jet substructure at the lhc with analytical methods

JET SUBSTRUCTURE AT THE LHCWITH ANALYTICAL METHODS

A THESIS

SUBMITTED TO THE UNIVERSITY OF MANCHESTER

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY (PHD)IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCES

ALESSANDRO VIRGINIO ARMANDO FREGOSO

SCHOOL OF PHYSICS AND ASTRONOMY

2013

CONTENTS

Abstract 19

Declaration 21

Copyright 23

Acknowledgements 25

List of publications 27

1 Introduction 29

2 Review of QCD 352.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 The QCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Running coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Running masses . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Basic features of QCD radiation . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Hadrons in the initial state . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Parton shower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6.1 The parton shower as a Markov process . . . . . . . . . . . . . . 50

2.6.2 Angular ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6.3 Ordering variables and the rest of the event . . . . . . . . . . . . 52

2.6.4 Monte Carlo tools . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Phenomenology of jets 553.1 Jet algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Exclusive mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.2 Inclusive mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.3 Incoming hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.4 A note on nomenclature . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Jet observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Large logarithms and resummation . . . . . . . . . . . . . . . . 59

3

CONTENTS

3.2.2 NLL in the expansion . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.3 NLL in the exponent . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 “Fat” jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 The origin of jet substructure . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Mass drop and filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Jet substructure today . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.7 Monte Carlo studies and open questions . . . . . . . . . . . . . . . . . . 68

4 Jet rates 714.1 EVENT2 parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Leading Order (LO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Single gluon emission . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 2- and 3-jet rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Next-to-Leading Order (NLO) . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 Exclusive kT algorithm . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.2 Inclusive algorithms . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 Resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.1 Exclusive kT algorithm . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.2 Inclusive algorithms . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4.3 Comparison with Monte Carlo . . . . . . . . . . . . . . . . . . . 90

4.4.4 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Jet substructure at fixed order 975.1 Plain jet mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 EVENT2 parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.2 LO calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.3 NLO calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.2 LO calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.4.3 NLO calculation: C2F channel . . . . . . . . . . . . . . . . . . . 105

5.4.4 NLO calculation: CFCA channel . . . . . . . . . . . . . . . . . . 108

5.4.5 Y-pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5 Mass Drop Tagger (MDT) . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5.2 LO calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4

CONTENTS

5.5.3 Logarithmic behaviour beyond LO . . . . . . . . . . . . . . . . . 1145.6 Modified Mass Drop Tagger (mMDT) . . . . . . . . . . . . . . . . . . . 118

5.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.6.2 NLO calculation: C2

F channel . . . . . . . . . . . . . . . . . . . 1185.6.3 NLO calculation: CFCA and CFNf channels . . . . . . . . . . . 1225.6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Jet substructure and resummation 1276.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1.1 The Lund Kinematic Diagram (LKD) . . . . . . . . . . . . . . . 1296.1.2 Limits of perturbative calculations . . . . . . . . . . . . . . . . . 1306.1.3 Details of MC generation and jet reconstruction . . . . . . . . . . 132

6.2 Plain jet mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.3 Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3.1 From fixed-order to resummation . . . . . . . . . . . . . . . . . 1356.3.2 Comparison with Monte Carlo . . . . . . . . . . . . . . . . . . . 137

6.4 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.4.1 Patterns in the pruning procedure . . . . . . . . . . . . . . . . . 1396.4.2 Y-pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.4.3 I-pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.4.4 Interplay of Y- and I-pruning . . . . . . . . . . . . . . . . . . . . 1426.4.5 Comparison with Monte Carlo . . . . . . . . . . . . . . . . . . . 143

6.5 MDT and mMDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.5.1 From fixed-order to resummation . . . . . . . . . . . . . . . . . 1446.5.2 Comparison with Monte Carlo . . . . . . . . . . . . . . . . . . . 147

6.6 Comparison between taggers . . . . . . . . . . . . . . . . . . . . . . . . 1486.7 Taggers and the signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7 Conclusions and outlook 155

A From Lie groups to colour charges 159

B Gluon jets 163B.1 Plain jet mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.2 Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.3 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.4 mMDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.5 Comparison with Monte Carlo results . . . . . . . . . . . . . . . . . . . 165

C Effects of a finite cut in the mMDT 169

Final word count: 35353

5

LIST OF FIGURES

1.1 Pictorial representation of the theoretical description of a typical LHCevent. This figure is taken from [4]. . . . . . . . . . . . . . . . . . . . . 30

1.2 Jet-mass distribution for an inclusive QCD jet sample generated for aproton-proton collision with outgoing particles generated with pT ≥ 2

TeV. Jets are defined using a variety of jet algorithms. This figure is takenfrom [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1 Diagrams contributing to the value of β0, Eq. (2.3.4) (ignoring ghosts).The first vertex correction is proportional to CF −CA/2, but the first termcancels with the one coming from the quark self-energy (the second dia-gram in the top row). The other vertex correction and one of the diagramscontributing to the gluon self-energy (centre row) have a colour factorproportional to CA. The other contribution to the gluon self-energy (bot-tom row) has a colour factor proportional to Nf . See App. A for moredetails of how to get these colour factors. . . . . . . . . . . . . . . . . . 38

2.2 Examples of the basic processes contributing to hard lepton-lepton, lepton-hadron and hadron-hadron collisions. . . . . . . . . . . . . . . . . . . . . 40

2.3 Feynman diagram for the pair production process. . . . . . . . . . . . . . 41

2.4 Kinematics for pair production process. . . . . . . . . . . . . . . . . . . 41

2.5 Soft gluon emission and virtual gluon exchange in e+e− annihilation. Theblack circle stands for the leptonic part of the process. . . . . . . . . . . . 42

2.6 Emission of a gluon off a quark. . . . . . . . . . . . . . . . . . . . . . . 43

2.7 One-loop correction to the Born amplitude. The black circle indicates thatthe photon is off-shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.8 Complex plane for ωg. The black circles indicate the various poles, whilethe dashed line represent the integration path. . . . . . . . . . . . . . . . 45

2.9 Contribution of one-loop corrections to the order αS . . . . . . . . . . . . 46

3.1 This figure, taken from [25], illustrates the mass-drop and filtering proce-dure. After the filtering step, the three hardest jets are considered in orderto catch the leading radiation from the bb pair and get a better reconstruc-tion of the Higgs mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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LIST OF FIGURES

3.2 Monte Carlo study of the mass-drop and filtering procedure. The signalconsists of ZH and WH production, with the Higgs boson (whose masswas set to mH = 115 GeV) decaying into a bb pair. The first three panelscorrespond to results in different decay channels of the electroweak bo-son that accompanies the Higgs boson production, while the bottom-rightpanel is their sum. The signal peak is clearly visible above the backgroundwith notable significance. Additional details can be found in [25], fromwhich this picture was taken. . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 A quick study of boosted Higgs bosons decaying into bb pairs using Her-wig++. We generated ZH events from proton-proton collisions at centre-of-mass energy of 14 TeV, with the highly-boosted Z boson (300 GeV< pT,Z < 350 GeV) decaying into electrons or muons. The mass of theHiggs boson was set to mH = 115 GeV. It is clear that plain jet recon-struction using the C/A algorithm (red line) gives no clear peak. Themass-drop procedure (green line) helps in uncovering the peak, whosequality is greatly improved by using filtering (blue line). Another proce-dure, pruning (purple line), performs comparably well. . . . . . . . . . . 67

3.4 Picture from [108] depicting the jet-mass distribution from hadronic topdecays before (in red) and after several jet-substructure treatments. Thequality of the top peak is clearly increased by all substructure methods.The W peak is also visible on the left of the histogram. . . . . . . . . . . 69

3.5 Picture from [109] which describes the top-tagging efficiency of severalsubstructure techniques. All of them do reasonably well over a large rangeof parameters and this makes boosted-top phenomenology using jet sub-structure a very attractive experimental endeavour. . . . . . . . . . . . . . 70

4.1 Visual description of a typical fixed-order calculation of jet rates: on theleft-hand side of the figure we have a particular Feynman diagram for theemission of n gluons and on the other side the kinematical configurationsof the partons, with jets symbolised by cones around the direction of theoriginating parton. One has to take into account the contributions to thevarious jet rates for each Feynman diagram, sorting them by how manyjets arise in different regions of the gluons’ phase space. . . . . . . . . . 73

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LIST OF FIGURES

4.2 The figure shows the coefficient of a/2 generated by EVENT2 for J2,defined by the inclusive anti-kT algorithm (red) and the exclusive (orDurham) kT algorithm (green). The black crosses represent the coeffi-cient of a/2 in the differences between EVENT2 and the correspondingcalculation in Eq. (4.2.10) and the first line of Eq. (4.2.15) respectively.The fact that these differences are constant is consistent with our claimof having achieved NLL-in-the-expansion accuracy at leading order in a,both for J2 and J3, since by unitarity J2 = −J3 at this order. EVENT2parameters were chosen according to the description in Sec. 4.1. . . . . . 76

4.3 Real and virtual contributions to the NLO jet fractions in the C2F channel. 77

4.4 This figure shows some features of the exclusive kT algorithm in theCFCA channel which are key to resummation. Above: the softer gluon(lighter in the picture) is first clustered to its parent and the combination isclustered to the quark. This contribution to the 2-jet fraction is completelycancelled by virtual corrections. Below: when the harder (darker) gluonis far enough from the quark, it forms a jet with its softer offspring, whichthus at NLL-in-the-expansion accuracy does not contaminate the quark jet. 79

4.5 Comparison between EVENT2 and NLO calculations at NLL in the ex-pansion. The differences between EVENT2 and our calculation can be fiton a parabola. The fitting is performed by gnuplot’s function fit. Thegoodness of the fit is demonstrated by the values of χ2

R reported on theplots. A value of χ2

R much smaller than unity indicates that the coefficientof L2, and possibly of L, are very small. We do not claim to control thoseanyway. The resulting differences are thus consistent with our statementof having achieved NLL accuracy in the expansion. EVENT2 parameterswere chosen according to the description in Sec. 5.2. . . . . . . . . . . . 81

4.6 Comparison between EVENT2 and NLO calculations at NLL in the ex-pansion in theCFCA channel. In all cases the differences between EVENT2and our calculation fit nicely on a parabola (the fit is performed by gnuplot’sfunction fit and the corresponding χ2

R is reported on each plot). This isconsistent with our claim of having correctly calculated the coefficients ofL4 and L3. EVENT2 parameters were chosen according to the descriptionin Sec. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7 Comparison between EVENT2 and NLO calculations at NLL in the ex-pansion in theCFNf channel (we explicitly set TF = 1/2). In all cases thedifferences between EVENT2 and our calculation fit nicely on a parabola(the fit is performed by gnuplot’s function fit and the correspondingvalues of χ2

R are reported on the plots). This is consistent with our claimof having correctly calculated the coefficients of L4 and L3. EVENT2parameters were chosen according to the description in Sec. 5.2. . . . . . 84

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LIST OF FIGURES

4.8 Comparison between Herwig++ and resummed jet rates using the exclu-sive kT (or Durham) algorithm. Results for Q = 1 TeV are shown onthe left, while results for Q = 10 TeV are shown on the right. The shapeof the Monte Carlo result (at parton level) is correctly reproduced in ouranalytical description, as well as the fact that the peak moves to smallervalues of the cutoff as the jet multiplicity increases and the centre-of-massenergy is increased. The heights of the peaks are also in fair agreement.Details of event generation and residual differences can be found in themain text. For the resummed distributions, we used αS(mZ) = 0.118,where mZ is the mass of the Z boson, mZ = 91.2 GeV. . . . . . . . . . . 91

4.9 Comparison between Herwig++ and resummed jet rates using the anti-kT inclusive algorithm. Results for Q = 1 TeV are shown on the left,while results for Q = 10 TeV are shown on the right. The shape ofthe Monte Carlo result (at parton level) is correctly reproduced in ouranalytical description, as well as the fact that the peak moves to smallervalues of the cutoff as the jet multiplicity increases and the centre-of-massenergy is increased. The heights of the peaks are also in fair agreement.Details of event generation and residual differences can be found in themain text. For the resummed distributions, we used αS(mZ) = 0.118,where mZ is the mass of the Z boson, mZ = 91.2 GeV. . . . . . . . . . . 92

4.10 Configuration that leads to exponentiation breaking for the JADE algo-rithm: the two gluons should be clustered with their parents for a 2-jetevent, while in a large portion of phase space they actually cluster to-gether in a third jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.11 When using the anti-kT algorithm in its exclusive mode with a kT dis-tance, the softer gluon can get clustered to the quark instead of its parent.This can happen even when the soft gluon is collinear to its parent, givinganother contribution to the 3-jet rate in addition to the one that can beexpected from resummability. Gluons are labelled as in the main text. . . 94

4.12 Comparison between EVENT2 and the NLO result at NLL in the expan-sion for the CFCA channel of J3 with the exclusive anti-kT algorithm.The difference between EVENT2 and our result is fit to a cubic curve bygnuplot’s fitting routine. The value of χ2

R shows the consistency of thisresult with our claim of having correctly calculated the coefficient of theleading logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1 Comparison between EVENT2 and the coefficient of aCF in Eq. (5.3.4)for two values of fcut. The correctness of the differential distribution oftrimmed jet masses we calculated is shown by the fact that both differ-ences (in black) tend to zero in the small-ρ limit. EVENT2 parameterswere chosen according to the description in Sec. 5.2. . . . . . . . . . . . 101

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LIST OF FIGURES

5.2 Comparison between EVENT2 and the coefficient of aC2F and aCFCA in

Eqs. (5.3.6) and (5.3.9) respectively, for two values of fcut. All differencestend to lie on a line in the small-ρ limit. This means we achieved NLLaccuracy in the expansion of the differential distributions for trimmed jetmasses at NLO. EVENT2 parameters were chosen according to the de-scription in Sec. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 Comparison between EVENT2 at LO and the coefficient of aCF in Eq. (5.4.9)for various values of zcut. EVENT2 shows a flat distribution for small val-ues of ρ which indicates a single-logarithmic behaviour for the integrateddistribution of pruned masses. The correctness of our result is shown bythe fact that all differences (in black) vanish in the small-ρ limit. EVENT2parameters were chosen according to the description in Sec. 5.2. . . . . . 106

5.4 Comparison between EVENT2 and the coefficient of a2C2F in Eq. (5.4.14)

for various values of zcut. The correctness of our result for the differentialdistribution of pruned jet masses in the C2

F channel is shown by the factthat all differences (in black) lie on a line in the small-ρ limit. EVENT2parameters were chosen according to the description in Sec. 5.2. . . . . . 108

5.5 Comparison between EVENT2 and the coefficient of a2CFCA in Eq. (5.4.17)for various values of zcut. The correctness of our result for the differen-tial distribution of pruned jet masses in the CFCA channel is shown bythe fact that all differences (in black) lie on a line in the small-ρ limit.EVENT2 parameters were chosen according to the description in Sec. 5.2. 109

5.6 This figure shows the typical configurations that gives rise to an I-prunedjet. Gluon 1 sets the pruning radius (the boundary is represented by adashed line) and then gets pruned away. The angular distance of emission2 from the quark is smaller than the pruning radius and thus it cannot bepruned away: this leads to a divergence as severe as the one correspond-ing to plain jet mass. Conversely, if gluon 1 is hard enough not to getpruned, we have a signal-like jet that we cannot discard. This is what wecall a Y-pruned jet. The final result in Eq. (5.4.11) includes real-virtualcontributions which, for clarity, we do not show here. Both gluons areassumed to be emitted inside the jet. . . . . . . . . . . . . . . . . . . . . 111

5.7 Comparison between EVENT2 and the coefficient of a2C2F in I2 as found

in Eq. (5.4.11) for various values of zcut. The correctness of our result forthe Y-pruned jet mass differential distribution in the C2

F channel is shownby the fact that all differences (in black) lie on a line in the small-ρ limit.EVENT2 parameters were chosen according to the description in Sec. 5.2. 112

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LIST OF FIGURES

5.8 Comparison between EVENT2 at LO and the coefficient of aC2F in Eq. (5.5.4)

for various values of ycut. EVENT2 shows a flat distribution for small val-ues of ρ which indicates a single-logarithmic behaviour for the integrateddistribution for MDT. The correctness of our result is shown by the factthat all differences (in black) vanish in the small-ρ limit. EVENT2 pa-rameters were chosen according to the description in Sec. 5.2. . . . . . . 115

5.9 This figure shows the configuration that leads to the WB effect. The quarkis unclustered first and, if the asymmetry condition is not met, gets dis-carded; this is because the MDT by definition follows the massive branchconsisting of gluon 1 and 2 clustered together. . . . . . . . . . . . . . . . 116

5.10 Comparison between EVENT2 at NLO and the coefficient of a2CFCA

and a2CFNf as reported in Eq. (5.5.9) for various values of ycut. Alldifferences (in black) are fitted by straight lines, proving that our calcu-lations correctly catch the most divergent behaviour of the distributionsfor the MDT in these colour channels. EVENT2 parameters were chosenaccording to the description in Sec. 5.2. . . . . . . . . . . . . . . . . . . 117

5.11 Comparison between EVENT2 at NLO and the coefficient of a2C2F in

Eq. (5.6.15) for various values of ycut and ρ < ycut1+ycut

∆2R. Each difference

(in black) reaches a constant value in the small-ρ limit, which means thatwe control the most divergent contribution to the mMDT differential dis-tribution at LO in this colour channel. EVENT2 parameters were chosenaccording to the description in Sec. 5.2. . . . . . . . . . . . . . . . . . . 122

5.12 Comparison between EVENT2 at NLO and the coefficients of a2CFCA

and a2CFNf in Eq. (5.6.15) for various values of ycut and ρ < ycut1+ycut

∆2R.

Each difference (in black) reaches a constant value in the small-ρ limit.This means that our result for the mMDT differential distribution in thesecolour channels is correct up to an unknown constant term. EVENT2parameters were chosen according to the description in Sec. 5.2. . . . . . 125

6.1 LKD for a generic observable. The integrated distribution of v is propor-tional to the shaded area, while the differential one is proportional to thelength of the lines where V (k) = v. . . . . . . . . . . . . . . . . . . . . 129

6.2 LKD for the integrated distribution of plain jet mass. The vertical line isthe cut on the emission angle induced by the clustering condition of theC/A algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.3 LKD for plain jet mass. Emissions in the shaded area are vetoed and giverise to the Sudakov form factor. Single-logarithmic contributions comefrom near the line of constant mass ρ and the left edge of the shaded region.134

6.4 LKD for trimmed jet mass for three different values of ρ, representedby the position of the thick black line. Emissions in the shaded area arevetoed and give rise to logarithmic enhancements as described in the text. 136

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LIST OF FIGURES

6.5 Comparison of trimmed mass distributions as obtained from Herwig++and Eq. (6.3.6) forRsub = 0.2 and different values of fcut. The ρ value cor-responding to mJ = 125 GeV (the approximate Higgs mass) and ωJ = 3

TeV is denoted by a dashed line for phenomenological reference. Eventgeneration and jet reconstruction were carried out as described in Sub-sec. 6.1.3. The transition points and all general features of the distribu-tions are confirmed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.6 LKD for pruned jet mass for three different values of ρ, represented by theposition of the thick black line. The dashed portion of this line representsI-pruning, while the solid part corresponds to Y-pruning. An emission onthe red line sets η, and thus D, but then gets pruned away because of itssoftness, leading to the possibility of getting an I-pruned jet. Emissionsin the shaded area are vetoed and give rise to logarithmic enhancementsas described in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.7 Comparison of pruned mass distributions as obtained from Herwig++ andEqs. (6.4.2) and (6.4.5) for zcut = 0.1. The plots show both total distri-butions and Y- and I-pruned components separately. The ρ value corre-sponding to mJ = 125 GeV (the approximate Higgs mass) and ωJ = 3

TeV is denoted by a dashed line for phenomenological reference. Eventgeneration and jet reconstruction were carried out as described in Sub-sec. 6.1.3. The transition points and all general features of the distribu-tions are confirmed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.8 Events generated by Herwig++ show that the WB effect is in practicevery small, and kicks in only for very small values of ρ . R2y3

cut, whichwith our choice of parameters is of order 10−3. The tagger follows awrong branch every time it chooses a prong with m2 +p2

T smaller than itspartner’s. We used ycut = 0.1 and µ = 0.67 for the MDT tagger. The restof the event generation follows that described in Subsec. 6.1.3. . . . . . . 144

6.9 LKD for mMDT. For ρ > ycutR2 we have the same situation as plain jet

mass, while for smaller values of ρ the vetoed region is simply a rectangle,which corresponds to a single-logarithmic leading divergence in ρ. . . . . 146

6.10 MC study of the impact of the value of the µ parameter of the mMDTdistribution. The distributions for µ = 0.67 and µ = 1 are on top of eachother. The ρ value corresponding to mJ = 125 GeV (the approximateHiggs mass) and ωJ = 3 TeV is denoted by a dashed line for phenomeno-logical reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

13

LIST OF FIGURES

6.11 Comparison of mMDT distributions as obtained from Herwig++ and Eq. (6.5.5),supplemented with finite-ycut effects (see App. C), for different values ofycut and µ = 0.67. The agreement between the two is reasonable. The ρvalue corresponding tomJ = 125 GeV (the approximate Higgs mass) andωJ = 3 TeV is denoted by a dashed line for phenomenological reference.Event generation and jet reconstruction were carried out as described inSubsec. 6.1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.12 Comparison of MC distributions for plain jet mass, trimming, pruning andmMDT for equivalent parameters, as reported on the plot. The choice ofycut is such that ycut = zcut/(1 − zcut). For the alternative version of themMDT described in the text we have chosen ycut = zcut. It is appar-ent that all taggers behave in a similar way between z2

cutR2 ' 10−2 and

zcutR2 ' 10−1, i.e. in between the two vertical dashed lines. In particular,

the alternative version of the mMDT looks just the same as pruning inthis region. Event generation and jet reconstruction were carried out asdescribed in Subsec. 6.1.3. . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.13 Signal efficiencies of several taggers (including plain jet mass) as a func-tion of the transverse momentum cut on W generation. Our choice ofparameters is fcut = 0.05, Rsub = 0.3, zcut = 0.1 and ycut = 0.11. Thefiltering radius is Rfilt = 0.3 and we kept the 3 hardest subjets, as de-scribed in [25]. The results for taggers other than filtering are in reason-able agreement with Eq. (6.7.1), while plain jet mass and filtering are bothdominated by NLO and non-perturbative effects and perform poorly. . . . 152

6.14 Significance of signal W bosons with quark (above) and gluon (below)backgrounds for several taggers and plain jet mass as a function of thegeneration cut on transverse momentum. Our choice of parameters isfcut = 0.05, Rsub = 0.3, zcut = 0.1 and ycut = 0.11. The filtering radius isRfilt = 0.3 and we kept the 3 hardest subjets, as described in [25]. Theseresults, obtained from Herwig++, confirm those reported in [152], whichare based on Pythia 6. For comparison, we also added plain jet mass andfiltering, both failing at getting an acceptable signal significance. Morecomments on these plots can be found in the main text. . . . . . . . . . . 153

A.1 Pictorial representation of squared matrix elements for tree-level branch-ings of quarks and gluons. . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.1 Comparison of trimmed mass distributions for gluon-initiated jets, as ob-tained from Herwig++ and Eq. (B.2.1) for Rsub = 0.2 and different valuesof fcut. The ρ value corresponding to mJ = 125 GeV (the approximateHiggs mass) and ωJ = 3 TeV is denoted by a dashed line for phenomeno-logical reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

14

LIST OF FIGURES

B.2 Comparison of pruned mass distributions for gluon-initiated jets, as ob-tained from Herwig++ and Eqs. (B.3.1) for zcut = 0.1. The plots showboth total distributions and Y- and I-pruned components separately. Theρ value corresponding to mJ = 125 GeV (the approximate Higgs mass)and ωJ = 3 TeV is denoted by a dashed line for phenomenological refer-ence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B.3 Comparison of mMDT distributions for gluon-initiated jets, as obtainedfrom Herwig++ and Eq. (B.4.1), partially supplemented with finite-ycut

effects (see [152] and App. C), for different values of ycut and µ = 0.67.The agreement between the two is reasonable. The ρ value correspondingto mJ = 125 GeV (the approximate Higgs mass) and ωJ = 3 TeV isdenoted by a dashed line for phenomenological reference. . . . . . . . . . 167

B.4 Comparison of MC distributions for plain jet mass, trimming, pruningand mMDT for equivalent parameters, as reported on the plot, and gluon-initiated jets. The choice of ycut is such that ycut = zcut/(1 − zcut). Forthe alternative version of the mMDT described in the text we have chosenycut = zcut. It is apparent that all taggers behave in a similar way betweenz2

cutR2 ' 10−2 and zcutR

2 ' 10−1, i.e. in between the two vertical dashedlines. In particular, the alternative version of the mMDT is very close topruning in this region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

15

LIST OF TABLES

2.1 Feynman rules for QCD. Spinor indices have been left implicit for clarity. 37

4.1 Integration results for the exclusive Durham algorithm and inclusive al-gorithms up to five jets and a3 at NLL accuracy in the expansion. We seta ≡ αS

πand L ≡ − ln εcut. . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1 Table summarising all basic features of the taggers we studied. L is de-fined as ln R2

ρ, the leading power ofL is calculated in the small-ρ limit. LP

is the estimate of the position of the peak in the differential distribution.NGLs stands for Non-Global Logarithms. The last column indicates theminimum estimated value of m2 below which Non-Perturbative effectskick in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

17

ABSTRACT

Jet substructure at the LHC with analytical methodsA thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy (PhD)Alessandro Virginio Armando Fregoso

January 16, 2014

Heavy boosted particles are being produced (or are expected, in the case of searchesfor new physics) in large quantities at the LHC. Unfortunately several search channelsare normally not viable because of huge QCD backgrounds and non-perturbative con-tamination. Since the decay products of such particles tend to get clustered in the samejet however, it is possible to exploit the jet’s substructure to enhance signal significance.Many Monte Carlo studies of many different substructure techniques have been carriedout so far, but little or no analytical insight into them is available. This work describes anoriginal research aimed at filling this gap.

As a first step towards a better understanding of jet substructure, we wish to investigatethe solidity of common preferences of jet algorithms. Our benchmarks are jet rates. Tostart with, we use an approximate fixed-order approach in order to study the impact of dif-ferent jet algorithms. The correctness of our results at the required accuracy is confirmedby checking against the program EVENT2. We then infer the resummation propertiesof these observables, allowing for a direct comparison of jet rates to the parton shower.Resummed results agree with Monte Carlo data from Herwig++. While carrying out thisoriginal work, a paper on a similar topic was published [1]. Overlapping results are foundto be in agreement. The important conclusion is that all inclusive algorithms (includinganti-kT ) behave in the same way up to the considered accuracy.

The next step is the study of substructure techniques, namely trimming, pruning andMDT. Again, we start with approximate fixed-order calculations. This provides enoughinformation to write down resummed distributions, which clearly highlight the differ-ences and similarities of the techniques under study. We propose several modificationsto some of these, aimed to improve their performances; our final results are compared toHerwig++, and they turn out to be consistent with our predictions.

A preliminary Monte Carlo study of signal efficiencies and significances completesthis work, which constitutes an important first step towards an analytical understandingof the effects of substructure techniques on both signals and backgrounds.

19

DECLARATION

The University of Manchester

Candidate Name: Alessandro Virginio Armando Fregoso

Faculty: Engineering and Physical Sciences

Thesis Title: Jet substructure at the LHC with analytical methods

Declaration to be completed by the candidate:

I declare that no portion of this work referred to in this thesis has been submitted insupport of an application for another degree or qualification of this or any other universityor other institute of learning.

Signed: Date: January 16, 2014

21

COPYRIGHT

The author of this thesis (including any appendices and/or schedules to this thesis)owns any copyright in it (the "Copyright")1 and he has given The University of Manch-ester the right to use such Copyright for any administrative, promotional, educationaland/or teaching purposes.

Copies of this thesis, either in full or in extracts, may be made only in accordance withthe regulations of the John Rylands University Library of Manchester. Details of theseregulations may be obtained from the Librarian. This page must form part of any suchcopies made.

The ownership of any patents, designs, trade marks and any and all other intellectualproperty rights except for the Copyright (the "Intellectual Property Rights") and any re-productions of copyright works, for example graphs and tables ("Reproductions"), whichmay be described in this thesis, may not be owned by the author and may be owned bythird parties. Such Intellectual Property Rights and Reproductions cannot and must not bemade available for use without the prior written permission of the owner(s) of the relevantIntellectual Property Rights and/or Reproductions.

Further information on the conditions under which disclosure, publication and ex-ploitation of this thesis, the Copyright and any Intellectual Property Rights and/or Repro-ductions described in it may take place is available from the Head of School of Physicsand Astronomy (or the Vice-President) and the Dean of the Faculty of Engineering andPhysical Sciences, for Faculty of Engineering and Physical Sciences candidates.

1This excludes material already printed in academic journals, for which the copyright belongs to saidjournal and publisher. Pages for which the author does not own the copyright are numbered differently fromthe rest of the thesis.

23

ACKNOWLEDGEMENTS

I dedicate this work to...

...my supervisors, Michael Seymour and Mrinal Dasgupta, for their patience and constantsupport.

...Gavin P. Salam and Simone Marzani, for their precious collaboration.

...all the people in the theory office, especially Alexander Powling, Lee Tomlinson andRené Angeles, for fruitful discussions and pleasant company.

...the HEP Group at the University of Manchester, full of competent people always readyto help.

...all proofreaders, you know who you are, and you know you are great!

...my friends all over the world, for making me feel at home.

...my family, for always being there.

...Paola, my greatest discovery.

25

LIST OF PUBLICATIONS

• M. Dasgupta, A. Fregoso, S. Marzani, and A. Powling, “Jet substructure withanalytical methods,” Eur. Phys. J. C, 73 11 (2013) 2623, arXiv:1307.0013[hep-ph].

• M. Dasgupta, A. Fregoso, S. Marzani, and G. P. Salam, “Towards an understanding ofjet substructure,” JHEP 1309 (2013) 029, arXiv:1307.0007 [hep-ph].

• A. Altheimer, S. Arora, L. Asquith, G. Brooijmans, J. Butterworth, et al., “JetSubstructure at the Tevatron and LHC: New results, new tools, new benchmarks,”J.Phys. G39 (2012) 063001, arXiv:1201.0008 [hep-ph].

27

http://dx.doi.org/10.1140/epjc/s10052-013-2623-3

http://arxiv.org/abs/1307.0013


http://dx.doi.org/10.1007/JHEP09(2013)029



CHAPTER

ONE

INTRODUCTION

Our current description of high-energy particle physics is based on the Standard Model(SM), which has been very successful in describing data collected up to now. The recentdiscovery of the Higgs boson at the Large Hadron Collider (LHC) completes the observa-tion of the particle content of the model [2, 3]. However, the SM cannot be regarded as afundamental theory of nature for several reasons: it does not describe gravity, dark energyand dark matter, it does not predict the right amount of CP-asymmetry necessary for ouruniverse to exist as we experience it, and it features several parameters, as well as otheropen questions.

Many theories have been proposed in an attempt to answer some or all of these ques-tions, and it is necessary to test them and eventually find signatures of new physics. Asfar as collider physics is concerned, no search can be carried out without a proper under-standing of QCD, the sector of the SM describing the constituents of nucleons, i.e. gluonsand quarks. This is particularly true for hadron colliders such as the Tevatron and theLHC, where QCD processes constitute by far the dominant background and are relevantfor the prediction of basically any accessible signal.

Fig. 1.1 shows a representation of a typical event at the LHC. The theoretical descrip-tion of such an event involves a lot of perturbative and Non-Perturbative (NP) ingredients.The former include the hard process, resummation of soft and collinear radiation, softand wide-angle radiation, jet algorithms (JetAlg); among the latter are parton densitiesf(x, µ), the Underlying Event (UE) and hadronisation. In this thesis we concentrate onthe perturbative side of the picture, and in particular on the clustering of final-state parti-cles into jets, which feature prominently both in QCD backgrounds and potential signalsignatures. The main challenge is then to be able to deduce the origin of a given jet fromits properties.

The simplest and most important property one can use to this end is jet mass. Themass of a jet is the first clue of its identity, as it is directly related to the mass of theparticle initiating it. However, QCD radiation can lead to a massive jet. To see this, wecan consider a jet made up of two massless constituents of four-momenta p1 and p2: thestandard recombination scheme (see Chap. 3) states that the four-momentum of the jet is

29

CHAPTER 1. INTRODUCTION

Figure 1.1: Pictorial representation of the theoretical description of a typical LHC event.This figure is taken from [4].

the sum of p1 and p2. This means that the squared mass of the jet m2J is

m2J = (p1 + p2)2 = p2

1 + p22 + 2p1 · p2 = 2ω1ω2(1− cos θ12) , (1.0.1)

where ωi are the energies and θ12 is the angle between the two constituents of the jet.In Chap. 6 we see how multiple QCD emissions give rise to highly non-trivial jet-massdistributions. An example for QCD jets with transverse momentum pT ≥ 2 TeV is shownin Fig. 1.2: clearly any signal of electroweak physics (recall that the Higgs boson has amass of around 125 GeV, while the W and Z bosons have masses of around 80 and 90GeV respectively) is bound to be swamped by the broad peak of QCD background. Abackground with such a non-trivial shape can potentially spoil any search for new parti-cles in this region of phase space. A better understanding of QCD jet-mass distributions istherefore critical to fully exploiting the potential of the LHC as a discovery tool. For thisreason, this topic has drawn a lot of attention in the latest years, both in the context of tra-ditional QCD [5, 6] and its effective approximations, chiefly Soft and Collinear EffectiveTheory (SCET [7–9]) [10–20].

An alternative approach is to only select events where the jets have a very high pT .While the inclusive jet cross-sections fall off as pT increases [21–24], several advan-tages can be exploited in this region: backgrounds fall faster, signal acceptance and massresolution get better [25], non-perturbative effects, which lack a clear theoretical com-prehension, have less of an impact on the final outcome of the measurement [26, 27].Additionally, as we will see in more detail later, decay products of heavy boosted reso-nances are more likely to get clustered into a single jet, whose substructure will be verydifferent from that of a pure QCD jet. Jet substructure is a relatively young field of re-

30

Figure 1.2: Jet-mass distribution for an inclusive QCD jet sample generated for a proton-proton collision with outgoing particles generated with pT ≥ 2 TeV. Jets are defined usinga variety of jet algorithms. This figure is taken from [21].

search which is concerned with precisely the use of differences between high-pT signaljets and QCD jets to enhance signals and to reject backgrounds.

The theoretical description of jet mass is based on essentially two tools: analyticalcalculations and numerical Monte Carlo (MC) simulations. The former mostly deal witha single observable at a time and are performed in the perturbative regime of QCD. Seefor instance [28, 29] and references therein for a review of the set of observables whichhave been treated so far.

On the other hand, MC simulations are completely exclusive on the final state andallow for measuring any number of observables simultaneously. This is achieved by usingMC methods to generate configurations that contribute at some fixed order in the strongcoupling and to calculate their weights. In this way, it is possible to get a full fixed-orderresult for the observable under study. EVENT2 and NLOJet++ both belong to this firstclass of MC simulators: EVENT2 exploits the Catani–Seymour subtraction method [30]to obtain predictions for e+e− collisions, while NLOJet++ extends the same method tohadron collisions1.

MC event generators give up full fixed-order accuracy to aim for a complete, if ap-proximate, simulation of a collision event as the one depicted in Fig. 1.1. Critical toachieving this goal is the parton shower picture, described in Chap. 2. The parton showeraccounts for the leading contribution of QCD radiation from initial- and final-state par-tons to the observable. Event generators also include fixed-order matrix elements forseveral hard processes and can simulate decays of bosons and hadrons; to complete thereconstruction, they probe the non-perturbative region of QCD through phenomenolog-

1EVENT2 and its documentation can be found athttp://hepwww.rl.ac.uk/theory/seymour/nlo/.

For NLOJet++, seehttp://www.desy.de/~znagy/Site/NLOJet++.html.

31

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

http://www.desy.de/~znagy/Site/NLOJet++.html

CHAPTER 1. INTRODUCTION

ical models of, among others, hadronisation and the underlying event. However, thisintroduces a set of parameters which need to be tuned to data with a certain residual de-gree of arbitrariness. In addition to this, the need to describe as completely as possible avery wide range of physical effects at the same time makes it harder to track the precisionand understand the range of validity of the final result. This is why analytical calculationsare still the clearest and most useful tools in the description of jet observables. However,analytical insight into substructure techniques is still very scarce. The main scope of thiswork is exactly to provide some material to fill this gap.

