# QCD resummation for high-pT jet shapes at hadron colliders

Kamel Khelifa-Kerfa

(2012)

###### Abstract

Exploiting the substructure of jets observed at the LHC to better understand
and interpret the experimental data has recently been a very active area of
research. In this thesis we study the substructure of high-$p_{T}$ QCD jets,
which form a background to many new physics searches. In particular, we
explore in detail the perturbative distributions of a certain class of
observables known as non-global jet shapes. More specifically, we identify and
present state-of-the-art calculations, both at fixed-order and to all-orders
in the perturbative expansion, of a set of large logarithms known as non-
global logarithms. Hitherto, these logarithms have been largely mis-treated,
and in many cases ignored, in the literature despite being first pointed out
more than a decade ago. Our work has triggered the interest of many groups,
particularly Soft and Collinear Effective Theory (SCET) groups, and led to a
flurry of papers on non-global logarithms and related issues.

Although our primary aim is to provide analytical results for hadron-hadron
scattering environments, it is theoretically instructive to consider the
simpler case of $e^{+}e^{-}$ annihilation. We thus examine, in chapters 4, 5
and 6, the the said jet shapes in the latter environment and compute the full
next-to-leading logarithmic resummation of the large logarithms present in the
distribution for various jet definitions. We exploit the gained experience to
extend our calculations to the more complex hadronic environment in chapter 7.
We provide state-of-the-art resummation of the jet mass observable in vector
boson + jet and dijet QCD processes at the LHC up to next-to-leading
logarithmic accuracy. The resultant distribution of the former (vector boson +
jet) process agrees well, after accounting for hadronisation corrections, with
standard Monte Carlo event generators and potential comparisons to data from
the LHC will be made soon.

###### Contents

  1. 1 Introduction
  2. 2 QCD phenomenology
    1. 2.1 $\rm SU(\mathrm{N_{c}})$ colour
      1. 2.1.1 Large $\mathrm{N_{c}}$ limit
    2. 2.2 Lagrangian and gauge invariance
      1. 2.2.1 Perturbation theory and Feynman rules
      2. 2.2.2 Optical theorem and cross-sections
    3. 2.3 Running of $\alpha_{s}$
      1. 2.3.1 Confinement and factorisation
  3. 3 Jet phenomenology
    1. 3.1 Jet clustering algorithms
      1. 3.1.1 Cone algorithms
      2. 3.1.2 IRC safety of jet algorithms
      3. 3.1.3 Sequential recombination algorithms
      4. 3.1.4 Jet grooming techniques
    2. 3.2 Jet shapes
      1. 3.2.1 Jet shape distributions
      2. 3.2.2 Non–global logarithms
      3. 3.2.3 Clustering logarithms
      4. 3.2.4 Resummation
      5. 3.2.5 Non-perturbative effects
    3. 3.3 Summary
  4. 4 Resummation of angularities in $e^{+}e^{-}$ annihilation
    1. 4.1 Introduction
    2. 4.2 High-$p_{T}$ jet-shapes/energy-flow correlation
      1. 4.2.1 Observable definition
    3. 4.3 Soft limit calculations
      1. 4.3.1 Two-gluon calculation and non-global logarithms
    4. 4.4 Resummation
      1. 4.4.1 Independent emission contribution
      2. 4.4.2 Non-global component
      3. 4.4.3 Numerical studies
    5. 4.5 Other jet algorithms
      1. 4.5.1 Revisiting the anti-kT algorithm at two-gluons
      2. 4.5.2 The kT algorithm and clustering logarithms
    6. 4.6 Conclusion
  5. 5 Non-global structure of jet shapes beyond leading log accuracy
    1. 5.1 Introduction
    2. 5.2 Jet-thrust distribution at $\mathcal{O}(\alpha_{s})$
      1. 5.2.1 Observable definition
      2. 5.2.2 LO distribution
    3. 5.3 Jet-thrust distribution at $\mathcal{O}(\alpha_{s}^{2})$
      1. 5.3.1 The anti-kT algorithm
      2. 5.3.2 The C/A algorithm
    4. 5.4 Resummation
      1. 5.4.1 The anti-kT algorithm
      2. 5.4.2 The C/A algorithm
      3. 5.4.3 All-orders numerical studies
    5. 5.5 Comparison to event2
    6. 5.6 Conclusion
  6. 6 On the resummation of clustering logarithms
    1. 6.1 Introduction
    2. 6.2 The jet mass distribution and clustering algorithms
    3. 6.3 Two-gluon emission calculation
      1. 6.3.1 Calculation in the kT algorithm
      2. 6.3.2 Calculation in the C/A algorithm
    4. 6.4 Three and four-gluon emission
      1. 6.4.1 Three-gluon emission
      2. 6.4.2 Four-gluon emission
    5. 6.5 All-orders result
    6. 6.6 Comparison to MC results
      1. 6.6.1 Non-global logs
    7. 6.7 Conclusion
  7. 7 Resummation of jet mass at hadron colliders
    1. 7.1 Introduction
    2. 7.2 General framework
    3. 7.3 The eikonal approximation and resummation
      1. 7.3.1 Exponentiation: the Z+jet case
      2. 7.3.2 Exponentiation: the dijet case
      3. 7.3.3 The constant term $C_{1}$
    4. 7.4 Non-global logarithms
      1. 7.4.1 Fixed order
      2. 7.4.2 All-orders treatment
    5. 7.5 Z+jet at the LHC
      1. 7.5.1 Different approximations to the resummed exponent
      2. 7.5.2 Matching to fixed order
      3. 7.5.3 Numerical estimate of constant term $C_{1}$
      4. 7.5.4 Comparison to Monte Carlo event generators
    6. 7.6 Dijets at the LHC
    7. 7.7 Conclusion
  8. 8 Conclusions
  9. A Colour algebra
    1. A.1 Fierz identities
  10. B Perturbative calculations in $e^{+}e^{-}$ annihilation
    1. B.1 Eikonal method
    2. B.2 pQCD: $e^{+}e^{-}$ $\rightarrow$ hadrons
      1. B.2.1 Generalities
      2. B.2.2 Born cross-section
      3. B.2.3 $\mathcal{O}(\alpha_{s})$ corrections
      4. B.2.4 $\mathcal{O}(\alpha_{s}^{2})$ corrections
  11. C Jet–thrust calculations
    1. C.1 Derivation of the LO distribution
    2. C.2 $G_{nm}$ coefficients
    3. C.3 Jet-thrust distribution in SCET
      1. C.3.1 Resummation
    4. C.4 Comparisons to event2 : single $R_{s}$ plots
  12. D Full $R$-dependence of clustering coefficients
  13. E Radiators
    1. E.1 Dipole calculations for the global part
      1. E.1.1 In-in dipole
      2. E.1.2 In-recoil dipoles
      3. E.1.3 Jet-recoil dipole
      4. E.1.4 In-jet dipoles
    2. E.2 Dipole calculations for the non-global contribution
      1. E.2.1 Dipoles involving the measured jet
      2. E.2.2 The remaining dipoles
    3. E.3 Resummation formulae

