2024

53

9. Quantum Chromodynamics[341] W. Bizoń et al., JHEP 12, 132 (2018), [arXiv:1805.05916].
[342] M. Bonvini and S. Marzani, Phys. Rev. Lett. 120, 20, 202003 (2018), [arXiv:1802.07758].
[343] M. Procura, W. J. Waalewijn and L. Zeune, JHEP 10, 098 (2018), [arXiv:1806.10622].
[344] G. Lustermans et al., JHEP 03, 124 (2019), [arXiv:1901.03331].
[345] P. F. Monni, L. Rottoli and P. Torrielli, Phys. Rev. Lett. 124, 25, 252001 (2020),[arXiv:1909.04704].
[346] T. Becher and T. Neumann, JHEP 03, 199 (2021), [arXiv:2009.11437].
[347] M. Dasgupta et al., JHEP 09, 029 (2013), [arXiv:1307.0007].
[348] A. J. Larkoski et al., JHEP 05, 146 (2014), [arXiv:1402.2657].
[349] A. J. Larkoski, I. Moult and B. Nachman, Phys. Rept. 841, 1 (2020), [arXiv:1709.04464].
[350] S. Marzani, G. Soyez and M. Spannowsky (2019), [Lect. Notes Phys.958,pp.(2019)],[arXiv:1901.10342].
[351] R. Kogler et al., Rev. Mod. Phys. 91, 4, 045003 (2019), [arXiv:1803.06991].
[352] Yu.L. Dokshitzer et al., “Basics of perturbative QCD,” Gif-sur-Yvette, France: Éditions frontières (1991), see also http://www.lpthe.jussieu.fr/~{}yuri/ BPQCD/cover.html.
[353] T. Sj ostrand et al., Comput. Phys. Commun. 135, 238 (2001), [hep-ph/0010017].
[354] T. Sj ostrand, S. Mrenna and P. Z. Skands, JHEP 05, 026 (2006), [hep-ph/0603175]; http://projects.hepforge.org/pythia6/.
[355] T. Sjostrand et al., Comput. Phys. Commun. 191, 159 (2015), [arXiv:1410.3012]; http://home.thep.lu.se/~torbjorn/Pythia.html.
[356] B. R. Webber, Nucl. Phys. B238, 492 (1984).
[357] G. Corcella et al., JHEP 01, 010 (2001), [hep-ph/0011363]; http://www.hep.phy.cam.ac.
uk/theory/webber/Herwig/.
[358] M. Bahr et al., Eur. Phys. J. C58, 639 (2008), [arXiv:0803.0883]; http://projects.
hepforge.org/herwig/.
[359] T. Gleisberg et al., JHEP 02, 007 (2009), [arXiv:0811.4622].
[360] L. Lonnblad, Comput. Phys. Commun. 71, 15 (1992).
[361] A. Buckley et al., Phys. Rept. 504, 145 (2011), [arXiv:1101.2599].
[362] J. Bellm et al., Eur. Phys. J. C 80, 2, 93 (2020), [arXiv:1903.12563].
[363] B. Andersson et al., Phys. Rept. 97, 31 (1983).
[364] T. Sjostrand, Nucl. Phys. B248, 469 (1984).
[365] T. Sjostrand and M. van Zijl, Phys. Rev. D36, 2019 (1987).
[366] S. Catani et al., JHEP 11, 063 (2001), [hep-ph/0109231].
[367] J. Alwall et al., Eur. Phys. J. C53, 473 (2008), [arXiv:0706.2569].
[368] S. Frixione and B. R. Webber, JHEP 06, 029 (2002), [hep-ph/0204244].
[369] P. Nason, JHEP 11, 040 (2004), [hep-ph/0409146].
[370] S. Alioli et al., JHEP 06, 043 (2010), [arXiv:1002.2581]; http://powhegbox.mib.infn.it/.
[371] S. Plätzer, JHEP 08, 114 (2013), [arXiv:1211.5467]; R. Frederix and S. Frixione, JHEP 12,
061 (2012), [arXiv:1209.6215]; K. Hamilton et al., JHEP 05, 082 (2013), [arXiv:1212.4504].
[372] P. F. Monni et al., JHEP 05, 143 (2020), [arXiv:1908.06987].
[373] S. Alioli et al., JHEP 06, 089 (2014), [arXiv:1311.0286].
31st May, 2024

54

9. Quantum Chromodynamics[374] K. Hamilton et al., JHEP 10, 222 (2013), [arXiv:1309.0017].
[375] A. Karlberg, E. Re and G. Zanderighi, JHEP 09, 134 (2014), [arXiv:1407.2940]; S. Höche,Y. Li and S. Prestel, Phys. Rev. D91, 7, 074015 (2015), [arXiv:1405.3607]; S. Höche, Y. Li
and S. Prestel, Phys. Rev. D90, 5, 054011 (2014), [arXiv:1407.3773]; S. Alioli et al., Phys.
Rev. D92, 9, 094020 (2015), [arXiv:1508.01475]; S. Alioli et al. (2021), [arXiv:2102.08390];
S. Alioli et al., JHEP 05, 128 (2023), [arXiv:2301.11875].
[376] W. Astill et al., JHEP 06, 154 (2016), [arXiv:1603.01620].
[377] W. Astill et al., JHEP 11, 157 (2018), [arXiv:1804.08141].
[378] E. Re, M. Wiesemann and G. Zanderighi, JHEP 12, 121 (2018), [arXiv:1805.09857].
[379] S. Alioli et al., Phys. Rev. D 100, 9, 096016 (2019), [arXiv:1909.02026].
[380] D. Lombardi, M. Wiesemann and G. Zanderighi (2020), [arXiv:2010.10478].
[381] D. Lombardi, M. Wiesemann and G. Zanderighi, Phys. Lett. B 824, 136846 (2022),[arXiv:2108.11315].
[382] S. Alioli et al., JHEP 04, 041 (2021), [arXiv:2010.10498].
[383] D. Lombardi, M. Wiesemann and G. Zanderighi (2021), [arXiv:2103.12077].
[384] S. Alioli et al., Phys. Lett. B 818, 136380 (2021), [arXiv:2103.01214].
[385] S. Alioli et al. (2022), [arXiv:2212.10489].
[386] J. M. Lindert et al., JHEP 11, 036 (2022), [arXiv:2208.12660].
[387] A. Gavardi, C. Oleari and E. Re, JHEP 09, 061 (2022), [arXiv:2204.12602].
[388] L. Buonocore et al., JHEP 01, 072 (2022), [arXiv:2108.05337].
[389] U. Haisch et al., JHEP 07, 054 (2022), [arXiv:2204.00663].
[390] S. Zanoli et al., JHEP 07, 008 (2022), [arXiv:2112.04168].
[391] J. Mazzitelli et al. (2020), [arXiv:2012.14267].
[392] J. Mazzitelli et al., JHEP 04, 079 (2022), [arXiv:2112.12135].
[393] J. Mazzitelli et al., Phys. Lett. B 843, 137991 (2023), [arXiv:2302.01645].
[394] M. Dasgupta et al., JHEP 09, 033 (2018), [Erratum:[arXiv:1805.09327].