The outline of this thesis is as follows. In Chap. 2 the main features of perturbativeQCD are reviewed, together with the MC tools that will be used. In Chap. 3 we give anoverview of jet physics: the class of sequential-recombination jet algorithms is describedand we show how logarithmic enhancements appear in jet observables. These enhance-ments originate from the infrared and collinear structure of QCD and threaten to spoil theperturbative expansion unless they can be resummed. The property of resummability is offundamental importance for this work, and we discuss its meaning and the different levelsof accuracy at which it can be attained. The rest of the chapter is dedicated to an accountof the current status of jet substructure.

Chap. 4 is concerned with the study of n-jet rates, i.e. the probability of getting an n-jet final state in a collision. Of particular interest is the effect of using different algorithmson the final value of the observable. This is intuitively connected to whether the clusteringsequence associated to a specific jet algorithm is able to reproduce the structure of QCDradiation. We show that, contrary to common belief, all algorithms behave in the sameway to our accuracy, meaning that the substructure of a jet is not drastically changed bychanging jet algorithm. While this original work was being completed, a similar studywas published [1] and overlapping results are found to be in agreement.

Chap. 5 is dedicated to a fixed-order study of substructure techniques, the first of itskind. We perform calculation up to Next-to-Leading-Order (NLO) of contributions en-hanced by soft and collinear radiation to the jet-mass distributions treated with substruc-ture techniques in e+e− annihilation into partons. The results show significant differencesin the behaviour of different techniques, namely trimming [31], pruning [32, 33] and theMass-Drop Tagger (MDT) [25]. Chap. 6 takes these results a step further to get resummed

jet-mass distributions for the same techniques. Our results constitute a quantitative esti-mate of the efficiency of each technique in rejecting QCD background. We propose sev-eral modifications of the algorithms to enhance these efficiencies. A MC study of signalefficiencies for the same set of substructure techniques is carried out.

In the last part of the thesis we summarise our results and describe some possibledirections for additional research on the topic. The most apparent task is to combinethese results with a study of signal distributions, the top quark and electroweak bosons(including the Higgs boson) being of primary interest. Of course these techniques arealso useful for searches beyond the SM, and again no analytical studies have been carriedout so far in this respect. A systematic application of the approach adopted in this work

32

to existing and future substructure techniques could also lead to a further reduction inredundancies and a better optimisation of the relevant parameters, making it easier to getthe best result in every kind of search.

We are now ready to move on to the next chapter for a brief review of QCD.

33

CHAPTER

TWO

REVIEW OF QCD

In this introductory chapter we will recall the basic features of QCD, and how toconstruct sensible observables for QCD processes. All the concepts reported and morecan be found in any QCD textbook, e.g. [34–36]. Another section is dedicated to a briefdescription of the software used throughout this work.

2.1 A brief history

We feel it useful to start our brief review of QCD with a historical perspective thathighlights the basic concepts of the theory and how they came to be discovered and con-firmed in experiments.

The extensive use of spark chambers and bubble chambers in the fifties led to thediscovery of a great number of particles called hadrons (from the greek ἁδρος, hadrós,meaning “stout, thick”). It was hard to believe all of these particles to be fundamental.

In the sixties, studies on the classification of hadrons, hadron mass spectra and hadronicinteractions strongly suggested that they were made of a few fundamental building blocks,called quarks, fermions with fractional electromagnetic charge of 2

3e or −1

3e [37–39]. It

was then natural to look for the dynamics obeyed by quark systems, which is responsiblefor the composition of hadrons as well as hadronic reactions.

In order to obtain experimental information on quark dynamics it seems to be mostsensible to probe the inside of hadrons (in practice, mostly protons) by applying a beam ofstructureless particles (usually electrons). This method of studying the structure of targetparticles is essentially the same as the one utilised a long time before by Geiger, Mars-den and Rutherford in clarifying the structure of atoms. For the study of the hadronicstructure we need much higher energies and larger momentum transfer to increase theresolution. The first series of such experiments to probe the structure of the proton wasinitiated in the sixties at the Stanford Linear Accelerator Center (SLAC), with the pro-cess known as Deep Inelastic electron-proton Scattering (DIS) [40]. The main discoverywas that of approximate Bjorken scaling [41]. This is a property of structure functions,which parametrise the structure of the target proton as it “appears” to the virtual pho-ton exchanged in the scattering of projectile electrons. Structure functions can be seen

35

CHAPTER 2. REVIEW OF QCD

to depend only on a dimensionless variable in the so-called Bjorken limit (which cor-responds to high momentum transfer). This means that projectile electrons scatter offsingle, almost-free point-like constituents inside the proton (which were called partons),otherwise the structure functions would depend on some length scale related to the sizeof the constituents.

The dynamics of partons was found in non-Abelian gauge theories, whose quantisa-tion had been achieved in 1967 by Faddeev and Popov [42] and their renormalisability hadbeen proven in 1972 by ’t Hooft and Veltman [43]. The extra symmetry of quark systemsrequired by this non-Abelian gauge field theory was identified with a three-fold symmetry,thus called colour. For that reason, the theory came to be named Quantum ChromoDy-

namics (QCD). The non-Abelian gauge field was called the gluon, as it is responsible forbinding the quarks together. We will shortly see how the presence of massless gluonsleads to infrared divergences.

Probably the most important feature of QCD is the property of asymptotic freedom atshort distances. This means that the constituents of hadrons behave as almost-free, point-like particles when probed at high energies, making it possible to give a perturbativedescription of QCD processes in this regime. On the other hand, infrared divergencesand the size of the QCD coupling at low energies suggest that partons are confined atlarge distances: the breakdown of the perturbative picture of QCD is consistent with thefact that we do not observe free quarks or gluons, but only their colourless bound states,hadrons.

The computation of higher-order effects and the introduction of a new kind of purelyshort-distance processes, i.e. jets from quarks and gluons, opened in 1977 [44] the secondphase in the development of perturbative QCD. This phase is still open today.

2.2 The QCD Lagrangian

It looks natural to formulate QCD in terms of a non-Abelian gauge theory of partons,a collective name for gluons and quarks. Assuming an unbroken gauge symmetry1, wecan write the following Lagrangian for an arbitrary number of colours N :

L = −1

4F aµνF

aµν +∑q

qi (iγµDµ −mq)ij qj . (2.2.1)

The sum runs over all quark flavours q. The spinor2 field corresponding to a quark ofcolour i is denoted by qi, with i running from 1 to N . Dirac matrices are representedby γµ. The gluon field strength F a

µν , the covariant derivative (Dµ)ij and the quark mass

1In the present work we will ignore any strong CP violation.2Spinor indices have been suppressed and the sum over repeated indices is assumed.

36

2.3. RUNNING COUPLING

Gluon propagatorp

a µ b ν δabgµν

p2 + iε

Quark propagatorp

i j δijm+ /p

m2 − p2 − iε

Gluon-quark vertex a µ

ij

gγµtaij

Three-gluon vertex a3 µ3a2 µ2

a1 µ1

p3

p1

p2 −igfa1a2a3Vµ1µ2µ3(p1, p2, p3)

Gluon-quark loop

p ∫d4p

i(2π)4

Table 2.1: Feynman rules for QCD. Spinor indices have been left implicit for clarity.

matrix (mq)ij are given by

F aµν = ∂µA

aν − ∂νAaµ − gfabcAbµAcν ,

(Dµ)ij = δij∂µ + igtaijAaµ ,

(mq)ij = mqδij .

(2.2.2)

taij and fabc are the SU(N) group generators in the fundamental and adjoint representationrespectively. Details of the group structure and colour charges are reported in App. A.

In order to ensure that the kinetic operator for the gluon is non singular, one adds tothe Lagrangian a gauge-fixing term, which in the Feynman gauge takes the form:

− 1

2(∂µA

µ)2 (2.2.3)

Then it is possible to derive the Feynman rules listed in Table 2.1, with the conventionsof [34]. The function Vµ1µ2µ3(p1, p2, p3) in the three-gluon vertex is defined as

Vµ1µ2µ3(p1, p2, p3) ≡ (p1 − p2)µ3gµ1µ2 + (p2 − p3)µ1gµ2µ3 + (p3 − p1)µ2gµ3µ1 . (2.2.4)

2.3 Running coupling

QCD is a renormalisable theory, which means that all ultraviolet divergences can bereabsorbed in a finite number of redefinitions of the coupling, the masses and the fieldswhich make up the theory. The redefinition of the coupling can be written as a powerexpansion of the bare one1. This introduces a dependence of the renormalised coupling

1For a thorough discussion of renormalisation, see e.g. [45].

37


Figure 2.1: Diagrams contributing to the value of β0, Eq. (2.3.4) (ignoring ghosts). Thefirst vertex correction is proportional to CF − CA/2, but the first term cancels with theone coming from the quark self-energy (the second diagram in the top row). The othervertex correction and one of the diagrams contributing to the gluon self-energy (centrerow) have a colour factor proportional to CA. The other contribution to the gluon self-energy (bottom row) has a colour factor proportional to Nf . See App. A for more detailsof how to get these colour factors.

on a finite energy scale µ:

αS(µ) ≡ g2(µ)

4π. (2.3.1)

The running of the coupling αS(µ) is determined by the Renormalisation Group Equation(RGE) [36, 46]

µ2dαS(µ)

dµ2= β[αS(µ)] . (2.3.2)

The β function can be expanded as a power series in αS(µ). It can be proven that thistakes the form1

β(αS) = −α2S(β0 + β1αS + . . . ) , (2.3.3)

whereβ0 ≡

11CA − 4TFNf

12π, (2.3.4)

andβ1 ≡

17C2A − 10CATFNf − 6CFTFNf

24π2. (2.3.5)

Nf is the number of “active” light flavours (see end of section). Fig. 2.1 shows the variousFeynman diagrams2 which contribute to the value of β0. The first diagram contributes afactor proportional to CF − CA/2. However, the first contribution cancels with the one

1When not strictly necessary, we leave the dependence of αS on µ implicit for clarity.2With the gauge fixing choice of Eq. (2.2.3) one has to add to the Feynman rules the contributions of

ghosts. These are intermediate particles which do not appear as asymptotic states, and will play no role inthe calculations discussed in this thesis.

38

2.3. RUNNING COUPLING

coming from the quark self-energy. The diagrams on the second row contribute to thecoefficient of CA and the one on the bottom row to TFNf (see also App. A).

β0 is positive, and the β function is negative, in contrast to what happens in QED.Eq. (2.3.2) makes it possible to relate the values of αS at different energy scales µ1 andµ2. At lowest order in αS , the solution is

1

αS(µ2)= β0 ln

µ22

µ21

+1

αS(µ1). (2.3.6)

This means that for any µ2 > µ1 we get αS(µ2) < αS(µ1). Then, as µ2 → +∞ thecoupling goes to zero, a phenomenon called asymptotic freedom [47, 48]. This is whatjustifies the perturbative approach in this regime. On the other hand, as the energy scaledecreases, the coupling gets bigger, up to a certain point where the perturbative expansionis not reliable anymore. We denote the energy scale at which this happens as Λ. Theperturbative approach is meaningless in the vicinity of this scale, so we can as well define

limµ1→Λ

αS(µ1) = +∞ . (2.3.7)

Then the running of the coupling at leading order takes the simple form:

αS(µ) =1

β0 ln µ2

Λ2

. (2.3.8)

In fact, β0 and β1 are the only coefficients in the perturbative expansion of the βfunction which are independent of the renormalisation scheme. Renormalisation involvesa certain degree of arbitrariness in that one has the freedom to choose how much of thefinite parts of UV-divergent diagrams are included in the renormalisation constants. Thisimplies that at higher orders the value of αS depends not only on the energy scale but alsoon the renormalisation scheme. In turn, the value of Λ depends on the renormalisationscheme too, as this parameter is defined as the value of µ for which αS diverges. Thevalue of Λ which is usually quoted nowadays is the one obtained in the modified minimalsubtraction scheme

(MS)

for five active quark flavours (see the next subsection), whichis around 200 MeV. However, this value does not have a great physical relevance, as thecoupling constant is big enough for the perturbative approach to fail long before reachingΛ.

2.3.1 Running masses

Throughout this work quark masses will be ignored. There is a valid theoretical basisfor this assumption. Quark masses appear just as another set of parameters in the La-grangian of Eq. (2.2.1). Like the coupling, they obey an RGE similar to Eq. (2.3.2). Itcan be shown [49] that the effect of masses on observables is suppressed by inverse pow-

39


e−

e+

q

q

γ/Z

(a)

e− e−

q q

γ/Z

(b)

e−

e+

q

q

γ/Z

(c)

Figure 2.2: Examples of the basic processes contributing to hard lepton-lepton, lepton-hadron and hadron-hadron collisions.

ers of the energy scale Q when m(Q2) Q. On the other hand, the effect of a quarkmass m Q on cross-sections is suppressed by inverse powers of m and can then beignored [50]. The value of Nf that appears in Eqs. (2.3.4) and (2.3.5) is equal to the num-ber of active light flavours, i.e. the number of quark flavours with masses much smallerthan the characteristic scale entering αS(Q). This is why we can safely assume masslessquarks and we have Nf = 5 throughout this work.

2.4 Basic features of QCD radiation

In this section we show some of the most basic elements of calculations in perturbativeQCD. To this end, imagine we have a high-energy electron-positron collider and we wantto predict the value of σ(e+e− → hadrons). The basic Feynman diagram for this processis given in Fig. 2.2a. Strictly speaking this is just an electroweak process. Crossingsymmetry makes it possible to easily adapt this result to two other important processesinvolving quarks: DIS, Fig. 2.2b, and the Drell–Yan process, Fig. 2.2c. It should alsobe noted that since the electroweak coupling is relatively small, α ≡ e2/4π ∼ 10−2 (ebeing the electron charge), diagrams involving single-photon exchange are sufficient foran accurate description of these processes. Since we are interested in the high-energylimit, we will also consider electrons to be massless. The quark charge will be denoted byeq. For simplicity, we ignore the running of the coupling for now and confine ourselvesto a single quark flavour throughout this section. Then the matrix element for e+e− →qq, which will be called T , is easily written down using the Feynman rules reported inTable 2.1 (for the labelling of momenta, see Fig. 2.3):

T = eq e v(k2, r2)γµu(k1, r1)gµνq2ui(p1, s1)δijγ

νvj(p2, s2) , (2.4.1)

where u and v are the various Dirac spinors involved, and r1, r2, s1 and s2 the corre-sponding spins (spinor indices are left implicit for clarity). Note the colour conservingδij , explicitly included to ensure that the qq pair forms a colour singlet (i and j are colour

40

2.4. BASIC FEATURES OF QCD RADIATION

e−

e+

q

q

q

k1

k2

p1

p2

Figure 2.3: Feynman diagram for the pair production process.

k1 k2

p1

p2

θ

Figure 2.4: Kinematics for pair production process.

indices that run from 1 to N ). Also, the electron and positron spinors correctly carry nocolour indices. Thus we have

T = eq e v(k2, r2)γµu(k1, r1)1

q2ui(p1, s1)δijγµvj(p2, s2) . (2.4.2)

It is most useful to describe the process in the centre-of-mass system (Fig. 2.4). Defining

Q2 ≡ (k1 + k2)2 = (p1 + p2)2 , (2.4.3)

the momenta turn out to be

k1 =Q

2(1, 0, 0, 1) ,

k2 =Q

2(1, 0, 0,−1) ,

p1 =Q

2(1, 0, sin θ, cos θ) ,

p2 =Q

2(1, 0,− sin θ,− cos θ) .

(2.4.4)

After averaging over initial-state quantum numbers, the cross-section reads

σ0 =1

8Q2

∑∫d3p1

(2π)32ωp1

d3p2

(2π)32ωp2(2π)4δ4(k1 + k2 − p1 − p2)|T |2 , (2.4.5)

41


(a) (b) (c)

Figure 2.5: Soft gluon emission and virtual gluon exchange in e+e− annihilation. Theblack circle stands for the leptonic part of the process.

where the sum is taken over spinor and colour indices. Carrying out sums, Dirac tracesand integration we get

σ0 = Ne2q

4πα2

3Q2. (2.4.6)

Notice how the only QCD element here is the number of colours N . The main featureof Eq. (2.4.6) is scaling, i.e. the dependence on 1/Q2 as the only scale of the process.This dependence can be tested in real experiments, where it is indeed confirmed. Onecould then wonder why a formula containing the lowest order contribution to the totalcross-section should also be able to describe the production of hadrons, since free quarksare never observed in the final state. To address this question we need to understand thebehaviour of QCD in the infrared region of phase space. Soft corrections of order αS tothe total cross-section can be easily computed: they are given by diagrams where a realgluon is emitted into the final state and diagrams in which a virtual gluon is exchanged.All contributing amplitudes are shown in Fig. 2.5 with their leptonic parts left implicit asit will be always done from now on. Let us start with the emission of a soft gluon from thequark. By soft we mean that the energy of the gluon ωg is negligibly small with respect tothe total energy Q. The corresponding amplitude T1 reads

T1 = − eq e v(l2, r2)γµu(l1, r1)gµνq2ui(p1, s1)gγαtaikδkj

(/k + /p1)

(k + p1)2γνvj(p2, s2)εα(k, s) .

(2.4.7)All momenta can be correctly assigned by looking at Fig. 2.6. g is the bare strong couplingconstant1, taij are the generators of the fundamental representation of the colour symmetrygroup SU(N) and εµ(k, s) is the gluon’s polarisation vector. The soft approximationallows us to neglect /k in the numerator of the quark propagator. From the Dirac equationwe have u(p1)/p1

= 0, then we can add /p1γα to the numerator of the quark propagator.

Since γα/p1+ /p1

γα = 2pα1 , expanding the denominator of the quark propagator we get

T1 = − eq e gtaikδkj v(l2, r2)γµu(l1, r1)1

q2ui(p1, s1)

pα1(p1 · k)

γµvj(p2, s2)εα(k, s) . (2.4.8)

A completely equivalent formula holds for the amplitude T2, corresponding to the emis-sion of a soft gluon off the antiquark, with the only difference being an additional minus

1For simplicity, we are ignoring the running of the coupling in this section.

42


p1 + k

k

p2

p1

q

Figure 2.6: Emission of a gluon off a quark.

sign arising from the antiquark propagator. Summing the two contributions gives

T = T1 + T2

= eq e gtaikδkj v(l2, r2)γµu(l1, r1)

1

q2ui(p1, s1)×

×[

pα2(p2 · k)

− pα1(p1 · k)

]γµvj(p2, s2)εα(k, s) .

(2.4.9)

Note that substituting εα(k, s) with kα gives a null result as one would expect from gaugeinvariance. We are now ready to write the cross-section for pair production with theemission of a soft gluon σqqg:

σqqg = CFg2σ0

∫d3k

2ωg(2π)3

2(p1 · p2)

(p1 · k)(p2 · k), (2.4.10)

where we left the scale of g implicit. We choose again to work in the centre-of-massframe of reference. Inserting the result in the cross-section we get

σqqg = CFαS2πσ0

∫d cos θ

dωgωg

4

(1− cos θ)(1 + cos θ), (2.4.11)

where we neglect the recoil of the quark-antiquark pair against the emitted gluon. Thismeans that θ is the angle between the quark and the soft gluon. The cross-section forproducing an extra gluon is divergent in three regions:

• When the emitted gluon is in the direction of the outgoing quark (θ → 0).

• When the emitted gluon is in the direction of the outgoing antiquark (θ → π).

• When the emitted gluon is soft (ωg → 0).

The first two divergences are called collinear divergences, while the last one is called asoft divergence; they are sometimes collectively referred to as infrared divergences. Phys-ically, this means that the cross-section is sensitive to long distance effects, like fermionmasses, hadronisation and so on. However, once we include virtual corrections, suchdivergences cancel.

We can see this explicitly at order αS . The virtual correction at this order is shownin Fig. 2.7. From now on we will omit the annihilation part in the amplitude as well, as

43


p1

p1 + k

p2 − k

p2

k

Figure 2.7: One-loop correction to the Born amplitude. The black circle indicates that thephoton is off-shell.

its presence has no role in the following calculations: we just need an off-shell photonproducing a qq pair. The corresponding amplitude T is given by

T = g2eq

∫d4k

i(2π)4ui(p1, s1)γαt

aikδkl

/p1+ /k

(p1 + k)2 + iεγµ

× /p2− /k

(p2 − k)2 + iε

δabgαβk2

tbljγβvj(p2, s2) .

(2.4.12)

Applying the usual techniques for colour algebra and Dirac matrix manipulation, anddefining the colourless Born factor:

T0 ≡ u(p1, s1)(−eqγµ)v(p2, s2) , (2.4.13)

we can rewrite T as

T = 4iCFg2T0

∫d3kdωk(2π)4

(p1 · p2)

(k2 + iε) [k2 + 2(p1 · k) + iε] [k2 − 2(p2 · k) + iε]. (2.4.14)

We choose the reference frame where the two quarks are emitted back-to-back along thez direction:

p1 =Q

2(1, 0, 0, 1) ,

p2 =Q

2(1, 0, 0,−1) ,

k = (ωg, kT , kz) .

(2.4.15)

Carrying out the scalar products we get

T = 4iCFg2T0

∫d3kdωg(2π)4

× (p1 · p2)

(ωg + |k| − iε)(ωg − |k|+ iε)

× 1

(ω2g − |k|2 +Qωg −Qkz + iε)(ω2

g − |k|2 −Qωg −Qkz + iε)=

44


−kz −Q+ iǫ

kz − iǫ |k| − iǫ kz +Q− iǫ

−kz + iǫ−|k|+ iǫ

ωg

Figure 2.8: Complex plane for ωg. The black circles indicate the various poles, while thedashed line represent the integration path.

= 4iCFg2T0

∫d3kdωg(2π)4

× (p1 · p2)

(ωg + |k| − iε)(ωg − |k|+ iε)

× 1

(ωg −Q− kz + iε)(ωg + kz − iε)(ωg − kz + iε)(ωg +Q+ kz − iε). (2.4.16)

The various poles are presented in Fig. 2.8 together with the integration path we choose.Note that the contribution of the poles depending on Q are negligible by virtue of the softapproximation. Then the relevant poles are ωg = |k| and ωg = kz. Using the residuetheorem then gives

T = CFg2T0

∫d3k

2ωg(2π)3

(−2w12(k)

ω2g

+ωg

(kz − iε)k2T

), (2.4.17)

where we introduced the radiator, which is defined as

w12(k) ≡ 1

2ω2g

(p1 · p2)

(p1 · k)(p2 · k)=

1

(1− cos θ)(1 + cos θ). (2.4.18)

The second addend of the last equation can be evaluated applying theSokhatsky–Weierstrass theorem which states that given a complex function f continu-ously defined on the real line and given two real constants a and b such that a < 0 < b,the following relation holds:

limε→0+

∫ b

a

dxf(x)

x± iε = ∓iπf(0) + pv

(∫ b

a

dxf(x)

x

), (2.4.19)

45


2

+ = + +

= + +|T0|2 T0T† TT

†0|T + T0|2

Figure 2.9: Contribution of one-loop corrections to the order αS .

where pv(x) denotes the Cauchy principal value. In the case at hand the final result is

T = CFg2T0

(−∫

d3k

ω3g(2π)3

w12(k) + iπ

∫dkT

(2π)2kT

). (2.4.20)

As can be plainly seen, there are two distinct contributions coming from the one-loopcorrection: the first is formally identical to the opposite of a real emission from thedipole formed by the quark–antiquark pair while the other is an imaginary part calledthe Coulomb (or Glauber) phase1. We need to interfere this amplitude with that of theBorn process to get the one-loop correction to the cross-section at order αS . The situationis represented pictorially in Fig. 2.9. In the figure, the square of the first element is oforder α2

S and thus beyond our accuracy, the square of the second addend gives the Borncross-section, and cross products contribute to order αS:

TT †0 + T0T† = CFg

2|T0|2[−∫

d3k

ω3g(2π)3

w12(k) + iπ

∫dkT

(2π)2kT

−∫

d3k

ω3g(2π)3

w12(k)− iπ∫

dkT(2π)2kT

]= −2CFg

2|T0|2∫

d3k

ω3g(2π)3

w12(k) .

(2.4.21)

In this case the Coulomb gluon contribution is completely erased. Constructing the cross-section from the leftover, it turns out to be identical to Eq. (2.4.10) but with a differentsign, hence the two completely cancel in the soft limit. The full result (see e.g. [34]) isfinite:

σ(e+e− → hadrons) = σ0

[1 +

3

4CF

αS (Q)

π+O(α2

S)

]. (2.4.22)

This cancellation of infrared and collinear divergences is the lowest-order manifestationof the Kinoshita–Lee–Nauenberg theorem [56, 57], that can be phrased as follows:

In a theory with massless fields, transition rates are free of infrared diver-

gences if the summation over the initial and final degenerate states is carried

out.

Roughly speaking, we could say that the final state with an extra soft gluon is degeneratewith the state with no additional gluons, since both these states belong to the same energy

1As we will see, this contribution will not play a part in this work, though it is currently under studyfor being connected to a breakdown of space-like collinear factorisation at higher orders in αS for certainnon-inclusive observables [51–55].

46

2.5. HADRONS IN THE INITIAL STATE

eigenstate in the limit of vanishing gluon energy. In the same way, a state with a quarkand a gluon with parallel momenta is degenerate with the state with no radiation at all.It is then the inclusion of virtual corrections that cancels infrared divergences order byorder.

2.5 Hadrons in the initial state

An additional complication arises in hadronic collisions. For example, at the LHC,two protons are made to collide head-on at a high centre-of-mass energy. The actual hardscattering happens between the constituent partons, possibly after QCD radiation (calledInitial-State Radiation or ISR). To complete the previous description one has to partitionthe total four-momentum of a hadron between its constituents. Each constituent i thuscarries fraction x of the total hadronic momentum with a probability density fi(x), theso-called Parton Density (or Distribution) Function (PDF), meaning that the probabilityfor x to fall between the range [x, x+dx] is given by fi(x)dx. From these assumptions thehadronic cross-sections can be calculated as a convolution of the partonic one (constructedby standard use of perturbation theory of partons) with the relevant PDFs.

Factorisation allows us to write the cross-section for the collision of hadrons H1 andH2 at centre-of-mass energy Q in the general form [58]

σH1,H2(p1, p2) =∑i,j

∫dx1dx2f

(H1)i (x1)f

(H2)j (x2)σij(x1p1, x2p2, µ) +O(1/Qp) ,

(2.5.1)where i and j label the hadrons’ constituents and σij is the partonic cross-section calcu-lated using partons i and j. µ is the renormalisation scale. Eq. (2.5.1) is called naïve

parton model formula. The last term, where p is a positive real number, indicates that fac-torisation holds up to corrections proportional to inverse powers of Q. In fact, this naïveparton model does not survive radiative corrections, as collinear divergences arise frominitial-state emission. However these divergences turn out to be universal, i.e. they are thesame for every process. Then they are a multiplicative factor in the naïve parton modelformula and can thus be reabsorbed in a redefinition of the PDFs. In this way the par-ton distribution functions acquire a dependence on a finite scale µF , called factorisation

scale, and the improved parton model formula, which does not contain any divergence,can be written as

σH1,H2(p1, p2) =∑i,j

∫dx1dx2f

(H1)i (x1, µF )f

(H2)j (x2, µF )σij(x1p1, x2p2, µF )

+O(1/Qp) .

(2.5.2)

The factorisation scale µF sets the boundary between perturbative and non-perturbativedynamics. For simplicity, we took µF = µ. As they both are arbitrary parameters, theirvalues can be varied to estimate the PDF uncertainties. The conventional choice is to set

47


µ2 = (CµF )2 and compare the results for C = 1/2, 1, 2.It is important to stress that the new PDFs depend exclusively on the hadrons involved

and not on the process their constituents undergo. Moreover, by definition the PDFsare determined by long-distance effects and thus cannot be calculated with perturbationtheory. Nonetheless, their universality makes it possible to measure them in a referencehard process (typically DIS) and use them in other observables.

Physical observables such as the cross-section σH1,H2(p1, p2) cannot depend on anarbitrary parameter such as µF . This means

∂σH1,H2(p1, p2)

∂ lnµF= 0 . (2.5.3)

Then it is possible to perturbatively calculate the dependence of the PDFs on µF . Theresult is the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equation[59–62]. Its standard form is [36]

∂f(Hn)i (xn, µF )

∂ lnµF=αS(µF )

2π

∑j

∫ 1

xn

dx

xf

(Hn)i (x, µF )Pij

[xnx, αS(µF )

]. (2.5.4)

The functions Pij [x, αS(µF )] are called splitting functions (or evolution kernels), as theyrepresent the rate of production of parton i from parton j with a fraction x of the longitu-dinal momentum of its parent and transverse momentum equal to µF . Charge-conjugationinvariance and the SU(Nf ) flavour symmetry encoded in the massless QCD Lagrangianrequire (we dropped the arguments for clarity)

Pqiqj = Pqiqj ,

Pqiqj = Pqiqj ,

Pqig = Pqig ≡ Pqg ,

Pgqi = Pgqi ≡ Pgq .

(2.5.5)

The splitting functions can be expanded in powers of the running coupling (whose argu-ment we leave implicit here):

Pij(x, αS) = P(0)ij (x) +

αS2πP

(1)ij (x) + . . . . (2.5.6)

At leading order, Pqiqj is non-zero only for i = j. In this case we define the splitting func-tion Pqq. Additionally Pqiqj = 0 for any choice of i and j. The leading-order contributions

48

2.5. HADRONS IN THE INITIAL STATE

are:

P (0)qq (x) = CF

[1 + x2

(1− x)+

+3

2δ(1− x)

],

P (0)qg (x) = TF

[x2 + (1− x)2

],

P (0)gq (x) = CF

[1 + (1− x)2

x

],

P (0)gg (x) = 2CA

[x

(1− x)+

+1− xx

+ x(1− x)

]+

11CA − 4TFNf

6δ(1− x) ,

(2.5.7)

where the subscript + stands for a regularisation prescription called the plus subtraction

scheme. This is defined by [63]:

F (x)+ ≡ F (x)− δ(1− x)

∫ 1

0

dyF (y) . (2.5.8)

The divergences in (2.5.7) are due to the emission of soft gluons. Only those for x → 1

need to be regulated, as the integration in Eq. (2.5.4) only extends to the lower limitxn > 0.

The colour factors in front of each splitting function in Eq. (2.5.7), show the connec-tion to the matrix elements in Fig. A.1. Splitting kernels are currently known up to orderα3S [64, 65]. Finding solutions to Eq. (2.5.4) involves the Mellin transform:

φ(N) =

∫ 1

0

dx xN−1f(x) . (2.5.9)

In the space of the moments N , Eq. (2.5.4) usually admits a solution which can be ob-tained with analytical methods. In other cases, and almost always when reverting tox-space, one has to rely on numerical methods.

We have just described the main elements of precise predictions in fixed-order per-turbative QCD. In the next section we will describe a different approach, which aims todefine an approximate picture where it is possible to take into account the effects of softand collinear enhancements at all orders. This is critical for a reliable description of ob-servables sensitive to this region of phase space, as we will see in the next chapter, andalso paves the way for the implementation of the resulting “parton shower” in computersimulations.

49


2.6 Parton shower

It is easy to carry out a calculation of the squared matrix elements in Fig. A.1 inthe collinear limit [36]. In doing this, one recovers the splitting functions of Eq. (2.5.7)in an unregularised form. In view of the following discussion, we strip them of theircolour factors and adopt a different normalisation. It will be also convenient to map allsoft divergences to x → 0. This is easily obtained by exploiting the symmetry of theunregularised splitting function Pgg(x) when exchanging x with 1 − x. The resultingsplitting functions read

Pgq(x) ≡ 1 + (1− x)2

2x,

Pgg(x) ≡ 1− xx

+x(1− x)

2,

Pqg(x) ≡ 1

2

[x2 + (1− x)2

].

(2.6.1)

Note how Pgq(x) and Pgg(x) reduce to 1/x as x → 0, reproducing the soft divergencewe found in Eq. (2.4.11). In contrast, Pqg(x) exhibits no soft divergence. In the follow-ing, our integrations over energy fractions will be explicitly regularised by our choice ofobservables, so that we can safely use the splitting functions in Eq. (2.6.1).

It can be shown [36] that in the collinear approximation the phase space for multipleemissions factorises. Let us then consider an ensemble of n partons, which includesparton i. Then, for a branching of parton i into parton j, we have, in a fixed-couplingapproximation:

dσn+1 = dσndt

tdxαSπCjiPji(x) , (2.6.2)

where we ignored the azimuthal angle φ (more on this in Subsec. 2.6.2). The termCjiPji(x) is the relevant splitting function multiplied by the corresponding colour fac-tor. The virtuality of the emitting parton is represented by t. Repeated application of thisequation makes it possible to describe the transition from the energy scale of the hardprocess to hadronisation in terms of a parton shower.

2.6.1 The parton shower as a Markov process

From Eq. (2.6.2) it is possible to define the probability dPi that parton iwith virtualitybetween t and t+ dt emits parton j. This is given by

dPi =αSπ

dt

t

∫ 1−t0/t

t0/t

dxPji(x) , (2.6.3)

where t0 is the cutoff needed to define the resolvability of partons, or equivalently whenhadronisation kicks in. We define Di(Q

2, t) as the probability of having no branchings

50

2.6. PARTON SHOWER

between Q2 and t. Then the following equation holds:

dDi(Q2, t)

dt= Di(Q

2, t)dPidt

. (2.6.4)

The solution in a fixed-coupling approximation is then

Di(Q2, t) = exp

[−αSπ

∫ Q2

t

dt′

t′

∫ 1−t0/t′

t0/t′dxPji(x)

]. (2.6.5)

This is called the Sudakov form factor. The fact that in this case we have an explicitcutoff t0 in the integration over x, is the reason why we can use unregularised splittingfunctions. It is possible to show [36] that using the parton branching formalism, the resultsof Eq. (2.5.7) can be recovered. As for the running of αS , a careful treatment [66] showsthat we should use the squared root of x(1 − x)t′ as its argument. This is essentially thetransverse momentum of the emission.

The formulation of the parton shower is intrinsically Markovian: the probability ofeach splitting depends only on the produced parton and its parent. In particular, it does notdepend on any other parton at any step in the shower. The parton shower then naturallylends itself to be implemented in a Monte Carlo generator. This is how it works in anutshell:

1. Generate a random number r ∈ [0, 1].

2. Solve Di(Q2, t) = r for t.

3. If t > t0 generate a branching with virtuality t, else the showering is over.

4. Generate z as a random variable distributed according to Pji(z).

5. Repeat the cycle.

This is the core functioning of most event generators. A few adjustments make it possi-ble to use the parton showering formalism also for the initial state, to describe the QCDbremsstrahlung of partons as the energy goes down from Q to the energy scale of hadro-nisation.

We still need to raise an important point. So far, only amplitudes with collinear en-hancements have been taken into account. We have to investigate what happens whennon-collinear amplitudes with soft enhancements are included in the mix.

2.6.2 Angular ordering

Let us go back for a moment to Eq. (2.4.18), the definition of the radiator in e+e− →qqg. For a generic pair of external lines i and j emitting a gluon g of momentum k(ωg, |k|)

51


we havewij(k) ≡ 1

2ω2g

(pi · pj)(pi · k)(pj · k)

=1− cos θij

2(1− cos θig)(1− cos θjg), (2.6.6)

which reduces to Eq. (2.4.18) when i and j are back-to-back. The radiator can also bewritten as the sum of two separate contributions, each containing one of the collineardivergences:

wij(k) = w[i]ij (k) + w

[j]ij (k) , (2.6.7)

wherew

[i]ij (k) ≡ 1

2

[wij(k) +

1

2(1− cos θig)− 1

2(1− cos θjg)

]. (2.6.8)

The solid angle element can be expressed in terms of θig and the azimuthal angle withrespect to the i direction φig as

dΩ = d cos θigdφig . (2.6.9)

If we now carry out the integration over the azimuthal angle [36] we get∫ 2π

0

dφig2π

w[i]ij (k) =

1

2(1− cos θig)Θ (θij − θig) . (2.6.10)

In words, after azimuthal averaging, the contribution from this term is confined to a conecentred on the direction of i and extending in angle up to the direction of j. Swapping iwith j, we recover the same answer for w[j]

ij (k). This property is called angular ordering,and it is an intrinsic feature of all gauge theories: it is present in QED as well, whereit gives rise to the Chudakov effect [67], i.e. a suppression of large-angle, soft bremm-strahlung off electron-positron pairs. The main complication arising when moving fromQED to QCD is that, due to the colour algebra, in the latter case the vector bosons of thetheory (the gluons) are charged. A non-trivial generalisation of the above result can befound in [68]. In words, it can be illustrated considering a parton k splitting into partonsi and j. When i and j are close in angle, their azimuthally-averaged incoherent radiationis limited to cones of half-angle θij . At larger angles, the radiation resulting from theinterference of contributions from i and j can be computed as if it was emitted from theparent k, considered on-shell.