Word count: 65,000

###### List of Figures

  1. 1.1 A typical LHC multijet event with associated various perturbative and non–perturbative effects. Former effects include: hard scattering subprocess, soft and collinear radiation (resummation), soft and wide–angle radiation, non-global logs (NGLs), effects of jet algorithms (JetAlg) with jet radius $R$ and recombination scheme $E$. Latter effects include: parton densities ($f(x,\mu_{F})$), underlying event (UE) and hadronisation. This thesis concerns perturbative effects.
  2. 1.2 Jet mass distribution for an inclusive QCD jet sample generated for the LHC with $p_{T,\min}$ for the hard scattering of $2\text{ TeV}$. Jets are defined using different jet algorithms with jet radius of $0.7$. The peak is around $125\text{ GeV}$ (in natural units, $c=\hbar=1$) (figure from [1]).
  3. 2.1 Graphical representation of the Fierz identity for a one–gluon exchange between a quark–antiquark pair ($t$–channel exchange).
  4. 2.2 Graph $(a)$ is an example of a planar graph which contributes at leading $\mathrm{N_{c}}$. Graph $(b)$ is an example of non–planar graph which contributes at subleading $\mathrm{N_{c}}$. The graphs represent _Feynman diagrams_ for the amplitude squared of $e^{+}e^{-}$ anihilation into $q\bar{q}gg$ (to be discussed in Sec. 2.2 and Appendix B).
  5. 2.3 Feynman rules for QCD in a covariant gauge: gluons (curly lines), fermions (solid lines) and ghosts (dashed lines).
  6. 2.4 Feynman diagrams contributing to QCD $\beta$ function. (top) fermion self–energy diagrams, (middle) gluon self–energy diagrams and (bottom) fermion–gluon vertex diagrams.
  7. 3.1 A schematic view of jet production at hadron colliders. The figure, modified from [1], shows a proton–proton collision at the LHC.
  8. 3.2 Figures $(a_{1}),(a_{2})$ illustrate the output of a an IR unsafe cone algorithm and figures $(b_{1}),(b_{2})$ illustrate the output of a collinear unsafe cone algorithm. The plane $(\mathbf{p_{t}},y)$ is the 2D projection of the 3D phase space $(\mathbf{p}_{t},y,\phi)$.
  9. 3.3 Sources of global (configuration 1) and non–global (configuration 2) logarithms in dijet events at fixed–order. For configuration 2, we show a real secondary emission (left) and its virtual correction (right).
  10. 3.4 An into-jet $\rm J_{1}$ coherent radiation from an ensemble of soft large-angle gluons (that are outside $\rm J_{1}$). Such configurations contribute to NGLs at higher-orders.
  11. 3.5 Fixed–order vs (matched) resummed calculations and comparison to experimental data for the total broadening shape variable (left) without hadronisation effects and (right) with hadronisation effects (figure from http://www.lpthe.jussieu.fr/~salam/talks/).
  12. 4.1 Diagrams representing the correlated emissions which give rise to the lowest-order non-global logarithms. $(A)$: the harder gluon $k_{1}$ lies outside both jets and the softest one $k_{2}$ is recombined with the measured jet and contributes to the angularity distribution. $(B)$: the harder gluon is inside the unmeasured jet and emits a softer gluon into the measured jet. $(C)$: the harder gluon is inside the unmeasured jet and emits a softer gluon outside both jets, which contributes to the $E_{0}$-distribution.
  13. 4.2 The angularity distribution $\textrm{\,d}\Sigma/\textrm{\,d}\tau_{a}$ for five values of the parameter $a$ given at $Q=500\text{ GeV}$ (about $250\,\mathrm{GeV}$ or so at a hadron collider) and different values of the jet radius and energy cut. The left column is for quark jets, while the right is for gluon gets. As shown in the legend, solid lines correspond to neglecting NGLs while dotted lines take them into account.
  14. 4.3 Non global contribution $1-S(t)$ from the measured jet (meas), the unmeasured jet (unmeas) and overall (tot) as a function of $E_{0}$ for $a=0,\tau_{0}=5\times 10^{-4}$ and $R=0.4$.
  15. 4.4 Diagrams contributing to independent two-gluon emission from a hard parton line.
  16. 5.1 Schematic representation of gluonic arrangement giving rise to NGLs. Here we assume that $k_{t1}>k_{t2}$. ($a$): harder gluon $k_{1}$ in interjet region emits softer gluon $k_{2}$ into $p_{R}$–jet, thus giving rise to leading NGLs contribution to the $p_{R}$–jet mass distribution. ($b$): $k_{1}$ inside $p_{R}$–jet emits $k_{2}$ into interjet region, thus giving rise to leading NGLs contribution to $E_{0}$ distribution. ($c$): $k_{1}$ inside $p_{L}$–jet emits $k_{2}$ into $p_{R}$–jet, thus leading to subleading NGLs contribution to $p_{R}$–jet mass. We have only shown the NGLs contributions to the $p_{R}$–jet, identical contributions apply to the $p_{L}$–jet.
  17. 5.2 A schematic representation of a three–jet final state after applying the C/A algorithm on real emission along with virtual correction diagrams. The two gluons are clustered in the E–Scheme (see Sec. 5.2). Identical diagrams hold for the $p_{L}$–jet.
  18. 5.3 A schematic representation of a two-jet final state after applying the C/A algorithm on real emission along with virtual correction diagrams. The two gluons are clustered in the E–Scheme (see Sec. 5.2). Identical diagrams hold for the left ($p_{L}$–) jet.
  19. 5.4 Leading non-global coefficient $\mathcal{S}_{22}$ in the anti-kT and C/A algorithms.
  20. 5.5 N3LL NGLs coefficients for both $\mathrm{C_{F}}\mathrm{C_{A}}$ and $\mathrm{C_{F}}\mathrm{T_{F}}n_{f}$ colour channels in anti-kT (solid) and C/A (kT) (dashed). Note that $\widetilde{S}^{a}_{21}=\mathcal{S}^{(a)}_{21}/\mathrm{C_{F}}\mathrm{C_{A}}$, $\widetilde{S}^{\mathrm{C_{A}}}_{21}=\mathcal{S}_{21}^{\mathrm{C_{A}}}/\mathrm{C_{F}}\mathrm{C_{A}}$ and $\widetilde{S}^{n_{f}}_{21}=\mathcal{S}_{21}^{n_{f}}/\mathrm{C_{F}}\mathrm{T_{F}}n_{f}$.
  21. 5.6 Comparison of analytical resummed differential distribution $\textrm{\,d}\Sigma/\textrm{\,d}\tau_{E_{0}}$ where the Sudakov, full anti-kT and full kT distributions are given, respectively, in Eqs. (5.78), (5.77), and (5.81). The plots are shown for a jet veto $E_{0}=60\text{ GeV}$ and a hard scale $Q=500\text{ GeV}$.
  22. 5.7 Variation of $1-S(t)$ as $E_{0}$ is varied for several jet radii. The smallest value of $E_{0}$ corresponds to the $E_{0}^{\min}=Q\tau_{E_{0}}/(2R_{s})$, derived from (5.30) where $e^{\Delta\eta}\sim 1/R_{s}$.
  23. 5.8 The difference between event2 and $\tau_{E_{0}}$ LO distribution for various jet radii in both anti-kT (left) and C/A (right) algorithms.
  24. 5.9 The difference between event2 and $\tau_{E_{0}}$ NLO distribution for various jet radii in both anti-kT (left) and C/A (right) algorithms.
  25. 6.1 Comparisons of the analytical result to the output of the Monte Carlo program in the kT algorithm for two values of the jet radius.
  26. 6.2 The output of the Monte Carlo program in the kT algorithm for various jet radii.
  27. 6.3 Comparisons of the Sudakov result, the correct primary result and the full result including non-global logarithms with and without clustering, as detailed in the main text.
  28. 7.1 Comparison between different approximations to the resummed exponent: jet functions (blue), with full resummation of the global contribution (green) and with non-global logarithms as well (red). The jet radius is $R=0.6$.
  29. 7.2 Comparison between different approximations to the resummed exponent: jet functions (blue), with full resummation of the global contribution (green) and with non-global logarithms as well (red). The jet radius is $R=1.0$.
  30. 7.3 Matching of the NLL resummed distribution to the LO one for $R=0.6$ (on the left) and $R=1.0$ (on the right).
  31. 7.4 The impact of the NLL constant $C_{1}$, for $R=0.6$ jets (on the left) and $R=1.0$ (on the right). The band is produced by suppressing the real radiation contributions with a quark or gluon jet form factor, as explained in the text.
  32. 7.5 Comparison of our resummed and matched result NLL+LO (in red) to standard Monte Carlo event generators, at the parton level.
  33. 7.6 Results for the $\zeta$ distributions obtained with standard Monte Carlo parton showers, with hadronisation corrections (dashed lines) compared to analytical resummation with non-perturbative shifts (shaded bands) as explained in the main text.
  34. 7.7 The NLL+LO jet mass distribution for dijets, with different approximation to the resummed exponent.
  35. A.1 Graphical representation of some important colour relations.
  36. A.2 Graphical representation of the Fierz identity.
  37. B.1 Eikonal Feynman rules for QCD: gluons are represented by curly lines and fermions by solid lines. For triple vertices, the colour is read clockwise.
  38. B.2 Feynman diagrams for $(a)\,e^{+}e^{-}\rightarrow q\,\overline{q}$, Born amplitude, $(b)\,e^{+}e^{-}\rightarrow q\overline{q}$, virtual corrections, and $(c)\,e^{+}e^{-}\rightarrow q\bar{q}g$, real emission.
  39. B.3 Dalitz plot for a three–body “decay” of a photon. All partons are assumed on–shell. The edges $x_{i}=1$ correspond to two partons being collinear and the corners $x_{i}=0$ correspond to one parton being soft ($p_{i}\rightarrow 0$). The different kinematical regions are: $(1)\,x_{1}>x_{2}>x_{3},(2)\,x_{2}>x_{1}>x_{3},(3)\,x_{2}>x_{3}>x_{1},(4)\,x_{3}>x_{2}>x_{1},(5)\,x_{3}>x_{1}>x_{2}$ and $(6)\,x_{1}>x_{3}>x_{2}$.
  40. B.4 Poles of the integrand of Eq. (B.48) in the complex plane.
  41. B.5 Feynman diagrams for $(a)\,e^{+}e^{-}\rightarrow q\overline{q}gg$ and $(b)\,e^{+}e^{-}\rightarrow q\bar{q}q^{\prime}\overline{q}^{\prime}$. There are three other diagrams corresponding to swapping the secondary gluons $g_{3},g_{4}$ in the top three diagrams.
  42. C.1 The difference between event2 and $\tau_{E_{0}}$ NLO distribution for various jet radii in both anti-kT (left) and C/A (right) algorithms. Plots are for $\mathrm{C_{F}^{2}}$ channel.
  43. C.2 The difference between event2 and $\tau_{E_{0}}$ NLO distribution for various jet radii in both anti-kT (left) and C/A (right) algorithms. Plots are for $\mathrm{C_{F}}\mathrm{C_{A}}$ channel.
  44. C.3 The difference between event2 and $\tau_{E_{0}}$ NLO distribution for various jet radii in both anti-kT (left) and C/A (right) algorithms. Plots are for $\mathrm{C_{F}}\mathrm{T_{F}}n_{f}$ channel.
  45. D.1 Full $R$-dependence of the two-gluon clustering coefficient $\mathcal{F}_{2}$.

###### List of Tables

  1. 5.1 Full $R_{s}$ numerical results for the leading and next-to-leading clustering logs’ coefficients.
  2. 5.2 Fitting values of the parameters $a,b$ and $c$ for $\mathcal{S}(t)$ factor in both anti-kT and kT . The coefficient $\mathbf{I}_{22}=\mathcal{S}_{22}/\mathrm{C_{F}}\mathrm{C_{A}}$.
  3. 5.3 Fitting values of the parameters $a,b$ and $c$ for $\mathcal{C}^{P}$ factor.
  4. D.1 Estimates of the clustering coefficients $\mathcal{F}_{n}$ for $n=2,3,4$ at different values of the jet radius $R$ in the $k_{t}$ and C/A algorithms. Recall that $\mathcal{F}_{2}^{\text{C/A}}=\mathcal{F}_{2}^{k_{t}}(\equiv\mathcal{F}_{2})$ and that $\mathcal{F}_{4}^{\text{C/A}}$ has not been computed. Note that $\mathcal{F}_{3}^{\text{C/A}}$ and $\mathcal{F}_{4}$ are more difficult to evaluate numerically than $\mathcal{F}_{3}$ (and obviously $\mathcal{F}_{2}$). Thus their final answers are subjected to larger errors.
  5. E.1 Numerical results for the coefficients that parametrize the resummation of non-global logarithms.
  6. E.2 Numerical results for the coefficients that parametrize the resummation of non-global logarithms. Note that the above results are valid in the case $|\Delta y|=2$.