JHEP 03, 083 (2020)],[395] L. Hartgring, E. Laenen and P. Skands, JHEP 10, 127 (2013), [arXiv:1303.4974].
[396] H. T. Li and P. Skands, Phys. Lett. B 771, 59 (2017), [arXiv:1611.00013].
[397] S. Höche and S. Prestel, Phys. Rev. D 96, 7, 074017 (2017), [arXiv:1705.00742].
[398] F. Dulat, S. Höche and S. Prestel, Phys. Rev. D 98, 7, 074013 (2018), [arXiv:1805.03757].
[399] A. Banfi, B. K. El-Menoufi and P. F. Monni, JHEP 01, 083 (2019), [arXiv:1807.11487].
[400] G. Bewick et al., JHEP 04, 019 (2020), [arXiv:1904.11866].
[401] M. Dasgupta et al., Phys. Rev. Lett. 125, 5, 052002 (2020), [arXiv:2002.11114].
[402] K. Hamilton et al., JHEP 03, 041, 041 (2021), [arXiv:2011.10054].
[403] J. R. Forshaw, J. Holguin and S. Plätzer, JHEP 09, 014 (2020), [arXiv:2003.06400].
[404] Z. Nagy and D. E. Soper, Phys. Rev. D 104, 5, 054049 (2021), [arXiv:2011.04773].
[405] Z. Nagy and D. E. Soper (2020), [arXiv:2011.04777].
[406] A. Karlberg et al., Eur. Phys. J. C 81, 8, 681 (2021), [arXiv:2103.16526].
[407] K. Hamilton et al., JHEP 03, 193 (2022), [arXiv:2111.01161].
31st May, 2024

55

9. Quantum Chromodynamics[408] F. Herren et al. (2022), [arXiv:2208.06057].
[409] Z. Nagy and D. E. Soper, Phys. Rev. D 106, 1, 014024 (2022), [arXiv:2204.05631].
[410] M. van Beekveld et al., JHEP 11, 019 (2022), [arXiv:2205.02237].
[411] M. van Beekveld et al., JHEP 11, 020 (2022), [arXiv:2207.09467].
[412] M. van Beekveld and S. Ferrario Ravasio (2023), [arXiv:2305.08645].
[413] M. Cacciari et al., JHEP 04, 068 (2004), [hep-ph/0303085].
[414] M. Cacciari and N. Houdeau, JHEP 09, 039 (2011), [arXiv:1105.5152].
[415] A. David and G. Passarino, Phys. Lett. B726, 266 (2013), [arXiv:1307.1843].
[416] E. Bagnaschi et al., JHEP 02, 133 (2015), [arXiv:1409.5036].
[417] C. Duhr et al., JHEP 09, 122 (2021), [arXiv:2106.04585].
[418] A. Ghosh et al. (2022), [arXiv:2210.15167].
[419] G. Soyez et al., Phys. Rev. Lett. 110, 16, 162001 (2013), [arXiv:1211.2811].
[420] M. Dasgupta and G. P. Salam, J. Phys. G30, R143 (2004), [hep-ph/0312283].
[421] C. Bierlich et al., SciPost Phys. 8, 026 (2020), [arXiv:1912.05451].
[422] S. Moretti, L. Lonnblad and T. Sjostrand, JHEP 08, 001 (1998), [hep-ph/9804296].
[423] G. P. Salam, Eur. Phys. J. C67, 637 (2010), [arXiv:0906.1833].
[424] S. D. Ellis et al., Prog. Part. Nucl. Phys. 60, 484 (2008), [arXiv:0712.2447].
[425] M. Cacciari, Int. J. Mod. Phys. A30, 31, 1546001 (2015), [arXiv:1509.02272].
[426] A. Abulencia et al. (CDF), Phys. Rev. D 74, 071103 (2006), [hep-ex/0512020].
[427] G. P. Salam and G. Soyez, JHEP 05, 086 (2007), [arXiv:0704.0292].
[428] S. Catani et al., Nucl. Phys. B406, 187 (1993).
[429] S. D. Ellis and D. E. Soper, Phys. Rev. D48, 3160 (1993), [hep-ph/9305266].
[430] Y. L. Dokshitzer et al., JHEP 08, 001 (1997), [hep-ph/9707323].
[431] M. Wobisch and T. Wengler, in “Monte Carlo generators for HERA physics. Proceedings,
Workshop, Hamburg, Germany, 1998-1999,” 270–279 (1998), [hep-ph/9907280].
[432] M. Cacciari, G. P. Salam and G. Soyez, JHEP 04, 063 (2008), [arXiv:0802.1189].
[433] S. Bethke et al., Nucl. Phys. B370, 310 (1992), [Erratum: Nucl. Phys.B523,681(1998)].
[434] M. Cacciari and G. P. Salam, Phys. Lett. B641, 57 (2006), [hep-ph/0512210]; M. Cacciari,
G. P. Salam and G. Soyez, Eur. Phys. J. C72, 1896 (2012), [arXiv:1111.6097].
[435] A. Banfi, G. P. Salam and G. Zanderighi, Eur. Phys. J. C 47, 113 (2006), [hep-ph/0601139].
[436] A. Banfi, G. P. Salam and G. Zanderighi, JHEP 07, 026 (2007), [arXiv:0704.2999].
[437] M. Czakon, A. Mitov and R. Poncelet, JHEP 04, 138 (2023), [arXiv:2205.11879].
[438] R. Gauld, A. Huss and G. Stagnitto, Phys. Rev. Lett. 130, 16, 161901 (2023),[arXiv:2208.11138].
[439] S. Caletti et al., JHEP 10, 158 (2022), [arXiv:2205.01117].
[440] F. Caola et al. (2023), [arXiv:2306.07314].
[441] S. Brandt et al., Phys. Lett. 12, 57 (1964).
[442] E. Farhi, Phys. Rev. Lett. 39, 1587 (1977).
[443] O. Biebel, Phys. Rept. 340, 165 (2001).
[444] S. Kluth, Rept. Prog. Phys. 69, 1771 (2006), [hep-ex/0603011].
31st May, 2024