2.6.3 Ordering variables and the rest of the event

Colour coherence means that some care is required when choosing the ordering vari-able for the Monte Carlo generation of the shower. This choice is actually made evenmore complicated by other factors, and there is no consensus on what the “best” order-ing variable is. For example, consider the most senior event generator, PYTHIA [69].Its Fortran version, PYTHIA 6 [70], is still very widely used. PYTHIA 6 can be used

52

2.6. PARTON SHOWER

with a virtuality-ordered shower where angular ordering is enforced separately, or witha shower ordered in transverse momentum. Another generator which is widely used isHERWIG [71], and its C++ version, Herwig++ [72]. They both use slightly differentangular-ordered showers.

There is also an alternative approach to parton shower called dipole showering, but itwill not be used in this work. For additional information about dipole showers, see [73]and references therein.

Though certainly of critical importance, parton showering is not enough for the fullsimulation of an event like that in Fig. 1.1. Some more ingredients are necessary:

• Fixed-order matrix elements.

• A showering prescription.

• Models of non-perturbative effects (e.g. hadronisation).

• Decay management.

This allows for a complete, if approximate, simulation of particle collisions, making itpossible to compare MC results with theoretical predictions. On the other hand, the out-come of such a simulation depends on the values of many free parameters which are tunedusing inputs from data. It is therefore crucial to distinguish between what is happeningin the generator and what can actually be measured, i.e. physical observables. Genera-tors usually employ lots of intermediate objects which are not physical observables bythemselves and thus should not be treated as such. Their specification is meaningful onlywhen all quantum interference effects between different processes that produce the samefinal state can be neglected. Some event generators, like SHERPA [74], take a radical ap-proach to avoid any ambiguity, forbidding completely user access to intermediate objects,while some others, like Herwig++ [72], have a different attitude towards this problem.It is important to bear in mind that in any case the output of any event generator alwayssuffers from a residual model dependence and should never be considered a proxy forexperimental data.

2.6.4 Monte Carlo tools

This work heavily relies on the use of Herwig++ [72]. This MC generator is basedon HERWIG [71] (Hadron Emission Reactions With Interfering Gluons). The new devel-opment features a new code written in C++ (the original one was written in Fortran) andseveral refinements of the underlying physics. The package and full documentation of thecode can be found at

https://herwig.hepforge.org/.

53

https://herwig.hepforge.org/


RIVET [75] (Robust Independent Validation of Experiment and Theory) is a veryuseful tool to carry out analyses on MC events. Along the lines of the previous discussion,analyses written with RIVET are not allowed to use generator-specific portions of eventrecords, as they are solely based on physical observables constructed from stable particlesand physical decayed particles. RIVET and its documentation can be found at

https://rivet.hepforge.org/.Another useful tool is EVENT2. This is not an event generator, but a program imple-

menting the Catani–Seymour subtraction algorithm [76] for calculating Next-to-Leading-Order (NLO) corrections to jet cross-sections in e+e− annihilation into partons. It usesMonte Carlo techniques to perform the integration over the phase space. It can be foundat

http://hepwww.rl.ac.uk/theory/seymour/nlo/.In all cases, jet reconstruction is made using FastJet [77], which is now the standard

in the field. It includes all main algorithms, as well as the possibility to define new onesthrough a plugin system. In our study of jet substructure, we have made extensive use ofthe FastPrune plugin. FastJet and its documentation can be found at

http://fastjet.fr/,and the FastPrune plugin is available at

http://www.phys.washington.edu/groups/lhcti/pruning/.In this chapter we have shown how the infrared and collinear enhancements of QCD

radiation lead to the parton-shower picture, which lends itself easily to be simulated interms of a Markov process. Given the factorisation of the hard process from hadronisation,the parton shower is also the main responsible for the angular and energy distributionsof final-state hadrons. In the next chapter we will see how these distributions can beexploited to reconstruct the hard process through the use of jets.

54

https://rivet.hepforge.org/


http://fastjet.fr/

http://www.phys.washington.edu/groups/lhcti/pruning/

CHAPTER

THREE

PHENOMENOLOGY OF JETS

A fundamental class of QCD observables is based on the concept of jets. As seen inChap. 1, most of the events at the LHC include a large number of jets, both in backgroundsand signals. We also mentioned the potential of jet mass as a discriminant of the originof a jet, especially in conjunction with jet-substructure techniques. There is however nounique definition of a jet, as it is just the outcome of a jet algorithm. We start this chapterby reviewing the original formulation of jets as cones enclosing fractions of energy inthe event. We then move on to the definition of sequential recombination jet algorithms,in their exclusive and inclusive variants. In the second part we show how they can beexploited to study jet substructure, and how this can help the search for new physics. Foran extensive review of jet physics, see [78].

3.1 Jet algorithms

From the discussion in the last chapter it is clear that any QCD observable has to beconstructed in a sensible way for it to have any predictive power. In particular, the degen-eracy of soft and collinear radiation has to be reflected in the behaviour of the observable.Whilst this is trivially true for fully inclusive observables like total cross-sections, it needsto be checked for other observables. Thus, a fundamental requirement is that a given setof partons and the same set accompanied by soft and/or collinear radiation give the samecontribution to the value of the observable. An observable that respects this condition iscalled InfraRed and Collinear (IRC) safe [79]. This requirement can be also phrased ina way that it is closer to experimental practice: the result of a measurement should notbe sensitive to changes in the detector’s energy resolution or granularity. Otherwise themeasurement is sensitive to long-distance physics which needs to be non-perturbativelymodelled, greatly limiting its validity. It is worth noting that it is not trivial to definesufficient conditions for IRC safety [29].

Due to the factorisation of hadronisation and the hard process, one could intuitivelythink that bunches of collinear particles in the final state correspond to single partons inthe hard process. This intuition was exploited to define one of the first examples of anobservable specifically designed to satisfy IRC safety: the Sterman–Weinberg jet defini-

55

CHAPTER 3. PHENOMENOLOGY OF JETS

tion. According to this definition, a hadronic e+e− event is defined to be a 2-jet event if afraction 1− ε of the centre-of-mass energy is contained within two back-to-back cones ofhalf angle δ. The values of ε and δ then define the “2-jettiness” of the event.

Jets defined by the Sterman–Weinberg clustering procedure are conical by construc-tion. This was the first example of a cone algorithm. A modern cone algorithm operatesin two steps: first, it searches for stable cones according to a specific recursive procedureand definition of stability; next, it runs a split-merge routine in order to separate overlap-ping cones and separate their contents, so that each object in the final-state is assigned toa single jet. At the moment the only IRC safe cone algorithm is the Seedless Infrared SafeCone (SISCone) algorithm [80]. For a review of cone algorithms, see [78].

Nowadays cone-based jet definitions are being replaced by sequential recombination

jet algorithms. In these algorithms, a distance εij between particles in the final state isdefined. Then the closest pair gets clustered together in a pseudoparticle, and the processis repeated until a predetermined condition is met. The remaining pseudoparticles areeither discarded or promoted to jets. Since in this thesis we will use only the latter kindof algorithms, it is worthwhile to spend some time to describe them in more depth.

For simplicity, we start with a discussion of sequential recombination jet algorithmsin e+e− annihilation into hadrons. A few complications arise in hadron collisions, and wewill comment on those at the end of this exposition.

Let us begin with an example: in the kT algorithm [81] (also known as the Durhamalgorithm) the distance εij between final-state objects i and j is defined by

εij = 2min

(ω2i , ω

2j

)Q2

(1− cos θij) ∼min

(ω2i , ω

2j

)Q2

θ2ij , (3.1.1)

for an e+e− collision at centre-of-mass energy Q. Another common choice is the anti-kTjet algorithm [82], which is defined by the distance1

εij =Q2

8min

(1

ω2i

,1

ω2j

)(1− cos θij) . (3.1.2)

These two algorithms belong to the continuous family of resolution measures:

εij =1

2

(Q

2

)−2p

min(ω2pi , ω

2pj

)(1− cos θij) . (3.1.3)

Of course for p = 1 we get the Durham (or kT ) algorithm, while for p = −1 the anti-kT al-gorithm; another important value is p = 0, which corresponds to the Cambridge/Aachen(C/A) algorithm [83, 84].

To fully define a jet algorithm, a recombination prescription is needed in order toassign momenta to merged particles. In the following, we will use the E-scheme, which

1Our choice of normalisation is slightly different from [82] in order to make our following parallelbetween exclusive and inclusive algorithms clearer. This choice leaves all physical features of the algorithmunchanged.

56

3.1. JET ALGORITHMS

assigns to the recombined pair of objects a four-momentum equal to the sum of the four-momenta of the original objects. This prescription allows for conservation of energyand momentum but the recombination of massless objects can be massive. It has beenadvocated as a standard in [85], it was the main scheme used during Run II of the Tevatronand it is now used by the LHC experiments.

Having specified how to merge particles together, we still have to describe when todo it. In this respect, sequential clustering algorithms split into two broad categories:exclusive mode and inclusive mode [86].

3.1.1 Exclusive mode

Clustering in exclusive mode is more suited for electron-positron annihilation, whereone has not to worry about soft initial-state jets. One defines an additional measure vijbetween final-state objects, belonging to the family described in (3.1.3), possibly identicalto εij . Given an arbitrary parameter εcut, one starts with a list of all the final-state objectsand then uses this algorithm:

1. If only one object is on the list, promote it to a jet and stop.

2. Pick the pair with the smallest εij .

3. If vij < εcut, merge i and j following the chosen prescription; if not, promote thesoftest in the pair to a jet and remove it from the list of final-state objects.

4. Go back to step 1.

If vij = εij the above procedure reduces to clustering the objects closest to each otheruntil εij ≥ εcut for any i and j, at which point each of the remaining objects is promotedto a jet.

3.1.2 Inclusive mode

Algorithms used in inclusive mode are more similar to their counterparts used inhadronic collisions (see next subsection), and thus their study can give information morerelevant to the LHC case. In addition to the distance between objects, one defines a pa-rameter εcut and beam distances εi for each object:

εi =εcut

4

(2ωiQ

)2p

, (3.1.4)

where p must be the same for all the εi and εij . Then, given a list of all the particles in thefinal state the algorithm proceeds as follows:

57


1. Pick the smallest from all the εij and εi.

2. If it is a εi and ω2i ≥ εcutQ

2 promote i to a jet and remove it from the list; ifω2i < εcutQ

2 discard i.

3. If it is a εij merge i and j following the chosen prescription.

4. Go back to step 1.

3.1.3 Incoming hadrons

The LHC is a proton-proton collider. For hadron collisions, it is standard to use vari-ables that are invariant under longitudinal boosts. This is due to the fact that the centre-of-mass energy of the hard process is unknown, since the participating partons take up arandom fraction of the energies of their respective hadrons, according to the distributionsdefined by the PDFs. In practice, one needs to substitute energies with transverse mo-menta with respect to the incoming beam direction p2

T i and replace 1− cos θij with ∆R2ij .

This is commonly defined as [87]

∆R2ij = (ri − rj)2 + (φi − φj)2 , (3.1.5)

where φi is the azimuthal angle of object i with respect to the beam direction and ri itsrapidity:

ri ≡1

2lnωi + pz,iωi − pz,i

. (3.1.6)

In the massless limit this is equal to the pseudorapidity ξi

ξi ≡1

2ln

1 + cos θi1− cos θi

= − ln

[tan

(θi2

)], (3.1.7)

where θi is the angle with the beam direction.We just mentioned that the total energy of the hard process is not well-defined in

hadron collisions. As a consequence, one has to use a dimensionful distance or definesome arbitrary normalisation. Also, QCD infrared divergences arise not just betweenpairs of outgoing particles, but also between an outgoing particle and the incoming beamdirection(s). The solution is to adopt beam distances, similar to the εis in the inclusivemode above. In this case they have the form [78]

εi = p2T i . (3.1.8)

Then one proceeds in finding the smallest distance. If it involves two pseudoparticles,then they are clustered together1. If it is a beam distance, in the exclusive mode it goes

1We are implying vij = εij , which is generally the case.

58

3.2. JET OBSERVABLES

into the beam jet; in the inclusive mode it becomes a jet provided it passes a given cut onthe transverse momentum.

3.1.4 A note on nomenclature

We close this section with a note on nomenclature. In the next chapter we will use εcut

as the name of the infrared cutoff. This is the traditional choice for exclusive algorithmsand we also adopted it for inclusive ones for the sake of compactness and internal consis-tency. Another common choice is to replace εcut with 2(1−cosR). This makes it possibleto interpretR as a sort of jet radius, though generally jets do not have exact conical shape.

3.2 Jet observables

As we already mentioned, jets play a prominent role at the LHC, and they can indeedbe used to construct a whole range of observables which in turn make it possible to re-construct and understand the hard process. In recent years, a new way to discriminatebetween jet origins has been found in studying the constituents of a jet: this field of re-search is called jet substructure. As this will be the main topic of this thesis, we dedicatethe next section to a review of its main features. For the moment, we move on to describesome general properties of jet observables.

3.2.1 Large logarithms and resummation

Let us consider a generic jet observable v for a QCD event with high-pT jets. Whenattempting a perturbative analytical calculation of it, particular care must be taken inchoosing the right method. We can identify two regimes which we call “small v” and“large v”. We consider ourselves to be in the regime of large v when v . 1. In thisregion, fixed-order calculations are sufficient for an accurate analytical prediction, and forany IRC safe observable such a prediction can be obtained using fixed-order Monte Carlossuch as [30, 88, 89]. In the small-v regime, the observable receives large enhancementsproportional to ln v. In this section we explain why the appearance of these logarithms is aserious threat to the accuracy of any fixed-order prediction, and show a possible treatment.

In the following we will be interested in integrated distributions Σ(v), i.e. the probabil-ity of the observable to acquire a value smaller than v. The general form of the integrateddistribution is

Σ(v) ≡∫ v

0

dv′1

σ0

dσ

dv′, (3.2.1)

where σ0 is the Born cross-section and σ includes both real and virtual contributions. Herewe can illustrate the emergence of large logarithms more clearly by using an explicit

59


example. Let us consider the 2-jet rate J2, i.e. the fraction of events with two jets, ine+e− annihilation in hadrons, up to order αS . We will use the exclusive kT algorithm ofEq. (3.1.1) to cluster partons into jets. The gluon and the quark will be clustered together if

the distance(ωgθqgQ

)2

is less than the cutoff εcut, which takes on the role of v in this case.

Then, using the soft and collinear approximation of Eq. (2.4.11) and a fixed coupling, wehave1

J2 = 1 +αSCFπ

∫ Q2

4

0

dω2g

ω2g

∫ 4

0

dθ2qg

θ2qg

[Θ(Q2εcut − ω2

gθ2qg

)− 1]. (3.2.2)

At order α0S , we have just J2 = 1, since without any emission from the primary qq pair

we get two jets for any value of εcut. The first term in square brackets represents thecontribution of a real gluon emission collinear to the primary quark: we want the quarkand gluon to cluster together. The second term corresponds to virtual corrections, which,as we have seen in the last chapter, come with a minus sign, and contribute to the 2-jetrate over the whole phase space. The result at order αS is then

J2 = 1− αSCF2π

L2 , (3.2.3)

whereL ≡ ln

1

εcut. (3.2.4)

We explicitly see the appearance of large logarithms in the jet parameter εcut. This comesfrom the fact that the definition of our observable leads to a partial mis-cancellation of realand virtual contributions. The meaningfulness of truncating the perturbative expansion ata certain power of αS is then strictly justified only when L 1√

αS. This is clearly not

true when εcut → 0.To restore the predictive power of the observable in the small-v regime, a resummation

of logarithmically-enhanced terms at all orders needs to be performed. This is a wellknown technique in perturbative QCD [90, 91], and it has been automated for severalobservables [28, 29]. The ideal situation is when logarithms exponentiate. This meansthat the observable in Eq. (3.2.1) takes the form [36]

Σ(v) = C(αS) exp[G(αS, L)] , (3.2.5)

where

C(αS) = 1 ++∞∑i=1

CiαiS ,

G(αS, L) = Lg1(αSL) + g2(αSL) + αSg3(αSL) + . . .

(3.2.6)

1Details of the parametrisation of the phase space and the limits of integration can be found in the nextchapter.

60

3.2. JET OBSERVABLES

The function g1 contains all leading logarithms (LL) that give rise to terms like αnSLn+1,

g2 contains the next-to-leading logarithms (NLL in the exponent1) αnSLn and all the other

subdominant corrections follow. We will encounter many observables whose leading be-haviour is

Σ(v) ∝ e−AL2+... , (3.2.7)

where A is some positive constant. The suppression coming from the exponent whenL → +∞ gives rise to a characteristic shape called the Sudakov peak. A clear exampleof this can be seen in the jet-mass distribution of Fig. 1.2.

In the next chapter we will aim for NLL accuracy in the expansion, which meanswe want to write an exponentiated formula that can be expanded at any order giving thecorrect coefficients to terms like αnSL

2n (double logarithms) and αnSL2n−1. In the next

two subsections we will list the effects that we need to take into account at NLL in theexpansion and in the exponent respectively. The latter will be useful in Chap. 5 and 6.

3.2.2 NLL in the expansion

So far we have only considered soft and collinear emissions: due to QCD infrareddivergences, in this regime the probability of each emission is enhanced by two powersof logarithms. If we relax one of the approximations while keeping the other, we can getcontributions to the coefficient of αnSL

2n−1. Using both approximations simultaneouslymeans that the integral over the energy of the emission is always weighted by 1/ω. Insteadof this weight, we can use the full splitting functions reported in Eq. (2.6.1) with thecorresponding colour factors we calculated in App. A. This allows us to calculate thehard-collinear contribution to the coefficient αnSL

2n−1. Given the form of the splittingfunctions, some of these contributions will be proportional to a power of the cut on theenergy. These are called power corrections and will be ignored in the limit where the cutis small. The observables we will consider are insensitive to soft, large-angle radiation atthis level of accuracy.

Another effect we need to take into account is the running of the strong couplingαS . We use Eq. (2.3.6), to relate the values of αS at two different scale µ1 and µ2, withµ2 µ1:

αS(µ1) =αS(µ2)

1− αS(µ2)β0 lnµ22µ21

, (3.2.8)

where β0 was defined in Eq. (2.3.4). We can expand in powers of αS(µ2) up to α2S(µ2)

and get

αS(µ1) = αS(µ2)

[1 + αS(µ2)β0 ln

µ22

µ21

]. (3.2.9)

1A condition called recursive InfraRed and Collinear Safety (rIRC) has been formulated in [29], whichensures resummability at this level of accuracy for global observables, i.e. observables that receive contri-butions from radiation anywhere in the phase space.

61


When we make this substitution in our result of order αnS , we get a contribution propor-tional to αn+1

S L2n+1, i.e. a NLL contribution in the expansion to the next order in αS .

3.2.3 NLL in the exponent

At NLL in the exponent we aim to control all the logarithms down to αnSLn (single

logarithms). This calls for a more accurate treatment of the running of the coupling. Wealso need to take into account two additional sources of single logarithms: Non-Global

Logarithms (NGLs) and Clustering Logarithms (CLs).NGLs [92, 93] appear in observables which are sensitive only to specific parts of the

phase space. Emissions outside these regions can still radiate into them or vice versa,complicating the form of the observable: such emissions are of non-Abelian nature andfree of collinear divergences. Resummation of NGLs has until very recently been re-stricted to the large-N limit [6], N being the number of quark colours. A resummationat finite N has been performed in [94], using an approach originally developed in [95].Some complications that arise beyond the leading N approximation have also been ex-plored in [51].

Clustering algorithms generally do not give rise to perfectly conical jets, with softclusterings far away from the jet axis impacting on the value of the observable. Thisgives rise to CLs in the Abelian sector of the matrix element for multiple emissions.They were also shown to reduce the impact of NGLs [96, 97]. When taking into accountCLs, the phase space quickly becomes intractable and generally not factorisable. Onlynumerical all-orders results for CLs have been obtained so far, but explicit calculations upto order α4

S have been shown to capture the all-orders behaviour [98]. Similar calculationswere performed in [99, 100]. It is worth noting here that cone jet algorithms, and moreimportantly the anti-kT algorithm, are free of CLs.

3.3 “Fat” jets

We already described the physical reasons behind using jets as observables in QCD;we have also reviewed many different ways of defining jets both in leptonic and hadroniccollisions. A large set of processes result in jets. In particular they can be very usefulin searches for resonances decaying hadronically, for example H → bb. Normally onewould expect one jet for each primary quark: measuring the dijet mass would then give adistribution peaked around the Higgs mass. Two elements complicate this simple picture.Firstly, a huge QCD background with a highly non-trivial shape makes it very hard toisolate the comparatively tiny signal. Another complication can be seen if one bears inmind that at the LHC such resonances are typically produced with transverse momentumpT ranging from a few hundred GeV to some TeV. This means that the decay products will

62

3.4. THE ORIGIN OF JET SUBSTRUCTURE

be very boosted, and as a result very close to each other in angle. To get an estimate of thisangular separation, let us consider a Higgs boson with transverse momentum pT decayinginto two massless partons with four-momenta p1 and p2 and an angle R12 between them.Then the relation between the mass of the Higgs mH and the mass of the jet is (seeEq. (1.0.1))

m2H ' x(1− x)p2

TR212 , (3.3.1)

where we assumed that p1 carries a fraction x of the Higgs boson’s pT and R12 is small.Solving for R12 we get

R12 'mH√

x(1− x)pT. (3.3.2)

It is clear that the minimum value of the jet radius R (or the corresponding value of εcut,see Subsec. 3.1.4) needed to cluster the decay products in a single jet (R > R12) getssmaller as pT increases at fixed x. As a result, we can expect a heavy boosted resonancewhich decays hadronically to produce a single, massive (“fat”) jet.

The mass resolution at a high-luminosity collider such as the LHC is also reducedby two other effects: the Underlying Event (UE) and Pile-Up (PU). The UE is whateverhappens to the constituents of the proton that do not take part in the primary process, butstill produce uncorrelated radiation and particles that end up in jets from the hard collision(for a review of recent developments on UE measurements and modelling, see [101,102]).PU is another source of uncorrelated contributions to the final state, this time comingfrom additional proton-proton collisions in the same bunch, or even in different bunches(out-of-time PU). Its incidence is particularly relevant in high-luminosity colliders like theLHC, and every effort is being made to reduce its impact on data (for recent measurementsof PU and of the effects of mitigation techniques, see [103, 104] and references therein).

In the last few years several techniques have been devised in an attempt to overcomethese problems: enough to generate a whole new field of research, called jet substructure.

3.4 The origin of jet substructure

The idea itself is not new: jet substructure was first mentioned as a gateway to dis-covery of boosted heavy particles in [105], and the first study is almost twenty yearsold [106]. The intuition is to extract information about the origin of a jet from the clus-tering sequence through which it is constructed. It is clear from the previous chapter thatdifferent algorithms give rise to different clustering sequences. For example, the anti-kT algorithm forms jets starting from hard cores in the event and clusters to them allsofter emissions without any direct connection to the parton shower dynamics: as a con-sequence, any substructure (from a decay, for example) tends to be washed away in theclustering sequence. On the other hand, if there are two hard prongs in a single jet, theytend to be clustered only in the final step when using the kT algorithm. The proceduredescribed in [106] is based on this fact and it is indeed very simple: undo the last clus-

63


tering step, and put a cut on the mass and angular separation of the two resulting subjetsand from the original jet to decide whether it is a signal or a background jet. The C/Aalgorithm holds a somehow intermediate position between the two previous algorithms:on one hand, it is constructed so as to reflect the angular ordering in the parton shower; onthe other hand, its last clustering steps feature large-angle, uncorrelated emissions whichcannot be directly connected to the origin of the jet. Nevertheless, the latter factor shouldjust mean that one has to dig a bit deeper in the clustering sequence to recover a mean-ingful jet substructure. Actually, in the next chapter we will find that, at least as far as jetrates are concerned, these differences between algorithms and the corresponding cluster-ing sequences do not show up at NLL-in-the-expansion accuracy; such differences mightthen turn out to impact in a subtler way than expected on jet substructure. Therefore wesuggest to carry out an extensive study of the performance of substructure techniques fordifferent jet algorithms.

3.5 Mass drop and filtering

Some years after [106], this idea of digging in the clustering sequence of a jet to getinformation about its origin was exploited in the pioneering paper by Butterworth J. M.,Davison A., Rubin M. and Salam G. P. (commonly referred to as BDRS) [25]. The maincontent of that paper is the description of a new technique used to remove contaminationfrom pile-up and the underlying event from a jet while optimising background rejectionand enhancing the signal. The case under study was Higgs-strahlung production off anelectroweak boson; the ambition, recovering the Higgs peak in the bb decay channel,which was widely considered hopeless in Higgs searches because of the huge QCD back-ground (whose shape is highly non-trivial). The algorithm used works as follows (recallthat ∆Rij was defined in Eq. (3.1.5)):

1. Identify events with a highly boosted electroweak boson and a big heavy jet Jobtained using the C/A algorithm. Because of the high boost, J likely containsthe decay products (i.e. the QCD radiation off the bb pair and contamination fromPU/UE) of the Higgs boson. J will then be the candidate Higgs jet.

2. Undo the last step of clustering to get the subjets J1 and J2, such that mJ1 > mJ2

3. Check for two conditions:

(a) Asymmetry:min(p2

T,J1, p2

T,J2)∆R2

J1,J2

m2J

> ycut . (3.5.1)

(b) Mass-drop:mJ1 < µmJ . (3.5.2)

4. If either of these conditions is not met, set J = J1 and start over.

64

3.5. MASS DROP AND FILTERING

Figure 3.1: This figure, taken from [25], illustrates the mass-drop and filtering procedure.After the filtering step, the three hardest jets are considered in order to catch the leadingradiation from the bb pair and get a better reconstruction of the Higgs mass.

5. If both conditions are met and the two subjets both have a b-tag, J is a Higgscandidate.

6. Filtering: recluster the constituents of J with a smaller radius

Rfilt = min

(0.3,

∆RJ1,J2

2

). (3.5.3)

7. If the two hardest jets have b tags, keep the three hardest jets, otherwise discard thecandidate.

This technique is clearly divided in two parts: the first is a tagging step, called the Mass-Drop Tagger (MDT), and its goal is to dig into the clustering sequence to find a sufficientlysymmetric splitting where the mass is evenly shared by the products, i.e. a structure whichwe can associate to a Higgs boson decaying into a bb pair; the second step is filtering, andits role is to remove PU/UE contamination from the Higgs candidate. µ and ycut are theparameters of the MDT, and they will be discussed in detail in Chaps. 5 and 6. Rfilt

is the parameter that defines filtering and its value is optimised in order to catch the b,the b and the leading gluon radiation each in one of the three hardest jets. A picturefrom [25] which depicts the procedure is reported in Fig. 3.1. In Fig. 3.2 the originalMonte Carlo study for the MDT with filtering is shown. It indicates that this procedurehas the potential of producing a significant Higgs-mass peak, thus recovering the bb decaychannel as a viable way to searching for the Higgs boson. In the spirit of [107], a smallextension to [25], we performed a quick Monte Carlo study of the Higgs mass distributionin the bb channel using Herwig++ 2.5.0. We generated proton-proton collisions at 14 TeVproducing a Higgs boson decaying in a bb pair together with a Z boson decaying intomuons or electrons. Hadronisation and the underlying event were included. We selectedboosted events, i.e. we required the pT of the reconstructed Z boson to be between 300and 350 GeV. The results are shown in Fig. 3.3. It is clear that the mass-drop step correctlyidentifies the signal peak, while filtering greatly helps in improving its quality. We alsoadded another important substructure procedure, pruning [32, 33], which at this pointcan be seen to perform comparably well. Pruning is designed in a slightly different waythan the MDT: rather than singling out jets with relevant substructure, it acts on all jets,removing soft and large angle constituents from the clustering sequence. The final result

65


Figure 3.2: Monte Carlo study of the mass-drop and filtering procedure. The signal con-sists ofZH andWH production, with the Higgs boson (whose mass was set tomH = 115GeV) decaying into a bb pair. The first three panels correspond to results in different decaychannels of the electroweak boson that accompanies the Higgs boson production, whilethe bottom-right panel is their sum. The signal peak is clearly visible above the back-ground with notable significance. Additional details can be found in [25], from whichthis picture was taken.

should be a reduction in spurious QCD contributions to jet mass and increased massresolution for resonances. Pruning [32,33] works as follows (recall that ∆Rij was definedin Eq. (3.1.5)):

1. Recluster the jet from scratch using any jet algorithm.

2. At each reclustering step, for the pair of clustering pseudojets J1 and J2 the follow-ing conditions are checked:

(a) ∆R2J1,J2

> D2.

(b)min (pT ,J1 , pT ,J2)

|pT ,J1 + pT ,J2|< zcut.

3. If both conditions are met, the softer pseudojet is discarded and then clusteringcontinues.

4. If either condition is not met, the clustering proceeds normally.

Pruning relies on two parameters: D and zcut. They will be discussed in detail in Chaps. 5and 6.

66

3.6. JET SUBSTRUCTURE TODAY

0

100

200

300

400

500

600

700

800

80 90 100 110 120 130 140 150 160

Nu

mb

er

of

eve

nts

Mass of the hardest jet (GeV)

C/A

C/A + mass drop

C/A + mass drop + filtering

C/A + pruning

Figure 3.3: A quick study of boosted Higgs bosons decaying into bb pairs using Her-wig++. We generated ZH events from proton-proton collisions at centre-of-mass energyof 14 TeV, with the highly-boosted Z boson (300 GeV < pT,Z < 350 GeV) decayinginto electrons or muons. The mass of the Higgs boson was set to mH = 115 GeV. Itis clear that plain jet reconstruction using the C/A algorithm (red line) gives no clearpeak. The mass-drop procedure (green line) helps in uncovering the peak, whose qual-ity is greatly improved by using filtering (blue line). Another procedure, pruning (purpleline), performs comparably well.

3.6 Jet substructure today

After the BDRS paper [25], a great surge of attention for the field has brought toseveral important advances: a large number of ways to exploit jet substructure have beenproposed, as reviewed in [108–111]. Some of these techniques have already proved them-selves useful in a number of experimental analyses [112–126]. It is thus clear that jetsubstructure will be an important component of the current and future particle physicsprogram.

Substructure methods are usually classified into two broad categories: taggers andgroomers. There is no general agreement on the exact separation between the two. Acommon definition is that a groomer is a procedure that somehow acts on the constituentsof a jet, but always returns a groomed jet from a given input jet; on the other hand, atagger can lead to rejections of input jets which then are completely removed from theanalysis. An alternative definition of grooming comes from [108], where it is defined as"elimination of uncorrelated UE/PU radiation from a target jet". With this definition, anyprocedure that through any cut rejects background jets more often than it rejects signaljets, even in the absence of showering, hadronisation, underlying event or pile-up, is a

67


tagging procedure. Then all the procedures we will consider here involve both groomingand tagging elements. We decide to collectively refer to all these techniques as taggers.

3.7 Monte Carlo studies and open questions

Given the vivacity and prolificacy of the field, it is probably the right time to addresssome of the questions that have been asked about the plethora of tools currently availableto investigate jet substructure. All these methods are specifically designed to enhance sig-nal jets arising from boosted heavy particles and discriminate against QCD backgrounds,using the fact that jet substructure is very different in the two cases. It is then naturalto look for information about their relative and absolute efficiency and robustness. Thisis mainly because while one wants to have a broad spectrum of tools to allow for someflexibility, it is also crucial to avoid duplication and redundancy as much as possible.Moreover, in order to fully exploit the available flexibility, a clear understanding of thebehaviour of each method is imperative. Such an understanding can only be obtained bydetailed and informed comparisons of the performance of different techniques over a widerange of values of jet masses, transverse momenta and all other parameters involved.

The currently available studies which compare the performance of substructure meth-ods mainly rely on MC generators. An example taken from [108] can be seen in Fig. 3.4.It is clear that the various techniques help in improving the resolution of the top peak.We contributed to [109] by generating samples for substructure studies. We used Her-wig++ 2.5.0 to generate QCD scatterings and tt production in proton-proton collisions ata centre-of-mass energy of 7 TeV (additional details can be found in [109]). One of theplots we produced is reported in Fig. 3.5. It is interesting to note that the performance ofthis top tagging for any substructure method is comparable to that of b-tagging, thoughonly information on the jet kinematics was used here.

However, insofar as MC studies can help in highlighting peculiar features, they do nottell us much about their origin. Additionally, it is unclear whether any difference in theperformances of taggers is due to their intrinsic properties or rather due to a particularchoice of parameters; furthermore, they do not help very much in understanding if suchdifferences are general or restricted to some specific kinematic region. It is also usuallyvery hard to disentangle the effects of different tunings and generators (see e.g. [109,113,116, 127]) from the actual physical description. All these difficulties cry out for someanalytical insight into jet substructure methods.

Such an achievement requires a good understanding of the behaviour of jet algorithms.In particular, we need to understand what is their impact on the substructure of the result-ing jets. Common sense suggests that algorithms like kT or C/A should be preferredover anti-kT , because the clustering sequences originating from the first two algorithmsshould by construction reflect more closely the dynamics of the parton shower; whereason the other hand, clustering sequences arising from the use of anti-kT are not expected tocarry any information about the history of physical splittings that brought to the final-state

68

3.7. MONTE CARLO STUDIES AND OPEN QUESTIONS

Figure 3.4: Picture from [108] depicting the jet-mass distribution from hadronic top de-cays before (in red) and after several jet-substructure treatments. The quality of the toppeak is clearly increased by all substructure methods. The W peak is also visible on theleft of the histogram.

objects.We investigate the legitimacy of this claim by carrying out a fixed order calculation

of logarithmically-enhanced contributions to n-jet rates, the normalised probabilities ofgetting an event with exactly n jets. From our results we infer resummed distributions,recovering the known results for the 2-jet rate [81] and finding those for inclusive al-gorithms. While carrying out this piece of research, some work on the same topic waspublished [1]. Overlapping results are found to be in agreement.

The next step is the direct study of substructure techniques: in Chap. 5 we startfrom fixed-order calculation of logarithmically-enhanced terms in the mass distributionfor trimming [31], pruning [32, 33] and the MDT [25]. We use our results in Chap. 6to infer all-order distributions. Along the way we find features of the taggers that wentunnoticed so far and we propose simple modifications that enhance their performance andmake their analytical description simpler.

69


Figure 3.5: Picture from [109] which describes the top-tagging efficiency of several sub-structure techniques. All of them do reasonably well over a large range of parametersand this makes boosted-top phenomenology using jet substructure a very attractive exper-imental endeavour.

70

CHAPTER

FOUR

JET RATES

Jets are basic observables in QCD, yet their definition is not unique, but rather it de-pends on the choice of a specific jet algorithm. Aside from IRC safety, no strong physicalreasons exist to prefer a particular jet algorithm over the others. It is then important tounderstand how the choice of an algorithm impacts on the value of observables. We studyn-jet rates (or fractions) Jn, the fraction of events with exactly n jets in the final state.In the first part of the chapter we perform a fixed-order calculation of logarithmically-enhanced terms up to the 5-jet rate for a variety of jet algorithms and test our resultsagainst EVENT2. The appearance of these logarithmic enhancements calls for a resum-mation of jet rates for small values of the jet parameter εcut. We show the known resultsfor the exclusive kT algorithm (also known as the Durham algorithm) [81] and extendthe formalism to inclusive algorithms. Our results agree reasonably well with events gen-erated by Herwig++. While carrying out this original research, some work on the sametopic was published [1] and overlapping results are found to be in agreement. In order tostress once again the fact that resummation is not a property trivially satisfied by any jetalgorithm, in the final part of the chapter we design an algorithm which, like the JADEalgorithm, is not resummable.

