14.6.1 Test of non-Abelian nature of QCD in e+e−→ 4 jets We have seen in section 14.3.1 how the colour factors associated with different QCD vertices (problem 14.5) play an important part in determining the rela-tive weights of different parton-level processes. The quark-gluon colour factor
14.6. Further developments 107
CF enters into the parton-level three-jet amplitude (14.67), but the triple-gluon vertex is not involved at order αs. This vertex is an essential feature of non-Abelian gauge theories, being absent in Abelian theories such as QED. A direct measurement of the triple-gluon vertex colour factor, CA, can be made in the process e e+ − → 4 jets.
4-jet events originate from the parton-level process e e+ − → q¯qg via three mechanisms: the emission of a second bremsstrahlung gluon, splitting of the first gluon into two gluons, and splitting of the first gluon into nf quark pairs.
As problem 14.5 shows, these three types of splitting vertices are characterized in cross sections by the colour factors CF , CA and nf TR, so that the cross section can be written as (Ali and Kramer 2011)
(αs
σ4−jet = )
CF [CF σbb + CAσgg + nf TRσq¯q]. (14.79) π
Measurements yield (Abbiendi et al. 2001)
CA/CF = 2.29 ± 0.06[stat.]± 0.14[syst.]
TR/CF = 0.38 ± 0.03[stat.]± 0.06[syst.], (14.80) in good agreement with the theoretical predictions CA/CF = 9/4 and TR/CF = 3/8 in QCD.
14.6.2 Jet algorithms
From the examples already discussed in this chapter, it is clear that jets are an essential element in making comparisons between experimental measurements involving final state particles in detectors, and theoretical calculations at the parton level using perturbative QCD. Conceptually, jets provide a common representation for these two classes of event – those at the detector level, and those at the parton level. For precision comparisons, it is necessary to have a rigorous definition of a jet – a jet algorithm – which should be equally applicable at the detector, and at the parton, level. In the more than thirty years that have passed since Sterman and Weinberg’s 1977 paper, many jet definitions have been developed and applied. All involve the basic notion of clustering together objects that are in some sense ‘near’ to each other. Two main classes of jet algorithm may be distinguished: cone algorithms based on proximity in coordinate space, as in the Sterman-Weinberg approach, and used extensively, until recently, at hadron colliders; and sequential recombination algorithms based on proximity in momentum space, as in the jet-mass criterion
+ −
of Kramer and Lampe (1987), and widely used at e e and e p colliders.
Recent general reviews of jet algorithms are provided by Salam (2010) and by Ali and Kramer (2011); see also Ellis et al. (2008), Campbell et al. (2007), and Kluth (2006). Here we shall give only a brief introduction to sequential recombination algorithms – all of which are IRC safe – since it seems likely that they will dominate future jet analyses.
Problems 109
Problems
14.1
(a) Show that the antisymmetric 3q combination of equation (14.2) is (i) a determinant, and (ii) invariant under the transformation (14.14) for each colour wavefunction.
(b) Suppose thatpαandqαstand for two SU(3)c colour wavefunctions, transforming under an infinitesimal SU(3)c transformation via
p'= (1 + iη·λ/2)p,
and similarly for q. Consider the antisymmetric combination of their components, given by
⎛
⎝ p2q3−p3q2
p3q1−p1q3
p1q2−p2q1
⎞
⎠≡
⎛
⎝ Q1
Q2
Q3
⎞
⎠;
that is, Qα = Eαβγpβqγ. Check that the three components Qα
transform as a 3∗c, in the particular case for which only the pa-rameters η1, η2, η3 and η8 are non-zero. [Note: you will need the explicit forms of the λmatrices (appendix M); you need to verify the transformation law
Q' = (1−iη·λ∗/2)Q.] 14.2
(a) Verify that the normally ordered QCD interaction ¯ˆqfγμ12λaqˆfAˆaμis C-invariant.