56

9. Quantum Chromodynamics[445] C. L. Basham et al., Phys. Rev. Lett. 41, 1585 (1978).
[446] A. Ali, E. Pietarinen and W. J. Stirling, Phys. Lett. 141B, 447 (1984).
[447] I. W. Stewart, F. J. Tackmann and W. J. Waalewijn, Phys. Rev. Lett. 105, 092002 (2010),[arXiv:1004.2489].
[448] A. Banfi, G. P. Salam and G. Zanderighi, JHEP 08, 062 (2004), [hep-ph/0407287].
[449] A. Banfi, G. P. Salam and G. Zanderighi, JHEP 06, 038 (2010), [arXiv:1001.4082].
[450] T. Becher, X. Garcia i Tormo and J. Piclum, Phys. Rev. D93, 5, 054038 (2016), [Erratum:
Phys. Rev.D93,no.7,079905(2016)], [arXiv:1512.00022].
[451] A. Gao et al., Phys. Rev. Lett. 123, 6, 062001 (2019), [arXiv:1901.04497].
[452] T. Aaltonen et al. (CDF), Phys. Rev. D83, 112007 (2011), [arXiv:1103.5143].
[453] G. Aad et al. (ATLAS), Eur. Phys. J. C72, 2211 (2012), [arXiv:1206.2135].
[454] G. Aad et al. (ATLAS), Phys. Rev. D88, 3, 032004 (2013), [arXiv:1207.6915].
[455] G. Aad et al. (ATLAS), Phys. Lett. B750, 427 (2015), [arXiv:1508.01579].
[456] G. Aad et al. (ATLAS), Eur. Phys. J. C76, 7, 375 (2016), [arXiv:1602.08980].
[457] M. Aaboud et al. (ATLAS), Eur. Phys. J. C77, 12, 872 (2017), [arXiv:1707.02562].
[458] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 9, 092004 (2018), [arXiv:1805.04691].
[459] G. Aad et al. (ATLAS), JHEP 07, 085 (2023), [arXiv:2301.09351].
[460] V. Khachatryan et al. (CMS), Phys. Lett. B699, 48 (2011), [arXiv:1102.0068].
[461] S. Chatrchyan et al. (CMS), Phys. Lett. B722, 238 (2013), [arXiv:1301.1646].
[462] V. Khachatryan et al. (CMS), JHEP 10, 87 (2014), [arXiv:1407.2856].
[463] A. M. Sirunyan et al. (CMS), JHEP 12, 117 (2018), [arXiv:1811.00588].
[464] A. Ali and G. Kramer, Eur. Phys. J. H 36, 245 (2011), [arXiv:1012.2288].
[465] S. Chatrchyan et al. (CMS), Phys. Lett. B730, 243 (2014), [arXiv:1310.0878].
[466] G. Aad et al. (ATLAS), Phys. Rev. D83, 052003 (2011), [arXiv:1101.0070].
[467] S. Chatrchyan et al. (CMS), JHEP 06, 160 (2012), [arXiv:1204.3170].
[468] B. B. Abelev et al. (ALICE), Phys. Rev. D91, 11, 112012 (2015), [arXiv:1411.4969].
[469] G. Aad et al. (ATLAS), Eur. Phys. J. C73, 12, 2676 (2013), [arXiv:1307.5749].
[470] C. Glasman (H1, ZEUS), Nucl. Phys. Proc. Suppl. 191, 121 (2009), [arXiv:0812.0757].
[471] T. Carli, K. Rabbertz and S. Schumann, in T. Schörner-Sadenius, editor, “The Large Hadron
Collider: Harvest of Run 1,” 139–194 (2015), [arXiv:1506.03239].
[472] P. Gras et al., JHEP 07, 091 (2017), [arXiv:1704.03878].
[473] A. Abdesselam et al., Eur. Phys. J. C71, 1661 (2011), [arXiv:1012.5412].
[474] D. Krohn, J. Thaler and L.-T. Wang, JHEP 02, 084 (2010), [arXiv:0912.1342].
[475] A. Altheimer et al., J. Phys. G39, 063001 (2012), [arXiv:1201.0008].
[476] A. Altheimer et al., Eur. Phys. J. C74, 3, 2792 (2014), [arXiv:1311.2708].
[477] P. C. Stichel and W. J. Zakrzewski, Eur. Phys. J. C75, 1, 9 (2015), [arXiv:1409.1363].
[478] D. Adams et al., Eur. Phys. J. C75, 9, 409 (2015), [arXiv:1504.00679].
[479] L. de Oliveira et al., JHEP 07, 069 (2016), [arXiv:1511.05190].
[480] F. A. Dreyer, G. P. Salam and G. Soyez, JHEP 12, 064 (2018), [arXiv:1807.04758].
[481] G. Aad et al. (ATLAS), Phys. Rev. Lett. 124, 22, 222002 (2020), [arXiv:2004.03540].
31st May, 2024

57

9. Quantum Chromodynamics[482] G. Louppe et al., JHEP 01, 057 (2019), [arXiv:1702.00748].
[483] D. Guest, K. Cranmer and D. Whiteson, Ann. Rev. Nucl. Part. Sci. 68, 161 (2018),[arXiv:1806.11484].
[484] A. Butter et al., SciPost Phys. 7, 014 (2019), [arXiv:1902.09914].
[485] P. T. Komiske, E. M. Metodiev and J. Thaler, JHEP 01, 121 (2019), [arXiv:1810.05165].
[486] T. Schorner-Sadenius,
J.C72,2133(2012)].

Eur. Phys. J. C72,

2060 (2012),[Erratum:Eur. Phys.

