9. QUANTUM CHROMODYNAMICS - Particle Data Group

9. Quantum chromodynamics 1

9. QUANTUM CHROMODYNAMICS

Written October 2009 by G. Dissertori (ETH, Zurich) and G.P. Salam (LPTHE, Paris).

9.1. Basics

Quantum Chromodynamics (QCD), the gauge field theory that describes thestrong interactions of colored quarks and gluons, is the SU(3) component of theSU(3)×SU(2)×U(1) Standard Model of Particle Physics.

The Lagrangian of QCD is given by

L =∑q

ψ̄q,a(iγμ∂μδab − gsγμtCabAC

μ − mqδab)ψq,b −14FA

μνFA μν , (9.1)

where repeated indices are summed over. The γμ are the Dirac γ-matrices. The ψq,a arequark-field spinors for a quark of flavor q and mass mq, with a color-index a that runsfrom a = 1 to Nc = 3, i.e. quarks come in three “colors.” Quarks are said to be in thefundamental representation of the SU(3) color group.

The ACμ correspond to the gluon fields, with C running from 1 to N2

c − 1 = 8, i.e.there are eight kinds of gluon. Gluons are said to be in the adjoint representation of theSU(3) color group. The tCab correspond to eight 3 × 3 matrices and are the generators ofthe SU(3) group (cf. the section on “SU(3) isoscalar factors and representation matrices”in this Review with tCab ≡ λC

ab/2). They encode the fact that a gluon’s interaction witha quark rotates the quark’s color in SU(3) space. The quantity gs is the QCD couplingconstant. Finally, the field tensor FA

μν is given by

FAμν = ∂μAA

ν − ∂νAAμ − gs fABCAB

μ ACν [tA, tB] = ifABCtC , (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.color-neutral) combinations of quarks, anti-quarks, and gluons.

Ab-initio predictive methods for QCD include lattice gauge theory and perturbativeexpansions in the coupling. The Feynman rules of QCD involve a quark-antiquark-gluon (qq̄g) vertex, a 3-gluon vertex (both proportional to gs), and a 4-gluon vertex(proportional to g2

s). A full set of Feynman rules is to be found for example in Ref. 1.

Useful color-algebra relations include: tAabtAbc = CF δac, where CF ≡ (N2

c − 1)/(2Nc) =4/3 is the color-factor (“Casimir”) associated with gluon emission from a quark;fACDfBCD = CAδAB where CA ≡ Nc = 3 is the color-factor associated with gluonemission from a gluon; tAabt

Bab = TRδAB , where TR = 1/2 is the color-factor for a gluon to

split to a qq̄ pair.

The fundamental parameters of QCD are the coupling gs (or αs =g2s

4π) and the quark

masses mq.

K. Nakamura et al., JPG 37, 075021 (2010) (http://pdg.lbl.gov)July 30, 2010 14:57

2 9. Quantum chromodynamics

This review will concentrate mainly on perturbative aspects of QCD as theyrelate to collider physics. Related textbooks include Refs. 1–3. Some discussion ofnon-perturbative aspects, including lattice QCD, is to be found in the reviews on “QuarkMasses” and “The CKM quark-mixing matrix” of this Review. Lattice-QCD textbooksand lecture notes include Refs. 4–6, while recent developments are summarized forexample in Ref. 7. For a review of some of the QCD issues in heavy-ion physics, see forexample Ref. 8.

9.1.1. Running coupling :In the framework of perturbative QCD (pQCD), predictions for observables are

expressed in terms of the renormalized coupling αs(μ2R), a function of an (unphysical)

renormalization scale μR. When one takes μR close to the scale of the momentumtransfer Q in a given process, then αs(μ2

R � Q2) is indicative of the effective strength ofthe strong interaction in that process.

The coupling satisfies the following renormalization group equation (RGE):

μ2R

dαs

dμ2R

= β(αs) = −(b0α2s + b1α

3s + b2α

4s + · · ·) (9.3)

where b0 = (11CA − 4nfTR)/(12π) = (33 − 2nf )/(12π) is referred to as the 1-loop beta-function coefficient, the 2-loop coefficient is b1 = (17C2

A − nfTR(10CA + 6CF ))/(24π2) =(153− 19nf )/(24π2), and the 3-loop coefficient is b2 = (2857− 5033

9 nf + 32527 n2

f )/(128π3).

The 4-loop coefficient, b3, is to be found in Refs. 9, 10†. The minus sign in Eq. (9.3)is the origin of asymptotic freedom, i.e. the fact that the strong coupling becomesweak for processes involving large momentum transfers (“hard processes”), αs ∼ 0.1 formomentum transfers in the 100 GeV –TeV range.

The β-function coefficients, the bi, are given for the coupling of an effective theory inwhich nf of the quark flavors are considered light (mq � μR), and in which the remainingheavier quark flavors decouple from the theory. One may relate the coupling for thetheory with nf + 1 light flavors to that with nf flavors through an equation of the form

α(nf +1)s (μ2

R) = α(nf )s (μ2

R)

(1 +

∞∑n=1

n∑�=0

cn� [α(nf )s (μ2

R)]n ln� μ2R

m2h

), (9.4)

where mh is the mass of the (nf +1)th flavor, and the first few cn� coefficients arec11 = 1

6π , c10 = 0, c22 = c211, c21 = 1924π2 , and c20 = − 11

72π2 when mh is the MS mass atscale mh (c20 = 7

24π2 when mh is the pole mass — mass definitions are discussed belowand in the review on “Quark Masses”). Terms up to c4� are to be found in Refs. 11, 12.Numerically, when one chooses μR = mh, the matching is a small effect, owing to thezero value for the c10 coefficient.

† One should be aware that the b2 and b3 coefficients arerenormalization-scheme-dependent, and given here in the MS scheme, as discussed below.

July 30, 2010 14:57


Working in an energy range where the number of flavors is constant, a simple exactanalytic solution exists for Eq. (9.3) only if one neglects all but the b0 term, givingαs(μ2

R) = (b0 ln(μ2R/Λ2))−1. Here Λ is a constant of integration, which corresponds

to the scale where the perturbatively-defined coupling would diverge, i.e. it is thenon-perturbative scale of QCD. A convenient approximate analytic solution to the RGEthat includes also the b1, b2, and b3 terms is given by (see for example Ref. 13),

αs(μ2R) � 1

b0t

(1 − b1

b20

ln t

t+

b21(ln2 t − ln t − 1) + b0b2

b40t2

−b31(ln

3 t − 52

ln2 t − 2 ln t +12) + 3b0b1b2 ln t − 1

2b20b3

b60t3

), t ≡ ln

μ2R

Λ2, (9.5)

again parametrized in terms of a constant Λ. Note that Eq. (9.5) is one of several possibleapproximate 4-loop solutions for αs(μ2

R), and that a value for Λ only defines αs(μ2R)

once one knows which particular approximation is being used. An alternative to the useof formulas such as Eq. (9.5) is to solve the RGE exactly, numerically (including thediscontinuities, Eq. (9.4), at flavor thresholds). In such cases the quantity Λ is not definedat all. For these reasons, in determinations of the coupling, it has become standardpractice to quote the value of αs at a given scale (typically MZ) rather than to quote avalue for Λ.

The value of the coupling, as well as the exact forms of the b2, c10 (and higher order)coefficients, depend on the renormalization scheme in which the coupling is defined,i.e. the convention used to subtract infinities in the context of renormalization. Thecoefficients given above hold for a coupling defined in the modified minimal subtraction(MS) scheme [14], by far the most widely used scheme.

A discussion of determinations of the coupling and a graph illustrating its scaledependence (“running”) are to be found in Section 9.3.4.

9.1.2. Quark masses :Free quarks are never observed, i.e. a quark never exists on its own for a time longer

than ∼ 1/Λ: up, down, strange, charm, and bottom quarks all hadronize, i.e. become partof a meson or baryon, on a timescale ∼ 1/Λ; the top quark instead decays before it hastime to hadronize. This means that the question of what one means by the quark mass isa complex one, which requires that one adopts a specific prescription. A perturbativelydefined prescription is the pole mass, mq , which corresponds to the position of thedivergence of the propagator. This is close to one’s physical picture of mass. However,when relating it to observable quantities, it suffers from substantial non-perturbativeambiguities (see e.g. Ref. 15). An alternative is the MS mass, mq(μ2

R), which depends onthe renormalization scale μR.

Results for the masses of heavier quarks are often quoted either as the pole mass oras the MS mass evaluated at a scale equal to the mass, mq(m2

q); light quark masses aregenerally quoted in the MS scheme at a scale μR ∼ 2 GeV . The pole and MS masses are

July 30, 2010 14:57


related by a slowly converging series that starts mq = mq(m2q)(1 +

4αs(m2q)

3π+ O(α2

s)),

while the scale-dependence of MS masses is given by

μ2R

dmq(μ2R)

dμ2R

=

[−αs(μ2

R)π

+ O(α2s)

]mq(μ2

R) . (9.6)

Quark masses are discussed in detail in a dedicated section of the Review, “QuarkMasses.”

9.2. Structure of QCD predictions

9.2.1. Inclusive cross sections :The simplest observables in QCD are those that do not involve initial-state hadrons

and that are fully inclusive with respect to details of the final state. One example is thetotal cross section for e+e− → hadrons at center-of-mass energy Q, for which one canwrite

σ(e+e− → hadrons, Q)σ(e+e− → μ+μ−, Q)

≡ R(Q) = REW(Q)(1 + δQCD(Q)) , (9.7)

where REW(Q) is the purely electroweak prediction for the ratio and δQCD(Q) is thecorrection due to QCD effects. To keep the discussion simple, we can restrict ourattention to energies Q � MZ , where the process is dominated by photon exchange(REW = 3

∑q e2

q , neglecting finite-quark-mass corrections),

δQCD(Q) =∞∑

n=1

cn ·(

αs(Q2)π

)n

+ O(

Λ4

Q4

). (9.8)

The first four terms in the αs series expansion are then to be found in Refs. 16, 17

c1 = 1 , c2 = 1.9857 − 0.1152nf , (9.9a)

c3 = −6.63694 − 1.20013nf − 0.00518n2f − 1.240η (9.9b)

c4 = −156.61 + 18.77nf − 0.7974n2f + 0.0215n3

f + Cη , (9.9c)

with η = (∑

eq)2/(3∑

e2q) and where the coefficient C of the η-dependent piece in the α4

sterm has yet to be determined. For corresponding expressions including also Z exchangeand finite-quark-mass effects, see Ref. 18.

A related series holds also for the QCD corrections to the hadronic decay width of theτ lepton, which essentially involves an integral of R(Q) over the allowed range of invariantmasses of the hadronic part of the τ decay (see e.g. Ref. 16). The series expansions forQCD corrections to Higgs-boson (partial) decay widths are summarized in Refs. 19, 20.