The main result of this chapter concerns the anti-kT algorithm: while it is usuallybelieved not to be able to correctly reproduce jet substructure [78], we show that it behavesin the same way as all the other inclusive algorithms, at least up to NLL-in-the-expansionaccuracy.

4.1 EVENT2 parameters

Exact numerical results up to NLO are obtained for e+e− collisions at the centre-of-mass energy Q = 1 TeV. For each comparison to analytical results, 109 events aregenerated. An infrared cutoff on the mass of any pair of partons i and j is includedin the program to ensure numerical accuracy. If m2

ij < CUTOFFQ2, the whole event isthrown away. The default value is CUTOFF = 10−8. We find this to be unsatisfactorywhen probing the small-εcut regime. For this reason, we set CUTOFF = 10−12. To improvethe convergence of the numerical results, we change the value of two parameters related

71

CHAPTER 4. JET RATES

to importance sampling1, NPOW1 and NPOW2, from their default value of 2 to 6. We findresults obtained with this choice of parameters to be stable and numerical errors are undercontrol.

4.2 Leading Order (LO)

We start by carrying out the calculation of the logarithmically-enhanced contributionsto jet fractions at Leading Order (LO) in αS . Obviously we get contributions to J2 and J3

only. Many conventions and techniques used throughout the thesis will be introduced inthis section. For clarity, we will omit the argument of the couplings in our results, exceptin the final resummed formulae.

4.2.1 Single gluon emission

When considering the process e+e− → qqg, it is clear that in the centre-of-masssystem the three produced particles will lie on a plane. Then we can integrate out theazimuth as it does not carry any useful information. Ignoring for now the running ofthe coupling, the spin-averaged, differential cross-section for e+e− → qqg can then bewritten as [35]

d2σ

dxqdxq= σ0a

CF2

x2q + x2

q

(1− xq)(1− xq), (4.2.1)

wherexi ≡

2ωiQ

, i ≡ q, q, g , (4.2.2)

anda ≡ αS

π. (4.2.3)

In the soft limit, the total cross-section becomes

σ = σ0aCF2

∫dθqgdxg

4

sin θqg

[1

xg− 1 +

xg4

(1 + cos2 θqg

)]. (4.2.4)

Here xg is the energy fraction of the gluon and θqg its angle with the quark. Let us nowstudy the collinear limit θqg → 0, π. Introducing the variable

tij ≡ 2(1− cos θij) , (4.2.5)

Eq. (4.2.4) becomes

σ = σ0aCF

∫dtqg

(1

tqg+

1

4− tqg

)∫dxgPgq(xg) . (4.2.6)

1For more information, see the documentation at http://hepwww.rl.ac.uk/theory/seymour/nlo/.

72



4.2. LEADING ORDER (LO)

Figure 4.1: Visual description of a typical fixed-order calculation of jet rates: on the left-hand side of the figure we have a particular Feynman diagram for the emission of n gluonsand on the other side the kinematical configurations of the partons, with jets symbolisedby cones around the direction of the originating parton. One has to take into accountthe contributions to the various jet rates for each Feynman diagram, sorting them by howmany jets arise in different regions of the gluons’ phase space.

The definition Pgq(xg) can be found in (2.6.1).In general, performing a fixed-order perturbative calculation of jet rates means con-

structing the cross-section for producing n partons, imposing constraints to get the con-tribution to each jet rate and taking into account virtual corrections to regularise infrareddivergences. This procedure is represented pictorially in Fig. 4.1.

4.2.2 2- and 3-jet rate

We start our discussion with the calculation of J2. Obviously J2 = 1 at order a0. Atorder a we get

J2 = 1 + aCF

∫dtqg

(1

tqg+

1

4− tqg

)∫dxgPgq(xg)∆2 , (4.2.7)

where ∆2 is a collection of step functions which restricts the integration on the phasespace to the region which contributes to the observable. It is most convenient to take aparticular collinear limit, say θqg → 0. This implies tqg → 0, then only the first addendin Eq. (4.2.7) is relevant. It can be shown that in the limit where θqg → π we get thesame result. We can then just ignore the second addend and double our answer. Let usbegin with the exclusive kT (or Durham) algorithm. We take the distance definition in

73


Eq. (3.1.3) setting p = 1 and vij = εij . Then in our case we are interested in the distance

εqg =tqg4x2g . (4.2.8)

In order to get a finite answer we need to put constraints on the real emission to get twojets and on the other hand take into account virtual corrections. Recalling the result ofEq. (2.4.20), this can be easily done by considering the equivalent of a real emission withopposite sign. Since the two quarks with a virtual gluon always become two jets, there isno need for any restriction of that phase space. The resulting constraints are

∆2 = Θ(4εcut

x2g

− tqg)− 1 = −Θ

(tqg −

4εcut

x2g

), (4.2.9)

and the jet fraction reads

J2 = 1− aCF2L2 + a

3CF2L . (4.2.10)

where we have defined L ≡ ln 1εcut

. We see here again the appearance of large logarithms.This comes from the fact that the definition of our observable leads to a partial mis-cancellation of real and virtual contributions. This is not necessarily a problem: ourobservable is still infrared safe and thus has predictive power. However we need to bewareof what happens when εcut becomes small: in this case the smallness of a itself mightnot be enough to guarantee that terms with higher powers in the coupling constant areindeed negligible. In fact, we expect this not to be the case: these logarithms come frominfrared and collinear terms, then at each order in a we can expect (leading) terms ofthe form C2na

nL2n. In Sec. 3.2.1 a way to take care of this problem is described. Forthe moment we carry on with our fixed-order treatment which aims to calculating thecoefficients C2n and C2n−1 (NLL in the expansion). We also work in the limit εcut → 0,which means that we do not care about power corrections (i.e. contributions proportionalto some power of εcut) of such coefficients. In particular, note how we set the lower limit ofintegration to zero for the regular part of Pqg(x) that describes hard and collinear radiationand contributes a NLL-in-the-expansion term.

The calculation for the leading-order 3-jet rate J3 is trivial. Here no virtual correctionsneed to be taken into account, as the definition of the jet algorithm itself takes care of theregularisation of the phase-space integrals. The result is

J3 = aCF2L2 − a3CF

2L , (4.2.11)

The first term in both Eq. (4.2.10) and Eq. (4.2.11) comes from soft and collinear radia-tion, while the second one comes from a hard and collinear gluon. Soft and large-angleradiation can be shown to contribute only power corrections and thus we ignore it. Ofcourse with just one gluon we cannot get more than three jets. It is reassuring that if we

74

4.2. LEADING ORDER (LO)

sum J2 and J3 up to order a we get 1. These results agree with those reported in [128].The procedure stays the same when using inclusive algorithms. In this case the rele-

vant distances are

εqg =tqg4

min(x2pg , 1) ,

εg =εcut

4x2pg ,

εq =εcut

4.

(4.2.12)

If the smallest distance is εqg then we get a clustering and thus two jets; alternatively, wehave a third jet if x2

g > 4εcut. Adding virtual corrections we get

∆2 = Θ(εcut − tqg

)+ Θ

(tqg − εcut

)Θ(

4εcut − x2g

)− 1

= −Θ(tqg − εcut

)Θ(x2g − 4εcut

).

(4.2.13)

Accordingly, for the 3-jet rate we get

∆3 = Θ(tqg − εcut

)Θ(x2g − 4εcut

). (4.2.14)

Notice how these functions are completely independent of p. This means that at thislevel of accuracy any choice of inclusive algorithm is equivalent as far as jet rates areconcerned. The results are

J2 = 1− aCFL2 + a3CF

2L ,

J3 = aCFL2 − a3CF

2L .

(4.2.15)

Fig. 4.2 compares our LO results to EVENT2. The differences (black crosses) betweenour results and EVENT2 show no significant deviation from a constant behaviour, whichis consistent with the level of accuracy we claim to have achieved.

75


-450

-400

-350

-300

-250

-200

-150

-100

-50

0

50

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a/2

log10 εcut

EVENT2 Inclusive J2

EVENT2 Durham J2

EVENT2 - NLL

Figure 4.2: The figure shows the coefficient of a/2 generated by EVENT2 for J2, de-fined by the inclusive anti-kT algorithm (red) and the exclusive (or Durham) kT algo-rithm (green). The black crosses represent the coefficient of a/2 in the differences be-tween EVENT2 and the corresponding calculation in Eq. (4.2.10) and the first line ofEq. (4.2.15) respectively. The fact that these differences are constant is consistent withour claim of having achieved NLL-in-the-expansion accuracy at leading order in a, bothfor J2 and J3, since by unitarity J2 = −J3 at this order. EVENT2 parameters were chosenaccording to the description in Sec. 4.1.

4.3 Next-to-Leading Order (NLO)

We now move to NLO. As before we start with the exclusive kT (or Durham) algo-rithm and then we move on to inclusive algorithms. Again all our results are compared toEVENT2.

4.3.1 Exclusive kT algorithm

The measures between partons now are:

εq1 =tq14x2

1 ,

εq2 =tq24x2

2 ,

ε12 =t12

4min

(x2

1, x22

).

(4.3.1)

76

4.3. NEXT-TO-LEADING ORDER (NLO)

k1 k2

(a)k1 k2

(b)k1 k2

(c)

k1 k2

(d)

Figure 4.3: Real and virtual contributions to the NLO jet fractions in the C2F channel.

where 1 and 2 are the gluons’ labels. Let us start with the simplest colour channel, corre-sponding to colour charge C2

F . This is related to two independent gluon emissions fromthe hard partons. The 2-jet fraction at this order is

J2,C2F

= 2a2C2F

∫dx1Pgq(x1)

∫dtq1tq1

∫dx2Pgq(x2)

∫dtq2tq2

∆2,C2F, (4.3.2)

where we included an overall factor of 12

due to the fact that the two gluons in the final stateare identical particles. We can ignore self clustering of the gluons as it only contributesterms beyond our approximation. We define

∆2,ij ≡ Θ

[4εcut

min(x2i , x

2j)− tij

]. (4.3.3)

The real and virtual contributions we need to take into account are shown in Fig. 4.3. It isclear then that this contribution is fully factorised:

∆2,C2F

= ∆2,q1∆2,q2 −∆2,q2 −∆2,q1 + 1 = ∆22 , (4.3.4)

where ∆2 is the same as in Eq. (4.2.9). The result is

J2,C2F

= 2a2C2F

[∫dxgPgq(xg)

∫dtqgtqg

∆2

]2

= a2C2F

8L4 − a2 3C2

F

4L3 . (4.3.5)

77


To get the C2F term in the 3-jet rate, we define

∆3,ij ≡ Θ

[tij −

4εcut

min(x2i , x

2j)

]. (4.3.6)

Then we can write

∆3,C2F

= ∆3,q1∆2,q2 + ∆3,q2∆2,q1 −∆3,q1 −∆3,q2 . (4.3.7)

It is worth spending some words to explain the equation above: the first two terms in thesum correspond to the situation depicted in Fig. 4.3a, i.e. both gluons are real; the othertwo terms correspond to Fig. 4.3b and Fig. 4.3c, which means that one of the gluons isvirtual, and that explains the minus sign. The first two terms ensure that one of the gluonseither clusters with the quark, or it is too soft to form a jet on its own, while the otherescapes the clustering and has enough energy to form a jet; the other two prevent the onlyreal gluon to cluster with the quark and make it hard enough to originate a new jet. Inthis way we will surely have three jets in the final state. A careful treatment of all the stepfunctions involved leads to

∆3,C2F

= −2∆22 , (4.3.8)

where ∆2 was defined in Eq. (4.2.9). The resulting jet fraction is

J3,C2F

= −4a2C2F

(∫dxgPgq(xg)

∫dtqgtqg

∆2

)2

= −a2C2F

4L4 + a2 3C2

F

2L3 . (4.3.9)

We also get the first contribution to J4 at this order . Unitarity and inspection of Eq. (4.3.5)and (4.3.9) make it clear that J4,C2

F= J2,C2

F.

Next we consider the emission of a gluon which in turn splits into two gluons. Thiscontributes to the CFCA channel. This splitting has both soft and collinear enhancements;another fundamental trait is angular ordering, which means that, after azimuthal averag-ing, the contribution of this radiation is confined to a cone centred around the direction ofthe emitting (harder) gluon and extending as far as the primary quark (see Subsec. 2.6.2).The relevant splitting function, Pgg(x), is defined in (2.6.1).

The features of g → gg splittings lead to a complete cancellation between real andvirtual contributions to the 2-jet rate. In this case then, all terms proportional to CFCAand CFNf come from the running of the coupling (see Subsec. 3.2.1). Looking at thedistances in (4.3.1) it is apparent that in the soft and collinear limit the smallest one is ε12.Then the softest gluon dominantly clusters with its parent. This behaviour, representedpictorially in Fig. 4.4, is critical for resumming large logarithms. The final result for J2 atorder a2 is

J2 = 1−aCFL2 +a3CF

2L+a2C

2F

8L4−a2

(3C2

F

4− 11CFCA − 2CFNf

36

)L3 . (4.3.10)

Next we calculate the value of J3 in the CFCA channel. We implicitly take into ac-

78


(a)

(b)

Figure 4.4: This figure shows some features of the exclusive kT algorithm in the CFCAchannel which are key to resummation. Above: the softer gluon (lighter in the picture) isfirst clustered to its parent and the combination is clustered to the quark. This contributionto the 2-jet fraction is completely cancelled by virtual corrections. Below: when theharder (darker) gluon is far enough from the quark, it forms a jet with its softer offspring,which thus at NLL-in-the-expansion accuracy does not contaminate the quark jet.

count angular (tq1 > t12) and energy (x1 > x2) ordering. With these conventions theconstraint on the phase space is

∆3,CFCA = ∆3,q1

(∆2,12 − 1

)= −∆3,q1∆3,12 . (4.3.11)

To get the full contribution to the coefficient of Nf in J3, we need to include the splittingfunction for a gluon splitting into a quark–antiquark pair, Pqg(x), defined in Eq. (2.6.1).

Then the final result for J3 up to a2 is given by

J3 = aCF2L2 − a3CF

2L

− a2

(C2F

4+CFCA

48

)L4 + a2

(3C2

F

2+

7CFCA12

− CFNf

12

)L3 .

(4.3.12)

By unitarity, the leading, hard collinear and g → qq contributions to J4 are equal andopposite to the corresponding ones in J3. Of course there are no running-coupling effectsto take into account at this order as J4 is null at order a, and all contributions to CFNf

come from the gluon splitting into a quark–antiquark pair. Then J4 reads

J4 = a2

(C2F

8+CFCA

48

)L4 + a2

(−3C2

F

4− 5CFCA

18+CFNf

36

)L3 . (4.3.13)

Higher orders and higher jet rates can be obtained by using the same techniques, butthe calculations soon become cumbersome. In Sec. 4.4 we will describe a way to resumall orders at NLL-in-the-expansion accuracy, as was originally done in [81].

79


4.3.2 Inclusive algorithms

We now move on to NLO for jet fractions defined with inclusive algorithms. Asbefore, we start with the C2

F channel. The distances are as follows:

εq1 =tq14

min(x2p1 , 1) , ε1 =

εcut

4x2p

1 ,

εq2 =tq24

min(x2p2 , 1) , ε2 =

εcut

4x2p

2 ,

ε12 =t12

4min(x2p

1 , x2p2 ) , εq =

εcut

4. (4.3.14)

Again, we can ignore self clustering of the gluons at our level of accuracy. Let ∆2,ij

be the condition that makes partons i and j cluster together, in its inclusive version:

∆2,ij ≡ Θ(εcut − tij

)+ Θ

(tij − εcut

)Θ(

4εcut −min(x2i , x

2j)). (4.3.15)

Taking p > 0, the full condition for getting two jets in the C2F channel reads

∆2,C2F

=[Θ(tq1x

2p1 − tq2x2p

2

)Θ(εcutx

2p1 − tq2x2p

2

)Θ(εcut − tq2

)− 1]

∆2,q1

+ Θ(tq1 − εcut

)Θ(tq2x

2p2 − εcutx

2p1

)Θ(x2p

2 − x2p1

)Θ(

4εcut − x21

)∆2,q2

+(

1←→ 2)

+ 1 .

(4.3.16)

If we now take x1 > x2 we get

∆2,C2F

=[Θ(εcut − tq2

)− 1]

∆2,q1

+ Θ(tq2 − εcut

)Θ(

4εcut − x22

)∆2,q1 −∆2,q2 + 1

= ∆22 ,

(4.3.17)

where ∆2 is the same as in Eq. (4.2.13). It can be easily seen that the same is true in theother three cases. This result is again independent of the value of p. After integrating overthe phase space, we get

J2,C2F

= a2C2F

2L4 − a2 3C2

F

2L3 . (4.3.18)

Something similar happens for the 3-jet rate. Defining the inclusive version of ∆3,ij

∆3,ij ≡ Θ(tij − εcut

)Θ(

min(x2i , x

2j)− 4εcut

). (4.3.19)

80


-7.5

-7.4

-7.3

-7.2

-7.1

-7

-6.9

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

2 F/4

×10

5

log10 εcut

-7.44

-7.42

-7.4

-7.38

-7.36

-7.34

-7.32

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

2 F/4

×10

5

log10 εcut

-12

-10

-8

-6

-4

-2

0

2

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

2 F/4

×10

4

log10 εcut

-2.5

-2

-1.5

-1

-0.5

0

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

2 F/4

×10

4log10 εcut

0

1

2

3

4

5

6

7

8

9

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

2 F/4

×10

4

log10 εcut

0

0.2

0.4

0.6

0.8

1

1.2

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

2 F/4

×10

4

log10 εcut

EVENT2 Inclusive J2EVENT2 - NLL

Parabolic fit

χ2R = 6.06× 10−4

EVENT2 Durham J2EVENT2 - NLL

Parabolic fit

χ2R = 1.82× 10−5


Parabolic fit

χ2R = 1.09


Parabolic fit

χ2R = 0.313


Parabolic fit

χ2R = 1.82


Parabolic fit

χ2R = 0.373

Figure 4.5: Comparison between EVENT2 and NLO calculations at NLL in the expan-sion. The differences between EVENT2 and our calculation can be fit on a parabola. Thefitting is performed by gnuplot’s function fit. The goodness of the fit is demonstratedby the values of χ2

R reported on the plots. A value of χ2R much smaller than unity indicates

that the coefficient of L2, and possibly of L, are very small. We do not claim to controlthose anyway. The resulting differences are thus consistent with our statement of havingachieved NLL accuracy in the expansion. EVENT2 parameters were chosen according tothe description in Sec. 5.2.

We get the familiar result:

∆3,C2F

= ∆3,q1

[Θ(tq2 − εcut

)Θ(tq1x

2p1 − εcutx

2p2

)Θ(x2p

1 − x2p2

)Θ(

4εcut − x22

)+ Θ

(tq2 − εcut

)Θ(tq2x

2p2 − εcutx

2p1

)Θ(x2p

2 − x2p1

)Θ(

4εcut − x22

)+ Θ

(εcut − tq2

)Θ(tq1x

2p1 − tq2x2p

2

)Θ(εcutx

2p1 − tq2x2p

2

)+Θ(εcut − tq2

)Θ(tq2x

2p2 − εcutx

2p1

)Θ(x2p

2 − x2p1

)− 1]

+(

1←→ 2)

= −2∆22 .

(4.3.20)81


The above result is expressed in terms of ∆22, as defined in Eq. (4.2.13). It was derived

for p > 0, and it obviously holds for p = 0. However, since self-clustering of the twogluons is again beyond our accuracy, ∆3,C2

Fis exactly the same for any p < 0. Then, for

any value of p, J3,C2F

reads

J3,C2F

= −a2C2FL

4 + 3a2C2FL

3 . (4.3.21)

Unitarity does not depend on the algorithm, so we get J4,C2F

= J2,C2F

for any value ofp. A comparison between the NLO results for the C2

F channel and EVENT2 is shown inFig. 4.5. We use the standard gnuplot’s function fit to fit a parabola to the differencesand observe that it is consistent with our claim of having achieved NLL accuracy in theexpansion.

Let us now move on to theCFCA channel. By inspection of the distances in Eq. (4.3.14)we see that there are no CFCA terms in the 2-jet rate. Also, for p > 0 the contribution tothe 3-jet fraction is completely analogous to the exclusive case. The case p ≤ 0 needs abit of extra care. Introducing the parameter r ≡ −p we get

∆3,CFCA = Θ(tq2 − tq1

)Θ( t12

x2r1

− tq1)

Θ(εcut − tq1

)∆3,q2

+ Θ(tq1 − tq2

)Θ( t12

x2r1

− tq2)

Θ(εcut − tq2

)∆3,q1

+ Θ(tq1 −

t12

x2r1

)Θ(tq2 −

t12

x2r1

)Θ(εcut −

t12

x2r1

)Θ(εcut − t12

)∆3,q1

+

[Θ( t12

x2r1

− εcut

)Θ(tq2 − εcut

)∆2,12 − 1

]∆3,q1 .

(4.3.22)

There are two relevant regimes here, namely t12 tq1, tq2 and t12 . tq1. In the first case

∆3,CFCA,t12tq1,tq2 =

[Θ(εcut −

t12

x2r1

)+ Θ

( t12

x2r1

− εcut

)]Θ(εcut − t12

)∆3,q1

+[Θ(tq2 − εcut

)Θ(t12 − εcut

)Θ(

4εcut − x22

)− 1]

∆3,q1

= −∆3,q1∆3,12 ,

(4.3.23)

and in the other case

∆3,CFCA,t12.tq1 =[Θ(εcut − tq2

)+ Θ

(tq2 − εcut

)∆2,12 − 1

]∆3,q1

= −∆3,q1∆3,12

[1−Θ

(εcut − tq2

)].

(4.3.24)

It can be shown that the additional term only contributes single logarithms to the 3-jetrate, which are irrelevant for our accuracy. Again the inclusive algorithms behave in thesame way as their exclusive counterparts. Including the effects described in Subsec. 3.2.2,

82


-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FC

A/4

×10

3

log10 εcut

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FC

A/4

×10

3

log10 εcut

0

2

4

6

8

10

12

14

16

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FC

A/4

×10

3

log10 εcut

0

0.2

0.4

0.6

0.8

1

1.2

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FC

A/4

×10

3

log10 εcut


Parabolic fit

χ2R = 0.415


Parabolic fit

χ2R = 0.252


Parabolic fit

χ2R = 1.34


Parabolic fit

χ2R = 0.207

Figure 4.6: Comparison between EVENT2 and NLO calculations at NLL in the expan-sion in the CFCA channel. In all cases the differences between EVENT2 and our cal-culation fit nicely on a parabola (the fit is performed by gnuplot’s function fit andthe corresponding χ2

R is reported on each plot). This is consistent with our claim of hav-ing correctly calculated the coefficients of L4 and L3. EVENT2 parameters were chosenaccording to the description in Sec. 5.2.

the final result for J3 is

J3 = aCFL2 − a3CF

2L

− a2

(C2F +

CFCA8

)L4 + a2

(3C2

F +7CFCA

4− CFNf

4

)L3 .

(4.3.25)

To extend the analysis to higher jet rates and higher orders in a is trivial. A summaryof up to 5-jet rates at order a3 for the exclusive kT algorithm and inclusive algorithms canbe found in Table 4.1. The plots reported in Fig. 4.6 and 4.7 support our claim of havingachieved NLL accuracy in the expansion of the 3- and 4-jet fractions.

83


-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FN

f/4

×10

3

log10 εcut

-2.5

-2

-1.5

-1

-0.5

0

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FN

f/4

×10

3

log10 εcut

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FN

f/4

×10

3

log10 εcut

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FN

f/4

×10

3

log10 εcut


Parabolic fit

χ2R = 0.282


Parabolic fit

χ2R = 1.25


Parabolic fit

χ2R = 0.741


Parabolic fit

χ2R = 1.22

Figure 4.7: Comparison between EVENT2 and NLO calculations at NLL in the expansionin the CFNf channel (we explicitly set TF = 1/2). In all cases the differences betweenEVENT2 and our calculation fit nicely on a parabola (the fit is performed by gnuplot’sfunction fit and the corresponding values of χ2

R are reported on the plots). This isconsistent with our claim of having correctly calculated the coefficients of L4 and L3.EVENT2 parameters were chosen according to the description in Sec. 5.2.

84


Lea

ding

loga

rith

ms

Har

d-co

lline

arra

diat

ion

Run

ning

coup

ling

g→

qqTo

tal

2je

ts

aL

Exc

l.kT

3CF/2

3CF/2

Incl

usiv

e3C

F/2

3CF/2

aL

2E

xcl.kT

−CF/2

−CF/2

Incl

usiv

e−CF

−CF

a2L

3E

xcl.kT

−3C

2 F/4

−11CFCA/3

6+CFNf/1

8−

3C2 F/4−

11CACF/3

6+CFNf/1

8

Incl

usiv

e−

3C2 F/2

−11CFCA/1

2+CFNf/6

−3C

2 F/2−

11CFCA/1

2+CFNf/6

a2L

4E

xcl.kT

C2 F/8

C2 F/8

Incl

usiv

eC

2 F/2

C2 F/2

a3L

5E

xcl.kT

3C3 F/1

611C

2 FCA/7

2−C

2 FNf/3

63C

3 F/1

6+

11CFCA/7

2−CFNf/3

6

Incl

usiv

e3C

3 F/4

11C

2 FCA/1

2−C

2 FNf/6

3C3 F/4

+11C

2 FCA/1

2−C

2 FNf/6

a3L

6E

xcl.kT

−C

3 F/4

8−C

3 F/4

8

Incl

usiv

e−C

3 F/6

−C

3 F/6

3je

ts

aL

Exc

l.kT

−3C

F/2

−3C

F/2

Incl

usiv

e−

3CF/2

−3C

F/2

aL

2E

xcl.kT

CF/2

CF/2

Incl

usiv

eCF

CF

a2L

3E

xcl.kT

3C2 F/2

+5C

FCA/1

811CFCA/3

6−CFNf/1

8−CFNf/3

63C

2 F/2

+7C

FCA/1

2−CFNf/1

2

Incl

usiv

e3C

2 F+

5CFCA/6

11CFCA/1

2−CFNf/6

−CFNf/1

23C

2 F+

7CFCA/4−CFNf/4

a2L

4E

xcl.kT

−C

2 F/4−CFCA/4

8−C

2 F/4−CFCA/4

8

Incl

usiv

e−C

2 F−CFCA/8

−C

2 F−CFCA/8

a3L

5E

xcl.kT

−9C

3 F/1

6−

49C

2 FCA/2

88−CFC

2 A/4

8−

11C

2 FCA/3

6−

11CFC

2 A/4

80+C

2 FNf/1

8+CFCANf/2

40C

2 FNf/7

2+CFCANf/4

80−

9C3 F/1

6−

137C

2 FCA/2

88−

7CFC

2 A/1

60+

5C2 FNf/7

2+CFCANf/1

60

Incl

usiv

e−

9C3 F/4−

49C

2 FCA/4

8−

5CFC

2 A/3

6−

11C

2 FCA/6−

11CFC

2 A/4

8+C

2 FNf/3

+CFCANf/2

4C

2 FNf/1

2+CFCANf/7

2−

9C3 F/4−

137C

2 FCA/4

8−

53CFC

2 A/1

44+

5C2 FNf/1

2+CFCANf/1

8

a3L

6E

xcl.kT

C3 F/1

6+C

2 FCA/9

6+CFC

2 A/9

60C

3 F/1

6+C

2 FCA/9

6+CFC

2 A/9

60

Incl

usiv

eC

3 F/2

+C

2 FCA/8

+CFC

2 A/7

2C

3 F/2

+C

2 FCA/8

+CFC

2 A/7

2

4je

ts

a2L

3E

xcl.kT

−3C

2 F/4−

5CFCA/1

8CFNf/3

6−

3C2 F/4−

5CFCA/1

8+CFNf/3

6

Incl

usiv

e−

3C2 F/2−

5CFCA/6

CFNf/1

2−

3C2 F/2−

5CFCA/6

+CFNf/1

2

a2L

4E

xcl.kT

C2 F/8

+CFCA/4

8C

2 F/8

+CFCA/4

8

Incl

usiv

eC

2 F/2

+CFCA/8

C2 F/2

+CFCA/8

a3L

5E

xcl.kT

9C3 F/1

6+

49C

2 FCA/1

44+

151C

FC

2 A/2

880

11C

2 FCA/7

2−C

2 FNf/3

6+

11CFC

2 ANf/4

80−CFCANf/2

40−

7C2 FNf/2

40−CFCANf/2

409C

3 F/1

6+

71C

2 FCA/1

44+

217C

FC

2 A/2

880−

41C

2 FNf/7

20−CFCANf/1

20

Incl

usiv

e9C

3 F/4

+49C

2 FCA/2

4+

111C

FC

2 A/2

8811C

2 FCA/1

2+

11CFC

2 A/4

8−C

2 FNf/6−CFCANf/2

4−

13C

2 FNf/7

2−CFCANf/3

69C

3 F/4

+71C

2 FCA/2

4+

177C

FC

2 A/2

88−

25C

2 FNf/7

2−

5CFCANf/7

2

a3L

6E

xcl.kT−C

3 F/1

6−C

2 FCA/4

8−

7CFC

2 A/2

880

−C

3 F/1

6−C

2 FCA/4

8−

7CFC

2 A/2

880

Incl

usiv

e−C

3 F/2−C

2 FCA/4−

5CFC

2 A/1

44−C

3 F/2−C

2 FCA/4−

5CFC

2 A/1

44

5je

ts

a3L

5E

xcl.kT

−3C

3 F/1

6−

49C

2 FCA/2

88−

91CFC

2 A/2

880

11C

2 FNf/7

20+CFCANf/4

80−

3C3 F/1

6−

49C

2 FCA/2

88−

91CFC

2 A/2

880

+11C

2 FNf/7

20+CFCANf/4

80

Incl

usiv

e−

3C3 F/4−

49C

2 FCA/4

8−

71CFC

2 A/2

887C

2 FNf/7

2+CFCANf/7

2−

3C3 F/4−

49C

2 FCA/4

8−

71CFC

2 A/2

88+

7C2 FNf/7

2+CFCANf/7

2

a3L

6E

xcl.kT

C3 F/4

8+C

2 FCA/9

6+CFC

2 A/7

20C

3 F/4

8+C

2 FCA/9

6+CFC

2 A/7

20

Incl

usiv

eC

3 F/6

+C

2 FCA/8

+CFC

2 A/4

8C

3 F/6

+C

2 FCA/8

+CFC

2 A/4

8

Tabl

e4.

1:In

tegr

atio

nre

sults

fort

heex

clus

ive

Dur

ham

algo

rith

man

din

clus

ive

algo

rith

ms

upto

five

jets

anda

3at

NL

Lac

cura

cyin

the

expa

nsio

n.W

ese

ta≡

αS π

andL≡−

lnε c

ut.

85


4.4 Resummation

In this section we collect all the results at fixed-order we obtained so far into an all-order answer that extends the reliability of the predictions for n-jet rates to the regionwhere ln 1

εcut' 1√

a. We report the procedure and the known results for the exclusive kT

(or Durham) algorithm [81], then we extend this procedure to inclusive algorithms. Wealso check our resummed results to events generated by Herwig++ and find reasonableaccord.

In particular, the anti-kT algorithm is commonly considered not to be able to reproducethe correct jet substructure as described by the parton shower. The main point of thissection is to show that it actually behaves in the exact same way as all the other inclusivejet algorithms, at least up to NLL accuracy in the expansion of jet rates.

4.4.1 Exclusive kT algorithm

It is known [81] that jet rates obtained with the exclusive kT algorithm (or equiva-lently with an exclusive version of the C/A algorithm [83]) can be resummed to NLL-in-the-expansion accuracy. We sketch how this resummation is carried out and check theresummed jet rates against Herwig++.

This calculation is based on the use of generating functions. Let Pp(n) represent theprobability of finding n objects of a particular kind in the final state of a given process p.In our case we are looking for n jets defined by the Durham algorithm for a given valueof εcut in e+e− annihilation at a certain centre-of-mass energy. The generating functionφp(u) is defined as

φp(u) ≡+∞∑n=0

unPp(n) . (4.4.1)

If the generating functions are known, the probability distribution Pp(n) can be recoveredby repeated differentiation at u = 0:

Pp(n) =1

n!

dnφpdun

∣∣∣∣u=0

. (4.4.2)

In our case, the Born configuration is a qq pair and therefore the generating function is

φe+e−(u) = φq(u)φq(u) , (4.4.3)

where φq (φq) is the generating function for the fragmentation of a (anti)quark into jets.Since φq = φq

φe+e−(u) = [φq(u)]2 . (4.4.4)

To write an expression for φq(u) it is necessary to take into account general elements of

86

4.4. RESUMMATION

QCD like splitting probabilities, running coupling, colour coherence; on the other hand italso involves implementing specific properties of the chosen algorithm. We are not goinginto the details of how this is achieved here, a full derivation can be found in [36]. Theimportant physical point is illustrated in Fig. 4.4: every emitted parton either creates anew jet by itself or gets clustered to its parent. In other words, at this level of accuracythe only relevant clustering parameter is the angle between two pseudojets. That is whythere is no possibility for an emission to end up in a jet initiated by a parton which is notits emitter. We report the results for up to five jets defined by the Durham algorithm (seealso [36] and [81]). In order to do this, it is useful to define the following functions:

Γq(k2T , k

2T1

)≡ CF

2

a (kT1)

k2T1

(lnk2T

k2T1

− 3

2

),

∆q

(εcut, k

2T

)≡ exp

[−∫ k2T

Q2εcut

dk2T1Γq

(k2T , k

2T1

)],

Γg(k2T , k

2T1

)≡ CA

2

a (kT1)

k2T1

(lnk2T

k2T1

− 11

6

),

Γf(k2T1

)≡ Nf

6

a (kT1)

k2T1

,

∆g

(εcut, k

2T

)≡ exp

−∫ k2T

Q2εcut

dk2T1

[Γg(k2T , k

2T1

)+ Γf

(k2T1

)],

∆f

(εcut, k

2T

)≡ [∆q (εcut, k

2T )]

2

∆g (εcut, k2T )

, (4.4.5)

where kT is the transverse momentum

kT ≡ ω sin θ , (4.4.6)

with ω the energy of the emitted parton and θ the angle with respect to its emitter. Thesedefinitions can be used to write the jet rates:

J2

(εcut, Q

2)

=[∆q

(εcut, Q

2)]2

,

J3

(εcut, Q

2)

= 2[∆q

(εcut, Q

2)]2 ∫ Q2

Q2εcut

dk2TΓq

(Q2, k2

T

)∆g

(εcut, k

2T

),

87


J4

(εcut, Q

2)

= 2[∆q

(εcut, Q

2)]2

[∫ Q2

Q2εcut

dk2TΓq

(Q2, k2

T

)∆g

(εcut, k

2T

)]2

+

∫ Q2

Q2εcut

dk2TΓq

(Q2, k2

T

)∆g

(εcut, k

2T

)×∫ k2T

Q2εcut

dk2T1

[Γg(k2T , k

2T1

)∆g

(εcut, k

2T1

)+ Γf

(k2T1

)∆f

(εcut, k

2T1

)],

J5

(εcut, Q

2)

=[∆q

(εcut, Q

2)]24

3

[∫ Q2

Q2εcut

dk2TΓq

(Q2, k2

T

)∆g

(εcut, k

2T

)]3

+ 4

[∫ Q2

Q2εcut

dk2TΓq

(Q2, k2

T

)∆g

(εcut, k

2T

)]2

×∫ k2T

Q2εcut

dk2T1

[Γg(k2T , k

2T1

)∆g

(εcut, k

2T1

)+ Γf

(k2T1

)∆f

(εcut, k

2T1

)]+

∫ Q2

Q2εcut

dk2TΓq

(Q2, k2

T

)∆g

(εcut, k

2T

)×[(∫ k2T

Q2εcut

dk2T1Γg

(k2T , k

2T1

)∆g

(εcut, k

2T1

))2

+ 2

∫ k2T

Q2εcut

dk2T1Γg

(k2T , k

2T1

)∆g

(εcut, k

2T1

)×∫ k2T1

Q2εcut

dk2T2

(Γg(k2T1, k

2T2

)∆g

(εcut, k

2T2

)+ 2Γf

(k2T2

)∆f

(εcut, k

2T2

) )+ 2

∫ k2T

Q2εcut

dk2T1Γf

(k2T1

)∆f

(εcut, k

2T1

)×∫ k2T1

Q2εcut

dk2T2

×(

2Γq(k2T1, k

2T2

)− Γg

(k2T1, k

2T2

)+ Γg

(k2T , k

2T2

) )∆g

(εcut, k

2T2

) ]. (4.4.7)

It is worth emphasising that this exponentiation is accurate at NLL order in the expansion,which means that we control terms of order anL2n and anL2n−1. NLL accuracy in the

exponent means the functions g1 and g2 as defined in Eq. (3.2.6) are known and so far theyhave been calculated only for the 2-jet rate [28]. The results for the Durham algorithmshown in Table 4.1 are verified by expanding the resummed formulae.