[487] J. M. Campbell, J. W. Huston and W. J. Stirling, Rept. Prog. Phys. 70, 89 (2007), [hepph/0611148].
[488] M. L. Mangano, Phys. Usp. 53, 109 (2010), [Usp. Fiz. Nauk180,113(2010)].
[489] J. M. Butterworth, G. Dissertori and G. P. Salam, Ann. Rev. Nucl. Part. Sci. 62, 387 (2012),[arXiv:1202.0583].
[490] T. Carli, T. Gehrmann and S. Hoeche, Eur. Phys. J. C67, 73 (2010), [arXiv:0912.3715].
[491] S. Chekanov et al. (ZEUS), Nucl. Phys. B792, 1 (2008), [arXiv:0707.3749].
[492] S. Chekanov et al. (ZEUS), Phys. Rev. D76, 072011 (2007), [arXiv:0706.3809].
[493] A. Aktas et al. (H1), Phys. Lett. B639, 21 (2006), [hep-ex/0603014].
[494] H. Abramowicz et al. (ZEUS), Eur. Phys. J. C71, 1659 (2011), [arXiv:1104.5444].
[495] H. Abramowicz et al. (ZEUS), Nucl. Phys. B864, 1 (2012), [arXiv:1205.6153].
[496] F. D. Aaron et al. (H1), Eur. Phys. J. C65, 363 (2010), [arXiv:0904.3870].
[497] F. D. Aaron et al. (H1), Eur. Phys. J. C54, 389 (2008), [arXiv:0711.2606].
[498] S. Chekanov et al. (ZEUS), Eur. Phys. J. C52, 515 (2007), [arXiv:0707.3093].
[499] S. Chekanov et al. (ZEUS), Phys. Rev. D78, 032004 (2008), [arXiv:0802.3955].
[500] H. Abramowicz et al. (ZEUS), Eur. Phys. J. C70, 965 (2010), [arXiv:1010.6167].
[501] H. Abramowicz et al. (ZEUS), Phys. Lett. B691, 127 (2010), [arXiv:1003.2923].
[502] S. Chekanov et al. (ZEUS), Phys. Rev. D85, 052008 (2012), [arXiv:0808.3783].
[503] F. D. Aaron et al. (H1), Eur. Phys. J. C67, 1 (2010), [arXiv:0911.5678].
[504] V. Andreev et al. (H1), Eur. Phys. J. C75, 2, 65 (2015), [arXiv:1406.4709].
[505] V. Andreev et al. (H1), Eur. Phys. J. C77, 11, 791 (2017), [Erratum: Eur. Phys. J. C81, 8,
738 (2021)], [arXiv:1709.07251].
[506] http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/SM.
[507] http://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined.
[508] A. Abulencia et al. (CDF), Phys. Rev. D75,
Rev.D75,119901(2007)], [hep-ex/0701051].

092006 (2007),[Erratum:Phys.

[509] T. Aaltonen et al. (CDF), Phys. Rev. D78,
Rev.D79,119902(2009)], [arXiv:0807.2204].

052006 (2008),[Erratum:Phys.

[510] V. M. Abazov et al. (D0), Phys. Rev. Lett. 101, 062001 (2008), [arXiv:0802.2400].
[511] V. M. Abazov et al. (D0), Phys. Rev. D85, 052006 (2012), [arXiv:1110.3771].
[512] B. Abelev et al. (ALICE), Phys. Lett. B722, 262 (2013), [arXiv:1301.3475].
[513] G. Aad et al. (ATLAS), Eur. Phys. J. C73, 8, 2509 (2013), [arXiv:1304.4739].
[514] G. Aad et al. (ATLAS), JHEP 02,[arXiv:1410.8857].

153 (2015),

31st May, 2024

[Erratum:

JHEP09,141(2015)],

58

9. Quantum Chromodynamics[515] M. Aaboud et al. (ATLAS), JHEP 09, 020 (2017), [arXiv:1706.03192].
[516] V. Khachatryan et al. (CMS), Eur. Phys. J. C76, 5, 265 (2016), [arXiv:1512.06212].
[517] S. Chatrchyan et al. (CMS), Phys. Rev. D87, 11, 112002 (2013), [Erratum:
Rev.D87,no.11,119902(2013)], [arXiv:1212.6660].

Phys.

[518] V. Khachatryan et al. (CMS), JHEP 03, 156 (2017), [arXiv:1609.05331].
[519] V. Khachatryan et al. (CMS), Eur. Phys. J. C76, 8, 451 (2016), [arXiv:1605.04436].
[520] M. Aaboud et al. (ATLAS), JHEP 05, 195 (2018), [arXiv:1711.02692].
[521] A. M. Sirunyan et al. (CMS), JHEP 12, 082 (2020), [arXiv:2005.05159].
[522] A. Tumasyan et al. (CMS), JHEP 02, 142 (2022), [Addendum: JHEP 12, 035 (2022)],[arXiv:2111.10431].
[523] J. Currie et al., JHEP 10, 155 (2018), [arXiv:1807.03692].
[524] A. Schwartzman, Int. J. Mod. Phys. A30, 31, 1546002 (2015), [arXiv:1509.05459].
[525] T.-J. Hou et al. (2019), [arXiv:1908.11238].
[526] R. Abdul Khalek et al., Eur. Phys. J. C 80, 8, 797 (2020), [arXiv:2005.11327].
[527] X. Jing et al. (2023), [arXiv:2306.03918].
[528] G. Aad et al. (ATLAS), JHEP 05, 059 (2014), [arXiv:1312.3524].
[529] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C77, 11, 746 (2017), [arXiv:1705.02628].
[530] T. Aaltonen et al. (CDF), Phys. Rev. D79, 112002 (2009), [arXiv:0812.4036].
[531] V. M. Abazov et al. (D0), Phys. Rev. Lett. 103, 191803 (2009), [arXiv:0906.4819].
[532] S. Chatrchyan et al. (CMS), JHEP 05, 055 (2012), [arXiv:1202.5535].
[533] V. Khachatryan et al. (CMS), Phys. Lett. B746, 79 (2015), [arXiv:1411.2646].
[534] A. M. Sirunyan et al. (CMS), JHEP 07, 013 (2017), [arXiv:1703.09986].
[535] G. Aad et al. (ATLAS), JHEP 01, 029 (2013), [arXiv:1210.1718].
[536] M. Aaboud et al. (ATLAS), Phys. Rev. D96, 5, 052004 (2017), [arXiv:1703.09127].
[537] V. M. Abazov et al. (D0), Phys. Rev. Lett. 94, 221801 (2005), [hep-ex/0409040].
[538] V. M. Abazov et al. (D0), Phys. Lett. B721, 212 (2013), [arXiv:1212.1842].
[539] G. Aad et al. (ATLAS), Phys. Rev. Lett. 106, 172002 (2011), [arXiv:1102.2696].
[540] V. Khachatryan et al. (CMS), Phys. Rev. Lett. 106, 122003 (2011), [arXiv:1101.5029].
[541] V. Khachatryan et al. (CMS), Eur. Phys. J. C76, 10, 536 (2016), [arXiv:1602.04384].
[542] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C78, 7, 566 (2018), [arXiv:1712.05471].
[543] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1902.04374].
[544] P. Kokkas, Int. J. Mod. Phys. A30, 31, 1546004 (2015), [arXiv:1509.02144].
[545] G. Aad et al. (ATLAS), JHEP 01, 188 (2021), [Erratum:[arXiv:2007.12600].