(b) Show thatλaFˆaμν transforms underCaccording to (14.36).
14.3Verify that the Lorentz-invariant ‘contraction’EμνρσFˆμνFˆρσ of two U(1) (Maxwell) field strength tensors is equal to 8E·B.
14.4Verify that the cross section for the exchange of a single massless scalar gluon between two quarks (or between a quark and an antiquark) contains no
‘1/ˆt2’ factor.
14.5This problem is concerned with the evaluation of various ‘colour factors’.
(a) Consider first the colour factor needed for equation (14.73). The
‘colour wavefunction’ part of the amplitude (14.59) is E
c
ac(c3)χ†(c1)λc
2 χ(c2) (14.83)
108 14. QCD I: Introduction, Tree Graph Predictions, and Jets The JADE algorithm (Bartel et al. 1986, Bethke et al. 1988) is a promi-nent early example of sequential recombination algorithms applied in e+e−
annihilation reactions. Particles are clustered in a jet iteratively as long as the quantity yij of (14.77) is less than some prescribed value yc. If for some pair (i, j), yij < yc, particles i and j are combined into a compound object (with the resultant 4-momentum, typically), and the process continues by pairing the compound with a new particle k. The procedure stops when all yij distances are greater than yc, and the compounds that remain at this stage are the jets, by definition.
One drawback with this scheme is that in higher orders of perturbation theory one meets terms of the form α2 s ln2n y (generalizations of the αs ln2 y term in (14.78)). Such terms can be large enough to invalidate a perturbative approach. Also, it is possible for two soft particles moving in opposite direc-tions to get combined in the early stages of clustering, which runs counter to the intuitive notion of a jet being restricted in angular radius. The kt -algorithm (Catani et al. 1991) avoids these problems by replacing the yij of (14.77) by
yij = 2min.[Ei 2, Ej 2](1 − cos θij )/Q2 . (14.81) This amounts to defining ‘distance’ by the minimum transverse momentum kt of the particles in nearby collinear pairs. The use of the minimum energy ensures that the distance between two soft, back-to-back particles is larger than that between a soft particle and a hard one that is close to it in angle.
The kt algorithm was widely used at LEP.
The basic idea of the kt algorithm was extended to hadron colliders (Ellis and Soper 1993, Catani et al. 1993), where the total energy of the hard scattering particles is not well defined experimentally. The distance measure yij is replaced by
2p 2p
dij = min.[pti , ptj ][(yi − yj)2 + (φi − φj )2]/R2 (14.82) where, for particle i, pti is the transverse momentum with respect to the (beam) z-axis, yi is the rapidity along the beam axis (defined by yi = 12 ln[(Ei+ pzi)/(Ei −pzi)]), φi is the azimuthal angle in the plane transverse to the beam, and R is a jet parameter. The variables yi, φi have the property that they are invariant under boosts along the beam direction. In addition, recombination with the beam jets is controlled by the quantity dij = kti 2p, which is included along with the dij ’s when recombining all the particles into (i) jets with non-zero transverse momentum, and (ii) beam jets. The power parameter p takes the value 1 in the (extended) kt algorithm, and -1 in the ‘anti-kt ’ algorithm (Cacciari et al. 2008). Whereas the former (and p = 0) leads to irregularly shaped jet boundaries, the latter leads to cone-like boundaries. The choice p= −1 was made in early LHC analyses.
108 14. QCD I: Introduction, Tree Graph Predictions, and Jets The JADE algorithm (Bartelet al. 1986, Bethkeet al. 1988) is a promi-nent early example of sequential recombination algorithms applied in e+e− annihilation reactions. Particles are clustered in a jet iteratively as long as the quantityyij of (14.77) is less than some prescribed value yc. If for some pair (i, j), yij < yc, particles i and j are combined into a compound object (with the resultant 4-momentum, typically), and the process continues by pairing the compound with a new particlek. The procedure stops when all yijdistances are greater thanyc, and the compounds that remain at this stage are the jets, by definition.