88

4.4. RESUMMATION

4.4.2 Inclusive algorithms

For inclusive algorithms, Fig. 4.4 is again a good pictorial representation of the clus-tering sequence, and then we can define equations in analogy1 with those in (4.4.5)

Γq(εcut, t, x

2, Q2;x21

)≡ CF

2

1

t

[∫ x2

4εcut

dx21

x21

a

(Qx1

2

√t

)− 3

2a

(Q

2

√t

)],

∆q

(εcut, t, x

2, Q2)≡ exp

[−∫ t

εcut

dt1Γq(εcut, t1, x

2, Q2;x21

)],

Γg(εcut, t, x

2, Q2;x21

)≡ CA

2

1

t

[∫ x2

4εcut

dx21

x21

a

(Qx1

2

√t

)− 11

6a

(Q

2

√t

)],

Γf(t, Q2

)≡ Nf

6

a

(Q

2

√t

)t

,

∆g

(εcut, t, x

2, Q2)≡ exp

−∫ t

εcut

dt1[Γg(εcut, t1, x

2, Q2;x21

)+ Γf

(t1, Q

2)]

,

∆f

(εcut, t, x

2, Q2)≡ [∆q (εcut, t, x

2, Q2)]2

∆g (εcut, t, x2, Q2), (4.4.8)

and then the resummed jet fractions take the form

J2

(εcut, Q

2)

=[∆q

(εcut, 4, 1, Q

2)]2

,

J3

(εcut, Q

2)

= 2[∆q

(εcut, 4, 1, Q

2)]2

,

×∫ 4

εcut

dtΓq(εcut, t, 1, Q

2;x2)

∆g

(εcut, t, x

2, Q2)

J4

(εcut, Q

2)

= 2[∆q

(εcut, 4, 1, Q

2)]2

×[∫ 4

εcut


2;x2)

∆g

(εcut, t, x

2, Q2)]2

+

∫ 4

εcut


2;x2)

∆g

(εcut, t, 1, x

2, Q2)

1The dependence of Γq and Γg on the energy fractions deserves some explanation. In theirdefinitions, x21 is just a dummy variable of integration. However, some jet rates have terms likeΓq

(εcut, t, 1, Q

2;x2)

∆g

(εcut, t, x

2, Q2). In this case, x2 in ∆g denotes the upper limit in the integra-

tion over the energy fraction of vetoed gluons. The presence of the same x2 among the arguments of Γq

serves as a reminder that this upper limit is integrated over in the final jet rate.

89


×[∫ t

εcut

dt1Γg(εcut, t1, Q

2;x21

)∆g

(εcut, t1, x

21, Q

2)

+

∫ x2

4εcut

dx21Γf

(t1, Q

2)

∆f

(εcut, 1, x

21, Q

2)]

, (4.4.9)

and so on. The same remark we made for the Durham algorithm holds here: the possibilityto write down these resummed distributions implies that, at this level of accuracy, the onlyrelevant parameter in deciding whether two pseudojets should cluster, is the angle betweenthem. In this way, the emission pattern of the parton shower is faithfully reproduced bythe jet-clustering sequence. These results agree with those reported in [1], where a fullcalculation using generating functionals can be found.

4.4.3 Comparison with Monte Carlo

We are now ready to compare our resummed results, Eqs. (4.4.7) and (4.4.9), to eventsgenerated by Herwig++. We use Herwig++ 2.5.0 to generate e+e− collisions and we countthe number of jets in the final state without including hadronisation (there is no underlyingevent in e+e− collisions). We do not expect a perfect accord with the analytical results.This is because the parton shower includes several subleading effects that can howeverhave a numerical impact on the final result. Non-perturbative parameters like the cutofffor the showering process and the masses of the quarks are also expected to modify theshape of the observable. Taking these effects into account is not trivial because theydepend on the specific model and thus on the particular event generator we are using.This is why we do not think such an effort is relevant to our current discussion, though itis worth noting that resummed results for the kT algorithm including quark masses wereobtained in [129].

We cannot expect a point-by-point difference to be meaningful as it was the case withfixed-order results and EVENT2. However, the MC results should be able to reproducethe rough position of the peak and its height, and we do not expect significant differencesin the shape of the observable, i.e. we expect a monotonic growth for J2, and a functionwith a single Sudakov peak and otherwise monotonic for the other jet fractions. As anadditional check, we plot the observable at two widely separated values of Q, and verifythat the effects on the jet rates are consistent.

Fig. 4.8 and Fig. 4.9 show the comparisons with Eqs. (4.4.7) and (4.4.9) respectively.The main features are indeed confirmed, with residual differences due to the reasons dis-cussed above. In particular, J4 and J5 for the anti-kT (inclusive) algorithm tend to zeromuch slower than their analytical counterparts as εcut → 0 at Q = 1 TeV. This can beattributed to non-perturbative effects induced by the cutoff that ends the parton shower:as we increase Q, this effect is less visible.

90

4.4. RESUMMATION

00.10.20.30.40.50.60.70.8

-5 -4.5 -4 -3.5 -3 -2.5 -2

J2

log10 εcut

Q = 1 TeV

00.10.20.30.40.50.60.70.8

-5 -4.5 -4 -3.5 -3 -2.5 -2

J2

log10 εcut

Q = 10 TeV

00.050.10.150.20.250.30.350.4

-5 -4.5 -4 -3.5 -3 -2.5 -2

J3

log10 εcut

Q = 1 TeV

00.050.10.150.20.250.30.350.4

-5 -4.5 -4 -3.5 -3 -2.5 -2J3

log10 εcut

Q = 10 TeV

0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4.5 -4 -3.5 -3 -2.5 -2

J4

log10 εcut

Q = 1 TeV

0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4.5 -4 -3.5 -3 -2.5 -2

J4

log10 εcut

Q = 10 TeV

0

0.05

0.1

0.15

0.2

0.25

-5 -4.5 -4 -3.5 -3 -2.5 -2

J5

log10 εcut

Q = 1 TeV

0

0.05

0.1

0.15

0.2

0.25

-5 -4.5 -4 -3.5 -3 -2.5 -2

J5

log10 εcut

Q = 10 TeV

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Figure 4.8: Comparison between Herwig++ and resummed jet rates using the exclusivekT (or Durham) algorithm. Results for Q = 1 TeV are shown on the left, while resultsfor Q = 10 TeV are shown on the right. The shape of the Monte Carlo result (at partonlevel) is correctly reproduced in our analytical description, as well as the fact that the peakmoves to smaller values of the cutoff as the jet multiplicity increases and the centre-of-mass energy is increased. The heights of the peaks are also in fair agreement. Details ofevent generation and residual differences can be found in the main text. For the resummeddistributions, we used αS(mZ) = 0.118, wheremZ is the mass of theZ boson,mZ = 91.2GeV.

91


0

0.1

0.2

0.3

0.4

0.5

0.6

-5 -4.5 -4 -3.5 -3 -2.5 -2

J2

log10 εcut

Q = 1 TeV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 -4.5 -4 -3.5 -3 -2.5 -2

J2

log10 εcut

Q = 10 TeV

00.050.10.150.20.250.30.350.4

-5 -4.5 -4 -3.5 -3 -2.5 -2

J3

log10 εcut

Q = 1 TeV

00.050.10.150.20.250.30.350.4

-5 -4.5 -4 -3.5 -3 -2.5 -2

J3

log10 εcut

Q = 10 TeV

0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4.5 -4 -3.5 -3 -2.5 -2

J4

log10 εcut

Q = 1 TeV

0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4.5 -4 -3.5 -3 -2.5 -2

J4

log10 εcut

Q = 10 TeV

0

0.05

0.1

0.15

0.2

0.25

-5 -4.5 -4 -3.5 -3 -2.5 -2

J5

log10 εcut

Q = 1 TeV

0

0.05

0.1

0.15

0.2

0.25

-5 -4.5 -4 -3.5 -3 -2.5 -2

J5

log10 εcut

Q = 10 TeV

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Herwig++Resummed

Figure 4.9: Comparison between Herwig++ and resummed jet rates using the anti-kTinclusive algorithm. Results for Q = 1 TeV are shown on the left, while results forQ = 10 TeV are shown on the right. The shape of the Monte Carlo result (at parton level)is correctly reproduced in our analytical description, as well as the fact that the peakmoves to smaller values of the cutoff as the jet multiplicity increases and the centre-of-mass energy is increased. The heights of the peaks are also in fair agreement. Details ofevent generation and residual differences can be found in the main text. For the resummeddistributions, we used αS(mZ) = 0.118, wheremZ is the mass of theZ boson,mZ = 91.2GeV.

92

4.4. RESUMMATION

Figure 4.10: Configuration that leads to exponentiation breaking for the JADE algorithm:the two gluons should be clustered with their parents for a 2-jet event, while in a largeportion of phase space they actually cluster together in a third jet.

4.4.4 Counterexamples

Resummability of jet fractions for a given algorithm is not a trivial property. The mostfamous example in this sense is given by the JADE algorithm [130,131], which is definedby the distance

εij =ωiωjtijQ2

. (4.4.10)

Jet reconstruction using the JADE algorithm is carried out in the spirit of an exclusivealgorithm: final state particles are clustered into pseudojets until the smallest distancebetween two pseudojets is above some fixed value εcut. Double logarithms for jet ratesdefined by this algorithm have been found analytically not to exponentiate [132]. Thephysical reason for this can be seen by considering the configuration in Fig. 4.10. Thereare two emissions collinear to the quark and anti-quark respectively. By inspection of(4.4.10) it is apparent that if these two emissions are soft, the distance between them caneasily be smaller than the distance between them and their parents. This means that aconfiguration that intuitively should give rise to a 2-jet configuration actually leads tothree jets in a large portion of phase space, and this is what spoils the exponentiation ofdouble logarithms.

To further emphasise our point, we now construct an exclusive version of the anti-kTalgorithm and we will show that no exponentiation is possible for the corresponding jetrates. Of course it is not possible to take vij = εij with p = −1 because this choice wouldnot be infrared safe. We can instead choose εij with p = −1 and vij with p = 1. Theanti-kT distance is then used to find the closest particles, while the kT distance is used tocheck when to stop clustering.

The leading order jet rates are obviously equivalent to the Durham case, and the sameholds for the C2

F channel in J2 at NLO. Problems arise when trying to calculate the con-tribution proportional to CFCA in J3. At NLO, the distances are given by

εq1 =tq14

, vq1 =tq14x2

1 ,

εq2 =tq24

, vq2 =tq24x2

2 ,

93


1

2

Figure 4.11: When using the anti-kT algorithm in its exclusive mode with a kT distance,the softer gluon can get clustered to the quark instead of its parent. This can happen evenwhen the soft gluon is collinear to its parent, giving another contribution to the 3-jet ratein addition to the one that can be expected from resummability. Gluons are labelled as inthe main text.

ε12 =t12

4x21

, v12 =t12

4x2

2 , (4.4.11)

where we put x2 < x1. Working out all the possible clustering sequences, the constraint∆′3,CFCA is

∆′3,CFCA = ∆3,CFCA + ∆3,q1Θ

[t12

x21

−min (tq1, tq2)

](∆2,q2 −∆2,12) , (4.4.12)

which can be compared to Eq. (4.3.11). The additional piece gives a contribution of−CFCA

24a2L4, which spoils the exponentiation. The reason for this is depicted in Fig. 4.11:

despite the softer gluon being collinear to its parent, it can still be clustered with the quark,deviating significantly from the physical picture of the parton shower. Fig. 4.12 shows theusual comparison with EVENT2. The difference with the analytical result is fit to a cubiccurve using gnuplot’s internal routine. The result is consistent with our claim of havingcorrectly calculated the coefficient of the leading term.

4.5 Conclusion

In this chapter we analysed the consequences of using different algorithms on n-jetrates, i.e. the probability of getting exactly a final-state with n jets in e+e− → qq. Westarted with a fixed-order calculation of logarithmically-enhanced terms, both for the ex-clusive kT (or Durham) algorithm and inclusive algorithms, at LO and NLO. From the re-sults we can infer the resummation of these observables. Doing this we recover the knownresults of [81,83] and we find the resummed jet rates for inclusive algorithms. While thisbit of original research was underway, a paper on the same topic was published [1]. Com-mon results have been checked and they have been found to be in agreement. We testedour resummed results against events generated by Herwig++ and we find them to be inreasonable agreement. In the last part of the chapter we described two counterexamples

94

4.5. CONCLUSION

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-6 -5.5 -5 -4.5 -4 -3.5 -3

Coeffi

cientof

a2C

FC

A/4

×10

3

log10 εcut

EVENT2

EVENT2 - NLL

Cubic fit

χ2R = 1.50

Figure 4.12: Comparison between EVENT2 and the NLO result at NLL in the expansionfor the CFCA channel of J3 with the exclusive anti-kT algorithm. The difference betweenEVENT2 and our result is fit to a cubic curve by gnuplot’s fitting routine. The value ofχ2R shows the consistency of this result with our claim of having correctly calculated the

coefficient of the leading logarithms.

which show that resummability is not shared by all jet algorithms.The main conclusion of this chapter is that all inclusive algorithms behave the same as

far as jet rates are concerned, at least up to NLL-in-the-expansion accuracy. In particular,this puts the anti-kT algorithm on the same footing as the other inclusive algorithms, incontrast with the common reasoning that attributes to the anti-kT a somewhat reducedability of reproducing the correct jet substructure. On the basis of these results, it may bepossible to reconcile the desirable properties of the anti-kT (inclusive) algorithm (namely,cone-like jets and no clustering logarithms) with recent substructure techniques. This iscertainly a topic worthy of further analytical investigation.

We now move on to the study of jet-substructure techniques proper. As a representa-tive set we choose trimming [31], pruning [32, 33], MDT [25]. We study the distributionof jet mass after the application of each technique to quantify their effectiveness on sup-pressing QCD background in hadronic collisions. This is the first in-depth analyticalstudy of jet substructure. In the next chapter, fixed-order calculations of logarithmically-enhanced terms will be carried out. We will identify the main features of each techniqueand propose modifications to enhance their efficiency. In Chap. 6 we switch to resummedmass distributions.

95

CHAPTER

FIVE

JET SUBSTRUCTURE AT FIXED ORDER

We already tried to convince the reader that jet substructure is a very promising fieldof research that could critically improve the chances of finding new particles and theprecision of many parameters of the Standard Model, chiefly the Higgs mass. We feelthat there is now general consensus on the importance of these techniques and it is timeto adopt a more systematic approach in describing their features. First of all we needan analytical comprehension to bring us beyond the picture we get from Monte Carlogenerators, ridden by all kinds of parametric uncertainties.

There have been mainly three attempts in this direction so far. In [133, 134] the massresolution of signal jets treated with filtering [25] was studied, together with an optimisa-tion of filtering parameters. Soft Collinear Effective Theory (SCET) has been applied tojet substructure methods in [135] to get constraints for optimal performance of some tag-gers. The N -subjettiness shape variable [136] for 2-body signal decays has been relatedto the thrust distribution in e+e− [12, 137–139] in [140]. Other studies include planarflow [141] and energy-energy correlations [142]. Jet rates studied in the last chapter andin [1] themselves can be seen as a jet substructure study: their predictions could be usedfor example to describe the number of subjets inside a fat jet, as has been done [143,144].Additionally, simple kinematical approximations were shown to give some insights intodifferent methods in [145].

The first question to settle is whether it is possible to perform the calculations neededfor an accurate enough description of observables such as jet masses after the applicationof boosted-object taggers. It is well-known that in the case of plain inclusive jet-mass dis-tributions there are large double logarithms 1/mJ α

nS ln2n−1 pT/mJ accompanied by less

singular but still logarithmically enhanced terms, in a way analogous to that describedin Sec. 3.2.1. Due to the very broad range of jet pT which the LHC can produce, wecan get mJ pT even for jet masses at the electroweak scale. Resummation of theselarge logarithms is therefore an inescapable requirement. It has been thoroughly studiedfor hadron-collider jet masses [5, 6, 146, 147]. In particular in [6] the inclusive jet-massdistribution was computed at NLL accuracy in the exponent for hadron collider jets. How-ever, due to the presence of non-global logarithms [92, 93], resummation of NLL termsin the exponent can only be carried out in the limit of large number of quark colours N ,

97

CHAPTER 5. JET SUBSTRUCTURE AT FIXED ORDER

which however is still well suited for comparison to experimental data. Recent work [94]suggests that it might be possible to go beyond this limit. The presence of clusteringeffects [96–98] forces the use of the anti-kT algorithm (where such effects are absent toNLL accuracy in the exponent) to define jets whose mass distribution can be resummed.This is potentially a problem, since substructure techniques are usually defined for jetsreconstructed with a specific algorithm, often C/A.

In all cases though, these methods aim to discriminate against QCD background byplacing cuts on soft and/or collinear QCD radiation inside the jet. This leads to an atleast partial removal of large logarithms. The first goal of our perturbative calculationsis then to point out whether fixed-order, all-order resummed or Monte Carlo techniquesmay provide the best theoretical description for this class of LHC observables.

In this chapter we consider in particular trimming [31], the Mass-Drop Tagger (MDT)[25] and pruning [32, 33]. Our aim is to describe the logarithmically-enhanced terms atNLO in the corresponding jet-mass distributions. We work with pure QCD backgroundjets produced in e+e− collisions. Each result is tested against EVENT2. This chapter isbased on [148].

5.1 Plain jet mass

In this chapter (and in the next) we consider differential distributions in the squaredjet mass normalised to the jet energy squared

1

σ0

dσ

dρ, ρ ≡ m2

J

ω2J

. (5.1.1)

The plain jet-mass distribution in the small-ρ limit develops large logarithms in a wayanalogous to that described in Sec. 3.2.1, with L ≡ ln R2

ρ, in an approximation where the

value of the jet radius1 R is small. For the case of plain jet mass one has [5]

ρ

σ0

dσ

dρ' a

(CFL−

3CF4

)+ a2

[−C

2F

2L3 +

3CF8

(3CF + 4b0)L2

], (5.1.2)

where b0 ≡ πβ0, and in turn β0 was defined in Eq. (2.3.4); the definition of a is Eq. (4.2.3).We have not reported the coefficient of a2L because its value depends on the contributionsof many complicated effects such as multiple emission effects, non-global logarithms,clustering logarithms, cross-talk between the resummed exponent and coefficient func-tions of order a, running coupling effects.

1In the present and in the following chapter we will consider a slightly modified version of the jetalgorithms used in the previous chapter, where instead of εcut we have 2(1− cosR). Clearly, in the small-R(or, equivalently, small-εcut) limit we have εcut ' R2. See also Subsec. 3.1.4.

98

5.2. EVENT2 PARAMETERS

5.2 EVENT2 parameters

Numerical results are obtained for e+e− collisions at the centre-of-mass energyQ = 1

TeV for comparison with analytical distributions. Jets are reconstructed using the C/Aalgorithm with R = 0.8. The output of EVENT2 is the average of the mass distributionsof the hardest and second-hardest jets. This is an infrared-safe selection which allows usto directly assess the correctness of our calculations by performing a point-by-point dif-ference of the analytical and the MC results. Each distribution results from the generationof 109 events. As before, the choice of parameters is

CUTOFF = 10−12

NPOW1 = NPOW2 = 6(5.2.1)

5.3 Trimming

5.3.1 Definition

We now proceed to calculate the jet-mass distribution obtained using trimming [31].We consider jets reconstructed with the C/A algorithm with radius R, though in principleany jet algorithm could be used. The trimming procedure for e+e− collisions is definedas follows:

1. recluster the constituents of the jet using Rsub < R.

2. Discard all jets with less than a fixed fraction fcut of the original jet energy.

3. Recombine the surviving jets to get the trimmed jet.

5.3.2 LO calculation

Here we get our result in the soft-collinear limit and for small values of fcut. Weconsider the emission of a gluon g from a qq pair produced in e+e− annihilation. We candefine the energies of the partons in terms of the fat-jet energy ωJ , i.e. ωg = xgωJ andωq = (1− xg)ωJ . For definiteness we assume the quark and the gluon to cluster together,which happens when cos θqg > cosR; in the collinear limit this is simply θ2

qg < R2. Thenif additionally θ2

qg < R2sub the gluon always contributes to the trimmed jet mass regardless

of its energy, otherwise the gluon gets discarded if its energy fraction is less than fcut. Thenormalised jet mass in the soft-collinear approximation can be written as

ρ = 2xg(1− xg)(1− cos θqg) ' xgθ2qg . (5.3.1)

99


The corresponding contribution is then

1

σ0

dσ

dρ

trimmed,LO

= aCF

∫ R2

0

dθ2qg

θ2qg

∫ 1

0

dxgxg

δ(ρ− xgθ2

qg

)×[Θ(R2

sub − θ2qg

)+ Θ

(θ2qg −R2

sub

)Θ (xg − fcut)

].

(5.3.2)

We can replace 1/xg with Pgq(x) (defined in Eq. (2.6.1)) in the above equation to in-clude the contribution of hard-collinear radiation. Ignoring power corrections, a simplecomputation leads to

1

σ0

dσ

dρ

trimmed,LO

=aCFρ

[lnR2

sub

ρΘ(fcutR

2sub − ρ

)+ ln

1

fcutΘ(ρ− fcutR

2sub

)Θ(fcutR

2 − ρ)

+ lnR2

ρΘ(ρ− fcutR

2)− 3

4

].

(5.3.3)

This result shows that in the small-ρ limit the trimmed jet mass distribution is double-logarithmic just like the case of plain jet mass. For greater values of ρ, it transitions to asingle-logarithmic region and then it becomes identical to the plain jet-mass distribution.

In order to compare this result with EVENT2 we need to go beyond the soft-collinearand small-fcut limit. The full calculation is carried out in [148]. The final result is

1

σ0

dσ

dρ

trimmed,LO,full

=aCFρ

[ln

(4 tan2

(Rsub

2

)ρ

e−34

)Θ(fcut∆

2Rsub− ρ)

+ ln

(1− fcut

fcute−

34

(1−2fcut)

)Θ(ρ− fcut∆

2Rsub

)Θ(fcut∆

2R − ρ

)],

(5.3.4)

where we defined∆2α ≡ 2(1− cosα) . (5.3.5)

We ignored the region ρ > fcut∆2R in Eq. (5.3.4), where the distribution is just the same

as the one for the plain jet mass. The comparison in Fig. 5.1 shows that we correctlycaptured the leading logarithmic behaviour at this order.

5.3.3 NLO calculation

We already observed that, in the small-ρ limit, trimming does not change the jet-massdistribution. It is easy to understand why: for ρ < fcutR

2sub an emission is inside the core

of the jet (θ2 < R2sub), where no cut on the energy is applied. The result for theC2

F channel

100

5.3. TRIMMING

-2

0

2

4

6

8

10

12

14

16

18

-5 -4 -3 -2 -1 0

Coeffi

cientof

aC

F

log10 ρ

Trimming at leading order (Rsub = 0.2 , R = 0.8)

fcut = 0.03

fcut = 0.15

EVENT2 - Analytic

Figure 5.1: Comparison between EVENT2 and the coefficient of aCF in Eq. (5.3.4) fortwo values of fcut. The correctness of the differential distribution of trimmed jet masseswe calculated is shown by the fact that both differences (in black) tend to zero in thesmall-ρ limit. EVENT2 parameters were chosen according to the description in Sec. 5.2.

in this region is

1

σ0

dσ

dρ

trimmed,C2F ,full

= −a2C2

F

ρ

1

2ln3 1

ρ+

3

2

[2 ln

(2 tan

Rsub

2

)− 3

4

]ln2 1

ρ

. (5.3.6)

The full calculation can be found in [148]. At this level of accuracy, the only contributionto the other colour channels is the running of the strong coupling. At one-loop level,this can be described by correctly taking into account a Running Coupling (RC) withargument kT in the leading order calculation. This yields

1

σ0

dσ

dρ

trimmed,RC

=a(ωJRsub)CF

ρ

1

1− λ lnR2

sub

ρ, (5.3.7)

whereλ ≡ b0a(ωJRsub) ln

R2sub

ρ, (5.3.8)

and b0 ≡ πβ0 (β0 was defined in Eq. (2.3.4)). At NLO we then get

1

σ0

dσ

dρ

trimmed,RC

=3a2CF b0

2ρln2 R

2sub

ρ. (5.3.9)

101


As we have seen before, many effects contribute to the coefficient of the a2L2 term in theintegrated distribution. We are not interested in such details as we already captured themain features of trimming in the above results. We just remark that non-global logarithmsought to be present here as well. They obviously arise from the configuration wherethe harder gluon is emitted outside the core of the jet and the softer one inside it; thenthe harder gluon is trimmed away but the other can survive no matter how much softerit is and gives a relevant contribution to the final jet mass. Ignoring clustering effects,these non-global logarithms are the same as those for plain jet mass with an energy vetocomputed for the anti-kT algorithm in [5]. Clustering effects in the C/A algorithm tend toreduce the impact of non-global logarithms [96]. However, due to the smallness of Rsub

in practice, this reduction should be negligible.The comparison of our results at NLO with EVENT2 is shown in Fig. 5.2. The fact

that the differences can be fit by lines shows that we control the most divergent part of theexact answer.

5.3.4 Summary

We collect the results obtained so far in the small-Rsub and small-ρ limit:

ρ

σ0

dσ

dρ= a

(CFL−

3CF4

)+ a2

[−C

2F

2L3 +

3CF8

(3CF + 4b0)L2

], (5.3.10)

with L ≡ lnR2

subρ

, b0 ≡ πβ0, and in turn β0 was defined in Eq. (2.3.4). Apart fromthis change of notation, it is clear that this is the same as Eq. (5.1.2). We also expectthe resummed result to be very similar to that of plain jet mass, but we postpone thisto the next chapter. We can however anticipate the presence of a Sudakov peak belowfcut∆

2Rsub

, i.e. in a phenomenologically crucial region for the vicinity to the electroweakscale, where one would like the background to be as featureless as possible. This seemsquite an undesirable trait of trimming.

5.4 Pruning

5.4.1 Definition

Now we move on to jet mass with pruning [32] using the same approach we usedfor trimming in the previous section. Again we consider a fat jet reconstructed using theC/A algorithm with radius1 R. Pruning starts from reclustering the jet using in principlean arbitrary jet algorithm. In practice the most common choice, and the one adopted by

1Recall that the jet parameter εcut in the previous chapter has been replaced with 2(1− cosR).

102

5.4. PRUNING

-600

-400

-200

0

200

400

600

800

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

2 F

log10 ρ

Trimming at NLO (Rsub = 0.2 , R = 0.8)

-150

-100

-50

0

50

100

150

200

250

300

350

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FC

A

log10 ρ


-120

-100

-80

-60

-40

-20

0

20

40

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FN

f

log10 ρ


fcut = 0.03

fcut = 0.15

Analytic - EVENT2

fcut = 0.03

fcut = 0.15

EVENT2 - Analytic

fcut = 0.03

fcut = 0.15

EVENT2 - Analytic

Figure 5.2: Comparison between EVENT2 and the coefficient of aC2F and aCFCA in

Eqs. (5.3.6) and (5.3.9) respectively, for two values of fcut. All differences tend to lie ona line in the small-ρ limit. This means we achieved NLL accuracy in the expansion ofthe differential distributions for trimmed jet masses at NLO. EVENT2 parameters werechosen according to the description in Sec. 5.2.

103


CMS [121] is again the C/A algorithm. We make that choice too. On the other hand,ATLAS [114] used the inclusive kT algorithm to perform the reclustering. A methodsimilar to ours could be used to study the effect of this difference, but we leave such aninvestigation to future work.

At each reclustering step, for the pair of clustering pseudojets J1 and J2 the followingquantity is calculated:

x =min (pT ,J1 , pT ,J2)

|pT ,J1 + pT ,J2 |, (5.4.1)

The recombination is aborted and the softer pseudojet is discarded if the followingcondition is met:

∆R2J1J2

> D2 ∧ x < zcut , (5.4.2)

where ∆R2J1J2

was defined in Eq. (3.1.5). We now need to introduce η, the plain (i.e. be-fore pruning) jet mass m2

plain, J normalised to the transverse momentum of the same jetpT ,J :

η ≡m2

plain,J

pT ,J. (5.4.3)

The pruning radius D is tuned to η:

D2 = 2Dfη , (5.4.4)

where Df is a fixed numerical factor which in this study is set to Df = 12.

For our purposes, we can use an adaptation of pruning for e+e− annihilation which in-volves the use of energies and angles rather than their boost-invariant counterparts. Hencewe can use the jet energy ωJ instead of pT,J and define

∆R2J1J2≡ 2(1− cos θJ1J2) , (5.4.5)

where θJ1J2 is the angle between the jet directions.Pruning looks similar to trimming, with the critical difference that the radius of the

core is dynamically chosen according to the mass of the original jet rather than fixed ab

initio.


At LO the unpruned jet is made up of a quark and a gluon from the splitting of a hardquark with respectively fractions xg and 1− xg of the energy of the original jet. Then

η = xg(1− xg)∆2θqg < ∆2

θqg . (5.4.6)

104

5.4. PRUNING

This means that in order to get a non-zero jet mass after pruning we need

min (xg,1− xg) > zcut , (5.4.7)

otherwise the softer parton will be pruned away and the jet will become massless. Thesoft-collinear result is then

1

σ0

dσ

dρ

pruned,LO

= aCF

∫ R2

0

dθ2qg

θ2qg

∫ 1−zcut

zcut

dxgxg

δ(ρ− xgθ2

qg

)=aCFρ

ln

(1− zcut

zcut

)Θ(zcutR

2 − ρ)

+ ln

[(1− zcut)

R2

ρ

]Θ(ρ− zcutR

2)

.

(5.4.8)

This result is single-logarithmic in the small-ρ limit, with the soft logarithm replaced bya logarithm of zcut, whose value in practice is chosen to be of order 10−1. Again it ispossible to go beyond the soft-collinear approximation. The full calculation can be foundin [148] and we report here the result for ρ < zcut∆

2R:

1

σ0

dσ

dρ

pruned,LO,full

=aCFρ

ln

[1− zcut

zcute−

34

(1−2zcut)

]. (5.4.9)

Now we can compare Eq. (5.4.9) with EVENT2. The result is shown in Fig. 5.3, andclearly demonstrates that we correctly computed the coefficient of the single logarithm.

5.4.3 NLO calculation: C2F channel

We now consider the contribution to the jet-mass distribution of real emissions andvirtual corrections in the C2

F channel (see Fig. 4.3) in the small-ρ limit. We start byexamining the double-real contribution. In this case we have η ' x1θ

2q1 + x2θ

2q2, where xi

is the fraction of the jet energy carried by parton i. We also keep xi,θqi 1. It is worthdividing the phase space into three regions:

1. Both emissions are at an angle greater than η.

2. Both emissions are at an angle smaller than η.

3. One emission has an angle greater than η and the other one is smaller.

The case of mixed real-virtual contributions is completely analogous to LO. Thismeans that we get no explicit constraint on the angle of the real emission other than therequirement for it to be in the original jet. Then we can divide up the whole real-virtualangular phase space in the same way as the double-real one, using η ' x1θ

2q1 + x2θ

2q2 for

both. Doing so greatly simplifies the process of real-virtual cancellation.

105


-1

0

1

2

3

4

5

6

7

-5 -4 -3 -2 -1 0

Coeffi

cientof

aC

F

log10 ρ

Pruning at leading order (R = 0.8)

zcut = 0.1

zcut = 0.2

zcut = 0.4

EVENT2 - Analytic

Figure 5.3: Comparison between EVENT2 at LO and the coefficient of aCF in Eq. (5.4.9)for various values of zcut. EVENT2 shows a flat distribution for small values of ρ whichindicates a single-logarithmic behaviour for the integrated distribution of pruned masses.The correctness of our result is shown by the fact that all differences (in black) vanishin the small-ρ limit. EVENT2 parameters were chosen according to the description inSec. 5.2.

It can be shown that in regions 1 and 2 no logarithmic divergences survive in thesmall-ρ limit. We turn then to region 3: for definiteness, let gluon 2 be emitted in the coreof the jet, i.e. within the pruning radius D. Then gluon 1 is emitted at an angular distancelarger than D. Adding real and virtual corrections in this angular region, the result forsmall zcut is

∆C2F = Θ

(η − θ2

q2

)Θ(θ2q1 − η

) [Θ (x1 − zcut) δ

(ρ− x1θ

2q1 − x2θ

2q2

)+ Θ (zcut − x1) δ

(ρ− x2θ

2q2

)−Θ (x1 − zcut) δ

(ρ− x1θ

2q1

)−Θ (x2 − zcut) δ

(ρ− x2θ

2q2

) ].

(5.4.10)

The first two lines in the last equation correspond to the real-emission contribution: emis-sion 2 is inside the core of the jet (θ2

q2 < η) and thus it can never be pruned; emission 1is outside the core so it will not be pruned only if it is hard enough (x1 > zcut). In thiscase both gluons will contribute to ρ, otherwise ρ = x2θ

2q2. As already noted, the virtual

contributions are just like LO. It is also worth pointing out that the first and the third linein Eq. (5.4.10) cancel in the soft-collinear limit of emission 2. The two integrals we need

106

5.4. PRUNING

to compute are then the following:

I1 ≡ a2C2F

∫ R2

0

dη

∫ zcut

0

dx1

x1

∫ 1

0

dx2

x2

∫ R2

η

dθ2q1

θ2q1

∫ η

0

dθ2q2

θ2q2

× δ(ρ− x2θ

2q2

)δ(η − x1θ

2q1 − x2θ

2q2

), (5.4.11)

I2 ≡ −a2C2F

∫ R2

0

dη

∫ 1

0

dx1

x1

∫ 1

zcut

dx2

x2

∫ R2

η

dθ2q1

θ2q1

∫ η

0

dθ2q2

θ2q2

× δ(ρ− x2θ

2q2

)δ(η − x1θ

2q1 − x2θ

2q2

). (5.4.12)

As mentioned before, we are working in the soft-collinear limit xi,θ2qi 1 and we are

ignoring power corrections in zcut. The result is

I1 =a2C2

F

6ρln3

(zcut

R2

ρ

)Θ(zcutR

2 − ρ),

I2 = −a2C2

F

2ρ

[(ln

1

zcutln2 R

2

ρ− ln2 1

zcutlnR2

ρ+

1

3ln3 1

zcut

)Θ(zcutR

2 − ρ)

−1

3ln3 R

2

ρΘ(ρ− zcutR

2)]

.

(5.4.13)

We can clearly see in I1 a term which is just as divergent as plain jet mass. This is dueto the emission of a soft gluon (gluon 1) that dominates the plain jet mass and then getspruned away for being too soft. The pruned jet then has no real hard substructure andsince it is dominated by a single hard prong we call it an I-pruned jet. It is possible to gobeyond the small-zcut limit to get the relevant power corrections to the results above andlift the simultaneous soft and collinear approximation to get the contributions from hardcollinear and soft wide-angle radiation. The full calculation can be found in [148]. Thefinal equation in the small-ρ limit is

1

σ0

dσ

dρ

pruned,C2F

=a2C2

F

ρ

1

6ln3 1

ρ

+

[1

2ln

z2cut

1− zcut− 5

4zcut +

z2cut

8+ ln

(2 tan

R

2

)]ln2 1

ρ

,

(5.4.14)

Note that we did not include any single logarithm. This is because its coefficient is subjectto a large number of physics effect including clustering logarithms. We then expect to findthat the difference between our result and EVENT2 can be fit by a line in the small-ρ limit.This is what actually happens, as can be seen in Fig. 5.4.