JHEP 12, 053 (2021)],[546] M. Aaboud et al. (ATLAS), Eur. Phys. J. C77, 6, 367 (2017), [arXiv:1612.03016].
[547] G. Aad et al. (ATLAS), Phys. Lett. B759, 601 (2016), [arXiv:1603.09222].
[548] R. Aaij et al. (LHCb), JHEP 08, 039 (2015), [arXiv:1505.07024].
[549] S. Chatrchyan et al. (CMS), JHEP 10, 132 (2011), [arXiv:1107.4789].
[550] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. 112, 191802 (2014), [arXiv:1402.0923].
[551] R. Aaij et al. (LHCb), JHEP 06, 058 (2012), [arXiv:1204.1620].
31st May, 2024

59

9. Quantum Chromodynamics[552] R. Aaij et al. (LHCb), JHEP 01, 155 (2016), [arXiv:1511.08039].
[553] R. Aaij et al. (LHCb), JHEP 09, 136 (2016), [arXiv:1607.06495].
[554] G. Aad et al. (ATLAS), Eur. Phys. J. C 80, 7, 616 (2020), [arXiv:1912.02844].
[555] A. M. Sirunyan et al. (CMS), JHEP 12, 061 (2019), [arXiv:1909.04133].
[556] S. Chatrchyan et al. (CMS), JHEP 10, 007 (2011), [arXiv:1108.0566].
[557] V. Khachatryan et al. (CMS), Eur. Phys. J. C75, 4, 147 (2015), [arXiv:1412.1115].
[558] G. Aad et al. (ATLAS), Phys. Lett. B725, 223 (2013), [arXiv:1305.4192].
[559] G. Aad et al. (ATLAS), JHEP 06, 112 (2014), [arXiv:1404.1212].
[560] G. Aad et al. (ATLAS), JHEP 08, 009 (2016), [arXiv:1606.01736].
[561] A. M. Sirunyan et al. (CMS), Submitted to: JHEP (2018), [arXiv:1812.10529].
[562] M. Aaboud et al. (ATLAS), JHEP 12, 059 (2017), [arXiv:1710.05167].
[563] S. Chatrchyan et al. (CMS), Phys. Rev. D90, 3, 032004 (2014), [arXiv:1312.6283].
[564] V. Khachatryan et al. (CMS), Eur. Phys. J. C76, 8, 469 (2016), [arXiv:1603.01803].
[565] G. Aad et al. (ATLAS), Submitted to: Eur. Phys. J. (2019), [arXiv:1904.05631].
[566] R. Aaij et al. (LHCb), JHEP 10, 030 (2016), [arXiv:1608.01484].
[567] G. Aad et al. (ATLAS), Eur. Phys. J. C76, 5, 291 (2016), [arXiv:1512.02192].
[568] V. Khachatryan et al. (CMS), JHEP 02, 096 (2017), [arXiv:1606.05864].
[569] A. M. Sirunyan et al. (CMS), JHEP 03, 172 (2018), [arXiv:1710.07955].
[570] M. Aaboud et al. (ATLAS), JHEP 05, 077 (2018), [arXiv:1711.03296].
[571] U. Blumenschein, Int. J. Mod. Phys. A30, 31, 1546007 (2015), [arXiv:1509.04885].
[572] G. Aad et al. (ATLAS) (2019), [arXiv:1907.06728].
[573] V. Khachatryan et al. (CMS), Phys. Rev. D95, 052002 (2017), [arXiv:1610.04222].
[574] A. M. Sirunyan et al. (CMS), Phys. Rev. D96, 7, 072005 (2017), [arXiv:1707.05979].
[575] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C78, 11, 965 (2018), [arXiv:1804.05252].
[576] G. Aad et al. (ATLAS), JHEP 06, 080 (2023), [arXiv:2205.02597].
[577] G. Aad et al. (ATLAS), JHEP 07, 032 (2013), [arXiv:1304.7098].
[578] Z. Bern et al., Phys. Rev. D88, 1, 014025 (2013), [arXiv:1304.1253].
[579] (2022), [arXiv:2204.12355].
[580] M. Voutilainen, Int. J. Mod. Phys. A30, 31, 1546008 (2015), [arXiv:1509.05026].
[581] G. Aad et al. (ATLAS), JHEP 08, 005 (2016), [arXiv:1605.03495].
[582] G. Aad et al. (ATLAS), JHEP 10, 203 (2019), [arXiv:1908.02746].
[583] S. Chatrchyan et al. (CMS), Phys. Rev. D84, 052011 (2011), [arXiv:1108.2044].
[584] S. Chatrchyan et al. (CMS), JHEP 06, 009 (2014), [arXiv:1311.6141].
[585] G. Aad et al. (ATLAS), Phys. Rev. D89, 5, 052004 (2014), [arXiv:1311.1440].
[586] M. Aaboud et al. (ATLAS), Phys. Lett. B780, 578 (2018), [arXiv:1801.00112].
[587] M. Aaboud et al. (ATLAS), Nucl. Phys. B918, 257 (2017), [arXiv:1611.06586].
[588] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 1, 20 (2019), [arXiv:1807.00782].
[589] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1907.08155].
[590] G. Aad et al. (ATLAS), JHEP 03, 179 (2020), [arXiv:1912.09866].
31st May, 2024

60

9. Quantum Chromodynamics[591] M. Aaboud et al. (ATLAS), Phys. Lett. B776, 295 (2018), [arXiv:1710.09560].
[592] J. H. Kuhn et al., JHEP 03, 059 (2006), [hep-ph/0508253].
[593] J. M. Lindert et al., Eur. Phys. J. C 77, 12, 829 (2017), [arXiv:1705.04664].
[594] V. Khachatryan et al. (CMS), JHEP 10, 128 (2015), [Erratum:[arXiv:1505.06520].