## Declaration

No portion of the work referred to in the thesis has been submitted in support
of an application for another degree or qualification of this or any other
university or other institute of learning.

## Copyright Statement

  1. i.

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## Acknowledgements

First and foremost all praises are due to Almighty ALLAH.

I would like to thank my supervisor, Mrinal Dasgupta, for his support and
continuous effort throughout the course of my PhD. I am indebted to Yazid
Delenda for intriguing discussions, particularly in chapter 6, reviewing and
proofreading the thesis, and to Simone Marzani for conceptual and technical
discussions as well as providing the numerical codes used in chapter 7. My
thanks extend to Andrea Banfi for assistance in chapter 5 and Mike Seymour for
aid with the numerical program event2 as well as helpful feedback on chapter
5. I would also like to thank Michael Spannowsky for collaborating on the work
presented in chapter 7, and in particular for providing the results of the
Monte Carlo event generators. The program of Ref. [2] with the implementation
of the kT clustering in [3], which has been heavily used in this thesis, has
not been made public yet and for that reason I would like to thank the
authors, Mrinal Dasgupta, Gavin P. Salam and Andrea Banfi, for approving its
usage in this thesis.

The encouragement of Apostolos Pilaftsis, my supervisor for the MPhys project,
is highly appreciated, and so is the help of Jeff Forshaw and Fred Loebinger
regarding administrative issues.

Moreover, I would like to express my gratitude to my brother Abdelkader for
the enormous support he offered me throughout my educational career.

The work presented in this thesis has been sponsored by both the University of
Manchester and the Algerian government. I would like to express my gratitude
to the Faculty of Engineering and Physical Sciences, the ministry of higher
education and scientific research and the Algerian consulate in London for
their financial and administrative support during my stay in the United
Kingdom.

## List of Publications

  * •

Y. Delenda and K. Khelifa-Kerfa, _On the resummation of clustering logarithms
for non-global observables_ , J. High Energy Phys. 09 (2012) 109 [hep-
ph/1207.4528].

  * •

M. Dasgupta, K. Khelifa-Kerfa, S. Marzani, M. Spannowsky, _On jet mass
distributions in Z+jet and dijet processes at the LHC_ , J. High Energy Phys.
10 (2012) 126 [hep-ph/1207.1640].

  * •

K. Khelifa-Kerfa, _Non-global logs and clustering impact on jet mass with a
jet veto distribution_ , J. High Energy Phys. 02 (2012) 072 [hep-ph/1111.2016]

  * •

A. Banfi, M. Dasgupta, K. Khelifa-Kerfa and S. Marzani, _Non-global logarithms
and jet algorithms in high-pT jet shapes_ , J. High Energy Phys. 08 (2010) 064
[hep-ph/1004.3483].

##  Chapter 1 Introduction

With the two Large Hadron Collider (LHC) experiments, ATLAS and CMS, running
and producing a proliferation of valuable data, a new era of physics is being
unfolded. Already at a centre–of–mass energy $7\text{ TeV}$, the Standard
Model (SM) of particle physics has been “re–discovered” [4]. The priority list
of the LHC programme includes a plethora of long–standing problems in High
Energy Physics (HEP) and related physics domains. Perhaps, at the top of the
list is the search for yet the only unobserved SM particle; the Higgs111Both
ATLAS and CMS have recently reported the observation of a Higgs–like particle
at a mass of $125\text{ GeV}$ [5, 6]. Investigations on its properties, such
as spin, are still ongoing.. The Beyond Standard Model (BSM) searches,
expected to reach their full potential as the machine hits its designed energy
$14\text{ TeV}$, cover a variety of fundamental ideas, such as SuperSymmetery
(SUSY), Extra Dimensions (ED), Dark Matter (DM), Dark Energy (DE) etc.

Whilst the prospects are high the challenges are alike. New particles are
generally expected to be heavy and likely to have too short of a lifetime to
be ever directly detected before they decay. The corresponding decay products
need be both stable and interact strongly enough to leave any signature at the
detectors222Weakly interacting particles such as neutrinos escape direct
detection and are treated as missing energy.. The last two conditions are
generic features of SM particles, which will thus form the bulk of the final
state products. Therefore, for a proton–proton collision at the LHC the final
state particles could have originated from either an “ordinary” SM process
(_background_) or a “more interesting” BSM process (_signal_). A reliable
discovery is only feasible once a detailed understanding of both processes is
established.

Figure 1.1: A typical LHC multijet event with associated various perturbative
and non–perturbative effects. Former effects include: hard scattering
subprocess, soft and collinear radiation (resummation), soft and wide–angle
radiation, non-global logs (NGLs), effects of jet algorithms (JetAlg) with jet
radius $R$ and recombination scheme $E$. Latter effects include: parton
densities ($f(x,\mu_{F})$), underlying event (UE) and hadronisation. This
thesis concerns perturbative effects.

At energies as high as those probed by the LHC, many interesting physics
signals will proceed via interactions involving low–$x$ partons333where $x$ is
the fraction of longitudinal momentum of initial protons., which are
predominantly gluons and sea quarks. As such they are characterised by both
enormous phase space and high radiation activity (due to the large colour
charge of gluons). Moreover, a significant fraction of those signals, which
are challenging to measure, will involve in their decay chain production of
quarks and gluons. The latter partons will emit radiation down to a
characteristic scale, $\Lambda_{\rm NP}$,444For Quantum Chromo–Dynamics (QCD)
this scale is $\Lambda_{\rm NP}\equiv\Lambda_{\rm
QCD}^{\overline{\textrm{MS}}}\sim 200\text{ MeV}$ [7]. before hadronising into
sprays of nearly collimated hadrons, or _jets_. Fig. 1.1 illustrates a typical
multijet event at the LHC. In fact, nearly all final states expected in
proton-proton collisions at $14\text{ TeV}$ will involve production of jets,
and for most channels they consist the dominant part of the detectable signal.
On the other hand, jets in SM processes, such as QCD $2\to 2$ jets and Z/W +
jets, will be produced with very high rates [1] and may thus swamp any signal
with similar signature. It is therefore necessary, particularly at the LHC
more than any previous colliders, to improve our understanding and use of
jets.

Although inclusive jet cross-sections fall off with transverse momentum
$p_{T}$ [1, 8, 9, 10, 11], high-$p_{T}$ jets have distinct features that
compensate for such smallness and thus offer an alternative promising channel
for discovery. For instance, there is a great possibility for the production
of heavy BSM particles that could subsequently decay to “boosted” lighter SM
particles, such as W, Z, top, Higgs (H) etc. The latter particles, having
transverse momenta that far exceed their rest masses, would then decay to even
lighter particles that are collimated and thus more likely to be clustered by
jet algorithms into a single “fat” jet. This is due to the fact that the
angular distance between decay products, $b$ and $c$, of a particle $a$ is
proportional to the inverse of the transverse momentum of $a$. Precisely

$\Delta
R_{bc}^{2}=\frac{m_{a}^{2}}{p_{Tb}\,p_{Tc}}=\frac{m_{a}^{2}}{z(1-z)p_{Ta}^{2}},$
(1.1)

where $z$ is the momentum fraction of, say, particle $b$. Thus for
$p_{Ta}^{2}\gg m_{a}^{2}$, and given that the decay is not too asymmetric,
i.e., $z$ neither too close to $0$ nor $1$, $\Delta R_{bc}$ will be small and
$bc$ will end up in a single jet. Moreover, owing to the fact that
non–perturbative corrections, particularly hadronisation, fall off with
$p_{T}$ [12, 13], high-$p_{T}$ jets are less sensitive to such corrections,
making them ideal for clean perturbative investigations.

The above mentioned high-$p_{T}$ (boosted) signal jets have _shape_ and
_substructure_ that are distinct from those of plain QCD jets initiated by
light quarks and gluons. Exploring such a rich substructure may prove very
useful in situations such as that depicted in Fig. 1.2, where QCD events may
have jet mass distributions that peak around $125\text{ GeV}$ (the current
experimentally measured Higgs’ mass) and thus form a strong background to
Higgs searches based on the use of jet masses. Such very high-$p_{T}$ jet
events are not uncommon at the LHC given the high momentum probed and hence
pose a great challenge. Therefore new and more powerful jet substructure
techniques may offer an indispensable tool in boosted signal searches (see e.g
[14, 15, 16, 17, 18, 19, 20, 21]). Note that the peak region in Fig. 1.2 is
well within the perturbative domain of QCD and _analytical “resummation”_
calculations are very efficient, as we shall be illustrating in this thesis.

Figure 1.2: Jet mass distribution for an inclusive QCD jet sample generated
for the LHC with $p_{T,\min}$ for the hard scattering of $2\text{ TeV}$. Jets
are defined using different jet algorithms with jet radius of $0.7$. The peak
is around $125\text{ GeV}$ (in natural units, $c=\hbar=1$) (figure from [1]).

The aforementioned substructure techniques may be divided into two broad
classes: grooming techniques and jet shapes.

Jets, the footprint of the underlying QCD partons, are not intrinsically
well–defined, and they thus ought to be “defined” before they can be used. A
modern jet definition consists of a jet algorithm and a recombination scheme
[22, 23, 24, 25]. We discuss these in more details in Chapter 3. Grooming
techniques, reviewed for instance in [26, 27], are based on jet algorithms and
primarily utilised to identify _subjets_ , within a fat jet, as well as to
mitigate isolated diffuse soft radiation coming from the underlying event (UE)
and pile–up (PU)555Underlying event refers to multiple soft/semi-hard
parton–parton interactions while pile–up refers to multiple soft/semi-hard
proton–proton interactions in the same bunch crossing.. Filtering [14],
pruning [28, 29] and trimming [30] are amongst the widely used jet grooming
algorithms. Applied to final state “ungroomed” jets, these algorithms produce
groomed jets with greatly enhanced features, such as high mass resolution. In
general, such tools and techniques have been shown to significantly reduce
contaminations from the aforementioned sources and, in many cases, recovering
otherwise buried search channels [14, 27]. We shall briefly revisit this topic
in Chapter 3.