One characteristic feature of the Eq. (9.8) is that the coefficients of αns increase rapidly

order by order: calculations in perturbative QCD tend to converge more slowly than

July 30, 2010 14:57


would be expected based just on the size of αs††. Another feature is the existence of an

extra “power-correction” term O(Λ4/Q4) in Eq. (9.8), which accounts for contributionsthat are fundamentally non-perturbative. All high-energy QCD predictions involve suchcorrections, though the exact power of Λ/Q depends on the observable.Scale dependence. In Eq. (9.8) the renormalization scale for αs has been chosenequal to Q. The result can also be expressed in terms of the coupling at an arbitraryrenormalization scale μR,

δQCD(Q) =∞∑

n=1

cn

(μ2

R

Q2

)·(

αs(μ2R)

π

)n

+ O(

Λ4

Q4

), (9.10)

where c1(μ2R/Q2) ≡ c1, c2(μ2

R/Q2) = c2 + πb0c1 ln(μ2R/Q2), c3(μ2

R/Q2) = c3 + (2b0c2π +b1c1π

2) ln(μ2R/Q2) + b20c1π

2 ln2(μ2R/Q2), etc.. Given an infinite number of terms in the

αs expansion, the μR 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.With just terms up to n = N , a residual μR dependence will remain, which implies

an uncertainty on the prediction of R(Q) due to the arbitrariness of the scale choice.This uncertainty will be O(αN+1

s ), i.e. of the same order as the neglected terms. Forthis reason it is standard to use QCD predictions’ scale dependence as an estimate of theuncertainties due to neglected terms. One usually takes a central value for μR ∼ Q, inorder to avoid the poor convergence of the perturbative series that results from the largelnn−1(μ2

R/Q2) terms in the cn coefficients when μR � Q or μR � Q.

9.2.1.1. Processes with initial-state hadrons:Deep Inelastic Scattering. To illustrate the key features of QCD cross sections inprocesses with initial-state hadrons, let us consider deep-inelastic scattering (DIS),ep → 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 virtualitiesQ2 ≡ −q2 far above the squared proton mass (but far below the Z mass), the differentialcross section in terms of the kinematic variables Q2, x = Q2/(2p · q) and y = (q · p)/(k · p)is

d2σ

dxdQ2=

4πα

2xQ4

[(1 + (1 − y)2)F2(x, Q2) − y2FL(x, Q2)

], (9.11)

where α is the electromagnetic coupling and F2(x, Q2) and FL(x, Q2) are proton structurefunctions, which encode the interaction between the photon (in given polarization states)and the proton (for an extended review, see Sec. 16).

Structure functions are not calculable in perturbative QCD, nor is any other crosssection that involves initial-state hadrons. To zeroth order in αs, the structure functions

†† The situation is significantly worse near thresholds, e.g. the tt̄ production threshold.An overview of some of the “effective field theory” techniques used in such cases is to befound for example in Ref. 21.

July 30, 2010 14:57


are given directly in terms of non-perturbative parton (quark or gluon) distributionfunctions (PDFs),

F2(x, Q2) = x∑q

e2qfq/p(x) , FL(x, Q2) = 0 , (9.12)

where fq/p(x) is the PDF for quarks of type q inside the proton, i.e. the number densityof quarks of type q inside a fast-moving proton that carry a fraction x of its longitudinalmomentum (the quark flavor index q, here, is not to be confused with the photonmomentum q in the lines preceding Eq. (9.11)). Since PDFs are non-perturbative, anddifficult to calculate in lattice QCD [22], they must be extracted from data.

The above result, with PDFs fq/p(x) that are independent of the scale Q, correspondsto the “quark-parton model” picture in which the photon interacts with point-like freequarks, or equivalently, one has incoherent elastic scattering between the electron andindividual constituents of the proton. As a consequence, in this picture also F2 and FLare independent of Q. When including higher orders in pQCD, Eq. (9.12) becomes

F2(x, Q2) =

x∞∑

n=0

αns (μ2

R)(2π)n

∑i=q,g

∫ 1

x

dz

zC

(n)2,i (z, Q2, μ2

R, μ2F ) fi/p

(x

z, μ2

F

)

+ O(Λ2

Q2

). (9.13)

Just as in Eq. (9.10), we have a series in powers αs(μ2R), each term involving a coefficient

C(n)2,i that can be calculated using Feynman graphs. An important difference relative to

Eq. (9.10) stems from the fact that the quark’s momentum, when it interacts with thephoton, can differ from its momentum when it was extracted from the proton, because itmay have radiated gluons in between. As a result, the C

(n)2,i coefficients are functions that

depend on the ratio, z, of these two momenta, and one must integrate over z. At zerothorder, C

(0)2,q = e2

qδ(1 − z) and C(0)2,g = 0.

The majority of the emissions that modify a parton’s momentum are actually collinear(parallel) to that parton, and don’t depend on the fact that the parton is destined tointeract with a photon. It is natural to view these emissions as modifying the proton’sstructure rather than being part of the coefficient function for the parton’s interactionwith the photon. The separation between the two categories is somewhat arbitrary andparametrized by a factorization scale, μF . Technically, one uses a procedure knownas factorization to give rigorous meaning to this distinction, most commonly throughthe MS factorization scheme, defined in the context of dimensional regularization. TheMS factorization scheme involves an arbitrary choice of factorization scale, μF , whosemeaning can be understood roughly as follows: emissions with transverse momenta aboveμF are included in the C

(n)2,q (z, Q2, μ2

R, μ2F ); emissions with transverse momenta below μF

are accounted for within the PDFs, fi/p(x, μ2F ).

July 30, 2010 14:57


The PDFs’ resulting dependence on μF is described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [23], which to leading order (LO) read∗

∂fi/p(x, μ2F )

∂μ2F

=∑j

αs(μ2F )

2π

∫ 1

x

dz

zP

(1)i←j(z)fj/p

(x

z, μ2

F

), (9.14)

with, for example, P(1)q←g(z) = TR(z2 + (1 − z)2). The other LO splitting functions are

listed in Sec. 16 of this Review, while results up to next-to-next-to-leading order (NNLO),α3

s, are given in Refs. 24, 25. The coefficient functions are also μF dependent, for

example C(1)2,i (x, Q2, μ2

R, μ2F ) = C

(1)2,i (x, Q2, μ2

R, Q2) − ln(μ2

FQ2 )

∑j

∫ 1x

dzz P

(1)i←j(z)C(0)

2,j (xz ).

For the electromagnetic component of DIS with light quarks and gluons they are knownto O(α3

s) (N3LO) [26]. For weak currents they are known fully to α2s (NNLO) [27] with

substantial results known also at N3LO [28]. For heavy quark production they are knownto O(α2

s) [29] (next-to-leading order (NLO) insofar as the series starts at O(αs)), withwork ongoing towards NNLO [30].

As with the renormalization scale, the choice of factorization scale is arbitrary, butif one has an infinite number of terms in the perturbative series, the μF -dependencesof the coefficient functions and PDFs will compensate each other fully. Given only Nterms of the series, a residual uncertainty O(αN+1

s ) is associated with the ambiguity inthe choice of μF . As with μR, varying μF provides an input in estimating uncertaintieson predictions. In inclusive DIS predictions, the default choice for the scales is usuallyμR = μF = Q.Hadron-hadron collisions. The extension to processes with two initial-state hadronsis straightforward, and for example the total (inclusive) cross section for W bosonproduction in pp̄ collisions can be written as

σ(pp̄ → W + X) =∞∑

n=0

αns (μ2

R)∑i,j

∫dx1dx2 fi/p

(x1, μ

2F

)fj/p̄

(x2, μ

2F

)

× σ̂(n)ij→W+X

(x1x2s, μ

2R, μ2

F

), (9.15)

where s is the squared center-of-mass energy of the collision. At LO, n = 0, thehard (partonic) cross section σ̂

(0)ij→W+X(x1x2s, μ

2R, μ2

F ) is simply proportional to

∗ LO is generally taken to mean the lowest order at which a quantity is non-zero. Thisdefinition is nearly always unambiguous, the one major exception being for the case of thehadronic branching ratio of virtual photons, Z, τ , etc., for which two conventions exist:LO can either mean the lowest order that contributes to the hadronic branching fraction,i.e. the term “1” in Eq. (9.7); or it can mean the lowest order at which the hadronicbranching ratio becomes sensitive to the coupling, n = 1 in Eq. (9.8), as is relevant whenextracting the value of the coupling from a measurement of the branching ratio. Becauseof this ambiguity, we avoided use of the term “LO” in that context.

July 30, 2010 14:57


δ(x1x2s−M2W ), in the narrow W -boson width approximation (see Sec. 40 of this Review

for detailed expressions for this and other hard scattering cross sections). It is non-zeroonly for choices of i, j that can directly give a W , such as i = u, j = d̄. At higherorders, n ≥ 1, new partonic channels contribute, such as gq, and there is no restrictionx1x2s = M2

W .Equation 9.15 involves a factorization between hard cross section and 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 pp̄predictions [31].

The fully inclusive hard cross sections are known to NNLO, α2s , for Drell-Yan

(DY) lepton-pair and vector-boson production [32,33], and for Higgs-boson production[33–36].Photoproduction. γp (and γγ) collisions are similar to pp collisions, with the subtletythat the photon can behave in two ways: there is “direct” photoproduction, in whichthe photon behaves as a point-like particle and takes part directly in the hard collision,with hard subprocesses such as γg → qq̄; there is also resolved photoproduction, in whichthe photon behaves like a hadron, with non-perturbative partonic substructure and acorresponding PDF for its quark and gluon content, fi/γ(x, Q2).

While useful to understand the general structure of γp collisions, the distinctionbetween direct and resolved photoproduction is not well defined beyond leading order, asdiscussed for example in Ref. 37.The high-energy limit. In situations in which the total center-of-mass energy

√s is

much larger than other scales in the problem (e.g. Q in DIS, mb for bb̄ production in ppcollisions, etc.), each power of αs beyond LO can be accompanied by a power of ln(s/Q2)(or ln(s/m2

b), etc.). This is known as the high-energy or Balitsky-Fadin-Kuraev-Lipatov(BFKL) limit [38–40]. Currently it is possible to account for the dominant and firstsubdominant [41,42] power of ln s at each order of αs, and also to estimate furthersubdominant contributions that are numerically large (see Refs. 43–45 and referencestherein).

Physically, the summation of all orders in αs can be understood as leading to a growthwith s of the gluon density in the proton. At sufficiently high energies this impliesnon-linear effects, whose treatment has been the subject of intense study (see for exampleRefs. 46, 47 and references thereto).

9.2.2. Non-inclusive cross-sections :QCD final states always consist of hadrons, while perturbative QCD calculations

deal with partons. Physically, an energetic parton fragments (“showers”) into manyfurther partons, which then, on later timescales, undergo a transition to hadrons(“hadronization”). Fixed-order perturbation theory captures only a small part of thesedynamics.

This does not matter for the fully inclusive cross sections discussed above: theshowering and hadronization stages are “unitary”, i.e. they do not change the overallprobability of hard scattering, because they occur long after it has taken place.

July 30, 2010 14:57


Non-inclusive measurements, in contrast, may be affected by the extra dynamics. Forthose sensitive just to the main directions of energy flow (jet rates, event shapes, cf.Sec. 9.3.1) fixed order perturbation theory is often still adequate, because showering andhadronization don’t substantially change the overall energy flow. This means that onecan make a prediction using just a small number of partons, which should correspondwell to a measurement of the same observable carried out on hadrons. For observablesthat instead depend on distributions of individual hadrons (which, e.g., are the inputsto detector simulations), it is mandatory to account for showering and hadronization.The range of predictive techniques available for QCD final states reflects this diversity ofneeds of different measurements.

While illustrating the different methods, we shall for simplicity mainly use expressionsthat hold for e+e− scattering. The extension to cases with initial-state partons willbe mostly straightforward (space constraints unfortunately prevent us from addressingdiffraction and exclusive hadron-production processes; extensive discussion is to be foundin Refs. 48, 49).