One drawback with this scheme is that in higher orders of perturbation theory one meets terms of the form α2sln2ny (generalizations of theαsln2y term in (14.78)). Such terms can be large enough to invalidate a perturbative approach. Also, it is possible for two soft particles moving in opposite direc-tions to get combined in the early stages of clustering, which runs counter to the intuitive notion of a jet being restricted in angular radius. The kt -algorithm (Catani et al. 1991) avoids these problems by replacing theyij of (14.77) by
yij= 2min.[Ei2, Ej2](1−cosθij)/Q2. (14.81) This amounts to defining ‘distance’ by the minimum transverse momentum kt of the particles in nearby collinear pairs. The use of the minimum energy ensures that the distance between two soft, back-to-back particles is larger than that between a soft particle and a hard one that is close to it in angle.
Thektalgorithm was widely used at LEP.
The basic idea of thektalgorithm was extended to hadron colliders (Ellis and Soper 1993, Catani et al. 1993), where the total energy of the hard scattering particles is not well defined experimentally. The distance measure yij is replaced by
dij = min.[p2pti, p2ptj][(yi−yj)2+ (φi−φj)2]/R2 (14.82) where, for particle i, pti is the transverse momentum with respect to the (beam)z-axis,yiis the rapidity along the beam axis (defined byyi =12ln[(Ei+ pzi)/(Ei−pzi)]),φiis the azimuthal angle in the plane transverse to the beam, andRis a jet parameter. The variablesyi, φi have the property that they are invariant under boosts along the beam direction. In addition, recombination with the beam jets is controlled by the quantitydij =k2pti, which is included along with thedij’s when recombining all the particles into (i) jets with non-zero transverse momentum, and (ii) beam jets. The power parameterptakes the value 1 in the (extended)kt algorithm, and -1 in the ‘anti-kt’ algorithm (Cacciariet al. 2008). Whereas the former (and p= 0) leads to irregularly shaped jet boundaries, the latter leads to cone-like boundaries. The choice p=−1 was made in early LHC analyses.
E 14.1
14.2
Problems 109
Problems
(a) Show that the antisymmetric 3q combination of equation (14.2) is (i) a determinant, and (ii) invariant under the transformation (14.14) for each colour wavefunction.
(b) Suppose that pα and qα stand for two SU(3) colour wavefunctions, c transforming under an infinitesimal SU(3)c transformation via
p ' = (1 + iη · λ/2)p,
and similarly for q. Consider the antisymmetric combination of their components, given by
⎛
⎝ p2q3 − p3q2 p3q1 − p1q3 p1q2 − p2q1
⎞
⎠≡
⎛
⎝ Q1 Q2
⎞
⎠; Q3
that is, Qα = Eαβγ pβ qγ . Check that the three components Qα transform as a 3 ∗ c , in the particular case for which only the pa-rameters η1, η2, η3 and η8 are non-zero. [Note: you will need the explicit forms of the λ matrices (appendix M); you need to verify the transformation law
Q' = (1 − iη · λ ∗ /2)Q.]
(a) Verify that the normally ordered QCD interaction ¯qˆf γμ 1 2 λaqˆf Aˆaμ is C-invariant.
(b) Show that λaFˆ aμν transforms under C according to (14.36).
Fˆμν ˆ
14.3 Verify that the Lorentz-invariant ‘contraction’ Eμνρσ F ρσ of two U(1) (Maxwell) field strength tensors is equal to 8E · B.
14.4 Verify that the cross section for the exchange of a single massless scalar gluon between two quarks (or between a quark and an antiquark) contains no
‘1/tˆ2’ factor.
14.5 This problem is concerned with the evaluation of various ‘colour factors’.
(a) Consider first the colour factor needed for equation (14.73). The
‘colour wavefunction’ part of the amplitude (14.59) is λc
ac(c3)χ†(c1) χ(c2) (14.83)
c 2
Problems 111 (b) The colour part for the triple gluon vertex g1→g2+ g3 is
E
c,d,e
a∗d(c2)a∗e(c3)fdecac(c1).