Jet shapes are event shape–like observables666See e.g., [7, 31, 32] and CAESAR
homepage (http://home.fnal.gov/~zanderi/Caesar/caesar.html) for a list of some
of the most extensively studied event shapes. that measure the energy flow
within individual jets in an event. A comprehensive list of recently
introduced jet shapes may be found in e.g., [26, 27]. The field of jet shapes,
and more generally jet substructure, has seen an intensive activity over the
very recent years from both theory and experimental groups, with a wealth of
impressive measurements coming from the LHC, e.g., [27, 33, 34, 35].
Substructure techniques have been used in various applications such as tagging
jets from decay of boosted particles, particularly in the Higgs and SUSY
searches [14, 36, 37]. In this thesis we rather focus on establishing a better
understanding of QCD jets through jet shapes, which would allow for an
efficient background subtraction. Specifically we study two observables:
angularities [38, 39] and jet mass, in typical QCD processes whereby final
states are clustered using a variety of infrared and collinear (IRC) safe jet
algorithms777More on IRC safe jet algorithms in Chapter 3.

In fact, much emphasis will be given to jet mass calculations. This is due to
a number of reasons. First, the jet mass is the simplest, yet
phenomenologically the most useful, shape observable to employ in the search
for decay of heavy particles [1, 27]. The first clue to a jet’s identity is
its mass, which is directly related to the mass of the particle initiating it.
Second, it has played a central role in the analytical _resummation_ programme
of numerous observables, including the most intensively studied ones such as
thrust [40]. Further, there has been a flurry of work on jet masses in the
recent years, both within traditional QCD and some of its modern effective
approximations such as soft and collinear effective theory (SCET [41, 42, 43])
[44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. Once a firm understanding of jet
mass distributions, resummation and other related issues is established, it
will provide a natural setting for similar work on other jet shapes and jet
substructure in general. However, such a task is delicate, especially in
complex environments like those typical at hadron colliders. As a matter of
fact, recent papers that addressed the latter task, such as Refs. [53, 54],
have omitted important contributions, as shall be explained in Chapter 7. In
this thesis, we set-out to accomplish the said task by carrying out careful
calculations of the many complex aspects involved. We focus on perturbative
calculations of jet shape _distributions_ in QCD.

There are two complementary theoretical tools to study shape distributions:
analytical calculations and numerical Monte Carlo (MC) simulations. The former
are inclusive over the final state and in most cases only deal with a single
observable at a time. The calculations are performed in the perturbative
region of QCD and have so far only treated a limited set of observables (see
for instance [31, 55] and references therein). Numerical simulations, on the
other side, are completely exclusive over the final state and thus any number
of observables can be measured (simultaneously). In addition to the
perturbative region, MC _event generators_ (cf. fixed–order Monte Carlos
discussed below) probe the non–perturbative region888The perturbative and
non–perturbative regions of QCD correspond to kinematical regions where the
strong coupling constant is small, $\alpha_{s}\ll 1$, and large,
$\alpha_{s}\sim 1$, respectively. of QCD through phenomenological models that
parametrise the effects of processes such as hadronisation, UE and PU, in
terms of a number tunable parameters. The state–of–the–art is that analytical
calculations, both at fixed– and all–orders (resummation), often have higher
precision than event generators999This stems from the fact that the formal
precision of event generators is leading logarithm (LL) accuracy. No higher
accuracy is guaranteed beyond that due to higher order and non–perturbative
contaminations.. As such, they can play a crucial role in advancing the
development of these generators as well as providing an indispensable tool for
the interpretation of new physics signals, should they show up at the LHC.
Moreover, analytical calculations are cleaner, offer a deeper insight into the
dynamics of perturbative QCD and can systematically be improved. For these,
and other reasons, the central focus in this thesis is on _analytical_
calculations of jet shape distributions. Comparisons of the analytical
findings to the output of various MCs are, whenever possible, provided though.

Perturbative analytical calculations of a generic jet shape $v$ of
high–$p_{T}$ QCD jets may be divided into two regimes: small $v$ and large
$v$. In the large $v$ region, $v\lesssim 1$, fixed–order calculations are
sufficient to capture the full features of the measured jet shape
distribution. Fixed–order predictions for an arbitrary IRC safe observable may
be obtained with the aid of fixed–order Monte Carlos (FOMCs), such as [56, 57,
58]. The small $v$ region, $v\ll 1$, on the other hand, receives large
enhancements (usually logarithmic) in ratios of the energy scales present in
the process. In an environment such that of the LHC (Fig. 1.1), the presence
of many scales is common for the majority of events. Such scales involve, for
instance: jet $p_{T}$, the measured value of the jet shape, jet radius $R$ and
other parameters of the jet algorithm, non–perturbative scale
$\Lambda_{\text{QCD}}$, and any other kinematical and selection cuts. The said
large logarithmic terms render fixed–order calculations unreliable and, more
seriously, threaten the validity of the perturbative expansion. To restore the
latter expansion, a summation of these logarithmic enhancements to all–orders
is inevitable. Such a task is dubbed _resummation_ , which may be regarded as
a consequence of more profound characteristic features of QCD [59]. Namely
coherence [60, 61] (and factorisation [62, 63, 64]). The small $v$ regime is
of prime interest, in this thesis, and will thus form the bulk of the
discussion presented herein.

In perturbative QCD (pQCD), resummation of large logarithms that occur in the
distribution of sufficiently inclusive shape observables, or _global_
observables, has become a typical exercise in modern phenomenology [65, 66].
For many such observables, the corresponding resummation has even been
automated [31, 55]. Resummation is most effective when the logarithms
_exponentiate_. This exponentiation, of jet shapes, follows primarily from the
corresponding (exponentiation) property of QCD matrix–elements in the soft and
collinear regions [40, 67, 68]. If we define the _jet shape fraction_ (or
integrated shape cross-section) $\Sigma(v)$ as

$\Sigma(v)=\int^{v}_{0}\textrm{\,d}v^{\prime}\,\frac{1}{\sigma^{(0)}}\frac{\textrm{d}\sigma}{\textrm{d}v^{\prime}},$
(1.2)

where $\sigma^{(0)}$ is the Born cross-section, then $\Sigma(v)$
exponentiates101010Not all shape observables exponentiate though. Examples of
observables that have been proven inconsistent with exponentiation include the
JADE jet–resolution thresholds [24]. For such type of observables even an LL
resummation does not exist [31, 69]. precisely means that it assumes the
following form at small $v$

$\Sigma(v)=C(\alpha_{s})\exp\left[L\,g_{1}(\alpha_{s}L)+g_{2}(\alpha_{s}L)+\alpha_{s}g_{3}(\alpha_{s}L)+\cdots\right]+D(\alpha_{s},v),$
(1.3)

with $L=\ln(1/v)$, $C(\alpha_{s})$ sums the loop constants and the remainder
function $D(\alpha_{s},v)$ vanishes in the limit $v\rightarrow 0$. The
function $g_{1}(\alpha_{s}L)$ resums leading logarithms (LL),
$\alpha_{s}^{n}\,L^{n+1}$, $g_{2}(\alpha_{s}L)$ resums next–to–leading
logarithms (NLL), $\alpha_{s}^{n}\,L^{n}$, $g_{3}(\alpha_{s}L)$ resums
next–to–next–to–leading logarithms (NNLL), $\alpha_{s}^{n}\,L^{n-1}$, and so
on. Moreover, one must establish, in addition to the matrix elements, that the
phase space defining the shape fraction for a particular shape variable
factorises (and hence exponentiate), possibly after a suitable integral
transformation [7]. We elaborate on this in more detail in Chapter 3 (Sec.
3.2.4). For the aforementioned global observables, $g_{1}$ is simply given by
the exponentiation of the single soft gluon emission result (known as
“Sudakov” form factor). Further, the resummation of subleading logarithms.
i.e., calculation of $g_{2},g_{3},\cdots$ can in principle be achieved
through, for example, inclusion of the running coupling at higher loops,
proper treatment of multiple emissions etc [31]. The current state–of–the–art
is N3LL resummation for thrust and other event/jet shapes [46, 49].

Nearly every measurement111111Not all of them require resummation though.
Often fixed–order calculations suffice. at the LHC will involve _non–global_
observables [2, 70]. Such observables are sensitive only to specific regions
of phase space. e.g., the invariant jet mass is only sensitive to radiation
into the jet region. Restricting the jet mass observable in the latter region
to be less than a specific value, say $\rm M_{J}$, induces large _non–global_
(single) logarithms (NGLs) in the ratio $p_{T}^{2}R^{2}/\rm M_{J}^{2}$, with
$R$ being the jet size determined by the jet algorithm used. The resummation
of these logarithms, which contribute to $g_{2}$ and higher functions
($g_{3},g_{4},\rm etc$) and start at $\mathcal{O}(\alpha_{s}^{2})$ in the
perturbative expansion, have not been possible analytically due to the
non–factorisation of the _non–abelian_ part of the multiple gluon emission
matrix–element. Nonetheless, two alternative methods have been developed to
resum NGLs. Namely: a Monte Carlo program [2] and a non–linear evolution
equation [71]. Their formal accuracy is leading NGLs, which is NLL for the jet
mass, and are only valid in the large-$\mathrm{N_{c}}$
limit121212$\mathrm{N_{c}}$ is the number of colour degrees of freedom. For
$\rm SU(3)$, the symmetry group of QCD, $\mathrm{N_{c}}=3$.. Neglected
subleading colour corrections contribute at the $10\%$ level. Phenomenological
impact of NGLs have been shown to lead to a reduction in the peak of the
corresponding Sudakov form factor that can be significant for some observables
[2, 70, 72]. Therefore, any reliable resummation of non–global observables
aiming at NLL accuracy, or beyond, should necessarily account for NGLs.