9.2.2.1. Preliminaries: 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 squaredtree-level matrix element |M2

n(p1, . . . , pn)| for the production of n partons with momentap1, . . . , pn, and a corresponding phase-space integration measure dΦn. If particle n isa gluon, and additionally it becomes collinear (parallel) to another particle i and itsmomentum tends to zero (it becomes “soft”), the matrix element simplifies as follows,

limθin→0, En→0

dΦn|M2n(p1, . . . , pn)|

= dΦn−1|M2n−1(p1, . . . , pn−1)|αsCi

π

dθ2in

θ2in

dEn

En, (9.16)

where Ci = CF (CA) if i is a quark (gluon). This formula has non-integrable divergencesboth for the inter-parton angle θin → 0 and for the gluon energy En → 0, which aremirrored also in the structure of divergences in loop diagrams. These divergences areimportant 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 tohard partons); secondly, they will determine which observables can be calculated withinperturbative QCD.

9.2.2.2. Fixed-order predictions:Let us consider an observable O that is a function Om(p1, . . . , pm) of the four-momenta

of the m particles in an event (whether partons or hadrons). In what follows, weshall consider the cross section for events weighted with the value of the observable,σO. As examples, if Om ≡ 1 for all m, then σO is just the total cross section; ifOm ≡ τ̂(p1, . . . , pm) where τ̂ is the value of the thrust for that event (see Sec. 9.3.1.2),then the average value of the thrust is 〈τ〉 = σO/σtot; if Om ≡ δ(τ − τ̂(p1, . . . , pm)) thenone gets the differential cross section as a function of the thrust, σO ≡ dσ/dτ .

July 30, 2010 14:57


In the expressions below, we shall omit to write the non-perturbative power correctionterm, which for most common observables is proportional to a single power of Λ/Q.LO. If the observable O is non-zero only for events with at least n particles, then the LOQCD prediction for the weighted cross section in e+e− annihilation is

σO,LO = αn−2s (μ2

R)∫

dΦn|M2n(p1, . . . , pn)| On(p1, . . . , pn) , (9.17)

where the squared tree-level matrix element, |M2n(p1, . . . , pn)|, includes relevant symmetry

factors, has been summed over all subprocesses (e.g. e+e− → qq̄qq̄, e+e− → qq̄gg) andhas had all factors of αs extracted in front. In processes other than e+e− collisions, thepowers of the coupling are often brought inside the integrals, with the scale μR chosenevent 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 programssuch as CompHEP [50], MadGraph [51], Alpgen [52], Comix/Sherpa [53], and Helac/Phegas [54]. Some of these (CompHEP, MadGraph) use formulae obtained from directevaluations of Feynman diagrams. Others (Alpgen, Helac/Phegas and Comix/Sherpa) usemethods designed to be particularly efficient at high multiplicities, such as Berends-Gielerecursion [55] (see also the review Ref. 56), which builds up amplitudes for complexprocesses from simpler ones.

The phase-space integration is usually carried out by Monte Carlo sampling, in orderto deal with the sometimes complicated cuts that are used in corresponding experimentalmeasurements. Because of the divergences in the matrix element, Eq. (9.16), the integralconverges only if the observable vanishes for kinematic configurations in which one of then particles is arbitrarily soft or it is collinear to another particle. As an example, thecross 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 ofenergy (or jets, see Sec. 9.3.1.1), each above some energy threshold and well separatedfrom each other in angle.

LO calculations can be carried out for 2 → n processes with n � 6 − 10. The exactupper 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 integrationcan be optimized to work around the large variations in the values of the matrix elements.NLO. Given an observable that is non-zero starting from n particles, its prediction atNLO involves supplementing the LO result with the (n + 1)-particle tree-level matrixelement (|M2

n+1|), and the interference of a n-particle tree-level and n-particle 1-loopamplitude (2Re(MnM∗

n,1−loop)),

σNLOO = σLO

O + αn−1s (μ2

R)∫

dΦn+1

|M2n+1(p1, . . . , pn+1)| On+1(p1, . . . , pn+1)

+ αn−1s (μ2

R)∫

dΦn 2Re(Mn(p1, . . . , pn)

M∗n,1−loop(p1, . . . , pn)

)On(p1, . . . , pn) . (9.18)

July 30, 2010 14:57


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 thatthey have yet to be fully automated, though considerable progress is being made in thisdirection (see Refs. 57–60 and references therein). Secondly, loop amplitudes are infinitein 4 dimensions, while tree-level amplitudes are finite, but their integrals are infinite, dueto the divergences of Eq. (9.16). These two sources of infinities have the same soft andcollinear origins and cancel after the integration only if the observable O satisfies theproperty of infrared and collinear safety,

On+1(p1, . . . , ps, . . . , pn) → On(p1, . . . , pn) if ps → 0On+1(p1, . . . , pa, pb, . . . , pn) → On(p1, . . . , pa + pb, . . . , pn)

if pa || pb . (9.19)

Examples of infrared safe quantities include event-shape distributions and jet crosssections (with appropriate jet algorithms, see below). Unsafe quantities include thedistribution of the momentum of the hardest QCD particle (which is not conserved undercollinear splitting), observables that require the complete absence of radiation in someregion of phase-space (e.g. rapidity gaps or 100% isolation cuts, which are affected by softemissions), or the particle multiplicity (affected by both soft and collinear emissions). Thenon-cancellation of divergences at NLO due to infrared or collinear unsafety compromisesthe usefulness not only of the NLO calculation, but also that of a LO calculation, sinceLO is only an acceptable approximation if one can prove that higher order terms aresmaller. Infrared and collinear unsafety usually also imply large non-perturbative effects.

As with LO calculations, the phase-space integrals in Eq. (9.18) are usually carried outby Monte Carlo integration, so as to facilitate the study of arbitrary observables. Variousmethods exist to obtain numerically efficient cancellation among the different infinities.The most widely used in current NLO computer codes is known as dipole subtraction [61];other methods that have seen numerous applications include FKS [62] and antenna [63]subtraction.

NLO calculations exist for nearly all 2 → n processes with n ≤ 3 (and for 1 → 4in e+e− → γ/Z →hadrons), as reviewed in Ref. 64. Some of the corresponding codesare public, and those that provide access to multiple processes include NLOJet++ [65]for e+e−, DIS, and hadron-hadron processes involving just light partons in the finalstate, MCFM [66] for hadron-hadron processes with vector bosons and/or heavy quarksin the final state, VBFNLO for vector-boson fusion processes [67], and the Phoxfamily [68] for processes with photons in the final state. The current forefront of NLOcalculations is 2 → 4 processes in pp scattering, for which results exist on tt̄bb̄ [59,60] andpp → W +3jets [57,58].NNLO. Conceptually, NNLO and NLO calculations are similar, except that one mustadd 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 ofthe 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.Arranging for their cancellation after numerical Monte Carlo integration is one of the

July 30, 2010 14:57


significant challenges of NNLO calculations, as is the determination of the relevant 2-loopamplitudes. The processes for which fully exclusive NNLO calculations exist include the3-jet cross section in e+e− collisions [69,70] (for which NNLO means α3

s), as well asvector- [71,72] and Higgs-boson [73,74] production in pp and pp̄ collisions (for whichNNLO means α2

s).

9.2.2.3. Resummation:

Many experimental measurements place tight constraints on emissions in the finalstate, for example, in e+e− events, that the thrust should be less than some value τ � 1,or in pp → Z events that the Z-boson transverse momentum should be much smaller thanits mass, pt,Z � MZ . A further example is the production of heavy particles or jets nearthreshold (so that little energy is left over for real emissions) in DIS and pp collisions.

In such cases the constraint vetoes a significant part of the integral over the softand collinear divergence of Eq. (9.16). As a result, there is only a partial cancellationbetween 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 alarge coefficient ∼ L2, where e.g. L = ln τ or L = ln(MZ/pt,Z). One ends up with aperturbative series whose terms go as ∼ (αsL

2)n. It is not uncommon that αsL2 � 1, so

that the perturbative series converges very poorly if at all.∗∗ In such cases one may carryout a “resummation,” which accounts for the dominant logarithmically enhanced termsto all orders in αs, by making use of known properties of matrix elements for multiplesoft and collinear emissions, and of the all-orders properties of the divergent parts ofvirtual corrections, following original works such as Refs. 75–84 (or more recently throughsoft-collinear effective theory, cf. the review in Ref. 85).

For cases with double logarithmic enhancements (two powers of logarithm per powerof αs), there are two classification schemes for resummation accuracy. Writing the crosssection including the constraint as σ(L) and the unconstrained (total) cross section asσtot, the series expansion takes the form

σ(L) � σtot

∞∑n=0

2n∑k=0

Rnkαns (μ2

R)Lk, L � 1 (9.20)

and leading log (LL) resummation means that one accounts for all terms with k = 2n,next-to-leading-log (NLL) includes additionally all terms with k = 2n − 1, etc.. Often

∗∗ To be precise one should distinguish 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). Nearly all perturbative series also suffer from “renormalon” divergencesαn

s n! (reviewed in Ref. 15), which however have an impact only at very high perturbativeorders and have a deep connection with non-perturbative uncertainties.

July 30, 2010 14:57


σ(L) (or its Fourier or Mellin transform) exponentiates ‡,

σ(L) � σtot exp

[ ∞∑n=1

n+1∑k=0

Gnkαns (μ2

R)Lk

], L � 1 , (9.21)

where one notes the different upper limit on k compared to Eq. (9.20). This is a morepowerful form of resummation: the G12 term alone reproduces the full LL series inEq. (9.20). With the form Eq. (9.21) one still uses the nomenclature LL, but this nowmeans that all terms with k = n + 1 are included, and NLL implies all terms with k = n,etc..

For a large number of observables, the state-of-the art for resummation is NLLin the sense of Eq. (9.21) (see Refs. 89–91 and references therein). NNLL has beenachieved for the DY and Higgs-boson pt distributions [92,93] (in addition the NLLResBos program [94] is still widely used), the back-to-back energy-energy correlation ine+e− [95], and the production of top anti-top pairs near threshold [96–100]. Finally,the parts believed to be dominant in the N3LL resummation are available for the thrustvariable in e+e− annihilations [101], and for Higgs- and vector-boson production nearthreshold [102,103] in hadron collisions. The inputs and methods involved in these variouscalculations are somewhat too diverse to discuss in detail here, so we recommend that theinterested reader consults the original references for further details.

9.2.2.4. 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 inhigh-energy collisions. However, one can factorize perturbative and non-perturbativecontributions via the concept of fragmentation functions. These are the final-stateanalogue of the parton distribution functions that are used for initial-state hadrons.

It should be added that if one ignores the non-perturbative difficulties and justcalculates the energy and angular spectrum of partons in perturbative QCD with somelow cutoff scale ∼ Λ (using resummation to sum large logarithms of

√s/Λ), then this

reproduces many features of the corresponding hadron spectra. This is often taken tosuggest that hadronization is “local” in momentum space.

Sec. 17 of this Review provides further information (and references) on these topics,including also the question of heavy-quark fragmentation.

‡ Whether or not this happens depends on the quantity being resummed. A classicexample involves jet rates in e+e− collisions as a function of a jet-resolution parameterycut. The logarithms of 1/ycut exponentiate for the kt (Durham) jet algorithm [86], butnot [87] for the JADE algorithm [88] (both are discussed below in Sec. 9.3.1.1).