107


-250

-200

-150

-100

-50

0

50

100

150

200

250

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

2 F

log10 ρ

Pruning at NLO (R = 0.8)

zcut = 0.1

zcut = 0.2

zcut = 0.4

Analytic - EVENT2

Figure 5.4: Comparison between EVENT2 and the coefficient of a2C2F in Eq. (5.4.14)

for various values of zcut. The correctness of our result for the differential distribution ofpruned jet masses in the C2

F channel is shown by the fact that all differences (in black)lie on a line in the small-ρ limit. EVENT2 parameters were chosen according to thedescription in Sec. 5.2.

5.4.4 NLO calculation: CFCA channel

We start by considering the usual splitting of gluon g into gluons 1 and 2. We assumea fairly hard parent gluon, with xg ' x1 > 1− zcut. Since 1 and 2 can be considered to becollinear to each other, the last recombination in the C/A algorithm is g with the quark,and since 1− x1 < zcut, the quark will get pruned away and the mass of the gluon jet getsmeasured. The corresponding contribution is1

1

σ0

dσ

dρ

pruned, CFCA= a2CFCA

∫ 1

1−zcut

dx1Pgq(x1)

∫ R2

0

dθ2q1

θ2q1

×∫ 1−zcut

zcut

dx2

x2

∫ θ2q1

0

dθ212

θ212

δ(ρ− x2x

21θ

212

).

(5.4.16)

1In this case it is more convenient to define x2 as the energy fraction of gluon 2 with respect to gluon 1,rather than the quark:

x2 ≡ω2

ω1. (5.4.15)

108

5.4. PRUNING

-60

-40

-20

0

20

40

60

80

100

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FC

A

log10 ρ


zcut = 0.1

zcut = 0.2

zcut = 0.4

Analytic - EVENT2

Figure 5.5: Comparison between EVENT2 and the coefficient of a2CFCA in Eq. (5.4.17)for various values of zcut. The correctness of our result for the differential distributionof pruned jet masses in the CFCA channel is shown by the fact that all differences (inblack) lie on a line in the small-ρ limit. EVENT2 parameters were chosen according tothe description in Sec. 5.2.

In the small-ρ limit the result for the CFCA channel is

1

σ0

dσ

dρ

pruned, CFCA=a2CFCA

2ρ

(ln

1

1− zcut− z2

cut

4− zcut

2

)ln2 R

2

ρ. (5.4.17)

This contribution vanishes in the zcut → 0 limit but it is indeed more than single-logarithmicin ρ for any non-null value of zcut. Again, the comparison with EVENT2 shown in Fig. 5.5proves that we control the most divergent part of the distribution.

Next we need to discuss Non-Global Logarithms (NGLs). Let us consider a jet madeup of a quark and two gluons labelled 1 and 2. Gluon 1 could very well be pruned buthas anyway the chance to emit the much softer gluon 2 in the core of the jet, i.e. withinan angle η with the hard quark. Then this softer gluon cannot be pruned away, givingrise to NGLs in the resulting jet-mass distribution. Then the conditions for NGLs can besummarised as follows:

∆NG = Θ (zcut − x1) Θ(θ2q1 − η

)Θ(η − θ2

q2

)Θ (x1 − x2) Θ

(θ2

12 − θ2q2

). (5.4.18)

The first two step functions ensure that gluon 1 gets pruned away, while the last one isto prevent gluon 1 and 2 to cluster first, which would make it impossible to get the non-

109


global configuration. In the collinear limit, we can write

θ212 = θ2

q1 + θ2q2 − 2θq1θq2 cosφ , (5.4.19)

where φ is an azimuthal angle. This means that the condition in the last step function inEq. (5.4.18) can be rewritten as

θ2q2 <

θ2q1

4 cos2 φ. (5.4.20)

On the other hand, in the small-ρ limit, from the third step function in Eq. (5.4.18) we get

θ2q2 < x1θ

2q1 . (5.4.21)

From the last two equations we can infer that clustering is avoided if

x1 <1

4 cos2 φ, (5.4.22)

which is always true if zcut < 1/4, since from the first step function in Eq. (5.4.18) wehave x1 < zcut. Physically, this comes from the fact that we want gluon 2 to be emitted inthe core of the jet. In the small-ρ limit, the radius of the core η is set by x1θ

21. Then, if x1

is too small, gluon 2 is forced to stay too close to the quark to be clustered with gluon 1.This derivation has been carried out in the small-angle limit, which means that the

actual value of zcut differs by terms of order R2, the square of the original jet radius. Inthe following we will focus on the small-zcut limit, meaning we will ignore the clusteringof the two gluons. In this case the integration over φ is trivially carried out and the non-global contribution is

1

σ0

dσ

dρ

pruned,NG

= 4a2CFCA

∫ 1

0

dx1

x1

∫ 1

0

dx2

x2

∫ 1

−1

d cos θq1

∫ 1

−1

d cos θq2

× Ω2∆NGδ(ρ− x2∆2

θq2

).

(5.4.23)

The definition of Ω2 is

Ω2 ≡2

(1− cos θq1)(1 + cos θq2)| cos θq1 − cos θq2|. (5.4.24)

The result of integrating Eq. (5.4.23) in the small-ρ limit is

1

σ0

dσ

dρ

pruned,NG

=4a2CFCA

ρLi2 (zcut) ln

1

ρ, (5.4.25)

where Li2(x) is the dilogarithm

Li2(x) =

∫ 0

x

dtln(1− t)

t. (5.4.26)

110

5.4. PRUNING

1

2

Figure 5.6: This figure shows the typical configurations that gives rise to an I-pruned jet.Gluon 1 sets the pruning radius (the boundary is represented by a dashed line) and thengets pruned away. The angular distance of emission 2 from the quark is smaller than thepruning radius and thus it cannot be pruned away: this leads to a divergence as severe asthe one corresponding to plain jet mass. Conversely, if gluon 1 is hard enough not to getpruned, we have a signal-like jet that we cannot discard. This is what we call a Y-prunedjet. The final result in Eq. (5.4.11) includes real-virtual contributions which, for clarity,we do not show here. Both gluons are assumed to be emitted inside the jet.

The pruning mass distribution then actually has non-global logarithms. However, Li2 (zcut)

vanishes as zcut → 0 and thus their impact can be expected not to be sizeable with respectto, for instance, the plain jet mass, where, in the small-R limit, the coefficient of NGLs isπ2

3[5]. Moreover, the effect of clustering logarithms is expected to reduce the size of the

non-global contribution as zcut increases. In any case, this further complicates the attemptto resum the pruned distribution at all orders and it surely is an unwanted feature of anyprocedure of this kind.

5.4.5 Y-pruning

We describe here a modification of pruning that eliminates the double-logarithmicstructure discussed in Subsec. 5.4.3. We add a condition to (5.4.2): the pruned jet J hasto have at least one constituent j outside its core (∆2

θqj> ∆2

η) which survived pruningby being hard enough (1 − zcut > xj > zcut), or be discarded as a whole. We call thismodified version of the tagger Y-pruning, because the additional condition ensures thatthe jet mass is dominated by two-prong, (semi)-hard radiation, with a pruning radius thatis then appropriate to account for the (lack of) jet substructure. Fig. 5.6 shows how twoemissions can give rise to either an I-pruned or a Y-pruned jet.

According to the definition in Sec. 3.6, pruning is a grooming procedure: given aninput jet, it always returns a (possibly unmodified) pruned jet. Conversely, Y-pruning is atagger, as it may or may not reject an input jet.

From the point of view of analytical results, I-pruning corresponds to I1 in (5.4.11),while I2 comes from the jet that pass the Y-pruning condition. As a consequence, theleading divergence in the integrated distribution at NLO for Y-pruning is a2L3, i.e. onelogarithm less than pruning and plain jet mass. The comparison with EVENT2 confirmsthis prediction and it can be found in Fig. 5.7.

111


-150

-100

-50

0

50

100

150

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

2 F

log10 ρ


zcut = 0.1

zcut = 0.2

zcut = 0.4

Analytic - EVENT2

Figure 5.7: Comparison between EVENT2 and the coefficient of a2C2F in I2 as found in

Eq. (5.4.11) for various values of zcut. The correctness of our result for the Y-pruned jetmass differential distribution in the C2

F channel is shown by the fact that all differences(in black) lie on a line in the small-ρ limit. EVENT2 parameters were chosen accordingto the description in Sec. 5.2.

5.4.6 Summary

Our results describe a rich structure for pruning. At NLO, pruning exhibits a dominanta2L3 divergence in its differential distribution (which translates into a double-logarithmica2L4 divergence in the integrated one) arising from the emission of a gluon dominatingthe original jet mass which then gets pruned away. We dub this situation I-pruning as allsubsequent emissions in such a jet are never checked for pruning and all contribute to thefinal jet mass. In other words, the first emission prevents pruning from acting on the restof the constituents of the jet, which is effectively left untouched. The result for pruningin the small-ρ and small-zcut limits is

ρ

σ0

dσ

dρ= aCF ln

e−3/4

zcut+ a2

[C2F

6L3 − C2

F ln1

zcutL2

]. (5.4.27)

As for the plain jet mass, many complicated effects contribute to single logarithms at NLOand we do not report the resulting coefficient here.

We propose Y-pruning, a modification for pruning which aims to explicitly veto anoma-lous configurations by asking that for each jet, at least one emission is outside the core

112

5.5. MASS DROP TAGGER (MDT)

and it is hard enough not to get pruned. In this case, the resummed formula can be writtenin a simple form:

ρ

σ0

dσ

dρ

Y-pruning,all-order

= aCF ln1

zcutexp

[−aCF

2ln2 R

2

ρ

], (5.4.28)

A full discussion of the resummation properties of both I- and Y-pruning will be carriedout in the next chapter. We just note that based on our results so far one can expect that forρ < z2

cutR2, I-pruning will give rise to a Sudakov peak in a region of phenomenological

relevance, where the background should be made as featureless as possible.

5.5 Mass Drop Tagger (MDT)

5.5.1 Definition

The Mass Drop Tagger (MDT) is defined for a hard jet J reconstructed with the C/Aalgorithm with radius R. The procedure is the following:

1. Undo the last step of clustering to get J1 and J2 from J , with mJ1 > mJ2 .

2. Define y =min(p2T ,J1

, p2T ,J2)∆R2J1J2

m2J

. If mJ1 < µmJ and y > ycut we have found asufficiently symmetric mass drop and the procedure stops.

3. Redefine J to be J1 and go back to step 1.

The behaviour of this tagger depends on two parameters: µ and ycut. The above algorithmcan be used in hadronic collisions. The definition of ∆RJ1,J2 is Eq. (3.1.5). However,we are still looking at e+e− collisions and we can thus adopt ∆R2

J1J2≡ ∆2

θJ1J2and

use energies instead of transverse momenta. We are mainly interested in the limit of acollinear parton splitting J → J1, J2. In this case, we can write

y =min (ωJ1 , ωJ2)

max (ωJ1 , ωJ2)=

min (x, 1− x)

max (x, 1− x), (5.5.1)

where x and 1− x are the energy fractions of the collinear partons.


Here we calculate the leading order contribution to the distribution defined in Eq. (5.1.1)after the application of the MDT. First of all we undo the clustering; we are left with twomassless partons, so the mass-drop condition is trivially satisfied; the asymmetry condi-tion is satisfied if

1

1 + ycut> xg >

ycut

1 + ycut. (5.5.2)

113


The result is then

1

σ0

dσ

dρ

MDT,LO

= aCF

∫ R2

0

dθ2qg

θ2qg

∫ 11+ycut

ycut1+ycut

dxgxg

δ(ρ− xgθ2

qg

)=aCFρ

[ln

1

ycutΘ

(ycut

1 + ycutR2 − ρ

)+ ln

R2

ρ(1 + ycut)Θ

(ρ− ycut

1 + ycutR2

)].

(5.5.3)

Note that for small jet masses only the first term survives. This means that in this limitthe result is single-logarithmic in ρ. The action of the tagger has been to replace a softlogarithm in ρ in the plain jet mass by a logarithm in ycut. The value of ycut can be chosenso that the size of these logarithms is modest and then, as intended, background is reducedcompared to the plain jet mass. For larger masses the result is still double-logarithmic. Itis also worth noting that since the mass-drop condition is always true, the final result doesnot depend on µ. We can once again use EVENT2 to check our results. Before being ableto do so however, we need to go beyond the soft-collinear limit we have exploited so far.The complete calculation can be found in [148]. Here we limit ourselves to reporting thefinal result for ρ < ycut

1+ycut∆2R:

1

σ0

dσ

dρ

MDT,LO,full

=aCFρ

ln

[1

ycute−

34( 1−ycut

1+ycut )], (5.5.4)

From this result it is clear that accounting for hard collinear emission by using the fullsplitting function changes the coefficient of single logarithms in ρ in the integrated dis-tribution. On the other hand, soft, large-angle emissions only produce subleading terms.Note that we get the same result as pruning if we replace ycut with zcut

1−zcut.

We are now in position to compare Eq. (5.5.4) with the fixed-order program EVENT2.This is shown in Fig. 5.8. It is clear that our result reproduces the full LO result in thesmall-ρ limit, for various values of ycut.

The interesting thing to do now is to go beyond leading order to see whether thissingle-logarithmic structure survives additional emissions.

5.5.3 Logarithmic behaviour beyond LO

We want now to investigate if the reduction of the logarithmic divergence that MDToperates on the jet-mass distribution is still effective at higher orders. In other words, wewant to understand if the MDT mass distribution is still single-logarithmic beyond LO.

Unfortunately, the following effect spoils this simple picture. Let us consider thebranching of a soft gluon g into offspring gluons 1 and 2. In the collinear limit, thesmallest angular distance is θ12, i.e. the distance between the two gluons. Then the C/A

114


-1

0

1

2

3

4

5

6

7

-5 -4 -3 -2 -1 0

Coeffi

cientof

aC

F

log10 ρ

MDT at leading order (R = 0.8)

ycut = 0.1

ycut = 0.2

ycut = 0.4

EVENT2 - Analytic

Figure 5.8: Comparison between EVENT2 at LO and the coefficient of aC2F in Eq. (5.5.4)

for various values of ycut. EVENT2 shows a flat distribution for small values of ρ whichindicates a single-logarithmic behaviour for the integrated distribution for MDT. The cor-rectness of our result is shown by the fact that all differences (in black) vanish in thesmall-ρ limit. EVENT2 parameters were chosen according to the description in Sec. 5.2.

algorithm would cluster these two gluons first, and then cluster the resulting pseudoparti-cle to the hard quark, giving rise to a fat jet. When we undo the last step of the clustering,we find two subjets: a massive jet J1 made up of the two gluons and a massless jet J2

which is just the quark. The mass-drop will be immediately satisfied in the soft limit,since the partons that make up the massive jet are much softer than the quark. We havenow two possibilities:

1. The asymmetry condition (step 2 in Sec. 5.5.1) is met. In this case no additionalproblems arise, in the sense that we measure the mass of the original jet.

2. The asymmetry condition is not met. In this case the MDT moves to J1 since it isthe only massive jet. We call this the Wrong-Branch (WB) effect.

We want to study the second case in more detail. The corresponding configuration isshown in Fig. 5.9. The failure of the asymmetry condition means

xg <ycut

1 + ycut∨ xg >

1

1 + ycut. (5.5.5)

We are interested in computing just the most divergent contribution to the jet-mass distri-bution, so we ignore the second condition in the last equation, which corresponds to the

115


1

2

Figure 5.9: This figure shows the configuration that leads to the WB effect. The quark isunclustered first and, if the asymmetry condition is not met, gets discarded; this is becausethe MDT by definition follows the massive branch consisting of gluon 1 and 2 clusteredtogether.

emission of a hard gluon by the quark. When the asymmetry condition fails we move toJ1 as described above. To study its mass we can consider gluon g as having undergonea collinear splitting into gluons 1 and 2, carrying fractions 1 − x2 and x2 of the parentgluon’s momentum respectively. Then we have

m2J1

ω2J

= x2gx2(1− x2)θ2

12 . (5.5.6)

In order to actually measure this mass, the splitting of g must be symmetric enough, whichimplies

ycut

1 + ycut< x2 <

1

1 + ycut. (5.5.7)

Putting everything together, the WB contribution is given by

1

σ0

dσ

dρ

MDT,WB

= a2CFCA

∫ 1

0

dxgxg

∫ R2

0

dθ2qg

θ2qg

Θ

(ycut

1 + ycut− xg

)×∫ 1

0

dx2

[Pgg(x2) +

TFNf

CAPqg(x2)

] ∫ θ2qg

0

dθ212

θ212

×Θ

(1

1 + ycut− x2

)Θ

(x2 −

ycut

1 + ycut

)δ[ρ− x2

gx2(1− x2)θ212

].

(5.5.8)

Pgg(x) and Pqg(x) are defined in Eq. (2.6.1). For ρ < R2y3cut/(1 + ycut)

3 the leading resultfor each colour channel is

1

σ0

dσ

dρ

MDT,WB,CFCA=a2CFCA

4ρ

[ln

1

ycut+

11y3cut + 9y2

cut − 9ycut − 11

12(1 + ycut)3

]ln2 1

ρ,

1

σ0

dσ

dρ

MDT,WB,CFNf=a2CFNf

ρ

1− y3cut

6(1 + ycut)3ln2 1

ρ.

(5.5.9)

As we have done before, we can now check these results against EVENT2. Fig. 5.10

116


-100

-50

0

50

100

-6 -5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FC

A

log10 ρ

MDT at NLO (R = 0.8)

-10

0

10

20

30

-6 -5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FN

f

log10 ρ

MDT at NLO (R = 0.8)

ycut = 0.1

ycut = 0.2

ycut = 0.4

EVENT2 - Analytic

ycut = 0.1

ycut = 0.2

ycut = 0.4

Analytic - EVENT2

Figure 5.10: Comparison between EVENT2 at NLO and the coefficient of a2CFCA anda2CFNf as reported in Eq. (5.5.9) for various values of ycut. All differences (in black)are fitted by straight lines, proving that our calculations correctly catch the most divergentbehaviour of the distributions for the MDT in these colour channels. EVENT2 parameterswere chosen according to the description in Sec. 5.2.

shows the comparisons we have carried out for both colour channels and for several valuesof ycut. When we subtract our calculation of the leading logarithms in ρ from EVENT2we are left with a straight line, which implies that our calculations correctly describes allcontributions which are more divergent than a single logarithm.

The important point of this subsection is that even in the small-ρ limit, the WB ef-fect spoils the simple single-logarithmic behaviour of the observable at LO. The leadinglogarithms are still less singular than the double logarithms which are found for plain jetmass, but they need to be resummed, with no guarantee that a compact resummed formulacan be easily found. In practice, we will see in the next chapter that the WB effect doesnot have a big impact on the mass spectrum of a parton shower. However, this effect isphysically undesirable, given that the aim of a tagger is to identify hard substructure. Itis also of interest to check whether it is possible to modify the tagger to make it purelysingle-logarithmic at all orders, making resummation much easier. For all these reasons,in the next section we will study a modified version of the MDT tagger (mMDT), and inparticular its logarithmic structure at NLO.

117


5.6 Modified Mass Drop Tagger (mMDT)

5.6.1 Definition

We want to study the effect on MDT of the following modification: replace the thirdstep in Sec. 5.5.1 with

3. Redefine J to be J1 if ωJ1 > ωJ2 , otherwise redefine J to be J2.

This modification is valid for colour-neutral initial states, in hadron colliders one wouldeither compare transverse momenta pT or transverse massesm2+p2

T . In general this is de-scribed as following the harder (rather than the more massive) subjet. There is obviouslyno difference from MDT at LO, so we immediately move on to NLO.

5.6.2 NLO calculation: C2F channel

We start our NLO calculation of the mMDT jet mass by addressing the independentemission of two gluons from a quark (or equivalently an antiquark), and the correspondingvirtual corrections. This is the colour channel with charge C2

F . The contributions we needto take into account are depicted in Fig. 4.3. Here we won’t need the double-virtualconfiguration because it gives no finite jet mass. We can also neglect the recoil of thequark against the soft gluons so that the jet axis is given by the original quark direction.We start by examining the region where θ2

q1, θ2q2 < R2 and θ2

q1 < θ2q2, θ

212. This means that

gluon 1 gets clustered to the quark first, followed by gluon 2. The integral we need tocompute is

1

σ0

dσ

dρ

mMDT,C2F

= a2C2F

∫ 1

0

dx1

x1

∫ 1

0

dx2

x2

∫ R2

0

dθ2q1

θ2q1

∫ R2

0

dθ2q2

θ2q2

×∫ 2π

0

dφ

2πΘ(θ2q2 − θ2

q1

)Θ(θ2

12 − θ2q1

)∆C2

F ,

(5.6.1)

where ∆C2F encodes the constraints given by the mMDT when measuring the jet mass.

We also have θ212 ' θ2

q1 + θ2q2− 2θq1θq2 cosφ, where φ is an azimuthal angle. At this level

of accuracy, we can actually ignore the condition that the two gluons should now clustertogether. A factor of 2 is included to take into account the case where θ2

q2 is the smallestangle, which by symmetry gives the same result. We can now start by undoing the laststep of clustering we then get gluon 2 and a massive jet J1. Now we need to check for themass-drop condition

mJ1 < µmJ . (5.6.2)

118

5.6. MODIFIED MASS DROP TAGGER (MMDT)

At our level of accuracy this is equivalent to

x2θ2q2 > fx1θ

2q1 , (5.6.3)

where f ≡ 1−µµ

. If this fails, the softer subjet will be discarded. If we want a non-zeromass, we need gluon 2 to be discarded. This means

1− x2 > x2 =⇒ x2 <1

2. (5.6.4)

Then we need to check the asymmetry condition just as in the LO case. The first constrainton the phase space for double-real emission is then given by

∆RR,I = Θ(fx1θ

2q1 − x2θ

2q2

)Θ

(1

2− x2

)×Θ

(1

1 + ycut− x1

)Θ

(x1 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1

).

(5.6.5)

On the other hand, if the mass-drop condition (5.6.3) is satisfied, we go to the asymmetrystep

ycut

1 + ycut< x2 <

1

1 + ycut. (5.6.6)

If this one fails due to gluon 2 being too soft, then surely J1 is the harder subjet and wefall again into the LO-like situation1. The second set of constraints is then given by

∆RR,II = Θ(x2θ

2q2 − fx1θ

2q1

)Θ

(ycut

1 + ycut− x2

)×Θ

(1

1 + ycut− x1

)Θ

(x1 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1

).

(5.6.7)

Finally if the condition in Eq. (5.6.6) is met, we measure the mass of the whole jet. Theresulting constraints in this case are

∆RR,III = Θ(x2θ

2q2 − fx1θ

2q1

)×Θ

(1

1 + ycut− x2

)Θ

(x2 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1 − x2θ

2q2

).

(5.6.8)

1If the asymmetry condition fails due to gluon 2 being too hard we get a null mass and in the other casethe constraint x2 < 1/2 is automatically satisfied for any ycut < 1.

119


The conditions for the whole double-real contribution can be obtained by summing theprevious ones

∆RR = ∆RR,I + ∆RR,II + ∆RR,III

= Θ(x2θ

2q2 − fx1θ

2q1

)Θ

(1

1 + ycut− x2

)Θ

(x2 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1 − x2θ

2q2

)+

[Θ(x2θ

2q2 − fx1θ

2q1

)Θ

(ycut

1 + ycut− x2

)+ Θ

(fx1θ

2q1 − x2θ

2q2

)Θ

(1

2− x2

)]×Θ

(1

1 + ycut− x1

)Θ

(x1 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1

).

(5.6.9)

The case of a real emission and a virtual correction is perfectly analogous (up to a minussign) to LO. Then we get

∆RV = −Θ

(1

1 + ycut− x1

)Θ

(x1 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1

)−Θ

(1

1 + ycut− x2

)Θ

(x2 −

ycut

1 + ycut

)δ(ρ− x2θ

2q2

).

(5.6.10)

The final constraints for the C2F channel are then

∆C2F = ∆RR + ∆RV

= −Θ

(1

1 + ycut− x1

)Θ

(x1 −

ycut

1 + ycut

)δ(ρ− x1θ

2q1

)×[Θ(x2θ

2q2 − fx1θ

2q1

)Θ

(x2 −

ycut

1 + ycut

)+ Θ

(fx1θ

2q1 − x2θ

2q2

)Θ

(x2 −

1

2

)].

(5.6.11)

Note that the first contribution in Eq. (5.6.9) cancels with the second one in Eq. (5.6.10)when either x1 or θ2

q1 tend to zero. This means that we won’t get any relevant logarithmfrom these two contributions combined.

Before plugging Eq. (5.6.11) into Eq. (5.6.1) and computing the integrals, we notethat we can further simplify the constraints on the angles. In particular, the condition onθ2

12 in Eq. (5.6.1) can be safely ignored and thus the integral over φ is trivial. Moreover,we can ignore the case where gluons 1 and 2 cluster together as it does not contribute tothe mMDT distribution at the level of accuracy we are interested in. Note that this is notthe case for the MDT, where the self clustering of the two gluons would produce a single-logarithmic contribution due to the WB effect described in Sec. 5.5.3, i.e. the selection ofthe more massive branch instead of the harder one. Then the result after integration over

120


the angles is

1

σ0

dσ

dρ

mMDT,C2F

= −a2C2

F

ρ

∫ 11+ycut

ycut1+ycut

dx1

x1

∫ 1

ycut1+ycut

dx2

x2

×[Θ (x2 − fx1) Θ

(R2 − ρ

x1

)lnx1R

2

ρ

+ Θ (fx1 − x2) Θ

(R2 − ρf

x2

)lnx2R

2

ρf

].

(5.6.12)

In the small-ycut limit the final result for ρ < ycut1+ycut

R2 is strikingly simple:

1

σ0

dσ

dρ

mMDT,C2F

= −a2C2

F

ρln2 e

− 34

ycutlnR2

ρ, (5.6.13)

where the −3/4 factor comes from hard collinear emissions, described by substituting1/xi in Eq. (5.6.12) with the full splitting functions Pgq(xi). If we consider the small-ycut

limit of Eq. (5.5.4), it is easy to write down the distribution in an exponentiated form:

1

σ0

dσ

dρ

mMDT,all-orders

=d

dρexp

[−a(ωJR)CF ln

e−34

ycutlnR2

ρ

], (5.6.14)

where we set the scale of the coupling to ωJR in accordance with [5]. It is also possible togo beyond the small-ycut limit by taking into account power corrections in hard collinearradiation in Eq. (5.6.12). The final result is

1

σ0

dσ

dρ

mMDT,C2F

= −a2C2

F

ρln fq (ycut)

ln [(1 + ycut)fq(ycut)]−

ycut (3ycut + 2)

4(1 + ycut)2

lnR2

ρ,

(5.6.15)where we have defined

fq(ycut) ≡1

ycute−

34( 1−ycut

1+ycut ) . (5.6.16)

It can be expected to find more finite-ycut terms in the CFCA and CFNf channels: wewill address them in the next subsection. In the next chapter we will show that all thesecontributions together conspire to a full exponentiation of the mMDT jet-mass distribu-tion. Before moving on though, it is necessary to test Eq. (5.6.15) against EVENT2. Theresults are shown in Fig. 5.11. It is clear that after removing Eq. (5.6.15), in the small-ρlimit we are left with a constant, which means that we control the dominant a2 ln ρ term.

121


-80

-60

-40

-20

0

20

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

2 F

log10 ρ

mMDT at NLO (R = 0.8)

ycut = 0.1

ycut = 0.2

ycut = 0.4

EVENT2 - Analytic

Figure 5.11: Comparison between EVENT2 at NLO and the coefficient of a2C2F in

Eq. (5.6.15) for various values of ycut and ρ < ycut1+ycut

∆2R. Each difference (in black)

reaches a constant value in the small-ρ limit, which means that we control the most di-vergent contribution to the mMDT differential distribution at LO in this colour channel.EVENT2 parameters were chosen according to the description in Sec. 5.2.

5.6.3 NLO calculation: CFCA and CFNf channels

The easiest contribution to take into account is the one coming from the running cou-pling. At NLO, this is given by

1

σ0

dσ

dρ

mMDT,NLO,RC

=a2CF b0

ρln [fq(ycut)] ln

R2

ρ, (5.6.17)

where b0 ≡ πβ0, and in turn β0 was defined in Eq. (2.3.4).We have also another effect which at first sight looks physically similar to the WB

effect in the MDT. If we consider a collinear splitting of the gluon emitted by the quark,there is still the possibility that the mMDT will follow the branch corresponding to themassive gluon jet. The crucial difference is that this happens when the parent gluon istoo hard, i.e. xg > 1

1+ycut. This means that this effect only contributes a single logarithm,

while the MDT produces an extra logarithm. This is because the MDT follows the gluonsubjet no matter how soft the parent gluon is, so that there is no effective cutoff on thecorresponding divergence. The colour charge associated with these contributions is CFC,where C is either CA or TFNf , depending on how the primary gluon splits. We can

122


calculate the impact of this effect on the mMDT distribution explicitly1:

1

σ0

dσ

dρ

mMDT, CFC

= a2CFCA

∫ 1

0

dxgxg

∫ R2

0

dθ2qg

θ2qg

Θ

(xg −

1

1 + ycut

)×∫ 1

0

dx2

[Pgg(x2) +

TFNf

CAPqg(x2)

] ∫ θ2qg

0

dθ212

θ212

×Θ

(1

1 + ycut− x2

)Θ

(x2 −

ycut

1 + ycut

)δ[ρ− x2

gx2(1− x2)θ212

].

(5.6.19)

If we restrict ourselves to the region ρ < ycut1+ycut

R2, the results for each colour channel are

1

σ0

dσ

dρ

mMDT, CFCA=a2CFCA

ρ

[ln

1

ycut+

11y3cut + 9y2

cut − 9ycut − 11

12(1 + ycut)3

]×[ln (1 + ycut)−

ycut(2 + 3ycut)

4(1 + ycut)2

]lnR2

ρ,

1

σ0

dσ

dρ

mMDT, CFNf=a2CFNf

ρ

1− y3cut

6(1 + ycut)3

[ln(1 + ycut)−

ycut(2 + 3ycut)

4(1 + ycut)2

]lnR2

ρ,

(5.6.20)

where we have used TF = 12. We can thus confirm that this effect only contributes a

single logarithm to the mMDT. Also of note is the fact that the above results vanish whenycut → 0. Values of ycut of order 10−1 are usually used, so we can expect the impact of theabove contributions on the jet-mass distribution to be very modest. In any case, they canbe easily resummed in a matrix form (see App. C).

The final possible issue we need to address is that of non-global logarithms [92, 93],which contribute to the CFCA channel. In the plain jet mass, non-global logarithms arisefrom gluons emitted just outside the jet which in turn emit inside the jet, which thenbecomes massive. The main contribution to these logarithms comes then from soft, large-angle radiation. This means that collinear logarithms are already removed, and thereforenon-global logarithms constitute a single-logarithmic contribution to the jet-mass distri-bution. On the other hand, in the case of the mMDT there is an additional cutoff ycut

on the energies of such gluons. This is supposed to remove any non-global contribution.Here we shall ignore the effect of soft-gluon clustering in the C/A algorithm, which hasbeen shown to further reduce non-global logarithms [96–98]. Our result is then to beconsidered an upper bound on the non-global contribution. The non-global contribution

1Here again we havex2 ≡

ω2

ω1. (5.6.18)

123


is given by

1

σ0

dσ

dρ

mMDT,NG

= 4a2CFCA

∫ 1

0

dx1

x1

∫ 1

0

dx2

x2

∫ 1

0

d cos θq1

∫ 1

0

d cos θq2

× Ω2∆NGδ(ρ− x2∆2

θq2

),

(5.6.21)

whereΩ2 ≡

2

(1− cos θq1)(1 + cos θq2)| cos θq1 − cos θq2|. (5.6.22)

As before, ∆NG in Eq. (5.6.21) encodes the constraints on the phase space needed to getthe non-global contribution. Since we want the harder gluon (numbered 1) to stay outsidethe jet and the other to go inside the jet, we get

∆NG = Θ (x1 − x2) Θ(

∆2θq1−∆2

R

)×Θ

(∆2R −∆2

θq2

)Θ

(1

1 + ycut− x2

)Θ

(x2 −

ycut

1 + ycut

).

(5.6.23)

Then the final result for ρ < ycut1+ycut

∆2R is

1

σ0

dσ

dρ

mMDT,NG

=a2CFCA

ρcot2

(R

2

)1 + ycut

ycut

ln

1

ycut− (1− ycut)[1− ln(1 + ycut)]

,

(5.6.24)

which shows that the mMDT distribution exhibits no non-global logarithms at order a2.For the same reason one may expect Abelian clustering logarithms [5,97,98] to be absenthere. We can turn this fixed-order statement into an all-order one by looking at the be-haviour of the tagger. On one hand, the absence of non-global logarithms at this order istrue also for the MDT; however, let us consider a soft gluon outside the jet which radiatesinside the jet. Then the soft and collinear branching of this gluon (we are now at order a3)generates a massive gluon jet, which the tagger follows due to the WB effect. This leadsto a non-global contribution in the integrated distribution proportional to a3 ln3 ρ. Thiseffect then generalises to higher orders. On the other hand, the same cannot happen withthe mMDT. This is yet another good reason to prefer the mMDT over the original MDT.

We can now once again check our results for theCFCA andCFNf channels, Eq. (5.6.17)and (5.6.20), against the exact calculations performed by EVENT2. The comparisons arein Fig. 5.12. The differences are again constant in the small-ρ limit, which means we fullycontrol the single logarithms.

124


-40

-20

0

20

40

60

80

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FC

A

log10 ρ


-15

-10

-5

0

5

-5 -4 -3 -2 -1 0

Coeffi

cientof

a2C

FN

f

log10 ρ


ycut = 0.1

ycut = 0.2

ycut = 0.4

EVENT2 - Analytic

ycut = 0.1

ycut = 0.2

ycut = 0.4

Analytic - EVENT2

Figure 5.12: Comparison between EVENT2 at NLO and the coefficients of a2CFCA anda2CFNf in Eq. (5.6.15) for various values of ycut and ρ < ycut

1+ycut∆2R. Each difference

(in black) reaches a constant value in the small-ρ limit. This means that our result forthe mMDT differential distribution in these colour channels is correct up to an unknownconstant term. EVENT2 parameters were chosen according to the description in Sec. 5.2.

5.6.4 Summary

To summarise the results for the mMDT differential jet-mass distribution, we writedown the result in the small-ycut limit:

ρ

σ0

dσ

dρ

mMDT

= aCF lne−3/4

ycut− a2

(C2F ln2 e

−3/4

ycut− CF b0 ln

e−3/4

ycut

)L . (5.6.25)

which can be compared to the analogous result for plain jet mass, Eq. (5.1.2). It is worthpointing out how much simpler this result is. In particular, we recall the fundamentalproperties of the mMDT we have uncovered:

• The contributions to the distribution are purely collinear, so that there is no need tostudy complicated soft large-angle colour structures.

• The absence of double-logarithmic terms means that there are no big Sudakov peaksin the background distribution.

• By tuning the value of ycut it is possible to completely cancel the coefficient ofa2L in Eq. (5.6.25). This means that the resummed distribution for mMDT is wellapproximated by just its leading order, i.e. we would get a flat distribution.

We will explore all these characteristics in more detail in the next chapter.

125

CHAPTER

SIX

JET SUBSTRUCTURE AND RESUMMATION

In the last chapter we carried out a fixed-order calculation at NLO of the logarithmically-enhanced contributions to differential jet-mass distributions employing several taggers. Itturned out that each tagging procedure acts very differently on background. Qualitatively,the main results of the last chapter can be summarised in a few points:

• Trimming shows the same kind of divergence as plain jet mass

• Pruning and MDT exhibit a flat mass spectrum at LO for ρ→ 0. The former thoughshows the same problematic features of trimming at NLO. A simple modificationof pruning greatly improves its behaviour.

• A slightly modified version of the MDT makes it possible to obtain an almost flatbackground jet-mass distribution

However, given the nature of logarithmic divergences in the small-mass limit, it is worthasking whether it is possible to go beyond a fixed-order description of the various jet-mass distributions. The scope of this chapter is exactly to show how to get resummed

jet-mass distributions on the basis of what was found in the previous chapter. This makesit possible to compare our results to Monte Carlo predictions. However, the situation hereis very different from when we were using EVENT2. There we had an exact, fixed-orderresult to compare to, here we have an approximate parton shower. On one hand, thismeans that, especially for small masses, we can get closer to what we would expect tohappen in an actual experiment; on the other hand, the amount of tuning in any general-purpose event generator makes it impossible to carry out a point-by-point comparisonas we previously did. Various mismatches in accuracy and the residual dependence ofthe generator on non-perturbative parameters mean we only are able to make qualitativecomparisons of our results with MC data and we focus more on critical characteristics likethe position of peaks and transition points rather than general shapes. However, while itis generally hard to trace the incidence of NP parameters and subleading contributions onthe MC results (not to mention that they vary with each different generator), our resultshave a well-defined analytical form, a clear connection to physical phenomena whichdetermines their accuracy and no dependence on arbitrary parameters. Therefore, if we

127

CHAPTER 6. JET SUBSTRUCTURE AND RESUMMATION

will be able to get a reasonable accord with MC data, we will have provided a much morereliable benchmark for jet substructure studies than anything currently available.