Jet clustering algorithms, other than cone–type [22] and anti-kT [73]
algorithms, when applied to final state partons lead to a more complicated
phase space due to the non–trivial role of clustering amongst soft partons.
Generally the resultant phase space is not factorisable (into single particle
phase space for each parton). Performing the integral in Eq. (1.2) with
modified phase space due to a jet algorithm yields two effects. First, new
large single logarithms in the abelian sector. These are termed _clustering_
logarithms (CLs) and were first pointed out in [74] for the interjet energy
flow distribution. Second, a reduction in the impact of NGLs, as was shown in
[75] for the same distribution. Since the phase space becomes increasingly
complex as the number of final state partons increases, analytical resummation
of CLs seems to be highly intricate. The MC program of Ref. [2] is at the
moment the only available method to resum both CLs and NGLs in the presence of
a jet algorithm. Nevertheless, it has been shown, through explicit fixed–order
calculations up to $\mathcal{O}(\alpha_{s}^{4})$ in the perturbative expansion
for the interjet energy flow distribution in $e^{+}e^{-}$ annihilation, that
CLs exponentiate [3]. The analytical results agreed well with the output of
the said MC program, indicating that the all–orders result is dominated by the
first few orders. We carry out analogous calculations for the jet mass in
Chapter 6.

As previously mentioned, resummation calculations are accurate for the
dominant part of the shape distribution, specifically in the small $v$ (peak)
region. For phenomenological analyses, including comparison to measurements,
other contributions and corrections ought to be taken into account. These
include (a) fixed–order calculations which can, as stated above, reliably
reproduce the tail of the jet shape distribution (large $v$ region). Hence, a
_matching_ of the two, resummation and fixed–order, calculations is necessary
for an accurate prediction over the full kinematical range of the jet shape
$v$; (b) non–perturbative (NP) corrections, such as hadronisation, UE and PU,
which affect the small $v$ region of the shape distribution. The latter are
non–calculable in pQCD and at present can only be quantified through models,
best implemented in Monte Carlo event generators (see [76] for a review).
Meanwhile, theoretical progress on estimating the impact of non–perturbative
corrections using perturbative methods such as renormalon–inspired techniques
and related approaches [12, 13, 77, 78] have yielded promising results [66,
79, 80, 81, 82]. Typically, these corrections, particularly hadronisation, may
be included as a shift, $v\rightarrow v+\langle\delta v\rangle_{\rm NP}$ where
$\langle\delta v\rangle$ is the mean of the change in $v$, in the jet shape
distribution131313Strictly speaking, the shift is only valid to the right of
the distribution peak, as shall be discussed in Chapter 3. [70, 83].

This thesis is divided into three main parts: physics background, results from
$e^{+}e^{-}$ annihilation and results from hadron-hadron collisions. The first
part includes Chapters 2 and 3. Chapter 2 surveys the theoretical basis of QCD
and serves as a background to the rest of the thesis. Important ingredients
which are essential to later discussions such as factorisation and
matrix–elements are provided. Most noticeably, the _eikonal_ approximation is
explained and the corresponding Feynman rules explicitly given. Since our
pivotal object in this thesis are jets, Chapter 3 is devoted to theoretical
exploration of jets, jet algorithms and jet shapes. A generic formulation of
the latter is presented and related topics such as IRC safety addressed. At
this point, the main conceptual basis is laid down and the computational
machinery is set up.

The actual calculations are then carried out in the other two parts. Whilst
our aim is to produce analytical results that can be compared to present
hadron collider measurements, it is theoretically instructive to start with
the simpler and cleaner $e^{+}e^{-}$ annihilation environment. Indeed, the
latter calculations form a natural setting for hadron collision calculations,
as we shall see. Consequently, the second part, which includes Chapters 4, 5
and 6, concerns jet shapes, specifically angularities, in chapter 4, and the
invariant jet mass, in the remaining two chapters 5 and 6, in $e^{+}e^{-}$
annihilation. Notice that Chapter 4 is similar in content to [84] but treats
the more general angularities shape observable. Chapter 5 largely expands [85]
and Chapter 6 is simply drawn from [86]. The final part, which includes
Chapter 7, extends the $e^{+}e^{-}$ annihilation findings to hadron-hadron
collisions, particularly addressing Z+jet and dijet processes at the LHC. Its
content is essentially the same as [87]. Once a next-to-leading order matching
is performed, the latter will represent the state-of-the-art calculations for
non–global observables at hadron colliders.

We consider final state clustering by various jet algorithms in $e^{+}e^{-}$
annihilation, and only one jet algorithm in hadron-hadron scattering. For each
jet algorithm, we perform the following:

  * •

Compute the full leading–order shape distribution accounting for soft
wide–angle radiation with full jet-radius and colour dependence in both
$e^{+}e^{-}$ and hadron-hadron scattering.

  * •

Compute the full next–to–leading order (two-gluon emission) shape distribution
up to NNLL in the _expansion_ (equivalent to $\alpha_{s}^{2}\,L^{2}$), with
full jet-radius and colour dependence. The two-gluon emission calculations are
generally performed within the eikonal framework. The latter is sufficient to
achieve the said NNLL accuracy in both $e^{+}e^{-}$ and hadron-hadron
scattering.

  * •

In $e^{+}e^{-}$ annihilation, we additionally compute the full N3LL terms in
the expansion (equivalent to $\alpha_{s}^{2}\,L$), for which a more accurate
emission amplitude is computed in Appendix B, and hence the only missing piece
to obtain the full two-gluon distribution is the two–loop constant $C_{2}$.
Our calculations represent the state-of-the-art, with full jet-radius
dependence of both NGLs and CLs coefficients.

  * •

Compute the full NLL resummation including numerical estimates of NGLs and CLs
(only in the $e^{+}e^{-}$ case) to all–orders with full jet-radius dependence
in the large-$\mathrm{N_{c}}$ limit.

  * •

In the hadron-hadron scattering case we perform matching to (leading)
fixed–order, estimate non–perturbative hadronisation corrections and compare
to MC event generators.

Finally, in Chapter 8 we draw our conclusions and highlight areas where more
work is needed in light of upcoming LHC measurements.

In the next chapter, we begin with a review of QCD phenomenology.

##  Chapter 2 QCD phenomenology

The contemporary Standard Model for elementary particles and forces
encompasses three, out of four, fundamental forces of nature: Electromagnetic,
Weak and strong forces. These interactions are elegantly described by the
mathematically consistent gauge field theory (GFT)111Quantum field theory with
the requirement of gauge invariance.. Within this framework, Quantum
Electrodynamics (QED) describes electromagnetic interactions, Electro–Weak
(EW) describes the unified electromagnetic and weak interactions and Quantum
Chromodynamics (QCD) describes the strong interactions. In this thesis we are
concerned with QCD. The latter theory is formulated, at the Lagrangian level,
in terms of quarks and gluons. Quarks are described in GFT by the Dirac spinor
fields and can exist in three different states of colour (usually referred to
as red, green and blue) and six flavours; up (u), down (d), strange (s), charm
(c), bottom (b) and top (t). Their electric charges are, in units of the
electron charge ($e$): $+2/3$ (u,c,t) and $-1/3$ (d,s,b). One striking
difference between QED and QCD is that quarks have never been seen in
isolation, i.e., as asymptotic states, as have electrons and photons. This QCD
phenomenon is known as _confinement_. What is instead observed at detectors
are hadrons, both baryons and mesons. The former are composites of three
spin–$\frac{1}{2}$ quarks, $qq^{\prime}q^{\prime\prime}$, while the latter are
composites of spin–$\frac{1}{2}$ quark–antiquark, $q\overline{q}^{\prime}$,
pairs222The idea that hadrons are not themselves elementary but bound states
of other more elementary states dates back to the early work of Fermi and Yang
[88]..

Gluons, in analogy to photons, are vector fields which arise as a result of
demanding the quark Lagrangian be invariant under local, or _gauge_ ,
rotations in the colour space. The corresponding mathematical group of such
rotations for QCD is the special unitary group $\rm SU(3)$ (see below).
Another sharp contrast between QED and QCD is the fact that gluons, which
mediate the strong interaction, are colourful, i.e., they are charged.
Consequently they can interact among themselves (in addition to their
interactions with quarks).

In this chapter, we begin by briefly reviewing the main features of the
general symmetry group $\rm SU(\mathrm{N_{c}})$, of which $\rm SU(3)$ is a
special case, on the way recalling some of the important relations and
identities which will be useful in later sections (and chapters). Most of the
material presented in this section is based on the references listed at the
end of this introductory section. After that we write the general form of the
QCD Lagrangian and discuss the various terms in it. To make contact with
experiments, it is necessary that one is able to extract predictions from the
latter abstract Lagrangian. At small coupling, $g_{s}$, this is achieved
through perturbation theory, equipped with some essential features of QCD such
as factorisation, to which we turn in Sec. 2.3. Calculations in perturbative
QCD are carried out (in the traditional method) via the use of Feynman
diagrams and Feynman rules. An important result of perturbation theory is the
running of the coupling $g_{s}$. Unlike QED, the QCD strong coupling decreases
as the energy scale at which it is measured increases. Therefore at small and
medium energies, the coupling approaches unity and perturbation theory breaks
down. Furthermore, when considering soft gluons dynamics, one may utilise a
practically simpler effective theory known as _eikonal_ theory [89]. We
elaborate on this, including the eikonal version of Feynman rules, in Appendix
B.1. Finally, in Appendix B.2, we present detailed perturbative calculations
of $e^{+}e^{-}$ annihilation into hadrons which will form the backbone of
Chapters 4, 5 and 6.

Further detailed discussions and in–depth treatment of QCD, and generally of
GFT, may be found in standard textbooks, e.g., [7, 90, 91, 92, 93, 94, 95].
Reviews such as those in [96, 97, 98] are also useful. Other references
specific to certain sections will be given therein.