July 30, 2010 14:57


9.2.2.5. Parton-shower Monte Carlo generators:Parton-shower Monte Carlo (MC) event generators like PYTHIA [104–106],

HERWIG [107–109], SHERPA [110], and ARIADNE [111] provide fully exclusivesimulations of QCD events. Because they provide access to “hadron-level” events theyare a crucial tool for all applications that involve simulating the response of detectors toQCD events. Here we give only a brief outline of how they work and refer the readerto [112] and references therein for a more complete overview.

The MC generation of an event involves several stages. It starts with the randomgeneration of the kinematics and partonic channels of whatever hard scattering processthe user has requested.

This is then followed by a parton shower, usually based on a resummed calculation ofthe probability Δ(Q0, Q1) for each parton, that it does not split into other partons (e.g.radiate a gluon) between the hard scale Q0 and some smaller scale Q1. Δ(Q0, Q1), knownas a Sudakov form factor, takes the form Δ(Q0, Q1) ∼ exp(−G12αs ln2(Q0/Q1) + . . .).By choosing a random number r uniformly in the range 0 < r < 1 and finding the Q1

value that solves r = Δ(Q0, Q1), the MC determines the scale of the first emission ofthe shower. The procedure is repeated to obtain Q2, the scale of the next emission, andso forth down to a scale ∼ 1 GeV that separates the perturbative and non-perturbativepart of the simulation.

Once it has generated a partonic configuration, the MC “hadronizes” it according tosome hadronization model. One widely-used model involves stretching a color “string”across quarks and gluons, and breaking it up into hadrons [113,114]. For a discussionof the implementation of this “Lund” model in the MC program PYTHIA, with furtherimprovements and extensions, see Ref. 104 and references therein. Another model breakseach gluon into a qq̄ pair and then groups quarks and anti-quarks into colorless “clusters”,which then give the hadrons. This cluster hadronization is implemented in the HERWIGevent generator [107–109].

For processes with initial-state hadrons, the showering off the incoming partons mustadditionally take into account the scale-dependence of the PDFs and the non-per-turbative part must account also for the proton remnants. In pp and γp scattering, thecollision between the hadron remnants generates an underlying event (UE), usually byimplementing additional 2 → 2 scatterings (“multiple parton interactions”) at a scale of afew GeV. The separation between the UE and other parts of the shower and hadronizationis somewhat ambiguous, because they are all interconnected in terms of their color flow.

Parton showers usually generate a correct distribution of soft and collinear emission,but they often fail to reproduce the pattern of hard wide-angle emissions that would begiven by the exact multi-parton matrix elements. In cases where this matters, it is usualto “merge” the parton showers with the generation of exact LO multi-parton matrixelements (Sec. 9.2.2.2), including a prescription to avoid double or under-counting of realand virtual corrections (e.g. CKKW [115] or MLM prescriptions [116]) .

MCs as described above generate cross sections for the requested hard process thatare correct at LO. For hadron-collider applications it is common to multiply thesecross sections by an inclusive K-factor, i.e. the ratio of (N)NLO to LO results for a

July 30, 2010 14:57


related inclusive cross section. For measurements with cuts, this may not always beadequate: higher-order corrections in a restricted phase-space region can be substantiallydifferent from those in the inclusive case. For a number of processes there also exist MCimplementations that are correct to NLO, using the MC@NLO [117] or POWHEG [118]prescriptions to avoid double counting the approximate NLO pieces already implicitlyincluded in the MCs through their showering.

9.2.3. Accuracy of predictions :LO calculations are often said to be accurate to within a factor of two. This is based on

the observed impact of scale variation across a range of observables and of the experiencewith NLO corrections in the cases where these are available. In processes involving newpartonic scattering channels at NLO and/or large ratios of scales (such as the productionof high-pt jets containing B-hadrons), the NLO to LO K-factors can be substantiallylarger than 2.

The accuracy of a given particular perturbative QCD prediction is usually estimatedby varying the renormalization and factorization scales around a central value Q that istaken close to the physical scale of the process.‡‡ A conventional range of variation isQ/2 < μR, μF < 2Q.

There does not seem to be a broad consensus on whether μR and μF should bekept identical or varied independently. One option is to vary them independently withthe restriction 1

2μR < μF < 2μR [119]. This limits the risk of misleadingly smalluncertainties due to fortuitous cancellations between the μF and μR dependence whenboth are varied together, while avoiding the appearance of large logarithms of μ2

R/μ2F

when both are varied completely independently.Calculations that involve resummations usually have an additional source of uncertainty

associated with the choice of argument of the logarithms being resummed, e.g. ln(2pt,ZMZ

)

as opposed to ln(12

pt,ZMZ

). In addition to varying renormalization and factorization scales,it is therefore also advisable to vary the argument of the logarithm by a factor of two ineither direction with respect to the “natural” argument.

The accuracy of QCD predictions is limited also by non-perturbative corrections,which typically scale as a power of Λ/Q. For measurements that are directly sensitiveto the structure of the hadronic final state the corrections are usually linear in Λ/Q.The non-perturbative corrections are further enhanced in processes with a significantunderlying event (i.e. in pp and pp̄ collisions) and in cases where the perturbative crosssections fall steeply as a function of pt or some other kinematic variable.

Non-perturbative corrections are commonly estimated from the difference betweenMonte Carlo events at the parton-level and after hadronization, though methods exist alsoto analytically deduce non-perturbative effects in one observable based on measurementsof other observables (see the reviews [15,120]) .

‡‡ A more conservative scheme is to take the uncertainty to be the size of the last knownperturbative order.

July 30, 2010 14:57


9.3. Experimental QCD

Since we are not able to directly measure partons (quarks or gluons), but onlyhadrons and their decay products, a central issue for every experimental test of QCDis establishing a correspondence between observables obtained at the partonic and thehadronic level. The only theoretically sound correspondence is achieved by means ofinfrared and collinear safe quantities, which allow one to obtain finite predictions at anyorder of perturbative QCD.

As stated above, the simplest case of infrared and collinear safe observables are totalcross sections. More generally, when measuring inclusive observables, the final state isnot 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 partonicor hadronic final state. In e+e− annihilation, widely used examples are the ratios ofpartial widths or branching ratios for the electroweak decay of particles into hadronsor leptons, such as Z or τ decays, (cf. Sec. 9.2.1). Such ratios are often favored overabsolute cross sections or partial widths because of large cancellations of experimentaland theoretical systematic uncertainties. The strong suppression of non-perturbativeeffects, O(Λ4/Q4), is one of the attractive features of such observables, however, at thesame time the sensitivity to radiative QCD corrections is small, which for example affectsthe statistical uncertainty when using them for the determination of the strong couplingconstant. In the case of τ decays not only the hadronic branching ratio is of interest, butalso moments of the spectral functions of hadronic tau decays, which sample differentparts of the decay spectrum and thus provide additional information. Other examples ofinclusive observables are structure functions (and related sum rules) in DIS. These areextensively discussed in Sec. 16 of this Review.

As soon as (parts) of the structure or composition of the final state are analyzedand cross section differential in one or more variables characterizing this structure areof interest, we talk about exclusive observables, such as jet rates, jet substructureand event-shape distributions. Furthermore, any cross section differential in somecharacteristic kinematic quantity of the final state falls into this category, such astransverse momentum distributions of jets or vector bosons in hadron collisions. The caseof fragmentation functions, i.e. the measurement of hadron production as a function ofthe hadron momentum relative to some hard scattering scale, is discussed in Sec. 17 ofthis Review.

It is worth mentioning that, besides the correspondence between the parton and hadronlevel, also a correspondence between the hadron level and the actually measured quantitiesin the detector has to be established. The simplest examples are corrections for finiteexperimental acceptance and efficiencies. However, measurements of exclusive observablessuch as jet rates require more involved corrections in order to relate, e.g. the energydeposits in a calorimeter to the jets at the hadron level. Typically detector simulationsare used in order to obtain these corrections. Care should be taken here in order tohave a clear separation between the parton-to-hadron level and hadron-to-detector levelcorrections, as well as to ensure the independence of the latter from the MC modelused in the simulations. Finally, it is strongly suggested to provide, whenever possible,measurements corrected for detector effects which then can be easily compared to the

July 30, 2010 14:57


results of other experiments and/or theoretical calculations.

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. These bunches are generically called jets. To a first approximation,a jet can be thought of as a hard parton that has undergone soft and collinear showeringand then hadronization. Jets are used both for testing our understanding and predictionsof high-energy QCD processes, and also for identifying the hard partonic structure ofdecays of massive particles like top quarks.

In order to map observed hadrons onto a set of jets, one uses a jet definition. Themapping involves explicit choices: for example when a gluon is radiated from a quark,for what range of kinematics should the gluon be part of the quark jet, or instead forma separate jet? Good jet definitions are infrared and collinear safe, simple to use intheoretical and experimental contexts, applicable to any type of inputs (parton or hadronmomenta, charged particle tracks, and/or energy deposits in the detectors) and lead tojets that are not too sensitive to non-perturbative effects. An extensive treatment ofthe topic of jet definitions is given in Ref. 121 (for e+e− collisions) and Refs. 122, 123(for pp or pp̄ collisions). Here we briefly review the two main classes: cone algorithms,extensively used at hadron colliders, and sequential recombination algorithms, morewidespread in e+e− and ep colliders.

Very generically, most (iterative) cone algorithms start with some seed particle i, sumthe momenta of all particles j within a cone of opening-angle R, typically defined in termsof (pseudo-)rapidity and azimuthal angle. They then take the direction of this sum as anew seed and repeat until the cone is stable, and call the contents of the resulting stablecone a jet if its transverse momentum is above some threshold pt,min. The parameters Rand pt,min should be chosen according to the needs of a given analysis.

There are many variants of cone algorithm, and they differ in the set of seeds theyuse and the 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 particlesis a problem w.r.t. infrared and collinear safety, and seeded algorithms are generallynot compatible with higher-order (or sometimes even leading-order) QCD calculations,especially in multi-jet contexts, as well as potentially subject to large non-perturbativecorrections and instabilities. Seeded algorithms (JetCLU, MidPoint, and various otherexperiment-specific iterative cone algorithms) are therefore to be deprecated. A modernalternative is to use a seedless variant, SISCone [124].

Sequential recombination algorithms at hadron colliders (and in DIS) are characterizedby a distance dij = min(k2p

t,i, k2pt,j)Δ

2ij/R2 between all pairs of particles i, j, where Δij

is their distance in the rapidity-azimuthal plane, kt,i is the transverse momentum w.r.t.the incoming beams, and R is a free parameter. They also involve a “beam” distancediB = k

2pt,i . One identifies the smallest 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

July 30, 2010 14:57


is removed from the list of particles and called a jet. As with cone algorithms, one usuallyconsiders only jets above some transverse-momentum threshold pt,min. The parameterp determines the kind of algorithm: p = 1 corresponds to the (inclusive-)kt algorithm[86,125,126], p = 0 defines the Cambridge-Aachen algorithm [127,128], while for p = −1we have the anti-kt algorithm [129]. All these variants are infrared and collinear safe toall orders of perturbation theory. Whereas the former two lead to irregularly shaped jetboundaries, the latter results in cone-like boundaries.