JHEP04,010(2016)],[595] G. Aad et al. (ATLAS), JHEP 09, 029 (2016), [arXiv:1603.01702].
[596] M. Aaboud et al. (ATLAS), Phys. Lett. B773, 354 (2017), [arXiv:1702.04519].
[597] M. Aaboud et al. (ATLAS) (2019), [arXiv:1905.04242].
[598] V. Khachatryan et al. (CMS), Eur. Phys. J. C76, 7, 401 (2016), [arXiv:1507.03268].
[599] G. Aad et al. (ATLAS), Phys. Lett. B 798, 134913 (2019), [arXiv:1903.10415].
[600] G. Aad et al. (ATLAS), Phys. Rev. Lett. 129, 6, 061803 (2022), [arXiv:2201.13045].
[601] A. M. Sirunyan et al. (CMS), Phys. Rev. Lett. 125, 15, 151802 (2020), [arXiv:2006.11191].
[602] G. Aad et al. (ATLAS), JHEP 01, 086 (2013), [arXiv:1211.1913].
[603] M. Aaboud et al. (ATLAS), Phys. Rev. D95, 11, 112005 (2017), [arXiv:1704.03839].
[604] G. Aad et al. (ATLAS), JHEP 11, 169 (2021), [arXiv:2107.09330].
[605] S. Chatrchyan et al. (CMS), Eur. Phys. J. C74, 11, 3129 (2014), [arXiv:1405.7225].
[606] M. Aaboud et al. (ATLAS), Phys. Lett. B781, 55 (2018), [arXiv:1712.07291].
[607] K. Kröninger, A. B. Meyer and P. Uwer, in T. Schörner-Sadenius, editor, “The Large Hadron
Collider: Harvest of Run 1,” 259–300 (2015), [arXiv:1506.02800].
[608] M. Aaboud et al. (ATLAS), Phys. Rev. D94, 9, 092003 (2016), [arXiv:1607.07281].
[609] M. Aaboud et al. (ATLAS), Eur. Phys. J. C77, 5, 292 (2017), [arXiv:1612.05220].
[610] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 5, 368 (2019), [arXiv:1812.10505].
[611] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C 80, 7, 658 (2020), [arXiv:1904.05237].
[612] G. Aad et al. (ATLAS), JHEP 07, 141 (2023), [arXiv:2303.15340].
[613] M. Czakon, D. Heymes and A. Mitov, Phys. Rev. Lett. 116, 8, 082003 (2016),[arXiv:1511.00549].
[614] S. Catani et al., JHEP 07, 100 (2019), [arXiv:1906.06535].
[615] M. Czakon et al., JHEP 05, 149 (2018), [arXiv:1803.07623].
[616] G. Aad et al. (ATLAS), Eur. Phys. J. C 83, 6, 496 (2023), [arXiv:2303.15061].
[617] G. Bevilacqua and M. Worek, JHEP 07, 111 (2012), [arXiv:1206.3064].
[618] S. Chatrchyan et al. (CMS), Phys. Lett. B728,
Lett.B738,526(2014)], [arXiv:1307.1907].

496 (2014),[Erratum:Phys.

[619] T. Klijnsma et al., Eur. Phys. J. C77, 11, 778 (2017), [arXiv:1708.07495].
[620] G. Aad et al. (ATLAS), Eur. Phys. J. C 74, 10, 3109 (2014), [Addendum: Eur.Phys.J.C 76,
642 (2016)], [arXiv:1406.5375].
[621] A. M. Sirunyan et al. (CMS), JHEP 09, 051 (2017), [arXiv:1701.06228].
[622] G. Aad et al. (ATLAS) (2023), [arXiv:2306.11379].
[623] M. Aaboud et al. (ATLAS), Phys. Lett. B786, 114 (2018), [arXiv:1805.10197].
[624] A. M. Sirunyan et al. (CMS), Phys. Lett. B792, 369 (2019), [arXiv:1812.06504].
[625] D. de Florian et al. (LHC Higgs Cross Section Working Group) (2016), [arXiv:1610.07922].
31st May, 2024