In more detail, our plan is to carry out the same study of the last chapter on backgroundjet-mass distributions from the point of view of resummation. As before, we focus ontrimming [31], pruning [32] and MDT [25] together with their modifications proposedin the last chapter. The variable under inspection is still ρ ≡ m2

J/p2T , mJ and pT being

the mass and transverse momentum of the jet respectively. Jets are again reconstructedusing the C/A algorithm [83] with radius R. A very helpful approximation is that ofindependent emission: we can ignore subsequent splittings of emission from primarypartons other than in the treatment of the running coupling. This is enough to reachNLL accuracy in the exponent for rIRC-safe observables [29] (see e.g. [137, 149]). As aconsequence, we aim to control terms like αnS lnn ρ in the integrated distribution at best.Often we will settle for less accuracy than NLL in the exponent, as different taggers willpresent different complications; we will however always aim to get jet-mass distributionswhich are accurate at NLL in the expansion. Every tagger includes some cutoff parameter.Practically these parameters assume values of order 10−1, so it is legitimate to ignore theirpower corrections. Since they are not usually taken to be parametrically small, there is noneed to resum them, though such a resummation is conceivable.

Our results are valid both for e+e− and hadron colliders. We are interested mainly incentral jets (zero rapidity), as a result transverse momenta and energies are interchange-able, as well as boost-invariant angular separations and angles. For simplicity of notationwe use energies and angles throughout this chapter. The small-R approximation makesit possible to ignore initial-state radiation. In practice, this approximation is known tohold up to quite large values of R ' 1 [6, 150]. For brevity only quark-initiated jets aredescribed in the main text: gluon jets are no more complicated and only involve trivialmodifications of the results for quark jets. The corresponding distributions can be foundin App. B.

A very valuable tool in describing the resummation properties of the various distri-butions under scrutiny is the Lund Kinematic Diagram (LKD) [151]. We give a quickdescription of it in the next section. The rest of the chapter is mainly devoted to obtainingresummed distributions for each tagger, comparing our results to MC data at each step;we conclude the chapter with a comparison of the performances of the taggers and weprovide some additional phenomenological considerations. Our MC results, generatedby Herwig++, complement and support those reported in [152], based on Pythia 6 anddifferent choices of parameters.

128

6.1. PRELIMINARIES

ln(k

T/Q

)

ξ

Hard collinear radiation

Har

d co

llinea

r rad

iatio

n

So

ft,

larg

e-a

ng

le r

ad

iatio

n

V(k) =

v

V(k) =

v

Figure 6.1: LKD for a generic observable. The integrated distribution of v is proportionalto the shaded area, while the differential one is proportional to the length of the lineswhere V (k) = v.

6.1 Preliminaries

6.1.1 The Lund Kinematic Diagram (LKD)

A useful tool to visualise the various resummation properties we will encounter through-out this chapter is the Lund Kinematic Diagram (LKD) [151]. It essentially consistsin plotting emissions in the ξ − ln kT

Qplane, where ξ is the pseudorapidity defined in

Eq. (3.1.7). Both this and the transverse momentum kT are to be considered with respectto the hard leg (in our case, the quark that initiates the jet). Q is the scale of the hard pro-cess. The contribution to a generic observable could then look like Fig. 6.1. Obviously,since we always have kT < Q, only the negative ordinate axis is physically relevant. Ac-tually, the whole area in the vicinity of the origin is badly described by the soft-collinearapproximation but its contribution is anyway negligible at our accuracy. For a soft emis-sion which is collinear to the primary parton with positive rapidity we can write

ξ = − ln

[tan

(θ

2

)]' − ln

θ

2=⇒ θ = 2e−ξ ,

2kTQ

= x sin θ ' xθ =⇒ x =kTQeξ ,

(6.1.1)

Because of conservation of energy we have that the energy fraction of any emission x hasto be less than one. This implies

lnkTQ

< −ξ . (6.1.2)

129


Since in our case the primary partons are back-to-back, this boundary is symmetric withrespect to the ordinate axis, as shown in Fig. 6.1. This means that only the region belowthe two bisectors is kinematically viable. Hard collinear radiation lies in the vicinity ofthese lines while soft and large-angle emissions are close to the ordinate axis. Similarly,rays parallel to the bisectors are loci of constant energy.

The remaining boundaries on the shaded area in Fig. 6.1 are given by the specific formof the observable. We are interested in the mass of one jet, thus we only use the right sideof the ξ − ln kT

Qplane. The limit imposed by the observable in the soft-collinear limit is

ρ ' xθ2 =⇒ lnkTQ

= ξ + lnρ

4, (6.1.3)

which is a line perpendicular to the bisector whose intercept depends on the value of ρ.The situation for jet mass is depicted in Fig. 6.2, where we also included the cut on theangle of the emission given by the clustering condition. Since we are using the C/Aalgorithm, this is just a vertical line. In fact, in the small-R limit we are not concernedwith large-angle radiation and thus we can ignore this limit and implicitly assume that the

value of ξ at the origin is − lnR

2. The value of the integrated distribution of ρ is then

proportional to the shaded area in Fig. 6.2. On the other hand, the value of the differentialdistribution is proportional to the length of the line of fixed ρ.

6.1.2 Limits of perturbative calculations

When including one-loop running-coupling effects in our resummed spectra we willget terms containing ln (1− λ), where

λ ≡ b0a(ωJR)L , (6.1.4)

where ωJ is the energy of the jet, its size is R and L is a logarithmic contribution tojet mass; b0 ≡ πβ0, and in turn β0 was defined in Eq. (2.3.4). Our results will then beperturbatively sensible as long as λ < 1. We can make sure this condition is satisfied byputting a cut on the argument of the running coupling, which is then a cut on transversemomentum of the leading emission with respect to its parent. Let us name this cut µNP.We can estimate the largest mass which is on the transition from the perturbative to thenon-perturbative domain by using what we learned about the various taggers in the lastchapter.

Let us consider an emission i just above the non-perturbative threshold, ωiθi = µNP.The plain jet mass is m2 = ωiωJθ

2i = µNPωJθi. Then the largest value of m is the one

when θi = R, so thatm2 = µNPωJR . (6.1.5)

130

6.1. PRELIMINARIES

ln(ρ/4)

-ln(R

/2)

ln(k

T/Q

)

ξ

Figure 6.2: LKD for the integrated distribution of plain jet mass. The vertical line is thecut on the emission angle induced by the clustering condition of the C/A algorithm.

It is worth noticing that this grows with the energy of the jet (its transverse momentum inboost-invariant coordinates), which means that even masses much larger than the hadroni-sation scale can still be dominated by non-perturbative effects. For example, a reasonablechoice of values for LHC physics, like µNP = 1 GeV, R = 1 and ωJ = 3 TeV, givesm = 55 GeV, dangerously close to the electroweak scale! If we trim the jet, we need tosusbtitute R with Rsub in the last equation. This gives a smaller threshold mass, but stillgrowing linearly with the jet energy. This line of reasoning would give the same answeralso for pruning, though in this case the fraction of events in the vicinity of the thresholdand the interplay between real and virtual contributions is very different from plain jetmass, and thus a proper estimate (which we do not give here) might not be quite close toEq. (6.1.5).

For the MDT tagger, non-perturbative effects can influence the magnitude of the WBeffect. If the wrong branch is followed, then the threshold mass scale is still the same as in

131


Eq. (6.1.5). However, one has to bear in mind that the WB effect is phenomenologicallyminor, and thus the impact of non-perturbative physics might be reduced. This of courseis not a problem for the mMDT. In this case, since we have an explicit cut ycut on theenergy of the emission, we get µ2

NP = (ycutωJθi)2. Then the threshold mass is given by

m2 = ycutω2Jθ

2i =

µ2NP

ycut. (6.1.6)

This is independent of the energy of the jet. It is also kept close to the QCD scale. TakingµNP = 1 GeV and ycut = 0.1, we get m = 3 GeV.

In the following we will test the accord between our analytical calculations and MCdata in the same kinematical range we used before, i.e. 10−5 < ρ < 1. In order to doso, all non-perturbative modelling in the event generator (chiefly the underlying eventand hadronisation) will be turned off; on the other hand, we need a prescription to get asensible analytical result beyond the non-perturbative threshold. Our choice is inspiredby analyses like [153] which suggest an infrared fixed point for QCD in Landau gauge,a claim which is also supported by results from lattice gauge theory [154]. We will thenassume no running of the coupling at all below the scale µNP, which will be set at 1 GeV.The typical integral over an emission will then become∫

dx

x

dθ2

θ2αS(xθωJ) −→

∫dx

x

dθ2

θ2αS(xθωJ)Θ (xθωJ − µNP)

+

∫dx

x

dθ2

θ2αS(µNP)Θ (µNP − xθωJ) .

(6.1.7)

6.1.3 Details of MC generation and jet reconstruction

Throughout this chapter we compare our analytical results to samples generated usingHerwig++ 2.5.0. The reference process is qq → qq scattering in proton-proton collisionsat 14 TeV.

Jets with pT,J ≥ 3 TeV are generated, and the average of the mass distributions ofthe hardest and second-hardest jets is compared to analytical results. These results areobtained for jets with a fixed value of ωJ . The sensibility of the comparison is ensuredby the fact that the pT,J spectrum of jets generated by Herwig++ is steeply falling. As aresult, the pT,J of almost all jets is very close to the generation cut. In addition to this,the hardness of the cut ensures that the rapidity of all jets is small, making pT,J and ωJinterchangeable.

All samples are parton-level and do not include any underlying event or pile-up. Jetsare reconstructed using the C/A algorithm as implemented in FastJet 2.4.2 with R = 0.8.

The plots presented in the rest of the chapter are made using these samples and com-plement the results presented in [152], where the MC jet-mass distributions used for

132

6.2. PLAIN JET MASS

comparisons with analytical data were generated by Pythia 6 with different choices ofparameters.

6.2 Plain jet mass

As before, we start from what is already known about plain jet mass to set the notationand show the basics on which we will build the results for the various taggers.

It is useful to define

D (ρ) ≡ CF

∫ R2

0

dθ2qg

θ2qg

∫ 1

0

dxg Pgq(xg) a(xgθqgωJ) Θ(xgθ2qg − ρ) , (6.2.1)

which is the opposite of the LO contribution to plain jet mass for quark-initiated jets withthe running of the coupling properly taken into account. If we set

ρ′ ≡ xgθ2qg , (6.2.2)

the previous equation can be written in a more instructive way as

D(ρ) = CF

∫ R2

ρ

dρ′

ρ′

∫ 1

ρ′/R2

dxPgq(x) a(ωJ√xρ′). (6.2.3)

Then the integrated distribution for plain jet mass at NLL accuracy in the exponent1 is

Σ(ρ) = e−D(ρ) · e−γED′(ρ)

Γ [1 +D′(ρ)]· C(ρ) . (6.2.4)

The first factor is double-logarithmic and accounts for the Sudakov suppression of emis-sions that would induce a jet mass greater than ρ. These emissions lie in the shadedregion in Fig. 6.3. The second factor includes the single-logarithmic effect of multipleemissions on the jet mass. They depend on D′(ρ) ≡ ∂LD(ρ), where L ≡ ln R2

ρ. γE is the

Euler–Mascheroni constant

γE ≡ limn→+∞

(n∑k=1

1

k− lnn

), (6.2.5)

and Γ(x) is the usual Gamma function

Γ(x) ≡∫ +∞

0

dt tx−1e−t . (6.2.6)

The last factor in Eq. (6.2.4) is also single-logarithmic and includes effects due to thefinite size of the jet: non-global logarithms and clustering logarithms. If we used the anti-

1This requires taking into account either the running of the coupling in Eq. (6.2.3) at two-loop in theCMW scheme [155], or equivalently the two-loop cusp anomalous dimension.

133


ln(ρ/4)

ln(k

T/Q

)

ξ

Vetoed emissionsCon

stan

t ρ

Figure 6.3: LKD for plain jet mass. Emissions in the shaded area are vetoed and give riseto the Sudakov form factor. Single-logarithmic contributions come from near the line ofconstant mass ρ and the left edge of the shaded region.

kT algorithm we would have had no clustering logarithms. Non-global logarithms couldthen be resummed in the large-N limit, although very recent work seems to have gonebeyond that approximation [94].

We can get a clearer feeling of the shape of the differential plain jet-mass distributionif we consider the fixed-coupling approximation of the first factor of Eq. (6.2.4) to get

ρ

σ0

dσ

dρ' aCF

(lnR2

ρ− 3

4

)e−aCF

(12

ln2 R2

ρ− 3

4ln R2

ρ

). (6.2.7)

This equation describes a linear growth in ln R2

ρas ρ decreases, which is then suppressed

by the Sudakov coefficient in the exponent. We can also estimate the position of the peak

134

6.3. TRIMMING

of the distribution. In general we have an integrated distribution of the form

Σ(ρ) ≡ exp(AL2 +BL+ C

), (6.2.8)

where L ≡ ln R2

ρand A, B and C are independent of ρ. The differential distribution is

ρ

σ0

dσ

dρ= (2AL+B) exp

(AL2 +BL+ C

). (6.2.9)

Then the position of the peak LP is given by

LP = − B

2A+

1√−2A

=1√aCF

+O(1) . (6.2.10)

6.3 Trimming

6.3.1 From fixed-order to resummation

We can write the result at LO of the previous chapter, Eq. (5.3.3), in a more instructivemanner:

ρ

σ0

dσ

dρ

trimmed,LO

= aCF

[lnR2

ρΘ(ρ− fcutR

2)

+ ln1

fcutΘ(fcutR

2 − ρ)

+ lnfcutR

2sub

ρΘ(fcutR

2sub − ρ

)− 3

4

].

(6.3.1)

We can visualise this result with the help of the LKD in Fig. 6.4. There are three distinctregions in which the distribution behaves differently:

1. For ρ > fcutR2 we have just plain mass: the triangular shape of the vetoing region

means that both width and height increase as ρ decreases. This translates into a sin-gle logarithm depending on ρ in the differential distribution and a double logarithmin the integrated one.

2. The region fcutR2sub < ρ < fcutR

2 has a rectangular shape, as a consequenceonly one of its dimensions increases as ρ decreases. This means we get a single-logarithmic integrated distribution and a constant differential logarithmic distribu-tion.

3. When ρ < fcutR2sub we have another triangular region, and the distribution again

behaves as for plain jet mass.

As already pointed out in the last chapter, no additional complications arise at NLO.it is then straightforward to extend the LO result to all orders. The best way to do this is

135


ln(k

T/Q

)ξ

ρ = f cu

tR2

ρ = f cu

tR2

sub

xg = fcut

θ qg

= R

sub

ln(k

T/Q

)

ξ

ρ = f cu

tR2

ρ = f cu

tR2

sub

xg = fcut

θ qg

= R

sub

ln(k

T/Q

)

ξ

ρ = f cu

tR2

ρ = f cu

tR2

sub

xg = fcut

θ qg

= R

sub

Figure 6.4: LKD for trimmed jet mass for three different values of ρ, represented by theposition of the thick black line. Emissions in the shaded area are vetoed and give rise tologarithmic enhancements as described in the text.

to use the momentum fraction of emission i

xi ≡ωiωJ

, (6.3.2)

and its contribution to the total, normalised jet mass

ρi ≡ xiθ2qi . (6.3.3)

Then the integrated distribution (in a fixed-coupling approximation) is

Σtrimming(ρ) =+∞∑n=0

anCnF

n!

n∏i=1

∫ R2

0

dρiρi

∫ 1

0

dxiPgq(xi)Θ(xiR

2 − ρi)

×[Θ (ρ− ρi) + Θ (ρi − ρ) Θ (fcut − xi) Θ

(ρi − xiR2

sub

)− 1],

(6.3.4)

where at our accuracy it is possible to use the approximation

Θ(O −∑i

Oi) '∏i

Θ (O −Oi) , (6.3.5)

for a generic variable O. This is necessary to write the total contribution of n emissionsin a factorised form and get the sum between square brackets in Eq. (6.3.4). Of the termsthat make up this sum, the third one corresponds to virtual corrections, while the other twoencode the trimming procedure: all emissions with ρi < ρ contribute to the observable,as well as all emissions with ρi > ρ that get trimmed away (recall that from the definition

136

6.3. TRIMMING

of ρi we get that ρi > xiR2sub is equivalent to θqi > Rsub). For all values of ρ, the sum of

the terms in square brackets is -1 in the shaded region of Fig. 6.4 and 0 elsewhere. Thestep function in the first line of Eq. (6.3.4) follows from the definition of ρi.

Eq. (6.3.4) can then be seen as the series expansion of an exponential, and thus it ispossible to write

Σtrimming(ρ) = exp

−D

[max

(fcutR

2, ρ)]−Θ

(fcutR

2 − ρ)S(ρ, fcutR

2)

− CFΘ(fcutR

2sub − ρ

) ∫ fcutR2sub

ρ

dρ′

ρ′

∫ fcut

ρ′/R2sub

dx

xa(ωJ√xρ′)

,

(6.3.6)

where D(ρ) was defined in Eq. (6.2.3) and

S(a, b) = CF

∫ b

a

dρ′

ρ′

∫ 1

fcut

dxPgq(x)a(ωJ√xρ′). (6.3.7)

It is clear that D(ρ) is associated with double logarithms and triangular regions in LKDs;S(a, b) corresponds to single logarithms in ρ and rectangular regions in LKDs. The lastterm in Eq. (6.3.6) is double-logarithmic in ρ and it is associated with the additionaltriangular region between Rsub and fcut in Fig. 6.4.


We are now in the position to test Eq. (6.3.6) against Herwig++. In both cases wehave used R = 0.8 and Rsub = 0.2. The outcome for two values of fcut are shown inFig. 6.5. Both Eq. (6.3.6) and Herwig++ consistently show three phenomenologicallydistinct regions:

1. For ρ > fcutR2 we have a linear rise of the distribution, proportional to ln R2

ρ

2. In the region fcutR2sub < ρ < fcutR

2 the distribution is almost constant: here wehave just a single-logarithmic contribution to the mass distribution. The height ofthis plateau increases as fcut decreases, because of the ln 1

fcutcoefficient.

3. As soon as ρ < fcutR2sub the linear rise starts again, soon to be suppressed by a

Sudakov form factor, i.e. the last term in Eq. (6.3.6).

137


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Analyti s , f ut

= 0.03

Herwig++ , f ut

= 0.03

Analyti s , f ut

= 0.15

Herwig++ , f ut

= 0.15

Figure 6.5: Comparison of trimmed mass distributions as obtained from Herwig++ andEq. (6.3.6) for Rsub = 0.2 and different values of fcut. The ρ value corresponding tomJ = 125 GeV (the approximate Higgs mass) and ωJ = 3 TeV is denoted by a dashedline for phenomenological reference. Event generation and jet reconstruction were carriedout as described in Subsec. 6.1.3. The transition points and all general features of thedistributions are confirmed.

We can also get an estimate of the peak position by writing Eq. (6.3.6) in a fixed-coupling approximation:

Σtrimming(ρ) ' exp

− aCF

2

[−3

2lnR2

ρ+ Θ

(ρ− fcutR

2)

ln2 R2

ρ

+ Θ(fcutR

2 − ρ)(

2 lnR2

ρln

1

fcut− ln2 1

fcut

)+Θ

(fcutR

2sub − ρ

)ln2 fcutR

2sub

ρ

].

(6.3.8)

The position of the peak LP is then given by

LP = − B

2A+

1√−2A

=1√aCF

− 2 lnRsub

R+O(1) . (6.3.9)

Note that we get back the position of the peak for plain jet mass if we set Rsub = R. If weplug LP back into Eq. (6.2.9) we immediately see that the height of the peak, unlike itsposition, depends heavily on the value of fcut. In particular, a smaller value of fcut impliesa greater Sudakov suppression of emissions between fcutR

2 and fcutR2sub. These results

138

6.4. PRUNING

ln(k

T/Q

)

ξ

ρ =

z cutR2

ρ =

z cutη

xg = z

cut

θqg =

η

ln(k

T/Q

)

ξ

ρ =

z cutR2

ρ =

z cutη

xg = z

cut

θqg =

η

ln(k

T/Q

)

ξ

ρ =

z cutR2

ρ =

z cutη

xg = z

cut

θqg =

η

Figure 6.6: LKD for pruned jet mass for three different values of ρ, represented by theposition of the thick black line. The dashed portion of this line represents I-pruning, whilethe solid part corresponds to Y-pruning. An emission on the red line sets η, and thus D,but then gets pruned away because of its softness, leading to the possibility of gettingan I-pruned jet. Emissions in the shaded area are vetoed and give rise to logarithmicenhancements as described in the text.

are in accord with those illustrated in Fig. 6.5.

6.4 Pruning

6.4.1 Patterns in the pruning procedure

We start our discussion of resummation for pruning by reporting here the leading orderresult. By solving the integrals in Eq. (5.4.8) in the small-zcut limit and taking into accounthard-collinear radiation we get

1

σ0

dσ

dρ=aCFρ

[ln

1

zcutΘ(zcutR

2 − ρ)

+ lnR2

ρΘ(ρ− zcutR

2)− 3

4

]. (6.4.1)

It is clear that at this order pruning does a good job in flattening out the backgroundspectrum of QCD jets. On the other hand, as we have already seen in the last chapter,things get more complicated at NLO, where I-pruned and Y-pruned components arise.We can understand how they are linked to an all-order formulation of the pruned jet-massdistribution by looking at the corresponding LKD diagram in Fig. 6.6. The left panelcorresponds to ρ > zcutR

2. Here nothing very interesting can happen, we only have Y-pruning and the result is the same as plain jet mass and trimming. In the central panelwe see a richer structure: an emission on the red line dominates the ungroomed jet mass

139


η and thus sets the pruning radius D. For this emission we have η ' x′θ′2. Then sincex′ < 1, this emission will always be at an angle larger than D. The interesting case iswhen it is pruned away, i.e. x′ < zcut. The final pruned mass ρ is set as usual by anemission along the thick line, with ρ ' xθ2. We have I-pruning when θ2 < η. In thiscase x > ρ

η. On the other hand, we have Y-pruning when θ2 > η. In this case, to avoid

this emission to be pruned away, we must set x > zcut. Then we get zcut < x < ρη. The

two cases are represented in Fig. 6.6 by a dashed line and a solid line respectively. In thepanel on the right there is no Y-pruning. This is because any emission on the thick linehas to have an angle which is smaller than η, or otherwise be pruned away for being toosoft. On the other hand, even emissions with x < zcut can contribute to I-pruning as theyall have θ2 < η. Since η < zcutR

2, this last contribution to I-pruning kicks in for valuesof ρ smaller than z2

cutR2.

As we did before, we can now construct a resummed formula by looking at the shapesin the LKD diagram and assign to each vetoed region the corresponding exponentiatedfunction D(ρ) or S(a, b), as defined in Eqs. (6.2.3) and (6.3.7) respectively1.

6.4.2 Y-pruning

Let us start from the simpler case of Y-pruning. The resummed formula for ρ < zcutR2

reads

ρ

σ0

dσ

dρ

Y-prune

= CF e−D(ρ)

∫ 1

zcut

dxPgq(x)a(ωJ√xρ)

+ C2F

∫ min(zcutR2,ρ/zcut)

ρ

dη

η

[e−D(η)

∫ zcut

η/R2

dx′

x′a(ωJ√x′η)]

× e−S(ρ,η)

∫ ρ/η

zcut

dxPgq(x)a(ωJ√xρ).

(6.4.2)

The first line comes from configurations where η ' ρ. In this case we get a triangularveto region, hence the exponentiated D(ρ). The term on the second and third line corre-sponds to an emission that sets a large η and then gets pruned away (x′ < zcut). As wehave just seen, to get Y-pruning we need η < ρ/zcut. Additionally, η needs to be smallerthan zcutR

2, otherwise the corresponding emission could not be pruned. The exponen-tiated D(η) function corresponds to the small triangle on the left of the central panelin Fig. 6.6, while the exponentiated S(ρ, η) function arises from the rectangular regionabove ρ. Additionally, in the factor between square brackets we integrate 1/x′ instead ofthe full splitting function because that would contribute just a power correction in zcut.In this approximation, the above expression captures terms proportional to anL2n−1 andanL2n−2 in the integrated distribution, where L ≡ ln R2

ρ.

1It is assumed that fcut in the definition of S(a, b) is substituted with the appropriate cutoff for the taggerin question, in this case zcut.

140

6.4. PRUNING

We can write the last result in a fixed-coupling approximation, neglecting terms withmore than one power of ln 1

zcut. For ρ < z2

cutR2 we obtain

ρ

σ0

dσ

dρ

Y-prune

' e−D(ρ)

[(eaCF ln 1

zcutln R2

ρ − 1

)1

ln R2

ρ

− 3

4aCF

]. (6.4.3)

By further assuming ln 1zcut

ln R2

ρ 1 we can write

ρ

σ0

dσ

dρ

Y-prune

' aCF e−D(ρ)

(ln

1

zcut− 3

4

), (6.4.4)

which is clearly consistent with the fixed-order results.

6.4.3 I-pruning

We can now move on to I-pruning. By looking at the last two panels of Fig. 6.6, it ispossible to write down the resummed distribution

ρ

σ0

dσ

dρ

I-prune

= C2F

∫ zcutR2

ρ

dη

ηe−D(η)

∫ zcut

η/R2

dx′

x′a(ωJ√x′η)

× e−S(ρ,η)

∫ 1

ρ/η

dxPgq(x)a(ωJ√xρ)

Θ

(ρ

η− zcut

)+ Θ

(zcut −

ρ

η

)exp

[−CF

∫ zcutη

ρ

dρ′

ρ′

∫ zcut

ρ′/η

dx′

x′a(ωJ√ρ′x′)]

.

(6.4.5)

As before, we need an emission that sets the unpruned jet mass to η and then gets prunedaway. This implies η < zcutR

2. The first line gives the distribution for η, and correspondsto the small triangle above this emission in the central panel of Fig. 6.6. On the secondline we have the same Sudakov suppression as before, but this time x is integrated fromρ/η to 1 because we are taking into account the shaded region above the dashed linein the central panel of Fig. 6.6. The last term, which is non-zero only for zcutη > ρ,is represented by the panel on the right in Fig. 6.6. Here we only have I-pruning, andthe additional Sudakov factor comes from the triangular vetoed region of emissions withenergy less than zcut that are not pruned because they have an angle greater than η. Asusual we have omitted the full splitting functions where the upper limit of integration overthe energy is not 1, i.e. we are ignoring power corrections in zcut. Then we control termsproportional to anL2n and anL2n−1 in the integrated distribution, because we have onepower of L more than for Y-pruning.

We can understand Eq. (6.4.5) better if we restrict ourselves to taking into account itsmost divergent part in the limit ρ→ 0. In this approximation we can consider a fixed and

141


ignore all terms containing zcut. The result is

ρ

σ0

dσ

dρ

I-prune

' a2C2F

∫ R2

ρ

dη

ηlnR2

ηlnη

ρe− 1

2aCF

(ln2 R2

η+ln2 η

ρ

)

' aCFL

2e−

12aCFL

2

+

√aCFπ

4e−

14aCFL

2

(−2 + aCFL2) Erf

(√aCFL

2

),

(6.4.6)

where Erf(x) is the Error function

Erf(x) ≡ 2√π

∫ x

0

e−t2

dt . (6.4.7)

The fixed-order result can be easily recovered by expanding Eq.(6.4.6) in powers of a.

6.4.4 Interplay of Y- and I-pruning

It is interesting to see what happens in the region z2cutR

2 < ρ < zcutR2. Here both

Y-pruning and I-pruning can happen and we thus have to sum up their contributions to thefinal distribution. In this region, the upper limit on the integration over η in Eq. (6.4.2)is zcutR

2; since ρ > z2cutR

2 and η < zcutR2, then surely ρ > zcutη, which means that

only the first addend in the curly brackets in Eq. (6.4.5) contributes; finally, the integralsover x simply sum up to a single integral from zcut to 1. For simplicity, we will stick to afixed-coupling approximation. The sum can then be written as

ρ

σ0

dσ

dρ

Prune

' aCF

∫ 1

zcut

dxPgq(x)

(e−D(ρ) + aCF

∫ zcutR2

ρ

dη

ηe−D(η)−S(ρ,η)

∫ zcut

η/R2

dx′

x′

).

(6.4.8)We can use a property of the Sudakov factors which is valid in the small-zcut limit:

D(ρ) = D(zcutR2) + S(ρ, zcutR

2) +aCF

2ln2 zcutR

2

ρ. (6.4.9)

Then we get

ρ

σ0

dσ

dρ

Prune

' aCF e−D(zcutR2)−S(ρ,zcutR2)

(ln

1

zcut− 3

4

). (6.4.10)

We can see that in the region z2cutR

2 < ρ < zcutR2 the resummed distribution is simply

obtained by adding the Sudakov suppression to the leading order result. The reason forthis can be seen in the central panel of Fig. 6.6, where the position of the pruned emissionthat sets η only modifies the relative contributions of Y- and I-pruning, but not their sum.We will see in Sec. 6.6 that all taggers behave in a similar way in this region, provided wemap the values of the parameters correctly.

142

6.4. PRUNING

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Analyti s , Pruning

Herwig++ , Pruning

Analyti s , I-pruning

Herwig++ , I-pruning

Analyti s , Y-pruning

Herwig++ , Y-pruning

Figure 6.7: Comparison of pruned mass distributions as obtained from Herwig++ andEqs. (6.4.2) and (6.4.5) for zcut = 0.1. The plots show both total distributions and Y-and I-pruned components separately. The ρ value corresponding to mJ = 125 GeV (theapproximate Higgs mass) and ωJ = 3 TeV is denoted by a dashed line for phenomeno-logical reference. Event generation and jet reconstruction were carried out as describedin Subsec. 6.1.3. The transition points and all general features of the distributions areconfirmed.


We compare our results with Herwig++ in Fig. 6.7. Again the agreement between thetwo is good. The transition points occur where they are expected, especially the one forρ = zcutR

2. The other transition (at ρ = z2cutR

2) is much smoother, as the new phase space(the small triangle in the third panel in Fig. 6.6) opens up slowly. Below this transitionpoint, the Sudakov peak from I-pruning is clearly visible. An estimate from Eq. (6.4.6)suggest that the position of the peak should be somewhere around LP = 2/

√aCF . This

is reasonably close to the MC result, though it is worth recalling that the position of theSudakov peak is strongly affected by subleading effects that we did not include.

143


0

0.05

0.1

0.15

0.2

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

MDT

WB

Figure 6.8: Events generated by Herwig++ show that the WB effect is in practice verysmall, and kicks in only for very small values of ρ . R2y3

cut, which with our choice ofparameters is of order 10−3. The tagger follows a wrong branch every time it chooses aprong with m2 + p2

T smaller than its partner’s. We used ycut = 0.1 and µ = 0.67 for theMDT tagger. The rest of the event generation follows that described in Subsec. 6.1.3.

6.5 MDT and mMDT

6.5.1 From fixed-order to resummation

As usual it is useful to start by reporting the LO result in the small-ycut limit:

ρ

σ0

dσ

dρ= aCF

[ln

1

ycutΘ(ycutR

2 − ρ)

+ lnR2

ρΘ(ρ− ycutR

2)− 3

4

]. (6.5.1)

This is identical to pruning provided we replace zcut with ycut (when the parameters aresmall). It is therefore interesting to verify whether in this case the desirable propertiesfound at LO survive at higher orders. A first possible issue we encountered in the lastchapter (see Subsec. 5.5.3) is the WB effect. As we anticipated earlier though, its numer-ical impact is rather small, as can be seen in the plot from Herwig++ in Fig. 6.8. Here wecan see that very rarely the tagger follows a prong whose m2 + p2

T is smaller than its part-ner’s. However it is worth stressing that although numerically small, this effect greatlycomplicates the task of finding a resummed formula for the MDT distribution and it isundesirable from a physical point of view. Therefore we immediately move to the mMDT

144

6.5. MDT AND MMDT

described in Sec. 5.6.1. Obviously the definition is the same, with energy replaced bytransverse mass mT

mT ≡√m2 + p2

T . (6.5.2)

It is relatively easy to calculate a resummed distribution for the mMDT. For ρ > ycutR2 it

is again identical to plain jet mass. We then implicitly focus on the region ρ < ycutR2. It

is convenient to work in an angular ordered formulation. This is the inherent ordering ofthe (un)clustering sequence, since it is based on the C/A algorithm. Let us then considern emissions, with θi+i θi. We also take xn > ycut and xi < ycut for all other emissions.The unclustering procedure starts from emission 1 and discards it together with all thesubsequent ones because of the asymmetry condition until it gets to emission n. Then thejet is tagged and returned to measure its mass. The equation that describes this is

1

σ0

dσ

dρ

mMDT

=+∞∑n=1

anCnF

∫ 1

0

dxnPgq(xn)

∫ R2

0

dθ2n

θ2n

Θ (xn − ycut) δ(ρ− xnθ2

n

)×

n−1∏i=1

∫ 1

0

dxiPgq(xi)

∫ R2

0

dθ2i

θ2i

[Θ (ycut − xi)− 1] Θ (θi − θi+1) ,

(6.5.3)

where we used a fixed coupling for easier readability. Since ycut 1 we can assume thatall emissions i < n only carry away a negligible fraction of the jet energy, so that all xisare energy fractions relative to the same, original jet energy. As usual, from unitarity wecan include virtual corrections by adding a −1 contribution in the square brackets. Wecan again substitute the integration over θ2

i with an integration over ρi, which inherits thesame strong ordering: ρi+1 ρi. We can then rewrite Eq. (6.5.3) as

1

σ0

dσ

dρ

mMDT

= −+∞∑n=1

anCnF

ρ

∫ 1

ycut

dxPgq(x)n−1∏i=1

∫ R2

ρi+1

dρiρi

∫ 1

max(ycut,ρi/R2)

dxiPgq(xi)

= −+∞∑n=1

anCnF

ρ (n− 1)!

∫ 1

ycut

dxPgq(x)

[∫ R2

ρ

dρ′

ρ′

∫ 1

max(ycut,ρ′/R2)

dxPgq(x)

]n−1

.

(6.5.4)

Reinstating the running of the coupling, the integrated distribution then clearly exponen-tiates to1

ΣmMDT(ρ) = exp

[−CF

∫ R2

ρ

dρ′

ρ′

∫ 1

max(ycut,ρ′/R2)

dxPgq(x)a(ωJ√xρ′)]

= exp−D

[max

(ρ, ycutR

2)]−Θ

(ycutR

2 − ρ)S (ρ, ycut)

.

(6.5.5)

In the above expression we have included the result for ρ > ycutR2, which is plain jet

mass again. As usual, we can understand the structure of this result better if we look atthe LKD diagram for mMDT, Fig. 6.9. In this case, the geometry of the vetoed region

1Remember to substitute fcut in the definition of S(a, b), Eq. (6.3.7), with ycut.