The kt algorithm in e+e− annihilations [86] uses yij = 2 min(E2i , E2

j )(1−cos θij)/Q2 asdistance measure and repeatedly merges the pair with smallest yij , until all yij distancesare above some threshold ycut, the jet resolution parameter. The (pseudo)-particles thatremain at this point are called the jets. Here it is ycut (rather than R and pt,min)that should be chosen according to the needs of the analysis. As mentioned above, thekt algorithm has the property that logarithms ln(1/ycut) exponentiate in resummationcalculations. This is one reason why it is preferred over the earlier JADE algorithm [88],which uses the distance measure yij = 2 Ei Ej (1 − cos θij)/Q2.

Efficient implementations of the above algorithms are available through the FastJetpackage [130], which is also packaged within SpartyJet [131].

9.3.1.2. Event Shapes:Event-shape variables are functions of the four momenta in the hadronic final state

that characterize the topology of an event’s energy flow. They are sensitive to QCDradiation (and correspondingly to the strong coupling) insofar as gluon emission changesthe shape of the energy flow.

The classic example of an event shape is the thrust [132,133] in e+e− annihilations,defined as

τ̂ = max�nτ

∑i |�pi · �nτ |∑

i |�pi| , (9.22)

where �pi are the momenta of the final-state particles and the maximum is obtainedfor the thrust axis �nτ . In the Born limit of the production of a perfect back-to-backqq̄ pair the limit τ̂ → 1 is obtained, whereas a perfectly symmetric many-particleconfiguration leads to τ̂ → 1/2. Further event shapes of similar nature have been definedand extensively measured at LEP and at HERA, and for their definitions and reviews werefer to Refs. 1,2,120,134,135. Some discussion of hadron-collider event shapes is given inRef. 136.

Event shapes are used for many purposes. These include measuring the strongcoupling, tuning the parameters of Monte Carlo showering programs, investigatinganalytical models of hadronization and distinguishing QCD events from events that mightinvolve decays of new particles (giving event-shape values closer to the spherical limit).

July 30, 2010 14:57


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 andhas been extensively used to study differences in the properties of quark and gluoninduced jets, strongly related to their different color charges. In general there is clearexperimental evidence that gluon jets are “broader” and have a softer particle spectrumthan (light-) quark jets, whereas b-quark jets are similar to gluon jets. As an examplefor an observable, the jet shape Ψ(r/R) is the fractional transverse momentum containedwithin a sub-cone of cone-size r for jets of cone-size R. It is sensitive to the relativefractions of quark and gluon jets in an inclusive jet sample and receives contributions fromsoft-gluon initial-state radiation and beam remnant-remnant interactions. Therefore, ithas been widely employed for validation and tuning of Monte Carlo models. CDF hasmeasured the jet shape Ψ(r/R) for an inclusive jet sample [137] as well as for b-jets [138].Similar measurements in DIS have been reported in Refs. 139, 140. Further discussions,references and, recent summaries can be found in Refs. 135, 141, 142.

The use of jet substructure has also been suggested in order to distinguish QCDjets from jets that originate from hadronic decays of boosted massive particles (high-pt

electroweak bosons, top quarks and hypothesized new particles). For a review anddetailed references, see sec. 5.3 of Ref. 122.

9.3.2. State of the art QCD measurements at colliders :There exists an enormous 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 anyjustice. Extensive reviews of the subject have been published in Refs. 134, 135 for e+e−colliders, whereas for hadron colliders comprehensive overviews are given in Refs. 123,143, and recent summaries can be found in, e.g. Refs. 144–146, 142. Below we concentrateour discussion on measurements that are most sensitive to hard QCD processes.

9.3.2.1. e+e− colliders: The analyses of jet production in e+e− collisions, mostly fromJADE data at center-of-mass energies between 14 and 44 GeV, as well as from LEPdata at the Z resonance and up to 209 GeV, covered the measurements of (differentialor exclusive) jet rates (with multiplicities typically up to 4, 5 or 6 jets), the study of3-jet events and particle production between the jets as a tool for testing hadronizationmodels, as well as 4-jet production and angular correlations in 4-jet events, useful formeasurements of the strong coupling constant and putting constraints on the QCD colorfactors, thus probing the non-abelian nature of QCD. There have also been extensivemeasurements of event shapes. The tuning of parton shower MC models, typicallymatched to matrix elements for 3-jet production, has led to good descriptions of theavailable, highly precise data. Especially for the large LEP data sample at the Z peak, thestatistical errors are mostly negligible, whereas the experimental systematic uncertaintiesare at the per-cent level or even below. These are usually dominated by the uncertaintiesrelated to the MC model dependence of the efficiency and acceptance corrections (oftenreferred to as “detector corrections”).

July 30, 2010 14:57


9.3.2.2. DIS and photoproduction: Multi-jet production in ep collisions at HERA, bothin the DIS and photoproduction regime, allows for tests of QCD factorization (oneinitial-state proton and its associated PDF versus the hard scattering which leads tohigh-pt jets) and NLO calculations which exist for 2- and 3-jet final states. Sensitivityis also obtained to the product of the coupling constant and the gluon PDF. By nowexperimental uncertainties of the order of 5− 10% have been achieved, mostly dominatedby jet energy scale uncertainties, whereas statistical errors are negligible to a large extent.For comparison to theoretical predictions, at large jet pt the PDF uncertainty dominatesthe theoretical error (typically of order 5 - 10%, in some regions of phase-space up to20%), therefore jet observables become useful inputs for PDF fits. In general, for Q2

above ∼ 100 GeV2 the data are well described by NLO matrix element calculations,combined with DGLAP evolution equations. Results at lower values (Q2 < 100 GeV2)point to the necessity of including NNLO effects. Also, at low values of Q2 and x, inparticular for large jet pseudo-rapidities, there are indications for the need of BFKL-typeevolution, though the predictions for such schemes are still limited. In the case ofphotoproduction, the data-theory comparisons are hampered by the uncertainties relatedto the photon PDF.

A few examples of recent measurements can be found in Refs. 147–150 for DIS and inRefs. 151–153 for photoproduction.

9.3.2.3. Hadron colliders: Jet measurements at the TEVATRON are now publishedfor data samples up to ∼ 2 fb−1. Among the most important cross sections measured isthe inclusive jet production as a function of the jet transverse energy (Et) or the jettransverse momentum (pt), now available for several rapidity regions and for pt up to 700GeV. Most notably, the TEVATRON experiments now have measurements based on theinfrared- and collinear-safe kt algorithm in addition to the more widely used Midpointand JetCLU algorithms of the past. Recent results by the CDF and D0 collaborationscan be found in Refs. 154, 155, where we observe a good description of the data by theNLO QCD predictions. The experimental systematic uncertainties are dominated by thejet energy scale error, by now quoted to less than 3% and thus leading to uncertaintiesof 10 to 60% on the cross section, increasing with pt. The PDF uncertainties dominatethe theoretical error. In fact, inclusive jet data are important inputs to global PDF fits,in particular for constraining the high-x gluon PDF.

A rather comprehensive summary, comparing NLO QCD predictions to data forinclusive jet production in DIS, pp, and pp̄ collisions, is given in Ref. 156 and reproducedhere in Fig. 9.1.

Dijet events are analyzed in terms of their invariant mass and angular distributions,which allow one to put stringent limits on deviations from the Standard Model, such asquark compositeness (two recent examples can be found in Refs. 158, 159). Furthermore,dijet azimuthal correlations between the two leading jets, normalized to the total dijetcross cross section, are an extremely valuable tool for studying the spectrum of gluonradiation in the event. As shown in Ref. 160, the LO (non-trivial) prediction for thisobservable, with at most three partons in the final state, is not able to describe thedata for an azimuthal separation below 2π/3, where NLO contributions (with 4 partons)

July 30, 2010 14:57


restore the agreement with data. In addition, this observable can be employed to tuneMonte Carlo predictions of soft gluon radiation in the final state.

Similarly important tests of QCD arise from measurements of vector boson (photon,W , Z) production together with jets. A recent analysis of photon+jet production byD0 [161] indicates that NLO calculations, combined with modern PDF sets, are unable todescribe the shape of the photon pt across the entire measured range, showing the needfor an improved and consistent theoretical description of this process.

In the case of Z+jets, the Z momentum can be precisely reconstructed using theleptons, allowing for a precise determination of the Z pt distribution, which is sensitiveto QCD radiation both at high and low scales and thus probes perturbative as wellas non-perturbative effects. For example, a recent D0 result [162] quotes experimentalstatistical and systematic uncertainties of the order of 10%, increasing up to 20% inthe lowest momentum range. The data are compared to predictions from NLO QCDand from different Monte Carlo models, where, for example, LO matrix elements for upto three partons are matched to a parton shower. Whereas the total cross section isunderestimated, the shape is well reproduced over a large phase-space region. Furtherexamples of recent results for Z (or W ) plus jets production are found in Refs. 161,163, 164. Among the most important recent developments is the completion of a NLOcalculation for W+3jet production [57,58], which will be relevant also for future LHCbackground estimations. This type of process is an example, where jets need to be foundwith an infrared and collinear safe jet algorithm, such as SISCone, in order to obtainfinite NLO predictions. This would not be possible with algorithms such as Midpoint orJetCLU. The latter is used for a CDF measurement [164], which is compared to the NLOQCD prediction with SISCone as jet algorithm. Besides this inconsistency, the agreementappears to be good.

Finally, TEVATRON measurements of heavy quark (b, c) jet production, inclusive orin association with vector bosons, have led to stringent tests of NLO predictions (seeRefs. 165–170 for examples of recent analyses).

9.3.3. Tests of the non-abelian nature of QCD :QCD is a gauge theory with SU(3) as underlying gauge group. For a general

gauge theory with a simple Lie group, the couplings of the fermion fields to thegauge fields and the self-interactions in the non-abelian case are determined by thecoupling constant and Casimir operators of the gauge group, as introduced in Sec. 9.1.Measuring the eigenvalues of these operators, called color factors, probes the underlyingstructure of the theory in a gauge invariant way and provides evidence of the gluonself-interactions. Typically, cross sections can be expressed as functions of the colorfactors, for example σ = f(αsCF , CA/CF , nfTR/CF ). Sensitivity at leading order inperturbation theory can be achieved by measuring angular correlations in 4-jet eventsin e+e− annihilation or 3-jet events in DIS. Some sensitivity, although only at NLO,is also obtained from event-shape distributions. Scaling violations of fragmentationfunctions and the different subjet structure in quark and gluon induced jets also giveaccess to these color factors. In order to extract absolute values, e.g. for CF and CA,certain assumptions have to be made for other parameters, such as TR, nf or αs, since

July 30, 2010 14:57


1

10

10 2

10 3

10 102

103

inclusive jet productionin hadron-induced processes

fastNLOhepforge.cedar.ac.uk/fastnlo

pp

DIS

pp-bar

√s = 200 GeV

√s = 300 GeV

√s = 318 GeV

√s = 546 GeV

√s = 630 GeV

√s = 1800 GeV

√s = 1960 GeV

STAR 0.2 < |y| < 0.8

H1 150 < Q2 < 200 GeV2

H1 200 < Q2 < 300 GeV2

H1 300 < Q2 < 600 GeV2

H1 600 < Q2 < 3000 GeV2

ZEUS 125 < Q2 < 250 GeV2

ZEUS 250 < Q2 < 500 GeV2

ZEUS 500 < Q2 < 1000 GeV2

ZEUS 1000 < Q2 < 2000 GeV2

ZEUS 2000 < Q2 < 5000 GeV2

CDF 0.1 < |y| < 0.7

DØ |y| < 0.5

CDF 0.1 < |y| < 0.7DØ 0.0 < |y| < 0.5DØ 0.5 < |y| < 1.0

CDF cone algorithmCDF kT algorithm

(× 400)

(× 100)

(× 35)

(× 16)

(× 6)

(× 3)

(× 1)

all pQCD calculations using NLOJET++ with fastNLO:

αs(MZ)=0.118 | CTEQ6.1M PDFs | μr = μf = pT jet

NLO plus non-perturbative corrections | pp, pp: incl. threshold corrections (2-loop)

pT (GeV/c)

data

/ th

eory

Figure 9.1: A compilation of data-over-theory ratios for inclusive jet crosssections as a function of jet transverse momentum (pT ), measured in differenthadron-induced processes at different center-of-mass energies; from Ref. 156,including some updates [157]. The various ratios are scaled by arbitrary numbers(indicated between parentheses) for better readability of the plot. The theoreticalpredictions have been obtained at NLO accuracy, for parameter choices (couplingconstant, PDFs, renormalization, and factorization scales) as indicated at thebottom of the figure. Color version at end of book.

typically only combinations (ratios, products) of all the relevant parameters appear inthe perturbative prediction. A recent compilation of results [135] quotes world averagevalues 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

July 30, 2010 14:57


expectations from SU(3) of CA = 3 and CF = 4/3. An overview of the history and thecurrent status of tests of asymptotic freedom, closely related to the non-abelian nature ofQCD, can be found in Ref. 171.