61

9. Quantum Chromodynamics[626] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 052005 (2018), [arXiv:1802.04146].
[627] A. M. Sirunyan et al. (CMS), JHEP 01, 183 (2019), [arXiv:1807.03825].
[628] G. Aad et al. (ATLAS), JHEP 08, 027 (2022), [arXiv:2202.00487].
[629] A. Tumasyan et al. (CMS), JHEP 07, 091 (2023), [arXiv:2208.12279].
[630] R. L. Workman et al. (Particle Data Group), PTEP 2022, 083C01 (2022).
[631] D. d’Enterria et al., in “Workshop on precision measurements of the QCD coupling constant(alphas-2019) Trento, Trentino, Italy, February 11-15, 2019,” (2019), [arXiv:1907.01435].
[632] D. d’Enterria et al. (2022), [arXiv:2203.08271].
[633] G. P. Salam, in A. Levy, S. Forte and G. Ridolfi, editors, “From My Vast Repertoire ...:
Guido Altarelli’s Legacy,” 101–121 (2019), [arXiv:1712.05165].
[634] A. Pich et al., in “13th Conference on Quark Confinement and the Hadron Spectrum (Confinement XIII) Maynooth, Ireland, July 31-August 6, 2018,” (2018), [arXiv:1811.11801].
[635] A. Pich, Prog. Part. Nucl. Phys. 117, 103846 (2021), [arXiv:2012.04716].
[636] Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG)), Eur. Phys. J. C 82, 10, 869 (2022),[arXiv:2111.09849].
[637] M. Beneke and M. Jamin, JHEP 09, 044 (2008), [arXiv:0806.3156].
[638] K. Maltman and T. Yavin, Phys. Rev. D78, 094020 (2008), [arXiv:0807.0650].
[639] S. Narison, Phys. Lett. B673, 30 (2009), [arXiv:0901.3823].
[640] I. Caprini and J. Fischer, Eur. Phys. J. C64, 35 (2009), [arXiv:0906.5211].
[641] A. Pich, Prog. Part. Nucl. Phys. 75, 41 (2014), [arXiv:1310.7922].
[642] D. Boito et al., Phys. Rev. D91, 3, 034003 (2015), [arXiv:1410.3528].
[643] G. Altarelli, PoS Corfu2012, 002 (2013), [arXiv:1303.6065].
[644] A. Pich and A. Rodríguez-Sánchez, Phys. Rev. D94, 3, 034027 (2016), [arXiv:1605.06830].
[645] M. Davier et al., Eur. Phys. J. C74, 3, 2803 (2014), [arXiv:1312.1501].
[646] D. Boito et al., Phys. Rev. D95, 3, 034024 (2017), [arXiv:1611.03457].
[647] D. Boito et al., Phys. Rev. D100, 7, 074009 (2019), [arXiv:1907.03360].
[648] A. H. Hoang and C. Regner, Phys. Rev. D 105, 9, 096023 (2022), [arXiv:2008.00578].
[649] A. H. Hoang and C. Regner, Eur. Phys. J. ST 230, 12-13, 2625 (2021), [arXiv:2105.11222].
[650] M. Golterman, K. Maltman and S. Peris, Phys. Rev. D 108, 1, 014007 (2023),[arXiv:2305.10386].
[651] M. A. Benitez-Rathgeb et al., JHEP 09, 223 (2022), [arXiv:2207.01116].
[652] M. A. Benitez-Rathgeb et al., JHEP 07, 016 (2022), [arXiv:2202.10957].
[653] D. Boito et al., Phys. Rev. D98, 7, 074030 (2018), [arXiv:1805.08176].
[654] D. Boito et al., Phys. Rev. D 103, 3, 034028 (2021), [arXiv:2012.10440].
[655] C. Ayala, G. Cvetic and D. Teca, J. Phys. G 50, 4, 045004 (2023), [arXiv:2206.05631].
[656] J.-M. Shen et al., JHEP 07, 109 (2023), [arXiv:2303.11782].
[657] V. Mateu and P. G. Ortega, JHEP 01, 122 (2018), [arXiv:1711.05755].
[658] C. Peset, A. Pineda and J. Segovia, JHEP 09, 167 (2018), [arXiv:1806.05197].
[659] S. Narison, Int. J. Mod. Phys. A 33, 10, 1850045 (2018), [Addendum: Int. J. Mod. Phys. A
33, 1850045 (2018)], [arXiv:1801.00592].
31st May, 2024

62

9. Quantum Chromodynamics[660] S. Narison, Int. J. Mod. Phys. A 33, 33, 1892004 (2018), [arXiv:1812.09360].
[661] D. Boito and V. Mateu, Phys. Lett. B 806, 135482 (2020), [arXiv:1912.06237].
[662] D. Boito and V. Mateu, JHEP 03, 094 (2020), [arXiv:2001.11041].
[663] J. Blümlein, H. Bottcher and A. Guffanti, Nucl. Phys. B774, 182 (2007), [hep-ph/0607200].
[664] P. Jimenez-Delgado and E. Reya, Phys. Rev. D89, 7, 074049 (2014), [arXiv:1403.1852].
[665] S. Alekhin et al., Phys. Rev. D96, 1, 014011 (2017), [arXiv:1701.05838].
[666] S. Alekhin, J. Blümlein and S. Moch, Eur. Phys. J. C78, 6, 477 (2018), [arXiv:1803.07537].
[667] T. Cridge et al. (2021), [arXiv:2106.10289].
[668] R. D. Ball et al. (NNPDF), Eur. Phys. J. C78, 5, 408 (2018), [arXiv:1802.03398].
[669] I. Abt et al. (H1, ZEUS), Eur. Phys. J. C 82, 3, 243 (2022), [arXiv:2112.01120].
[670] R. S. Thorne and G. Watt, JHEP 08, 100 (2011), [arXiv:1106.5789].
[671] S. Alekhin, J. Blümlein and S. Moch, Eur. Phys. J. C71, 1723 (2011), [arXiv:1101.5261].
[672] R. D. Ball et al. (NNPDF), Phys. Lett. B704, 36 (2011), [arXiv:1102.3182].
[673] R. D. Ball et al. (NNPDF), Phys. Lett. B723, 330 (2013), [arXiv:1303.1189].
[674] R. S. Thorne, PoS DIS2013, 042 (2013), [arXiv:1306.3907].
[675] G. Dissertori et al., JHEP 08, 036 (2009), [arXiv:0906.3436].
[676] G. Abbiendi et al. (OPAL), Eur. Phys. J. C71, 1733 (2011), [arXiv:1101.1470].
[677] S. Bethke et al. (JADE), Eur. Phys. J. C64, 351 (2009), [arXiv:0810.1389].
[678] G. Dissertori et al., Phys. Rev. Lett. 104, 072002 (2010), [arXiv:0910.4283].
[679] J. Schieck et al. (JADE), Eur. Phys. J. C73, 3, 2332 (2013), [arXiv:1205.3714].
[680] A. Verbytskyi et al., JHEP 08, 129 (2019), [arXiv:1902.08158].
[681] A. Kardos et al., Eur. Phys. J. C78, 6, 498 (2018), [arXiv:1804.09146].
[682] R. A. Davison and B. R. Webber, Eur. Phys. J. C59, 13 (2009), [arXiv:0809.3326].
[683] R. Abbate et al., Phys. Rev. D83, 074021 (2011), [arXiv:1006.3080].
[684] T. Gehrmann, G. Luisoni and P. F. Monni, Eur. Phys. J. C73, 1, 2265 (2013),[arXiv:1210.6945].
[685] A. H. Hoang et al., Phys. Rev. D91, 9, 094018 (2015), [arXiv:1501.04111].
[686] G. Luisoni, P. F. Monni and G. P. Salam, Eur. Phys. J. C 81, 2, 158 (2021),[arXiv:2012.00622].
[687] F. Caola et al., JHEP 01, 093 (2022), [arXiv:2108.08897]; F. Caola et al., JHEP 12, 062(2022), [arXiv:2204.02247].
[688] P. Nason and G. Zanderighi, JHEP 06, 058 (2023), [arXiv:2301.03607].
[689] R. Frederix et al., JHEP 11, 050 (2010), [arXiv:1008.5313].
[690] P. Bolzoni, B. A. Kniehl and A. V. Kotikov, Nucl. Phys. B875, 18 (2013), [arXiv:1305.6017].
[691] J. Currie et al., Phys. Rev. Lett. 119, 15, 152001 (2017), [arXiv:1705.10271].
[692] M. Czakon et al. (2019), [arXiv:1907.12911].
[693] S. Dittmaier, A. Huss and C. Speckner, JHEP 11, 095 (2012), [arXiv:1210.0438].
[694] R. Frederix et al., JHEP 04, 076 (2017), [arXiv:1612.06548].
[695] M. Czakon et al., JHEP 10, 186 (2017), [arXiv:1705.04105].
31st May, 2024