Show that the modulus squared of this, averaged over the initial gluon colours and summed over the final gluon colours, is
1 8
E
c,d,e
fdecfdec,
where each of c, d, e runs from 1 to 8. Deduce using (12.84) that this expression can be written as gen-erators of SU(3) in the 8-dimensional (adjoint) representation (see section 12.2). The expression (E
dG(8)d G(8)d ) is the SU(3) Casimir operator ˆC2 in the adjoint representation, which from (M.67) has the value CA18, where 18 is the 8×8 unit matrix, andCA = 3.
Hence show that the (averaged, summed) triple gluon vertex colour factor isCA= 3.
(c) The colour part of the g→q + ¯q vertex is χ∗r(c3)(λc
2 )rsχs(c2)ac(c1).
Show that the modulus squared of this, averaged over the initial gluon colours and summed over the final quark colours is
1 This number is usually denoted byTR. 14.6Verify equation (14.60).
14.7Verify equation (14.72).
14.8Verify that expression (14.68) becomes the factor in large parentheses in equation (14.73), when expressed in terms of the xi’s.
E
E
E
E )
110 14. QCD I: Introduction, Tree Graph Predictions, and Jets where c1, c2 and c3 label the colour degree of freedom of the quark,
by analogy with the spin wavefunctions of SU(2). The cross section is obtained by forming the modulus squared of (14.83) and summing over the colour labels ci: (cf equation (8.62)). We proceed to evaluate it as follows:
(i) Show that
χs(c2)χ∗ l (c2) = δsl.
c2
(ii) Assuming the analogous result
ac(c3)a∗ d(c3) = δcd section M.5 in appendix M) for SU(3) in the fundamental represen-tation 3, which from (M.67) has the value CF 13, where 13 is the unit 3 ×3 matrix, and CF = 4/3. Hence show that the colour factor for (14.73) is 4.
Note that if we averaged over the colours of the initial quark, or considered one particular colour, the colour factor would be CF .
110 14. QCD I: Introduction, Tree Graph Predictions, and Jets wherec1, c2andc3 label the colour degree of freedom of the quark, antiquark and gluon respectively, and the sum on the index c has been indicated explicitly. The χ’s are the colour wavefunctions of the quark and antiquark, and are represented by three-component column vectors; a convenient choice is
by analogy with the spin wavefunctions of SU(2). The cross section is obtained by forming the modulus squared of (14.83) and summing over the colour labelsci: where summation is understood on the matrix indices on theχ’s and λ’s, which have been indicated explicitly. In this form the expres-sion is very similar to thespinsummations considered in chapter 8 (cf equation (8.62)). We proceed to evaluate it as follows:
(i) Show that
where the (implied) sum on rruns from 1 to 3.
(iii) The expression E
cλc
2 λc
2 is just the Casimir operator ˆC2 (see section M.5 in appendix M) for SU(3) in the fundamental represen-tation 3, which from (M.67) has the value CF13, where 13 is the unit 3×3 matrix, andCF = 4/3. Hence show that the colour factor for (14.73) is 4.
Note that if we averaged over the colours of the initial quark, or considered one particular colour, the colour factor would beCF.
E = gen-erators of SU(3) in the 8-dimensional (adjoint) representation (see
(8) (8)
section 12.2). The expression (E
d Gd Gd ) is the SU(3) Casimir operator Cˆ2 in the adjoint representation, which from (M.67) has the value CA18, where 18 is the 8 × 8 unit matrix, and CA = 3.
Hence show that the (averaged, summed) triple gluon vertex colour factor is CA = 3.
14.8 Verify that expression (14.68) becomes the factor in large parentheses in equation (14.73), when expressed in terms of the xi’s.