9.3.4. Measurements of the strong coupling constant :If the quark masses are fixed, there is only one free parameter in the QCD Lagrangian,

the strong coupling constant αs. The coupling constant in itself is not a physicalobservable, but rather a quantity defined in the context of perturbation theory, whichenters predictions for experimentally measurable observables, such as R in Eq. (9.7).

Many experimental observables are used to determine αs. Considerations in suchdeterminations include:

• The observable’s sensitivity to αs as compared to the experimental precision.For example, for the e+e− cross section to hadrons (cf. R in Sec. 9.2.1), QCDeffects are only a small correction, since the perturbative series starts at orderα0

s ; 3-jet production or event shapes in e+e− annihilations are directly sensitiveto 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 order term is ∝ α3

s .• The accuracy of the perturbative prediction, or equivalently of the relation between

αs and the value of the observable. The minimal requirement is generally consideredto be an NLO prediction. Some observables are predicted to NNLO (many inclusiveobservables, 3-jet rates and event shapes in e+e− collisions) or even N3LO (e+e−hadronic cross section and τ branching fraction to hadrons). In certain cases,fixed-order predictions are supplemented with resummation. The precise magnitudeof theory uncertainties is usually estimated as discussed in Sec. 9.2.3.

• The size of uncontrolled non-perturbative effects (except for lattice-based determina-tions of αs). Sufficiently inclusive quantities, like the e+e− cross section to hadrons,have small non-perturbative uncertainties ∼ Λ4/Q4. Others, such as event-shapedistributions, have uncertainties ∼ Λ/Q.

• The scale at which the measurement is performed. An uncertainty δ on a measurementof αs(Q2), at a scale Q, translates to an uncertainty δ′ = (α2

s(M2Z)/α2

s(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.In this review, we make no attempt to compile a full list of measurements of αs or

to produce a new world average value from them. We rather prefer to quote a recentanalysis by Bethke [172], which incorporates results with recently improved theoreticalpredictions and/or experimental precision. For detailed comments on the selected set ofrecent results we refer to Ref. 172. Here we quote the main inputs:

• Several re-analyses of the hadronic τ decay width [16,174–179], based on the newN3LO predictions, have been performed, with different approaches towardsthe detailed treatment of the perturbative (fixed order or contour improved

The time evolution of αs combinations can be followed by consulting Refs. 171, 173as well as earlier editions of this Review.

July 30, 2010 14:57


perturbative expansions) and non-perturbative contributions. In Ref. 172 a value ofαs(M2

Z) = 0.1197 ± 0.0016 is quoted as average, where the uncertainty spans thedifference of those recent analyses.

• The N3LO calculation of the hadronic Z decay width was used in a recentrevision of the global fit to electroweak precision data [180], resulting inαs(M2

Z) = 0.1193+0.0028−0.0027 ± 0.0005, where the first error is of experimental and the

second of theoretical origin.• A combined analysis of non-singlet structure functions from DIS [181], based on

QCD predictions up to N3LO, gives αs(M2Z) = 0.1142 ± 0.0023. This uncertainty

includes a theoretical error of ±0.0008.• A recent re-analysis of event shapes, measured by ALEPH at the Z peak and LEP2

energies up to 209 GeV, using NNLO predictions matched to NLL resummation, hasresulted in αs(M2

Z) = 0.1224± 0.0039 [182], with a dominant theoretical uncertaintyof 0.0035. Similarly, an analysis of JADE data [183] at center-of-mass energiesbetween 14 and 46 GeV gives αs(M2

Z) = 0.1172 ± 0.0051, with contributions fromhadronization model (perturbative QCD) uncertainties of 0.0035 (0.0030).

• A new combination [184] of precision measurements at HERA, based on NLO fitsto inclusive jet cross sections in neutral current DIS at high Q2, quotes a combinedresult of αs(M2

Z) = 0.1198 ± 0.0032, which includes a theoretical uncertainty of±0.0026.

• An improved extraction of the strong coupling constant from a NLO analysis ofradiative Υ decays [185] resulted in αs(MZ) = 0.119+0.006

−0.005.• The HPQCD collaboration [186] computes Wilson loops and similar short-

distance quantities with lattice QCD and analyzes them with NNLO perturbativeQCD. This yields a value for αs, but the lattice scale must be related to aphysical energy/momentum scale. This is achieved with the Υ′-Υ mass difference,however, many other quantities could be used as well [187]. HPQCD obtainsαs(M2

Z) = 0.1183 ± 0.0008, where the uncertainty includes effects from truncatingperturbation theory, finite lattice spacing and extrapolation of lattice data.An independent perturbative analysis of the same lattice-QCD data yieldsαs(M2

Z) = 0.1192 ± 0.0011 [188]. The HPQCD value [186] is taken for the average.It is the most precise of all inputs used in Ref. 172. It is worth noting that there isa more recent result in Ref. 189, which avoids the staggered fermion treatment ofRef. 186. There a value of αs(M2

Z) = 0.1205 ± 0.0008 ± 0.0005 +0.0000−0.0017 [189] is found,

where the first uncertainty is statistical and the others are from systematics. Sincethis approach uses a different discretization of lattice fermions and a different generalmethodology, it provides an important cross check of other lattice extractions of αs.

A non-trivial exercise consists in the evaluation of a world-average value for αs(M2Z).

A certain arbitrariness and subjective component is inevitable because of the choice ofmeasurements to be included in the average, the treatment of (non-Gaussian) systematicuncertainties of mostly theoretical nature, as well as the treatment of correlations amongthe various inputs, again mostly of theoretical origin. In Ref. 172 an attempt has beenmade to take account of such correlations, using methods as proposed, e.g., in Ref. 190.

July 30, 2010 14:57


The central value is determined as the weighted average of the individual measurements.For the error an overall, a-priori unknown, correlation coefficient is introduced anddetermined by requiring that the total χ2 of the combination equals the number ofdegrees of freedom. The world average quoted in Ref. 172 is

αs(M2Z) = 0.1184 ± 0.0007 ,

with an astonishing precision of 0.6%. It is worth noting that a cross check performed inRef. 172, consisting in excluding each of the single measurements from the combination,resulted in variations of the central value well below the quoted uncertainty, and in amaximal increase of the combined error up to 0.0012. Most notably, excluding the mostprecise determination from lattice QCD gives only a marginally different average value.Nevertheless, there remains an apparent and long-standing systematic difference betweenthe results from structure functions and other determinations of similar accuracy. Thisis evidenced in Fig. 9.2 (left), where the various inputs to this combination, evolved tothe Z mass scale, are shown. Fig. 9.2 (right) provides strongest evidence for the correctprediction by QCD of the scale dependence of the strong coupling.

0.11 0.12 0.13α (Μ )s Z

Quarkonia (lattice)

DIS F2 (N3LO)

τ-decays (N3LO)

DIS jets (NLO)

e+e– jets & shps (NNLO)

electroweak fits (N3LO)

e+e– jets & shapes (NNLO)

Υ decays (NLO)

QCD α (Μ ) = 0.1184 ± 0.0007s Z

0.1

0.2

0.3

0.4

0.5

αs (Q)

1 10 100Q [GeV]

Heavy Quarkoniae+e– AnnihilationDeep Inelastic Scattering

July 2009

Figure 9.2: Left: Summary of measurements of αs(M2Z), used as input for the

world average value; Right: Summary of measurements of αs as a function of therespective energy scale Q. Both plots are taken from Ref. 172.

July 30, 2010 14:57


9.4. Acknowledgments

We are grateful to S. Bethke, J. Butterworth, M. Cacciari, L. del Debbio, P. Gambino,A. Kronfeld, M. d’Onofrio, S. Sharpe, D. Treille, N. Varelas, M. Wobisch, W.M. Yao,C.P. Yuan, and G. Zanderighi for their suggestions and comments.

References:1. R.K. Ellis, W.J. Stirling, and B.R. Webber, “QCD and collider physics,” Camb.

Monogr. Part. Phys. Nucl. Phys. Cosmol. 81 (1996).2. G. Dissertori, I.G. Knowles, and M. Schmelling, “High energy experiments and

theory,”” Oxford, UK: Clarendon (2003).3. R. Brock et al., [CTEQ Collaboration], Rev. Mod. Phys. 67, 157 (1995), see also

http://www.phys.psu.edu/∼cteq/handbook/v1.1/handbook.pdf.

4. J. Smit, “Introduction to quantum fields on a lattice: A robust mate,” CambridgeLect. Notes Phys. 15 (2002), p. 1.

5. T. DeGrand and C.E. Detar, Lattice methods for quantum chromodynamics,WorldScientific (2006).

6. “Perspectives in Lattice QCD,” Proceedings of the Workshop, Nara InternationalSeminar House (2005), Ed. Y. Kuramashi, World Scientific 2008.

7. T. Onogi, arXiv:0906.2344 [hep-ph].8. D. d’Enterria, arXiv:0902.2488 [nucl-ex].9. T. van Ritbergen, J.A.M. Vermaseren, and S.A. Larin, Phys. Lett. B400, 379

(1997) [arXiv:hep-ph/9701390].10. M. Czakon, Nucl. Phys. B710, 485 (2005) [arXiv:hep-ph/0411261].11. Y. Schroder and M. Steinhauser, JHEP 0601, 051 (2006) [arXiv:hep-

ph/0512058].12. K.G. Chetyrkin, J.H. Kuhn, and C. Sturm, Nucl. Phys. B744, 121 (2006)

[arXiv:hep-ph/0512060].13. K.G. Chetyrkin, B.A. Kniehl, and M. Steinhauser, Nucl. Phys. B510, 61 (1998)

[arXiv:hep-ph/9708255].14. See for example section 11.4 of M.E. Peskin and D.V. Schroeder, “An Introduction

To Quantum Field Theory,” Reading, USA: Addison-Wesley (1995), 842 p.15. M. Beneke, Phys. Reports 317, 1 (1999) [arXiv:hep-ph/9807443].16. P.A. Baikov, K.G. Chetyrkin, and J.H. Kuhn, Phys. Rev. Lett. 101, 012002 (2008)

[arXiv:0801.1821 [hep-ph]].17. P.A. Baikov, K.G. Chetyrkin, and J.H. Kuhn, arXiv:0906.2987 [hep-ph].18. K.G. Chetyrkin, J.H. Kuhn, and A. Kwiatkowski, Phys. Reports 277, 189 (1996).19. M. Spira, Fortsch. Phys. 46, 203 (1998) [arXiv:hep-ph/9705337].20. A. Djouadi, Phys. Reports 457, 1 (2008) [arXiv:hep-ph/0503172].21. A.H. Hoang, PoS TOP2006 032, (2006) [arXiv:hep-ph/0604185].22. J.M. Zanotti, PoS LC2008 051 (2008).23. V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972);

G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977);Yu.L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977).