145


ln(k

T/Q

)

ξ

ρ = y cu

tR2

xg = y

cut

Figure 6.9: LKD for mMDT. For ρ > ycutR2 we have the same situation as plain jet mass,

while for smaller values of ρ the vetoed region is simply a rectangle, which correspondsto a single-logarithmic leading divergence in ρ.

is strikingly simple: for ρ > ycutR2 we still have the same triangular region as in plain

jet mass. Below that, the triangular part "freezes" and we just have a rectangle of fixedwidth from ycut to 1 and height given by the distance between ρ and ycutR

2. Since onlyone dimension is increasing as ρ gets smaller and smaller, we expect (and we find) asingle-logarithmic leading divergence in the distribution. This is the best performanceat rejecting background so far (excluding Y-pruning), whereas divergences in the othertaggers are more severe and give rise to additional Sudakov peaks. In the fixed-couplingapproximation Eq. (6.5.5) takes this simple form:

ΣmMDT(ρ) = exp

[−aCF

(ln

1

ycutlnR2

ρ− 3

4lnR2

ρ− 1

2ln2 1

ycut

)]. (6.5.6)

This is just the exponentiation of the integrated LO result. We have included terms whichare subleading with respect to single-logarithmic accuracy, but still enhanced for smallvalues of ycut. The reason is that mass ordering, used in the derivation of Eq. (6.5.5), isvalid for all terms

(a ln2 ycut

)m(a ln ρ)n. To prove this point, let us consider two emis-

sions, 1 and 2. If x1 > ycut then there is a probability of around a ln2 ycut that x2 > ycut

and we also get θ1 > θ2 and ρ1 < ρ2, i.e. a configuration which violates mass ordering.In this case a gets unclustered first, and since its contribution to the jet mass is small, no

146

6.5. MDT AND MMDT

0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

µ = 0.1

µ = 0.2

µ = 0.4

µ = 0.67

µ = 1

Figure 6.10: MC study of the impact of the value of the µ parameter of the mMDTdistribution. The distributions for µ = 0.67 and µ = 1 are on top of each other. The ρvalue corresponding to mJ = 125 GeV (the approximate Higgs mass) and ωJ = 3 TeV isdenoted by a dashed line for phenomenological reference.

mass-drop happens, so that this emission gets discarded. This implies that the tagger isactually insensitive to the presence of emissions like a, which means that mass-orderingviolation can be ignored in our calculation without losing accuracy. In the context ofstrong ordering, this statement holds also for µ = 1. In this case emission a always trig-gers the mass-drop condition, but its contribution to the final jet mass is so small in mostof the phase space that it does not produce any relevant effect in the distribution.


First of all, it is important to note that our all-order results do not depend at all onthe mass-drop parameter µ. This is because we have assumed a strong ordering in ρ,so that at each unclustering step the mass-drop condition is automatically satisfied andonly the asymmetry condition needs to be checked. Of course, for finite µ there is stillthe possibility that the mass of a subjet is greater than µ times the mass of the parent.However, this would replace the collinear logarithm with lnµ. We have already shownthat we have control of single logarithms, as a consequence this effect is beyond the levelof our accuracy. Fig. 6.10 shows a simple MC study to assess the impact of powers oflnµ on the mMDT distribution.

147


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Analyti s , y ut

= 0.1

Herwig++ , y ut

= 0.1

Analyti s , y ut

= 0.13

Herwig++ , y ut

= 0.13

Analyti s , y ut

= 0.2

Herwig++ , y ut

= 0.2

Analyti s , y ut

= 0.4

Herwig++ , y ut

= 0.4

Figure 6.11: Comparison of mMDT distributions as obtained from Herwig++ andEq. (6.5.5), supplemented with finite-ycut effects (see App. C), for different values of ycut

and µ = 0.67. The agreement between the two is reasonable. The ρ value correspondingto mJ = 125 GeV (the approximate Higgs mass) and ωJ = 3 TeV is denoted by a dashedline for phenomenological reference. Event generation and jet reconstruction were carriedout as described in Subsec. 6.1.3.

Fig. 6.11 shows the full comparison of Eq. (6.5.5) with Herwig++ for different valuesof ycut and µ = 0.67. Additional finite-ycut effects have been taken into account in theanalytical predictions. Details can be found in [152]. The agreement is reasonable.

6.6 Comparison between taggers

Using the information gathered so far, we can compare all the taggers we studied andhighlight features, similarities and differences. Fig. 6.12 shows all the distributions to-gether with plain jet mass as generated by Herwig++. A consistent set of parameters waschosen. As we already noted several times, all distributions are close to plain jet mass forbig values of ρ. Then there is a plateau region where all the taggers seem to behave in thesame way. This is due to the interplay between Y- and I-pruning that makes the distribu-tion equal to mMDT in the region z2

cutR2 < ρ < zcutR

2, in accord with the MC result, oncethe parameters are tuned to ycut = zcut

1−zcut. Eq. (6.3.4) shows that also trimming behaves in

the same way in the region fcutR2sub < ρ < fcutR

2, provided we choose fcut = zcut. Smalldifferences between taggers in this region remain due to subleading effects. In particular,while in this region the distributions for trimming and pruning are relatively close to each

148

6.6. COMPARISON BETWEEN TAGGERS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Plain jet mass

Trimming

Pruning

mMDT

Alternative mMDT

R = 0.8 Rsub

= 0.2

f ut

= z ut

= 0.1

y ut

= 0.11 µ = 0.67

Figure 6.12: Comparison of MC distributions for plain jet mass, trimming, pruning andmMDT for equivalent parameters, as reported on the plot. The choice of ycut is such thatycut = zcut/(1 − zcut). For the alternative version of the mMDT described in the text wehave chosen ycut = zcut. It is apparent that all taggers behave in a similar way betweenz2


2 ' 10−1, i.e. in between the two vertical dashed lines. In par-ticular, the alternative version of the mMDT looks just the same as pruning in this region.Event generation and jet reconstruction were carried out as described in Subsec. 6.1.3.

other, the one for the mMDT is further apart. By looking at how the taggers work, thiscan be traced back to differences in the cuts on the energy of emissions. Trimming andpruning directly cut on the pT of emissions, though with small differences in normalisa-tion; the mMDT on the other hand cuts on a ratio of pT -distance to mass, and this onlyindirectly translates into a cut on the pT itself. It is possible to define an alternative ver-sion of the mMDT where ycut is compared to min(pT,J1 , pT,J2)/(pT,J1 + pT,J2). Then bychoosing ycut = zcut one sees in Fig. 6.12 that the distributions for pruning and this alter-native version of the mMDT go on top of each other in the region z2

cutR2 < ρ < zcutR

2. Itis also clear that this modification does not change the analytical results.

One of the main advantages of having an analytical understanding of jet substructuremethods is exactly this ability to relate parameters of different taggers in a meaningfulway, allowing for consistent comparisons of their performances. This is a step forwardwith respect to current studies (such as [108]) where taggers with non-equivalent choicesof parameters were compared.

Being able to describe and predict backgrounds is crucial in all physics searches atparticle colliders, and it is particularly important when the signal-to-background ratio

149


is expected to be small. Our results, merged with fixed-order calculations, represent thestate-of-the-art for analytical prediction of background in searches that involve the taggerswe have studied. Alternatively, one could use data-driven methods. This means measuringthe mass spectrum on the left and right of some expected resonance and interpolate themeasurement to get the distribution in the resonance region. It is also possible to take thebackground shape for low pT jets and use it to predict its behaviour at higher pT . In allcases it is desirable to have a background that looks as featureless as possible. Sudakovpeaks are always a complication: their position is difficult to establish beyond a certainprecision and it usually depends on pT . This, together with the fact that the Sudakovpeak could very well lie in the resonance region, really spoils the reliability of data-drivenmethods.

By looking at Fig. 6.12, one can easily see that in phenomenologically-relevant re-gions of ρ such as, for example, the Higgs mass, the background due to plain QCDjet mass is huge and dominated by a very broad Sudakov peak; trimming makes thingsslightly better, though it still develops a quite prominent Sudakov peak; pruning too ex-hibits a peak of moderate height at smaller values of ρ, which can still be a source ofproblems for searches and makes the accurate description of the background harder. ThemMDT looks very interesting from this point of view. First of all, it is free of Sudakovpeaks. Another interesting property can be seen by expanding Eq. (6.5.5) to second orderin a. For ρ < ycutR

2 this is

ρ

σ0

dσ

dρ

mMDT

= aCF lne−

34

ycut

[1 + a ln

1

ρ

(b0 − CF ln

e−34

ycut

)]. (6.6.1)

where b0 ≡ πβ0, and in turn β0 was defined in Eq. (2.3.4). We see that the dependence onρ of the spectrum is the result of two effects: the running coupling tends to make it rise asρ becomes smaller, while the exponentiation suppresses the spectrum. As already notedat the end of Sec. 5.6, it is then possible to choose ycut such that these two effects cancelcompletely, leaving a distribution which is independent of ρ. This can be seen clearly inFig. 6.11, where the optimal value for ycut = 0.13 has been chosen to be a bit above theestimate one would get from Eq. (6.6.1) to take into account finite-ycut effects. Table 6.1summarises the main features of taggers we found so far.

6.7 Taggers and the signal

Though the present work is mainly concerned with the effect of taggers on backgroundjets, it is of course necessary to understand how well they are able to reconstruct signaljets for a complete assessment of their performance. In line with [152], we present herea preliminary MC study of tagging efficiencies and significances for several taggers. Asbefore, our generator of choice is Herwig++; these results are meant to complement thosein [152], which were obtained using Pythia 6.

150

6.7. TAGGERS AND THE SIGNAL

Leading logs Transitions (ρ) LP NGLs NP

Plain mass anL2n - 1/√aCF yes µNPpTR

Trimming anL2n fcutR2sub, fcutR

2 1/√aCF − 2 ln(Rsub/R) yes µNPpTRsub

Pruning anL2n z2cutR

2, zcutR2 2/

√aCF yes µNPpTR

Y-pruning anL2n−1 zcutR2 - yes µNPpTR

MDT anL2n−1 y3cutR

2, ycutR2 - yes µNPpTR

mMDT anLn ycutR2 - no µ3

NP/ycut

Table 6.1: Table summarising all basic features of the taggers we studied. L is definedas ln R2

ρ, the leading power of L is calculated in the small-ρ limit. LP is the estimate

of the position of the peak in the differential distribution. NGLs stands for Non-GlobalLogarithms. The last column indicates the minimum estimated value of m2 below whichNon-Perturbative effects kick in.

At tree level, the efficiency εS of a tagger like the mMDT can be estimated to be

εS =

∫ 1−ycut

ycut

dxPH→qq(x) = 1− 2ycut , (6.7.1)

since PH→qq(x) = 1, which means that the decay products of the Higgs boson have a flatenergy distribution. The same result holds for pruning, ycut and zcut being interchangeablein the small-ycut limit. For trimming, the result depends on ρ: it is 1−2fcut for ρ > fcutR

2sub

and is 1 for smaller values of ρ. Obviously, no tagging at all would result in perfectefficiency at tree level. Any tagging has to happen in a mass window, and we assumethat this window is always wide enough to prevent substantial loss of mass resolution dueto the underlying event, pile-up and ISR (the width of the mass window was extensivelystudied for the MDT with filtering in [133]). All these effects also contribute correctionsto our estimate of the signal efficiency.

We generated signal samples of full (i.e. including the underlying event and hadro-nisation) WZ events with Herwig++1 to test Eq. (6.7.1). We set the Z to decay lep-tonically and the W to decay hadronically. The tagger is applied to the hardest jet inthe event, which is tagged if its final mass mJ is within 20 % of the nominal W mass,i.e. 64 GeV < mJ < 96 GeV. Fig. 6.13 shows the results as a function of the pT genera-tion cut on theW boson. They can be seen to be in reasonable agreement with Eq. (6.7.1).Residual differences are due to NLO effects and the fact that the tagging efficiency of Wbosons depends on their polarisation, which in turn is influenced by their pT .

The conclusion one can draw from this is that there are no big difference in the varioustaggers’ efficiencies2, and thus their performance is driven by their effect on background,which is an a posteriori justification for our focusing on jets induced by pure QCD pro-

1We switched to Herwig++ 2.6.3 for this last study to take advantage of a more advanced model of theunderlying event.

2Though we do not show it in Fig. 6.13, we verified that the efficiency of the MDT does not exhibit anyrelevant deviation from the one of the mMDT.

151


0

0.2

0.4

0.6

0.8

1

1.2

1.4

300 30001000

ε S

pT generation ut (GeV)

Plain

Trimming

Pruning

Y-Pruning

mMDT

Filtering (3 subjets)

Figure 6.13: Signal efficiencies of several taggers (including plain jet mass) as a functionof the transverse momentum cut on W generation. Our choice of parameters is fcut =0.05, Rsub = 0.3, zcut = 0.1 and ycut = 0.11. The filtering radius is Rfilt = 0.3 and wekept the 3 hardest subjets, as described in [25]. The results for taggers other than filteringare in reasonable agreement with Eq. (6.7.1), while plain jet mass and filtering are bothdominated by NLO and non-perturbative effects and perform poorly.

cesses.On the other hand, NLO and non-perturbative effects seem to impact very heavily on

the performance of plain jet mass, which monotonically decreases as the pT cut on thegenerated W boson increases. Using filtering [25] alone does not drastically improve theefficiency either, especially at high pT .

To conclude this introductory study on the effects of taggers on signals, we plot thesignificance S of the taggers we studied (including plain jet mass) as a function of the pTgeneration cut. The significance S is defined as

S ≡ εS√εB

, (6.7.2)

and quantifies the predominance of a signal peak over a background fluctuation. Fig. 6.14shows the results obtained by using samples generated by Herwig++.

Let us give some comments about the significance associated with each tagging pro-cedure. First of all, note that in the quark case all the taggers we analysed in the previouschapters have comparable behaviours up to around 800 GeV. After that, the significancefor trimming starts to fall, and that for pruning ceases to increase. This can be tracedto the Sudakov peaks that this two taggers develop as ρ decreases, the one for trimming

152

6.7. TAGGERS AND THE SIGNAL

0

1

2

3

4

5

6

7

300 30001000

ε S/√

ε B


0

1

2

3

4

5

6

7

300 30001000

ε S/√

ε B


Plain

Trimming

Pruning

Y-Pruning

mMDT


Plain

Trimming

Pruning

Y-Pruning

mMDT


Figure 6.14: Significance of signalW bosons with quark (above) and gluon (below) back-grounds for several taggers and plain jet mass as a function of the generation cut on trans-verse momentum. Our choice of parameters is fcut = 0.05, Rsub = 0.3, zcut = 0.1 andycut = 0.11. The filtering radius is Rfilt = 0.3 and we kept the 3 hardest subjets, as de-scribed in [25]. These results, obtained from Herwig++, confirm those reported in [152],which are based on Pythia 6. For comparison, we also added plain jet mass and filtering,both failing at getting an acceptable signal significance. More comments on these plotscan be found in the main text.

153


happening earlier and being more pronounced than that for pruning. These effects aremitigated in the gluon case, because of the additional background suppression providedby swapping CF with CA which reduces the height of the Sudakov peaks (see App. B)1.

It is surely worth mentioning the good performance of Y-pruning, which takes advan-tage of the double-logarithmic suppression of background in the limit ρ→ 0. This is evenmore apparent in the gluon case, where the suppression is enhanced by the colour factor,as described above and in App. B. A residual problem in Y-pruning is that the classifica-tion of a pruned jet as containing a single or double prong is affected by the underlyingevent and pile-up contamination. It is desirable to develop a tagger which enjoys the samedouble-logarithmic suppression of background without suffering from this flaw.

Surprisingly, the significance of filtering alone is even worse than plain jet mass athigh pT . This effect still needs investigation.

1As expected, we found no relevant differences between the MDT and the mMDT, and that is why weonly show the latter in Fig. 6.14.

154

CHAPTER

SEVEN

CONCLUSIONS AND OUTLOOK

In this thesis we studied jet physics, focusing on jet rates and jet substructure. Jetsubstructure is a relatively new and prolific field in jet physics, and could hold the keyto a much more efficient use of jet reconstruction in the description of QCD backgroundand searches for new particles decaying hadronically. Given the attractiveness of theseprospects, it is perhaps not surprising that a great number of substructure techniques(which we collectively name taggers) has flourished in the last few years. However, onlyMonte Carlo studies, sometimes supported by rough analytical estimates, have been usedto test their performance. The lack of an analytical understanding of the behaviour ofthese taggers led to a general difficulty in comparing, in order to establish connectionsbetween the various parameter spaces and optimise their implementation.

Filling the gap in the analytical understanding of these taggers is the main objectiveof this thesis. As a warm-up, we turned our attention to a specific class of observables, jet

rates. These are classic tools of jet physics, and their study is very helpful in getting togrips with the analytical tools exploited throughout the whole body of research reported inthis thesis. Jet rates are sensitive to the algorithm used for jet reconstruction in principle,and we therefore extensively studied the effects of using different incarnations of bothexclusive (i.e. those who assign every single particle in the final state to some jet) andinclusive (which throw away soft particles from the event) algorithms.

We started with a fixed-order calculation of logarithmically-enhanced terms (NLL in

the expansion) up to NLO in e+e− annihilation into qq pairs, comparing our results step-by-step with EVENT2. Our main finding is that at this level of accuracy all inclusivealgorithms give the same jet rates, in accordance with newly published work [1]. Usingthese results, we built up resummed jet rates, obtaining again the same results as in theliterature, which were also confirmed by Monte Carlo runs using Herwig++. We closedthe chapter by turning our attention to two cases where the way in which one reconstructsthe jets prevents the corresponding jet rates from being resummable. Firstly the veryfamous JADE algorithm, which inappropriately clusters together soft gluons which arealmost back-to-back. The other is a specifically-designed counterexample: an exclusiveversion of the anti-kT which uses the kT distance to check when to stop the clusteringsequence (choosing the anti-kT for this task as well would result in an infrared-unsafe

155

CHAPTER 7. CONCLUSIONS AND OUTLOOK

observable). This leads to an excessive tendency to cluster correlated emission to the hardcore of the jet rather than its parent in the parton shower, thus spoiling the possibility ofany resummation at NLL-in-the-expansion accuracy.

It is interesting to note that jet rates could actually be used as a substructure tool, forexample for counting the number of jets inside a fat jet. One would expect this numberto depend on whether the fat jet comes from a hadronic decay of some heavy resonanceor from QCD radiation. Some work in this sense has been already carried out (see [143],[144] and [156]), and it should be possible to build on that for future progress.

From Chap. 5 on we undertook a study of proper substructure taggers, namely trim-ming, pruning and MDT. The observable of choice was the distribution of ρ, (the jet massnormalised to its transverse momentum) for QCD (i.e. background) jets. Being able toaccurately describe the mass spectrum of background jets is crucial to any search forparticles, so it is important to see whether any of these taggers is able to suppress thebackground and aid in the derivation of analytical predictions.

In Chap. 5 we focused on fixed-order calculations in e+e− annihilation into qq pairs.We pushed the logarithmic accuracy of our results as far as the complexity of each taggerallowed for. Bear in mind that any tagger is supposed to reduce the logarithmic enhance-ments to the jet-mass spectrum, and as a consequence there is not much point in re-doing afull fixed-order calculation in regions of phase space where the tagger does not do its job:the plain jet-mass distribution is already relatively well-known and confirming its featuresis not among the aims of this thesis. We discovered that all taggers behave in a similar wayat LO, but they are all very different from each other already at NLO. Trimming gives riseto a distribution which looks just as singular as that corresponding to plain jet mass. TheNLO result for pruning is made up of two components: I-pruning, where the jet mass isdominated by a single hard prong, and Y-pruning, where two semi-hard partons make upthe pruned jet. It is easy to amend the pruning procedure to reject I-pruned jets and thusreduce the logarithmic enhancement of the distribution. MDT looks more promising fromthe start, but it is affected by the Wrong-Branch (WB) effect, which results in the taggerfollowing a soft branch of correlated radiation inside the jet. Although the phenomeno-logical impact of this effect is tiny, it is fairly easy to define a modified MDT (mMDT)which leads to an effectively single-logarithmic distribution in ρ, making it much easier toget a reliable description of background. All the results of this chapter were successfullycompared to EVENT2.

In Chap. 6 we used the results from the previous chapter to get resummed distribu-tions for quark jets from proton-proton collisions and compare them to Herwig++. Boththe analytical predictions and the MC data agreed in confirming what we anticipated inthe previous chapter: trimmed jets are basically the same as plain jets with a smaller ra-dius; pruning still features a Sudakov peak of moderate height and, with a careful choiceof parameters, the mMDT background can be made to be flat. This is of course a great ad-vantage from the point of view of predicting QCD background shapes in searches for newparticles. Terms which are enhanced by logarithms of ycut, the cutoff parameter relevant

156

to mMDT, and subleading in ρ are also under control. It should be possible to also controlthese terms in trimming and pruning, but we leave the study of this conjecture to futurework. We also confirmed the mapping between the parameters of different taggers, en-abling consistent comparisons between taggers. A preliminary study of signal efficienciesand significances is carried out in the final part of Chap. 6.

Additional insight into the signal distributions is desirable. It would also be in-teresting to try and understand the effect (if any) of choosing different algorithms forjet reconstruction and/or tagging on the performance of these substructure techniques.It is surely worthwhile to explore whether a fixed-order calculation is actually enoughto describe the behaviour of less-divergent taggers like the mMDT. Some preliminarywork on this was carried out in [152], but there is still much to do to settle this ques-tion, such as comparing the fixed-order distribution to MC data enhanced by NLO toolssuch as POWHEG [88] or MC@NLO [157] (or alternatively matching schemes such asCKKW [158] or MLM [159]).

Our analytical method should be readily extendable to a range of variables such as N -subjettiness [136], energy correlations [142], or even combinations of variables, as wasdone in [160] and [161]. Other possible candidates include pull [162], dipolarity [163]and template tagging [164].

The final goal is obviously to compare analytical results to experimental data. Exper-imental analyses are already available for many of the taggers we mentioned [112–126].The present situation seems to be favourable for the next step in jet physics: finally go-ing beyond MC simulation and testing our analytical comprehension of jet substructuredirectly to accelerator data.

157

APPENDIX

A

FROM LIE GROUPS TO COLOUR CHARGES

Here we define the fundamental concepts necessary to understand how the SU(N)

colour symmetry of QCD gives rise to the various colour factors in parton splittings.More details of the concepts borrowed from group theory can be found in e.g. [165].

Let us start by recalling some basic features of Lie groups. Each Lie group is deter-mined by a Lie algebra, i.e. a vector space whose generators T a obey the relation1:

[T a, T b

]= ifabcT c , (A.0.1)

where fabc are called structure constants and are completely antisymmetric tensors. Thenormalisation of the generators is given by

TrT aT b

≡ TRδ

ab . (A.0.2)

TR is the Dynkin index for representation R. At this point, it is an arbitrary normalisationconstant. However, the choice of index in a given representation R determines the valuesof the indices in all other representations. We will see an example of this in a moment.Two basic properties of Dynkin indices for two representations R1 and R2 are

TR1⊕R2 = TR1 + TR2 ,

TR1⊗R2 = dimR1 TR2 + dimR2 TR1 ,(A.0.3)

where dimR is the dimension of representation R. Another important parameter is thequadratic Casimir operator:

T 2 ≡ T aT a† . (A.0.4)

It can be shown that the Casimir operator commutes with all the generators. Then, bySchur’s first lemma, the Casimir operator is proportional to the identity. We will use CRto denote the constant of proportionality for a given representation R.

We can now restrict ourselves to the case of SU(N), the symmetry group of QCD. Inthe fundamental representation, the generators can be represented by theN×N Hermitian

1We will always assume a sum over repeated indices.

159

APPENDIX A. FROM LIE GROUPS TO COLOUR CHARGES

matrices taij . Recalling the dimension of SU(N), it is clear that a runs from 1 to N2 − 1.The Dynkin index is given by

Trtatb

= TF δab . (A.0.5)

If we set a = b and sum over a, we get

Tr tata = TF (N2 − 1) . (A.0.6)

The Casimir operator ist2ik = taijt

ajk = CF δik . (A.0.7)

If we now set i = k and sum over i we get

Tr tata = CFN , (A.0.8)

which implies

CF = TFN2 − 1

N. (A.0.9)

Now we want to do the same for the adjoint representation. In this case the generatorscan be represented by the (N2 − 1)× (N2 − 1) traceless Hermitian matrices F a

bc, with

F abc ≡ −ifabc . (A.0.10)

In this case the Dynkin index is given by

TrF aF b

= TAδ

ab . (A.0.11)

As before, we set a = b and sum over a:

Tr F aF a = TA(N2 − 1) . (A.0.12)

On the other hand, the Casimir operator is

F 2bc = F a

bcFacd = CAδbd . (A.0.13)

Again we set b = d and sum over b to get

Tr F aF a = CA(N2 − 1) , (A.0.14)

which givesCA = TA . (A.0.15)

We know from group theory that the direct product of the fundamental and anti-fundamentalrepresentations is equal to the direct sum of the singlet representation and the adjoint rep-

160

2

(a)

2

(b)

2

(c)

Figure A.1: Pictorial representation of squared matrix elements for tree-level branchingsof quarks and gluons.

resentation. Using the properties in Eq. (A.0.3) we then get

TA = 2NTF . (A.0.16)

The most common convention, and the same we use in this thesis, is to choose TF = 12.

As anticipated, this forces TA = N and

CA = N ,

CF =N2 − 1

2N.

(A.0.17)

It is straightforward to get the new values for CF and CA in case a different value for TF ischosen. It is then necessary to also rescale the value of the strong coupling g accordingly.

We can now use the Feynman rules in Table 2.1 to get the colour factors associatedwith the matrix elements for the branchings shown in Fig. A.1. In Fig. A.1a the emissionof a gluon from a quark is shown. Its corresponding colour factor Cq→qg can be calculated(assuming sums over repeated indices) to be

Cq→qg =1

NTr[tata]

=1

2Nδaa

=N2 − 1

2N= CF ,

(A.0.18)

where we averaged over the N incoming colours. CF is the colour charge associated witha gluon splitting into a quark and a gluon.

We can go on, calculating the colour factor Cg→gg corresponding to Fig. A.1b:

Cg→gg =1

N2 − 1fabcfabc

=N

N2 − 1δaa = CA ,

(A.0.19)

since the number of incoming colours is now N2 − 1. CA is the colour charge associatedwith a gluon splitting into gluons.

161

APPENDIX A. FROM LIE GROUPS TO COLOUR CHARGES

Finally, we can get the colour factor Cg→qq corresponding to the process in Fig. A.1c:

Cg→qq =1

N2 − 1Tr[tata]

= TF .

(A.0.20)

It is clear that the colour charge associated with a gluon splitting into a quark–antiquarkpair is TF , which in our case would be 1

2.

162

APPENDIX

B

GLUON JETS

In the main text we described the effect of taggers on quark jets. The results we gotcan be easily extended to the case of gluon jets. The first thing to do is replace the colourfactor CF with CA. Next we need to describe gluon splittings rather than quark splittings.This means we need to carry out the substitution:

Pgq(x) −→ Pgg(x) +TFNf

CAPqg(x) . (B.0.1)

The splitting functions were defined in Eq. (2.6.1). Then, instead of Eq. (6.2.3) andEq. (6.3.7) one simply gets

Dg(ρ) = CA

∫ R2

ρ

dρ′

ρ′

∫ 1

ρ′/R2

dx

[Pgg(x) +

TFNf

CAPqg(x)

]a(ωJ√xρ′),

Sg(a, b) = CA

∫ b

a

dρ′

ρ′

∫ 1

fcut

dx

[Pgg(x) +

TFNf

CAPqg(x)

]a(ωJ√xρ′).

(B.0.2)

We can now write the resummed distributions for all the taggers. As before, we assumethat power corrections associated with the parameters of tagging are negligible.

B.1 Plain jet mass

The resummed expression for the integrated distribution of plain jet mass of gluon-initiated jets is

Σg(ρ) = e−Dg(ρ) · e−γED′g(ρ)

Γ[1 +D′g(ρ)

] · Cg(ρ) , (B.1.1)

where Cg(ρ) contains the corresponding non-global and clustering logarithms. The aboveexpression can be compared to the case of quark-initiated jets, Eq. (6.2.4).

163

APPENDIX B. GLUON JETS

B.2 Trimming

The resummed expression for the integrated distribution of trimmed jet mass of gluon-initiated jets is

Σtrimmingg (ρ) = exp

−Dg

[max

(fcutR

2, ρ)]−Θ

(fcutR

2 − ρ)Sg(ρ, fcutR

2)

− CAΘ(fcutR

2sub − ρ

) ∫ fcutR2sub

ρ

dρ′

ρ′

∫ fcut

ρ′/R2sub

dx

xa(ωJ√xρ′)

,

(B.2.1)

which is to be compared to the quark case, Eq.(6.3.6).

B.3 Pruning

The resummed distribution for pruned jet mass is more clearly written in its differ-ential form. For ρ < zcutR

2, the Y and I components are given by (σ0,g is the Borncross-section for producing the primary gluons)

ρ

σ0,g

dσ

dρ

Y-prune

= CAe−Dg(ρ)

∫ 1

zcut

dx

[Pgg(x) +

TFNf

CAPqg(x)

]a(ωJ√xρ)

+ C2A

∫ min(zcutR2,ρ/zcut)

ρ

dη

η

[e−Dg(η)

∫ zcut

η/R2

dx′

x′a(ωJ√x′η)]

× e−Sg(ρ,η)

∫ ρ/η

zcut

dx

[Pgg(x) +

TFNf

CAPqg(x)

]a(ωJ√xρ),

ρ

σ0,g

dσ

dρ

I-prune

= C2A

∫ zcutR2

ρ

dη

ηe−Dg(η)

∫ zcut

η/R2

dx′

x′a(ωJ√x′η)

× e−Sg(ρ,η)

∫ 1

ρ/η

dx

[Pgg(x) +

TFNf

CAPqg(x)

]a(ωJ√xρ)

Θ

(ρ

η− zcut

)+ Θ

(zcut −

ρ

η

)exp

[−CA

∫ zcutη

ρ

dρ′

ρ′

∫ zcut

ρ′/η

dx′

x′a(ωJ√x′ρ′)]

.

(B.3.1)

The above expressions can be compared to the results for quark-initiated jets, Eqs. (6.4.2)and (6.4.5) respectively.

B.4 mMDT

Finally, the integrated mass distribution for the mMDT with gluon jets is

ΣmMDTg (ρ) = exp

−Dg

[max

(ρ, ycutR

2)]−Θ

(ycutR

2 − ρ)Sg (ρ, ycut)

, (B.4.1)

164

B.5. COMPARISON WITH MONTE CARLO RESULTS

0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Analyti s , f ut

= 0.03

Herwig++ , f ut

= 0.03

Analyti s , f ut

= 0.15

Herwig++ , f ut

= 0.15

Figure B.1: Comparison of trimmed mass distributions for gluon-initiated jets, as obtainedfrom Herwig++ and Eq. (B.2.1) for Rsub = 0.2 and different values of fcut. The ρ valuecorresponding to mJ = 125 GeV (the approximate Higgs mass) and ωJ = 3 TeV isdenoted by a dashed line for phenomenological reference.

which is to be compared with the corresponding result for quark jets, Eq. (6.5.5).

B.5 Comparison with Monte Carlo results

The results for trimming, pruning and the mMDT with gluon jets are reported inFig. B.1, Fig. B.2 and Fig. B.3 respectively, compared with the spectra generated by Her-wig++. The agreement is as good as in the quark case. Fig. B.4 shows a summary jet-massdistributions analogous to the one shown in Fig. 6.12, this time for gluon jets. As before,we find that by modifying the mMDT as described in Sec. 6.6 brings its distribution closerto that of pruning in the region y2

cutR2 < ρ < ycutR

2.

165


0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Analyti s , Pruning

Herwig++ , Pruning

Analyti s , I-pruning

Herwig++ , I-pruning

Analyti s , Y-pruning

Herwig++ , Y-pruning

Figure B.2: Comparison of pruned mass distributions for gluon-initiated jets, as obtainedfrom Herwig++ and Eqs. (B.3.1) for zcut = 0.1. The plots show both total distributionsand Y- and I-pruned components separately. The ρ value corresponding to mJ = 125GeV (the approximate Higgs mass) and ωJ = 3 TeV is denoted by a dashed line forphenomenological reference.

166

B.5. COMPARISON WITH MONTE CARLO RESULTS

0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Analyti s , y ut

= 0.1

Herwig++ , y ut

= 0.1

Analyti s , y ut

= 0.2

Herwig++ , y ut

= 0.2

Analyti s , y ut

= 0.33

Herwig++ , y ut

= 0.33

Analyti s , y ut

= 0.4

Herwig++ , y ut

= 0.4

Figure B.3: Comparison of mMDT distributions for gluon-initiated jets, as obtained fromHerwig++ and Eq. (B.4.1), partially supplemented with finite-ycut effects (see [152] andApp. C), for different values of ycut and µ = 0.67. The agreement between the two isreasonable. The ρ value corresponding to mJ = 125 GeV (the approximate Higgs mass)and ωJ = 3 TeV is denoted by a dashed line for phenomenological reference.

167


0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0

1 σ0

dσ

dlog10ρ

log10 ρ

Plain jet mass

Trimming

Pruning

mMDT

Alternative mMDT

R = 0.8 Rsub

= 0.2

f ut

= z ut

= 0.1

y ut

= 0.11 µ = 0.67

Figure B.4: Comparison of MC distributions for plain jet mass, trimming, pruning andmMDT for equivalent parameters, as reported on the plot, and gluon-initiated jets. Thechoice of ycut is such that ycut = zcut/(1− zcut). For the alternative version of the mMDTdescribed in the text we have chosen ycut = zcut. It is apparent that all taggers behave ina similar way between z2


2 ' 10−1, i.e. in between the two verticaldashed lines. In particular, the alternative version of the mMDT is very close to pruningin this region.

168

APPENDIX

C

EFFECTS OF A FINITE CUT IN THE mMDT

The resummed integrated distribution of Eq. (6.5.5) was obtained assuming a smallcut on the energy of emissions, i.e. ycut 1. If we want to go beyond this approximation,we need to take into account the possibility that in a q → qg splitting, the gluon is theharder parton; as a consequence, the unclustering procedure follows the latter rather thanthe quark. It is then necessary to include g → gg and g → qq splittings, even for quarkjets.

First of all, we give a more precise integration over the energy fraction over the usualgluon energy fraction, defining

Fq ≡ CF

∫ 1

0

dxPgq(x)Θ

(x

1− x − ycut

)Θ

(1− xx− ycut

). (C.0.1)

This reduces to the choice of integration limits of Eq. (6.5.3) for small x. In addition tothis, it takes into account the fact that if the gluon carries a fraction x of the parent energy,then the quark carries a fraction 1 − x. The second step function is used when the quarkis softer than the gluon, i.e. x > 1

2. We can define a similar function for gluon splittings:

Fg ≡ CA

∫ 1

0

dx

[Pgg(x) +

TFNf

CAPqg(x)

]Θ

(x

1− x − ycut

)Θ

(1− xx− ycut

).

(C.0.2)If otherwise the splitting fails the asymmetry cut, we need to take into account the flavourswitching. When the mMDT follows a quark in a q → qg splitting, this is done by defining

Fq→g ≡ CF

∫ 1

0

dxPgq(x)Θ

(ycut −

1− xx

). (C.0.3)

Finally, the flavour always changes when a gluon splits into a quark–antiquark pair. Inthis case we need

Fg→q ≡ TFNf

∫ 1

0

dxPqg(x)

[Θ

(ycut −

1− xx

)+ Θ

(ycut −

x

1− x

)]. (C.0.4)

We can now include all these flavour-changing effects in our resummed integrated distri-bution for the mMDT. This implies it acquires a matrix structure in flavour space. In a

169

APPENDIX C. EFFECTS OF A FINITE CUT IN THE mMDT

fixed-coupling approximation, we get

ρ

σ

dσ

dρ= a

(Fq Fg

)exp

a ln1

ρ

−Fq − Fq→g Fg→q

Fq→g −Fg − Fg→g

F0 . (C.0.5)

F0 is the vector describing the initial fractions of quarks and gluons. If we are interestedin quark jets, then

F0 =

1

0

, (C.0.6)

and σ = σ0. For gluon jets we have instead

F0 =

0

1

, (C.0.7)

and σ = σ0,g.It is worth mentioning here another subtlety in the derivation of Eq. (6.5.5). To get

there, we assumed that the maximum value of xn for any n is 1. However, the energy ofparton n is scaled by a factor fn with respect to the original jet, with

fn ≡ (1− x1)(1− x2) . . . (1− xn−1) , (C.0.8)

where xi < ycut for any i < n. If ycut 1 it is safe to set fn = 1. It may no longer bethe case if we relax this approximation. Suppose there is a probability p(f, aL) for thereto be a modification by a factor f of the pT of the tagged jet, and consequently of its ρ.Then we can recover the “true” integrated distribution Σt(ρ) by taking into account thisscaling in Σ(ρ), as calculated in Eq. (6.5.5). Doing this yields

Σt(ρ) = Σ(ρ) +

∫df[Σ( ρx2

)− Σ(ρ)

]p(f, aL) , (C.0.9)

which shows that any such modification is beyond single-logarithmic accuracy.

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