###  2.1 $\rm SU(\mathrm{N_{c}})$ colour

In group theory (see e.g., Refs. [99, 100, 101]), $\rm SU(\mathrm{N_{c}})$ is
the group of special unitary $\mathrm{N_{c}}\times\mathrm{N_{c}}$ complex
matrices $\rm U$ such that:

$\det\rm U=e^{\operatorname{Tr}\ln\rm
U}=1\;\;\;\;\rm{and}\;\;\;\;\;U^{\dagger}U=\mathbb{1}.$ (2.1)

The above relations leave $\mathrm{N_{c}^{2}}-1$ independent real parameters
to characterise $\rm U$. The matrices $\rm U$ may be parametrised by the
exponential form

$\rm U=e^{\imath\,\omega^{a}\,t^{a}},\;\;\;a=1,\cdots,\mathrm{N_{c}^{2}}-1,$
(2.2)

where $\omega^{a}$ are colour rotation angles, $t^{a}$ are the _generators_ of
the the group and summation over repeated indices is (henceforth always)
implied. From Eqs. (2.1) and (2.2), it follows that the generators are both
hermitian and traceless,

$t^{a}=\left(t^{\dagger}\right)^{a},\;\;\;\operatorname{Tr}\left(t^{a}\right)=0.$
(2.3)

The set of generators $\\{t^{a}\\}$ forms a basis of a vector space over the
field of complex numbers $\mathbb{C}$ (or simply a Lie algebra [99]):

$\displaystyle[t^{a},t^{b}]=t^{a}.\,t^{b}-t^{b}.\,t^{a}$ $\displaystyle=$
$\displaystyle\imath f_{abc}t^{c},$ (2.4a)
$\displaystyle\left[t^{a},[t^{b},t^{c}]\right]+\left[t^{c},[t^{a},t^{b}]\right]+\left[t^{b},[t^{c},t^{a}]\right]$
$\displaystyle=$ $\displaystyle 0.$ (2.4b)

The real expansion coefficients $f_{abc}$ are known as the _structure_
constants of the group and the second equality (2.4b) is known as the _Jacobi_
identity. The structure constants satisfy analogous relations to Eqs. (2.4a)
and (2.4b). The generators $t^{a}$ can be _represented_ by either
$D_{F}(t^{a})=\mathrm{N_{c}}\times\mathrm{N_{c}}$ matrices, in the
_fundamental_ representation, or by the structure constant,
$[D_{A}(t^{a})]_{bc}\equiv\left[T^{a}\right]_{bc}=-\imath
f_{abc}=(\mathrm{N_{c}^{2}}-1)\times(\mathrm{N_{c}^{2}}-1)$ matrices, in the
_adjoint_ representation333Note that for the sake of brevity we shall write
$D_{F}(t^{a})=t^{a}$ and $D_{A}(t^{a})=T^{a}$ throughout. A widely used
representation of $\rm SU(3)$ is that provided by Gell–Mann matrices. Their
explicit expressions can be found in, e.g., [7].. For a given representation,
$R$, the normalisation of the corresponding matrices, $D_{R}(t^{a})$, is such

$\operatorname{Tr}\left[D_{R}(t^{a})\,D_{R}(t^{b})\right]=T_{R}\,\delta^{ab}.$
(2.5)

The $\rm SU(\mathrm{N_{c}})$ (and hence $\rm SU(3)$) fundamental and adjoint
representations have $T_{F}=1/2$ and $T_{A}=\mathrm{N_{c}}$, respectively .
For a given representation, $R$, the casimir operator (or “colour charge”),
$T_{R}^{2}$, which commutes with all other generators in the representation
$R$ is given by [99]

$(T_{R}^{2})_{ij}=\sum_{a=1}^{\mathrm{N_{c}^{2}}-1}\sum_{k=1}^{d(R)}\left[D_{R}(t^{a})\right]_{ik}\left[D_{R}(t^{a})\right]_{kj}=C_{R}\,\delta_{ij},$
(2.6)

where $d(R)$ is the dimension of the representation $R$, namely
$d(F)=\mathrm{N_{c}}$ and $d(A)=\mathrm{N_{c}^{2}}-1$. Eqs. (2.5) and (2.6)
yield

$(T_{R}^{2})_{ii}=T_{R}\,\left(\mathrm{N_{c}^{2}}-1\right)=C_{R}\,d(R).$ (2.7)

We can then read off the $\rm SU(\mathrm{N_{c}})$ casimirs (for the
fundamental and adjoint representations),

$C_{F}=\frac{\mathrm{N_{c}^{2}}-1}{2\mathrm{N_{c}}},\;\;\;C_{A}=\mathrm{N_{c}}.$
(2.8)

In Appendix A we provide some important colour identities as well as some
useful relations that will prove indispensable when performing calculations in
QCD.

####  2.1.1 Large $\mathrm{N_{c}}$ limit

An alternative way to view the Fierz identity (A.5) is as a one–gluon exchange
between two quarks, or quark and antiquark, expressed in terms of plain quark
lines (Fig. 2.1),

$\left(t^{a}\right)^{i}_{j}\left(t^{a}\right)^{\ell}_{k}=\frac{1}{2}\,\delta^{i}_{k}\,\delta^{\ell}_{j}-\frac{1}{2\mathrm{N_{c}}}\,\delta^{i}_{j}\,\delta^{\ell}_{k}.$
(2.9)

Figure 2.1: Graphical representation of the Fierz identity for a one–gluon
exchange between a quark–antiquark pair ($t$–channel exchange).

If one considers the limit $\mathrm{N_{c}}\rightarrow+\infty$, with
$\left(\mathrm{N_{c}}\,g_{s}^{2}\right)$ held fixed, then Eq. (2.9) (and Fig.
2.1) suggests that the colour structure of a gluon is equal to that of a quark
and an antiquark. This approximation444It is an approximation because for QCD
the number of colours is $\mathrm{N_{c}}=3$, which is hardly large enough.,
first introduced by ’t Hooft [102] in the context of strong interactions,
turned out to be a powerful one in simplifying perturbative calculations by
limiting the number of graphs contributing to a specific process. To leading
order in $\mathrm{N_{c}}$ only topologically _planar_ graphs need to be
computed. _Non–planar_ graphs’ contributions to physical cross sections are
suppressed by $1/\mathrm{N_{c}^{2}}$.

An example of planar and non–planar graphs is shown in Fig. 2.2, which occurs
in the $e^{+}e^{-}$ annihilation into $q\,\bar{q}$ accompanied with the
emission of two soft gluons. Utilising the colour machinery developed in the
previous section (Sec. 2.1) and Appendix A, in particular Fig. A.1 and Eq.
(A.3a), one obtains, to leading $\mathrm{N_{c}}$, an overall factor of
$(\mathrm{N_{c}}\,g_{s}^{2})^{2}$ and $(g_{s}^{2})^{2}$ for graphs $(a)$ and
$(b)$ respectively. In other words, graph $(b)$’s contribution to the
amplitude squared is suppressed by $1/\mathrm{N_{c}^{2}}$ factor relative to
graph $(a)$’s contribution.

Figure 2.2: Graph $(a)$ is an example of a planar graph which contributes at
leading $\mathrm{N_{c}}$. Graph $(b)$ is an example of non–planar graph which
contributes at subleading $\mathrm{N_{c}}$. The graphs represent _Feynman
diagrams_ for the amplitude squared of $e^{+}e^{-}$ anihilation into
$q\bar{q}gg$ (to be discussed in Sec. 2.2 and Appendix B).

It is worthwhile noting that this approximation is the basis of the “colour
dipole evolution” formalism that stands at the heart of some numerical Monte
Carlo programs such as ariadne [103] and the program developed in [2]. In the
said formalism, the probability of a (soft) gluon emission off an ensemble of
harder partons (quarks, antiquarks and gluons) is simply proportional to the
sum of the independent colour dipoles formed by the ensemble (see eikonal
method in Appendix B.1).

###  2.2 Lagrangian and gauge invariance

The full QCD Lagrangian density may be written as (see e.g., Refs. [7, 104]):

$\mathcal{L}_{\rm QCD}=\mathcal{L}_{\rm Dirac}+\mathcal{L}_{\rm Yang-
Mills}+\mathcal{L}_{\rm gauge-fixing}+\mathcal{L}_{\rm ghost},$ (2.10)

where the various terms are defined below:

  * •

The quark content of QCD is described by the Dirac Lagrangian density

$\mathcal{L}_{\rm
Dirac}=\sum_{f=1}^{n_{f}}\,\overline{\mathbf{q}}_{f}\left(\imath\,D\\!\\!\\!\\!/-m_{f}\right)\mathbf{q}_{f},$
(2.11)

where $\mathbf{q}$ labels a quark field of flavour $f$ and mass $m_{f}$. In
QCD there are $n_{f}=6$ flavours of quark fields and their conjugates,
$\overline{\mathbf{q}}_{f}=\mathbf{q}_{f}^{\dagger}\,\gamma^{0}$. The quark
field forms a three–vector in the colour space;
$\mathbf{q}=\left(q_{1},q_{2},q_{3}\right)$ with $q_{a}$ being Dirac spinor
fields555The three values of the colour index $a$ are usually referred to as
red, blue and green, as mentioned in the introductory part.. The symbol
$D\\!\\!\\!\\!/$ denotes the product $\gamma^{\mu}\,D_{\mu}$ where
$\gamma^{\mu}$ are the traceless Dirac matrices satisfying the Clifford
algebra $\\{\gamma^{\mu},\gamma^{\nu}\\}=2\,g^{\mu\nu}$, with the latter
metric given by $g^{\mu\nu}=\rm{diag}\left(+1,-1,-1,-1\right)$666Especially
useful representation of the Dirac $\gamma$ matrices is the Weyl (or chiral)
representation. Explicit expressions of the latter may be found in e.g.,
[93]., and $D_{\mu}$ is the covariant derivative given by

$D_{\mu}=\partial_{\mu}-\imath\,g_{s}\,A_{\mu}.$ (2.12)

The $\rm SU(3)$ gluon fields are given by
$A_{\mu}=\sum_{a=1}^{8}A_{\mu}^{a}\,t^{a}$ when $D_{\mu}$ is acting on the
quark field (colour triplet) and $A_{\mu}=\sum_{a=1}^{8}A_{\mu}^{a}\,T^{a}$
when acting on the gluon field (colour octet), with $t^{a}$ and
$T^{a}\left(\equiv\left(T^{a}\right)_{bc}\right)$ being, respectively, the
generators of $\rm SU(3)$ in the fundamental and adjoint representations
discussed previously. The strong coupling parameter, $g_{s}$, measures the
strength of the strong interaction between quarks and gluons or gluons amongst
themselves.