July 30, 2010 14:57


24. A. Vogt, S. Moch, and J.A.M. Vermaseren, Nucl. Phys. B691, 129 (2004)[arXiv:hep-ph/0404111].

25. S. Moch, J.A.M. Vermaseren, and A. Vogt, Nucl. Phys. B688, 101 (2004)[arXiv:hep-ph/0403192].

26. J.A.M. Vermaseren, A. Vogt, and S. Moch, Nucl. Phys. B724, 3 (2005)[arXiv:hep-ph/0504242].

27. E.B. Zijlstra and W.L. van Neerven, Phys. Lett. B297, 377 (1992).28. S. Moch, J.A.M. Vermaseren, and A. Vogt, arXiv:0812.4168 [hep-ph].29. E. Laenen et al., Nucl. Phys. B392, 162 (1993);

S. Riemersma, J. Smith, and W.L. van Neerven, Phys. Lett. B347, 143 (1995)[arXiv:hep-ph/9411431].

30. I. Bierenbaum, J. Blumlein, and S. Klein, Nucl. Phys. B820, 417 (2009)[arXiv:0904.3563 [hep-ph]].

31. J.C. Collins, D.E. Soper, and G. Sterman, Nucl. Phys. B261, 104 (1985).32. R. Hamberg, W.L. van Neerven, and T. Matsuura, Nucl. Phys. B359, 343 (1991);

Erratum ibid., B 644 403, (2002).33. R.V. Harlander and W.B. Kilgore, Phys. Rev. Lett. 88, 201801 (2002) [arXiv:hep-

ph/0201206].34. C. Anastasiou and K. Melnikov, Nucl. Phys. B646, 220 (2002) [arXiv:hep-

ph/0207004].35. V. Ravindran, J. Smith, and W.L. van Neerven, Nucl. Phys. B665, 325 (2003)

[arXiv:hep-ph/0302135].36. R.V. Harlander and K. J. Ozeren, arXiv:0909.3420 [hep-ph].37. M. Greco and A. Vicini, Nucl. Phys. B415, 386 (1994).38. L.N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976) [Yad. Fiz. 23, 642 (1976)].39. E.A. Kuraev, L.N. Lipatov, and V.S. Fadin, Sov. Phys. JETP 45, 199 (1977) [Zh.

Eksp. Teor. Fiz. 72, 377 (1977)].40. I.I. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978) [Yad. Fiz. 28,

1597 (1978)].41. V.S. Fadin and L.N. Lipatov, Phys. Lett. B429, 127 (1998) [arXiv:hep-

ph/9802290].42. M. Ciafaloni and G. Camici, Phys. Lett. B430, 329 (1998) [arXiv:hep-

ph/9803389].43. S. Marzani et al., Nucl. Phys. B783, 143 (2007)

[arXiv:0704.2404 [hep-ph]].44. M. Ciafaloni et al., JHEP 0708, 046 (2007) [arXiv:0707.1453 [hep-ph]].45. C.D. White and R.S. Thorne, Phys. Rev. D75, 034005 (2007) [arXiv:hep-

ph/0611204].46. I. Balitsky, Nucl. Phys. B463, 99 (1996) [arXiv:hep-ph/9509348].47. Y.V. Kovchegov, Phys. Rev. D60, 034008 (1999) [arXiv:hep-ph/9901281].48. A. Hebecker, Phys. Reports 331, 1 (2000) [arXiv:hep-ph/9905226].49. A.V. Belitsky and A.V. Radyushkin, Phys. Reports 418, 1 (2005) [arXiv:hep-

ph/0504030].

July 30, 2010 14:57


50. E. Boos et al., [CompHEP Collaboration], Nucl. Instrum. Methods A534, 250(2004) [arXiv:hep-ph/0403113]; http://comphep.sinp.msu.ru/.

51. J. Alwall et al., JHEP 0709, 028 (2007) [arXiv:0706.2334 [hep-ph]];http://madgraph.hep.uiuc.edu/.

52. M.L. Mangano et al., JHEP 0307, 001 (2003) [arXiv:hep-ph/0206293];http://cern.ch/mlm/alpgen/.

53. T. Gleisberg and S. Hoche, JHEP 0812, 039 (2008) [arXiv:0808.3674 [hep-ph]].54. A. Cafarella, C.G. Papadopoulos, and M. Worek,arXiv:

0710.2427 [hep-ph];http://cern.ch/helac-phegas/.55. F.A. Berends and W.T. Giele, Nucl. Phys. B306, 759 (1988).56. L.J. Dixon, arXiv:hep-ph/9601359.57. C.F. Berger et al., arXiv:0907.1984 [hep-ph].58. R. Keith Ellis, K. Melnikov, and G. Zanderighi,

arXiv:0906.1445 [hep-ph].59. G. Bevilacqua et al., arXiv:0907.4723 [hep-ph].60. A. Bredenstein et al., arXiv:0905.0110 [hep-ph]].61. S. Catani and M.H. Seymour, Nucl. Phys. B485, 291 (1997) [Erratum-ibid. B 510

(1998) 503] [arXiv:hep-ph/9605323].62. S. Frixione, Z. Kunszt, and A. Signer, Nucl. Phys. B467, 399 (1996) [arXiv:hep-

ph/9512328].63. D.A. Kosower, Phys. Rev. D57, 5410 (1998) [arXiv:hep-ph/9710213];

J.M. Campbell, M.A. Cullen, and E.W.N. Glover, Eur. Phys. J. C9, 245 (1999)[arXiv:hep-ph/9809429];D.A. Kosower, Phys. Rev. D71, 045016 (2005) [arXiv:hep-ph/0311272].

64. Z. Bern et al., [NLO Multileg Working Group],arXiv:0803.0494 [hep-ph].

65. Z. Nagy, Phys. Rev. D68, 094002 (2003) [arXiv:hep-ph/0307268];http://cern.ch/nagyz/Site/NLOJet++.html.

66. J.M. Campbell and R.K. Ellis, Phys. Rev. D62, 114012 (2000) [arXiv:hep-ph/0006304]; http://mcfm.fnal.gov/.

67. K. Arnold et al., Comput. Phys. Commun. 180 1661, (2009) [arXiv:0811.4559[hep-ph]]; http://www-itp.particle.uni-karlsruhe.de/∼vbfnloweb/.

68. T. Binoth et al., Eur. Phys. J. C16, 311 (2000) [arXiv:hep-ph/9911340];http://lappweb.in2p3.fr/lapth/PHOX FAMILY/.

69. A. Gehrmann-De Ridder et al., JHEP 0712, 094 (2007) [arXiv:0711.4711[hep-ph]].

70. S. Weinzierl, Phys. Rev. Lett. 101, 162001 (2008)[arXiv:0807.3241 [hep-ph]];JHEP 0906, 041 (2009) [arXiv:0904.1077 [hep-ph]].

71. K. Melnikov and F. Petriello, Phys. Rev. D74, 114017 (2006) [arXiv:hep-ph/0609070].

72. S. Catani et al., arXiv:0903.2120 [hep-ph].

July 30, 2010 14:57


73. C. Anastasiou, K. Melnikov, and F. Petriello, Nucl. Phys. B724, 197 (2005)[arXiv:hep-ph/0501130].

74. S. Catani and M. Grazzini, Phys. Rev. Lett. 98, 222002 (2007) [arXiv:hep-ph/0703012].

75. Y.L. Dokshitzer, D.Diakonov, and S.I. Troian, Phys. Reports 58, 269 (1980).76. G. Parisi and R. Petronzio, Nucl. Phys. B154, 427 (1979).77. G. Curci, M. Greco, and Y. Srivastava, Nucl. Phys. B159, 451 (1979).78. A. Bassetto, M. Ciafaloni, and G. Marchesini, Nucl. Phys. B163, 477 (1980).79. J.C. Collins and D.E. Soper, Nucl. Phys. B193, 381 (1981) [Erratum-ibid. B213,

545 (1983)].80. J.C. Collins and D.E. Soper, Nucl. Phys. B197, 446 (1982).81. J. Kodaira and L. Trentadue, Phys. Lett. B112, 66 (1982).82. J. Kodaira and L. Trentadue, Phys. Lett. B123, 335 (1983).83. J.C. Collins, D.E. Soper, and G. Sterman, Nucl. Phys. B250, 199 (1985).84. S. Catani, et al., Nucl. Phys. B407, 3 (1993).85. S. Fleming, arXiv:0907.3897 [hep-ph].86. S. Catani, et al., Phys. Lett. B269, 432 (1991).87. N. Brown and W.J. Stirling, Phys. Lett. B252, 657 (1990).88. W. Bartel, et al., [JADE Collaboration], Z. Phys. C33, 23 (1986).89. N. Kidonakis, G. Oderda, and G. Sterman, Nucl. Phys. B531, 365 (1998)

[arXiv:hep-ph/9803241].90. R. Bonciani et al., Phys. Lett. B575, 268 (2003) [arXiv:hep-ph/0307035].91. A. Banfi, G.P. Salam, and G. Zanderighi, JHEP 0503, 073 (2005) [arXiv:hep-

ph/0407286].92. D. de Florian and M. Grazzini, Phys. Rev. Lett. 85, 4678 (2000) [arXiv:hep-

ph/0008152].93. G. Bozzi et al., Phys. Lett. B564, 65 (2003) [arXiv:hep-ph/0302104].94. C. Balazs and C.P. Yuan, Phys. Rev. D56, 5558 (1997) [arXiv:hep-ph/9704258].95. D. de Florian and M. Grazzini, Nucl. Phys. B704, 387 (2005) [arXiv:hep-

ph/0407241].96. S. Moch and P. Uwer, Phys. Rev. D78, 034003 (2008) [arXiv:0804.1476

[hep-ph]].97. N. Kidonakis and R. Vogt, Phys. Rev. D78, 074005 (2008) [arXiv:0805.3844

[hep-ph]].98. M. Beneke, P. Falgari, and C. Schwinn, arXiv:0907.1443 [hep-ph].99. M. Czakon, A. Mitov, and G. Sterman, arXiv:0907.1790 [hep-ph].

100. A. Ferroglia et al., arXiv:0907.4791 [hep-ph].101. T. Becher and M.D. Schwartz, JHEP 0807, 034 (2008) [arXiv:0803.0342

[hep-ph]].102. S. Moch and A. Vogt, Phys. Lett. B631, 48 (2005) [arXiv:hep-ph/0508265].103. E. Laenen and L. Magnea, Phys. Lett. B632, 270 (2006) [arXiv:hep-ph/0508284].104. T. Sjostrand et al., Comput. Phys. Commun. 135 238, (2001) [arXiv:hep-

ph/0010017].