63

9. Quantum Chromodynamics[696] M. Reyer, M. Schönherr and S. Schumann, Eur. Phys. J. C 79, 4, 321 (2019),[arXiv:1902.01763].
[697] M. Johnson and D. Maître, Phys. Rev. D97, 5, 054013 (2018), [arXiv:1711.01408].
[698] A. Gehrmann-De Ridder et al., JHEP 07, 133 (2016), [arXiv:1605.04295].
[699] S. Forte and Z. Kassabov, Eur. Phys. J. C 80, 3, 182 (2020), [arXiv:2001.04986].
[700] S. Alekhin et al., Eur. Phys. J. C 75, 304 (2015), [arXiv:1410.4412].
[701] F. Lorkowski (ZEUS), Acta Phys. Polon. Supp. 16, 5, 35 (2023).
[702] A. Hayrapetyan et al. (CMS) (2024), [arXiv:2402.13864].
[703] CMS Collaboration (CMS), CMS Physics Analysis Summary CMS-PAS-SMP-22-005 (2023),
URL http://cds.cern.ch/record/2868568.
[704] D. d’Enterria and A. Poldaru, JHEP 06, 016 (2020), [arXiv:1912.11733].
[705] S. Camarda, G. Ferrera and M. Schott, Eur. Phys. J. C 84, 1, 39 (2024), [arXiv:2203.05394].
[706] G. Aad et al. (ATLAS) (2023), [arXiv:2309.12986].
[707] T. Affolder et al. (CDF), Phys. Rev. Lett. 88, 042001 (2002), [hep-ex/0108034].
[708] S. Chekanov et al. (ZEUS), Phys. Lett. B 649, 12 (2007), [hep-ex/0701039].
[709] V. M. Abazov et al. (D0), Phys. Rev. D80, 111107 (2009), [arXiv:0911.2710].
[710] B. Malaescu and P. Starovoitov, Eur. Phys. J. C72, 2041 (2012), [arXiv:1203.5416].
[711] V. Khachatryan et al. (CMS), Eur. Phys. J. C75, 6, 288 (2015), [arXiv:1410.6765].
[712] D. Britzger et al., Eur. Phys. J. C79, 1, 68 (2019), [arXiv:1712.00480].
[713] S. Chekanov et al. (ZEUS), Eur. Phys. J. C44, 183 (2005), [hep-ex/0502007].
[714] S. Chatrchyan et al. (CMS), Eur. Phys. J. C73, 10, 2604 (2013), [arXiv:1304.7498].
[715] V. M. Abazov et al. (D0), Phys. Lett. B718, 56 (2012), [arXiv:1207.4957].
[716] V. Khachatryan et al. (CMS), Eur. Phys. J. C75, 5, 186 (2015), [arXiv:1412.1633].
[717] V. Andreev et al. (H1), Eur. Phys. J. C 77, 4, 215 (2017), [arXiv:1611.03421].
[718] J. Haller et al., Eur. Phys. J. C78, 8, 675 (2018), [arXiv:1803.01853].
[719] J. de Blas et al., Phys. Rev. Lett. 129, 27, 271801 (2022), [Supplemental Material:
Phys.Rev.Lett. 129, 271801 (2022)], [arXiv:2204.04204].
[720] J. de Blas et al., Phys. Rev. D 106, 3, 033003 (2022), [arXiv:2112.07274].
[721] T. Aaltonen et al. (CDF), Science 376, 6589, 170 (2022).
[722] S. Schael et al. (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group,
SLD Electroweak Group, SLD Heavy Flavour Group), Phys. Rept. 427, 257 (2006), [hepex/0509008].
[723] L. Del Debbio and A. Ramos, Physics Reports 920, 1 (2021), ISSN 0370-1573,[arXiv:2101.04762].
[724] S. Aoki et al. (Flavour Lattice Averaging Group), Eur. Phys. J. C 80, 2, 113 (2020),[arXiv:1902.08191].
[725] M. Bruno et al. (ALPHA), Phys. Rev. Lett. 119, 10, 102001 (2017), [arXiv:1706.03821].
[726] S. Aoki et al. (PACS-CS), JHEP 10, 053 (2009), [arXiv:0906.3906].
[727] C. McNeile et al., Phys. Rev. D82, 034512 (2010), [arXiv:1004.4285].
[728] K. Maltman et al., Phys. Rev. D78, 114504 (2008), [arXiv:0807.2020].
31st May, 2024

64

9. Quantum Chromodynamics[729] B. Chakraborty et al., Phys. Rev. D91, 5, 054508 (2015), [arXiv:1408.4169].
[730] C. Ayala, X. Lobregat and A. Pineda, JHEP 09, 016 (2020), [arXiv:2005.12301].
[731] A. Bazavov et al. (TUMQCD), Phys. Rev. D 100, 11, 114511 (2019), [arXiv:1907.11747].
[732] S. Cali et al., Phys. Rev. Lett. 125, 242002 (2020), [arXiv:2003.05781].
[733] M. Dalla Brida et al. (ALPHA), Phys. Lett. B 807, 135571 (2020), [arXiv:1912.06001].

31st May, 2024