July 30, 2010 14:57


105. T. Sjostrand, S. Mrenna, and P. Skands, JHEP 0605, 026 (2006) [arXiv:hep-ph/0603175]; http://projects.hepforge.org/pythia6/.

106. T. Sjostrand, S. Mrenna, and P. Skands, Comput. Phys. Commun. 178 852, (2008)[arXiv:0710.3820 [hep-ph]];http://home.thep.lu.se/∼torbjorn/ Pythia.html.

107. B.R. Webber, Nucl. Phys. B238, 492 (1984).108. G. Corcella et al., JHEP 0101, 010 (2001) [arXiv:hep-ph/0011363];

http://hepwww.rl.ac.uk/theory/seymour/herwig/.

109. M. Bahr et al., Eur. Phys. J. C58, 639 (2008) [arXiv:0803.0883 [hep-ph]]; http://projects.hepforge.org/herwig/.

110. T. Gleisberg et al., JHEP 0902, 007 (2009) [arXiv:0811.4622 [hep-ph]];http://projects.hepforge.org/sherpa/.

111. L. Lonnblad, Comput. Phys. Commun. 71 15, (1992).112. T. Sjostrand, arXiv:hep-ph/0611247.113. B. Andersson et al., Phys. Reports 97, 31 (1983).114. T. Sjostrand, Nucl. Phys. B248, 469 (1984).115. S. Catani et al., JHEP 0111, 063 (2001) [arXiv:hep-ph/0109231].116. J. Alwall et al., Eur. Phys. J. C53, 473 (2008) [arXiv:0706.

2569 [hep-ph]].117. S. Frixione and B.R. Webber, JHEP 0206, 029 (2002) [arXiv:hep-ph/0204244].118. P. Nason, JHEP 0411, 040 (2004) [arXiv:hep-ph/0409146].119. M. Cacciari et al., JHEP 0404, 068 (2004) [arXiv:hep-ph/0303085].120. M. Dasgupta and G.P. Salam, J. Phys. G30, R143 (2004) [arXiv:hep-

ph/0312283].121. S. Moretti, L. Lonnblad, and T. Sjostrand, JHEP 9808, 001 (1998) [arXiv:hep-

ph/9804296].122. G.P. Salam, arXiv:0906.1833 [hep-ph].123. S.D. Ellis et al., Prog. in Part. Nucl. Phys. 60, 484 (2008) [arXiv:0712.2447

[hep-ph]].124. G.P. Salam and G. Soyez, JHEP 0705, 086 (2007) [arXiv:0704.0292 [hep-ph]].125. S. Catani et al., Nucl. Phys. B406, 187 (1993).126. S.D. Ellis and D.E. Soper, Phys. Rev. D48, 3160 (1993) [arXiv:hep-ph/9305266].127. Y.L. Dokshitzer et al., JHEP 9708, 001 (1997) [arXiv:hep-ph/9707323].128. M. Wobisch and T. Wengler, arXiv:hep-ph/9907280.129. M. Cacciari, G.P. Salam, and G. Soyez, JHEP 0804, 063 (2008) [arXiv:0802.1189

[hep-ph]].130. M. Cacciari and G.P. Salam, Phys. Lett. B641, 57 (2006) [arXiv:hep-

ph/0512210];M. Cacciari, G.P. Salam, and G. Soyez, http://fastjet.fr/.

131. P.A. Delsart, K. Geerlins, and J. Huston,http://www.pa.msu.edu/∼huston/SpartyJet/SpartyJet.html.

July 30, 2010 14:57


132. S. Brandt et al., Phys. Lett. 12, 57 (1964).133. E. Farhi, Phys. Rev. Lett. 39, 1587 (1977).134. O. Biebel, Phys. Reports 340, 165 (2001).135. S. Kluth, Rept. on Prog. in Phys. 69, 1771 (2006) [arXiv:hep-ex/0603011].136. A. Banfi, G.P. Salam, and G. Zanderighi, JHEP 0408, 062 (2004) [arXiv:hep-

ph/0407287].137. D.E. Acosta et al., [CDF Collaboration], Phys. Rev. D71, 112002 (2005)

[arXiv:hep-ex/0505013].138. T. Aaltonen et al., [CDF Collaboration], Phys. Rev. D78, 072005 (2008)

[arXiv:0806.1699 [hep-ex]].139. C. Adloff et al., [H1 Collaboration], Nucl. Phys. B545, 3 (1999) [arXiv:hep-

ex/9901010].140. S. Chekanov et al., [ZEUS Collaboration], Nucl. Phys. B700, 3 (2004) [arXiv:hep-

ex/0405065].141. C. Glasman [H1 Collaboration and ZEUS Collaboration], Nucl. Phys. (Proc. Supp.)

191, 121 (2009) [arXiv:0812.0757 [hep-ex]].142. M. Martinez, arXiv:0905.2727 [hep-ex].143. J.M. Campbell, J.W. Huston, and W.J. Stirling, Rept. on Prog. in Phys. 70, 89

(2007) [arXiv:hep-ph/0611148].144. C. Royon, arXiv:0811.1544 [hep-ex].145. V.D. Elvira [D0 Collaboration and CDF Collaboration], arXiv:0808.0901

[hep-ex].146. C. Glasman, arXiv:0810.3570 [hep-ex].147. F.D. Aaron et al., [H1 Collaboration], arXiv:0904.3870 [hep-ex].148. F.D. Aaron et al., [H1 Collaboration], Eur. Phys. J. C54, 389 (2008)

[arXiv:0711.2606 [hep-ex]].149. S. Chekanov et al., [ZEUS Collaboration], Eur. Phys. J. C52, 515 (2007)

[arXiv:0707.3093 [hep-ex]].150. S. Chekanov et al., [ZEUS Collaboration], Phys. Rev. D78, 032004 (2008)

[arXiv:0802.3955 [hep-ex]].151. S. Chekanov et al., [ZEUS Collaboration], Nucl. Phys. B792, 1 (2008)

[arXiv:0707.3749 [hep-ex]].152. S. Chekanov et al., [ZEUS Collaboration], Phys. Rev. D76, 072011 (2007)

[arXiv:0706.3809 [hep-ex]].153. A. Aktas et al., [H1 Collaboration], Phys. Lett. B639, 21 (2006) [arXiv:hep-

ex/0603014].154. A. Abulencia et al., [CDF - Run II Collaboration], Phys. Rev. D75, 092006 (2007)

[Erratum-ibid. 119901] [arXiv:hep-ex/0701051].155. V.M. Abazov et al., [D0 Collaboration], Phys. Rev. Lett. 101, 062001 (2008)

[arXiv:0802.2400 [hep-ex]].156. T. Kluge, K. Rabbertz, and M. Wobisch, arXiv:hep-ph/0609285.157. K. Rabbertz, private communication, Aug 2009.158. T. Aaltonen et al., [CDF Collaboration], Phys. Rev. D79, 112002 (2009)

[arXiv:0812.4036 [hep-ex]].

July 30, 2010 14:57


159. V.M. Abazov et al., [D0 Collaboration], arXiv:0906.4819 [hep-ex], submitted toPhys. Rev. Lett.

160. V.M. Abazov et al., [D0 Collaboration], Phys. Rev. Lett. 94, 221801 (2005)[arXiv:hep-ex/0409040].

161. V.M. Abazov et al., [D0 Collaboration], Phys. Rev. Lett. 678, 45 (2009)[arXiv:0903.1748 [hep-ex]].

162. V.M. Abazov et al., [D0 Collaboration], Phys. Lett. B669, 278 (2008)[arXiv:0808.1296 [hep-ex]].

163. T. Aaltonen et al., [CDF - Run II Collaboration], Phys. Rev. Lett. 100, 102001(2008) [arXiv:0711.3717 [hep-ex]].

164. T. Aaltonen et al., [CDF Collaboration], Phys. Rev. D77, 011108 (2008)[arXiv:0711.4044 [hep-ex]].

165. CDF Collaboration, public note 8418, July 2006; see also http://www-cdf.fnal.gov/physics/new/qcd/abstracts/bjet 05.html.

166. V.M. Abazov et al., [D0 Collaboration], Phys. Rev. Lett. 102, 192002 (2009)[arXiv:0901.0739 [hep-ex]].

167. V.M. Abazov et al., [D0 Collaboration], Phys. Lett. B666, 23 (2008)[arXiv:0803.2259 [hep-ex]].

168. T. Aaltonen et al., [CDF Collaboration], Phys. Rev. Lett. 100, 091803 (2008)[arXiv:0711.2901 [hep-ex]].

169. A. Abulencia et al., [CDF Collaboration], Phys. Rev. D74, 032008 (2006)[arXiv:hep-ex/0605099].

170. T. Aaltonen et al., [CDF collaboration], arXiv:0812.4458 [hep-ex], submitted toPhys. Rev. D..

171. S. Bethke, Prog. in Part. Nucl. Phys. 58, 351 (2007) [arXiv:hep-ex/0606035].172. S. Bethke, arXiv:0908.1135 [hep-ph].173. S. Bethke, J. Phys. G26, R27 (2000) [arXiv:hep-ex/0004021].174. M. Beneke and M. Jamin, JHEP 0809, 044 (2008) [arXiv:0806.3156 [hep-ph]].175. M. Davier et al., Eur. Phys. J. C56, 305 (2008) [arXiv:0803.

0979 [hep-ph]].176. K. Maltman and T. Yavin, Phys. Rev. D78, 094020 (2008) [arXiv:0807.0650

[hep-ph]].177. S. Menke, arXiv:0904.1796 [hep-ph].178. S. Narison, Phys. Lett. B673, 30 (2009) [arXiv:0901.3823 [hep-ph]].179. I. Caprini and J. Fischer, arXiv:0906.5211 [hep-ph], acc. by Eur. Phys. J. C..180. H. Flacher et al., Eur. Phys. J. C60, 543 (2009) [arXiv:0811.

0009 [hep-ph]].181. J. Blumlein, H. Bottcher, and A. Guffanti, Nucl. Phys. B774, 182 (2007)

[arXiv:hep-ph/0607200].182. G. Dissertori et al., JHEP 0908, 036 (2009) [arXiv:0906.3436 [hep-ph]].183. S. Bethke et al., [JADE Collaboration], arXiv:0810.1389 [hep-ex].184. C. Glasman [H1 Collaboration and ZEUS Collaboration], J. Phys. Conf. Ser. 110

022013 (2008) [arXiv:0709.4426 [hep-ex]].

July 30, 2010 14:57


185. N. Brambilla et al., Phys. Rev. D75, 074014 (2007) [arXiv:hep-ph/0702079].186. C.T.H. Davies et al., [HPQCD Collaboration], Phys. Rev. D78, 114507 (2008)

[arXiv:0807.1687 [hep-lat]].187. C.T.H. Davies et al., [HPQCD Collaboration, UKQCD Collaboration, and MILC

Collaboration], Phys. Rev. Lett. 92, 022001 (2004) [arXiv:hep-lat/0304004].188. K. Maltman, et al., Phys. Rev. D78, 114504 (2008) [arXiv:0807.2020 [hep-

lat]].189. S. Aoki, et al., [PACS-CS Collaboration], arXiv:0906.3906 [hep-lat].190. M. Schmelling, Phys. Scripta 51, 676 (1995).

July 30, 2010 14:57

Documents

9. QUANTUM CHROMODYNAMICS - Particle Data Group