  * •

The Yang–Mills Lagrangian density describes the free propagation of the gluon
field and reads

$\mathcal{L}_{\rm Yang-
Mills}=-\frac{1}{2}\operatorname{Tr}\left[F_{\mu\nu}^{2}(A)\right]^{2},$
(2.13)

where the non–abelian, i.e., non–commutative, $\rm SU(3)$ field strength
tensor, $F_{\mu\nu}=F_{\mu\nu}^{a}\,t^{a}$, is defined by the commutator

$\left[D_{\mu},D_{\nu}\right]=-\imath\,g_{s}F_{\mu\nu}.$ (2.14)

Explicitly written in terms of the gluon fields,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-\imath\,g_{s}\left[A_{\mu},A_{\nu}\right]$.
The sum of the Dirac and Yang–Mills Lagrangians is the _classical_ QCD
Lagrangian. The latter is invariant under the local gauge transformations
(represented by the $\rm SU(3)$ matrix $\rm U$ given in Eq. (2.2)):

$\displaystyle\mathbf{q}(x)$ $\displaystyle\rightarrow$
$\displaystyle\mathbf{q}^{\prime}(x)=\mathrm{U}(x)\,\mathbf{q}(x),$
$\displaystyle A_{\mu}(x)$ $\displaystyle\rightarrow$ $\displaystyle
A_{\mu}^{\prime}(x)=\mathrm{U}(x)\left[A_{\mu}(x)+\frac{\imath}{g_{s}}\,\partial_{\mu}\right]\mathrm{U}^{\dagger}(x).$
(2.15)

The field strength tensor (2.14) thus transforms under the latter finite
transformations (2.15) as

$F_{\mu\nu}\rightarrow F^{\prime}_{\mu\nu}=\mathrm{U}\,F_{\mu\nu}\,\rm
U^{\dagger}.$ (2.16)

Considering infinitesimal gauge transformations, for which the $\rm U$ matrix
may be written as an expansion around the unit matrix; $\rm
U=\mathbb{1}+\imath\,\omega^{a}t^{a}$, then it follows from Eq. (2.16) that
$F_{\mu\nu}$ is not gauge invariant,

$F_{\mu\nu}^{a}\rightarrow
F_{\mu\nu}^{a\,^{\prime}}=F_{\mu\nu}^{a}-f_{abc}\,F_{\mu\nu}^{b}\,\omega^{c}+\mathcal{O}(\omega^{2}).$
(2.17)

This result is in contrast to the QED case where the photon field strength
tensor is gauge invariant. The immediate consequence of this is that gluons
are coloured and cannot thus be seen as asymptotic states. Notice also that
since a mass term $A_{\mu}\,A_{\nu}$ breaks the gauge invariance then it
cannot be included in the Lagrangian density (2.13). It follows that gluons
are massless.

  * •

To eliminate the gauge freedom in the Yang–Mills Lagrangian one must fix a
gauge to work in. Without the gauge fixing term the gluon propagator is not
well defined (see e.g., Refs. [91, 105]). One possible class of gauge
conditions is that of _covariant_ gauges, defined by

$\mathcal{L}_{\rm gauge-
fixing}=-\frac{1}{2\,\xi}\left(\partial_{\mu}\,A^{\mu}\right)^{2},$ (2.18)

where $\xi$ is the gauge parameter. Any value of $\xi$ can be used and any
gauge–invariant calculation is independent of the choice of latter. Amongst
the commonly used choices are $\xi=1,0$ for Feynman and Landau gauges
respectively.

  * •

To remove the unphysical polarisation states of the gluon vector field, one
introduces an unphysical Faddeev–Popov _ghost_ term in the QCD Lagrangian
[106]. This is best illustrated using the path integral formalism of quantum
field theory (QFT) [107]. For the above mentioned covariant gauges the ghost
Lagrangian reads

$\mathcal{L}_{\rm
ghost}=\partial_{\mu}\,\eta^{a\dagger}D^{\mu}_{ac}\,\eta^{c},$ (2.19)

with $\eta$ being a complex scalar field with anti–commuting properties of a
spinor field777A sign that is an unphysical field. Such a term is not even
needed in certain gauges such as axial gauges..

####  2.2.1 Perturbation theory and Feynman rules

The QCD Lagrangian888We use the terminologies Lagrangian and Lagrangian
density interchangeably. The former is the three–volume integral of the
latter. (2.10) may be written as the sum of _free_ , $\mathcal{L}_{0}$, and
_interaction_ , $\mathcal{L}_{\rm I}$, Lagrangians:

$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{\rm I}.$ (2.20)

$\mathcal{L}_{0}$ contains terms that are bilinear in fields whilst
$\mathcal{L}_{\rm I}$ include all remaining terms of the full Lagrangian.
Perturbation theory is constructed on the basis that the interaction part,
$\mathcal{L}_{\rm I}$, can be treated as small modifications, or
_perturbations_ , to the free theory, described by $\mathcal{L}_{0}$. We can
conceptually sketch a scattering process $i\rightarrow f$ as starting with an
initial state $\left\lvert i\right\rangle$ of freely propagating particle
waves at time $t_{0}$ and ending with a final state $\left\lvert
f\right\rangle$ of freely propagating particle waves at time $t$. The
interaction of the free particles with the force carrying particles take place
at a very short instant within the interval $\left(t-t_{0}\right)$. The
corresponding time evolution operator of the state $\left\lvert
i\right\rangle$ is given by999We work in the _interaction_ picture of QFT
throughout. For more details on the latter consult, for instance, [93].

$\mathcal{T}\left(t,t_{0}\right)=\mathrm{T}\left\\{\exp\left[-\imath\int_{t_{0}}^{t}\textrm{\,d}^{4}x\,\mathcal{L}_{\rm
I}(x)\right]\right\\},$ (2.21)

where $\rm T\\{\cdots\\}$ stands for the _time ordered_ product of fields in
$\mathcal{L}_{\rm I}$. In the limit where the initial and final states take
place at a long time in the past, $t_{0}\rightarrow-\infty$, and a long time
in the future, $t\rightarrow+\infty$, respectively, the evolution operator
(2.21) is simply the $S$–matrix,

$S\equiv\lim_{t_{0}/t\rightarrow\mp\infty}\mathcal{T}(t,t_{0}).$ (2.22)

The $S$–matrix encodes a plethora of information about the scattering process.
The entries of the $S$–matrix, or the matrix–elements $S_{fi}$, are related to
the $i\rightarrow f$ transition amplitude (or invariant matrix elements)
$\mathcal{M}_{fi}$ through,

$S_{fi}=\left\langle f\right\rvert S\left\lvert
i\right\rangle=\delta_{fi}-\imath T_{fi},$ (2.23)

with the $T$–matrix elements given by

$\imath
T_{fi}=\imath\left(2\pi\right)^{4}\delta^{(4)}\left(P_{i}-P_{f}\right)\,\mathcal{M}_{fi},$
(2.24)

where $P_{i(f)}$ is the total four–momentum of initial (final) state
particles. The delta term $\delta_{fi}$ in (2.23) is the zeroth order in the
expansion of (2.22) and describes the state where no interaction happens. The
transition amplitude $\imath T_{fi}$, or $\imath\mathcal{M}_{fi}$, consists of
the timed–ordered expansion of $S$, starting at first order and adding an
extra interaction term $\mathcal{L}_{\rm I}(x)$ at each higher order.

To compute $\mathcal{M}_{fi}$ one employs a set of _Feynman rules_ , along
with a graphical visualisation of the process via _Feynman graphs_ , derived
from the Lagrangian (2.10). These rules are represented by _external_ lines
for incoming (initial) or outgoing (final) particles, _internal_ lines for
propagators (that are neither initial nor final) and _vertices_ for the
interaction terms. The set of Feynman rules for QCD are depicted in Fig. 2.3.

Figure 2.3: Feynman rules for QCD in a covariant gauge: gluons (curly lines),
fermions (solid lines) and ghosts (dashed lines).

####  2.2.2 Optical theorem and cross-sections

An important feature of the $S$–matrix that is useful in checking the
consistency of Feynman amplitude calculations is _unitarity_. This follows
from the fact that the probability of producing a final state $\left\lvert
f\right\rangle$ given an initial state $\left\lvert i\right\rangle$ is simply
$\left|S_{fi}\right|^{2}$. Summing over all possible final states, the total
probability must be equal to unity. Schematically

$\sum_{f}\left|S_{fi}\right|^{2}=1\;\;\Rightarrow\;\;S^{\dagger}S=1.$ (2.25)

Substituting for $\imath T$ from Eq. (2.23), the unitarity relation (2.25)
implies

$-\imath\left(T_{i^{\prime}i}-T^{\star}_{i^{\prime}i}\right)=\sum_{f}T_{fi^{\prime}}^{\star}T_{fi}.$
(2.26)

In the special case of elastic, $\left\lvert
i^{\prime}\right\rangle=\left\lvert i\right\rangle$, two–particle scattering,
Eq. (2.26) leads to the optical theorem:

$\mathfrak{Im}\left(\mathcal{M}_{ii}\right)=\lambda^{1/2}(s,m_{a}^{2},m_{b}^{2})\,\sigma_{\rm
tot}\left(i\rightarrow X\right),$ (2.27)

which relates the forward scattering amplitude to the total cross section for
production of all possible final states $X$. The normalisation factor
$\lambda(s,m_{a}^{2},m_{b}^{2})$ given in terms of the centre–of–mass energy
squared $s$ and the masses of the scattering particles $a$ and $b$ has the
form $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2\,xy-2\,xz-2\,yz$. The total cross-
section $\sigma_{\rm tot}$ is given by:

$\sigma_{\rm tot}(i\rightarrow
X)=\sum_{f}\,\int\,\mathrm{F}\,\left\lvert\overline{\mathcal{M}_{fi}}\right\rvert^{2}\,\textrm{\,d}\Pi_{f},$
(2.28)

with the sum being over all possible final states $\left\lvert
f\right\rangle$, $\rm F$ is the flux of initial particles,
$\left\lvert\overline{\mathcal{M}_{fi}}\right\rvert^{2}$ is the invariant
matrix elements squared summed (averaged) over final (initial) state spin,
polarisation and colour and $\textrm{\,d}\Pi_{f}$ is the corresponding final
state phase space. Let $\left\lvert f\right\rangle$ be composed of $n$
particles with momenta
$p_{\ell}^{\mu}=\left(E_{\ell},\mathbf{p}_{\ell}\right)$, then the
$n$–particle Lorentz invariant phase space is defined by

$\displaystyle\textrm{\,d}\Pi_{n}$ $\displaystyle=$
$\displaystyle\left(2\pi\right)^{4}\delta^{(4)}\left(P_{f}-p_{a}-p_{b}\right)\prod_{\ell=1}^{n}\frac{\textrm{\,d}^{3}\mathbf{p}_{\ell}}{\left(2\pi\right)^{3}2E_{\ell}}\,\frac{1}{\rm
m\,!},$ (2.29) $\displaystyle\rm F$ $\displaystyle=$
$\displaystyle\left(4E_{a}E_{b}\left|\mathbf{v}_{a}-\mathbf{v}_{b}\right|\right)^{-1}=\frac{1}{2\lambda^{1/2}(s,m_{a}^{2},m_{b}^{2})},$
(2.30)

where $\mathbf{v}_{a}$ and $\mathbf{v}_{b}$ are the three–velocities of the
two particles ($a$ and $b$) in the initial state and $\rm m$ is the number of
identical particles in the final state.

###  2.3 Running of $\alpha_{s}$

QCD is a _renormalisable_ 101010Renormalisation may be summarised, very
briefly, as a procedure for treating ultra–violet (UV) divergences of Feynman
amplitudes. For an intensive discussion of renormalisation the reader is
referred to e.g., Ref. [108]. field theory. This implies that the coupling
$\alpha_{s}\equiv g_{s}^{2}/4\pi$ (defined with analogy to the fine structure
constant of QED $\alpha\equiv e^{2}/4\pi$) must be defined at the
renormalisation scale, which we denote $\mu$. The renormalisation flow of the
coupling $\alpha_{s}(\mu)$ is determined by the renormalisation group equation
(RGE), which can be cast in the form [7, 95]

$\mu^{2}\frac{\textrm{\,d}\alpha_{s}(\mu)}{\textrm{\,d}\mu^{2}}=\beta(\alpha_{s}(\mu)).$
(2.31)

The $\beta$ function is computed in perturbation theory as a power series in
$\alpha_{s}(\mu)$, from Feynman diagrams involving fermion self–energy, gluon
self–energy and fermion–gluon vertex, as shown in Fig. 2.4.

Figure 2.4: Feynman diagrams contributing to QCD $\beta$ function. (top)
fermion self–energy diagrams, (middle) gluon self–energy diagrams and (bottom)
fermion–gluon vertex diagrams.

We thus write

$\beta(\alpha_{s})=-\alpha_{s}^{2}\left(\beta_{0}+\beta_{1}\,\alpha_{s}+\cdots\right),$
(2.32)

where $\beta_{0}=\left(11\mathrm{C_{A}}-2n_{f}\right)/12\pi$,
$\beta_{1}=\left(17\mathrm{C_{A}}^{2}-5\mathrm{C_{A}}n_{f}-3\mathrm{C_{F}}n_{f}\right)/24\pi^{2}$
and the minus sign on the right–hand side of (2.32) distinguishes QCD from
QED, for which one has (see e.g., [109])

$\beta^{\rm QED}(\alpha)=+\alpha^{2}\left(\beta_{0}^{\rm QED}+\beta_{1}^{\rm
QED}\alpha+\cdots\right),$ (2.33)

with $\beta_{0}^{\rm QED}=n_{f}/3\pi$ and $\beta_{1}^{\rm QED}=1/4\pi^{2}$.
The solution to Eq. (2.31) determines the value of the coupling at a scale
$\mu_{2}$ in terms of its value at another scale $\mu_{1}$111111Usually the
magnitude of $\alpha_{s}$ in QCD is given at the mass of the $Z$ boson:
$\alpha_{s}(\mu=m_{Z})\sim 0.118$ [110]. The magnitude at any other scale is
then deduced from the solution of Eq. (2.31).. At lowest order, one obtains
for QCD an expression of the form

$\alpha_{s}(\mu_{2})=\frac{\alpha_{s}(\mu_{1})}{1+\beta_{0}\,\alpha_{s}(\mu_{1})\ln\left(\mu_{2}^{2}/\mu_{1}^{2}\right)}=\frac{1}{\beta_{0}\,\ln\left(\mu_{2}^{2}/\Lambda_{\text{QCD}}^{2}\right)},$
(2.34)

where the last equality follows from rewriting Eq. (2.31) as,

$\ln\left(\frac{\mu_{2}^{2}}{\mu_{1}^{2}}\right)=\int_{\alpha_{s}(\mu_{1})}^{\alpha_{s}(\mu_{2})}\frac{\textrm{\,d}\alpha_{s}}{\beta(\alpha_{s})},$
(2.35)

and setting $\mu_{1}\rightarrow\Lambda_{\text{QCD}}$. Provided the first
coefficient $\beta_{0}$ is positive, which is only true if the number of
flavours $n_{f}$ is less than about $16.5$, then the coupling weakens
(diverges) logarithmically at larger (smaller) renormalisation masses $\mu$.
Indeed, $\alpha_{s}$ approaches zero in the limit $\mu\rightarrow\infty$ and
blows up in the limit $\mu\rightarrow\Lambda_{\text{QCD}}$, as can be seen
from Eq. (2.34). The former limit is known as _asymptotic_ freedom [111, 112]
and justifies the validity of perturbation theory in this regime. The latter
limit signals the failure of the lowest–order approximation (2.34) where the
theory becomes strongly coupled and thus essentially _non–perturbative_. To
the contrary, the QED coupling $\alpha$ is small at low energies,
$\alpha(\mu=0)\sim 1/137$, and grows logarithmically at high energies,
$\alpha(\mu=m_{Z})\sim 1/128$.

####  2.3.1 Confinement and factorisation

Probed at sufficiently high energies, or momentum transfer $\gg\Lambda_{\rm
QCD}$, hadrons may reliably be considered as “bags” of weakly interacting
quarks and gluons. Consequently, collisions between hadrons can effectively be
regarded as scattering amongst their “free” constituent partons (quarks and
gluons). The _hard_ scattering process, to which mainly two partons, one from
each of the colliding hadrons, participate with the rest of the “spectator”
partons interfering at lower energies and thus forming hadron remnants, is
fully calculable in perturbation theory. Such a hard scattering occurs, by the
uncertainty principle, at very short distances.

As the energy falls to $\sim\Lambda_{\text{QCD}}$ or below, the quarks and
gluons inside a hadron become strongly coupled to the extent that it is
impossible to knock an individual quark out of the hadron. This phenomenon is
know as _confinement_. Starting with a meson (quark–antiquark bound state), if
one attempts to pull apart its constituents then instead of the meson breaking
into two quarks it breaks into two mesons. The new quark–antiquark pair is
created from quantum mechanical fluctuations once the energy in the “flux
tube”121212This concept arises in the investigation of QCD confinement from
the point of view of a non–abelian analogue to the Meissner effect in
superconductors [104]., formed between the original quark–antiquark pair,
becomes higher than the rest mass of the pair (recall that the masses of the
light quarks, $\mathcal{O}(5\text{ MeV})$, are far below the QCD
non–perturbative scale $\Lambda_{\text{QCD}}\sim 200\text{ MeV}$). Therefore
at low energies, or equivalently long distances, individual quarks cannot be
observed as they immediately confine, or _hadronise_ , into hadronic bound
states.

A typical event at the LHC proceeds as follows: First, two protons are made to
collide head–on at a high centre–of–mass energy. The actual hard scattering
occurs between the constituent partons, one from each proton. As the quarks
travel towards the outer parts of the detector they lose energy by emitting
gluons, which in turn radiate more gluons and/or split into quark–antiquark
pairs. At each step the energy falls and the number of quarks and gluons grows
up leading to the formation of a “shower” of partons. Eventually, at low
energies, quarks and gluons undergo the inevitable process of hadronisation
(confinement) into mesons and baryons, which are observed at detectors as
collimated energy depositions, or equivalently “jets”. The latter will be the
pivotal subject of the next chapter (Chapter 3).

Two processes that take place at two disparately different scales, long and
short distances, are quantum mechanically incoherent. i.e., their interference
vanishes, up to inverse powers of the hard scale of the system, as shown in
Eq. (2.36). This allows for a separate treatment of each process. Such a
separation is known as _factorisation_. The general form of a factorised
physical cross section for a process initiated by two hadrons with momenta
$P_{1}$ and $P_{2}$ is [62, 63, 97, 113]

$\sigma\left(P_{1},P_{2}\right)=\sum_{i,j}\int\textrm{\,d}x_{1}\textrm{\,d}x_{2}\,f_{i}(x_{1},\mu_{F})f_{j}(x_{2},\mu_{F})\,\hat{\sigma}_{ij}\left(p_{1},p_{2},\alpha_{s}(\mu_{F}),Q/\mu_{F}\right)+\mathcal{O}(1/Q^{p}),$
(2.36)

where $p_{i}=x_{i}P_{i}$ is the momentum of parton $i$ participating in the
hard scattering, as schematically depicted in Fig. 1.1. $\hat{\sigma}$ is a
short distance, hard scattering partonic cross section, which is _infrared
safe_ 131313Infrared (and collinear) safety, which refers to the insensitivity
to soft emissions and collinear splittings, is crucial to the finiteness of
perturbative calculations. This topic will be addressed in the next chapter
(Chapter 3). and calculable in perturbation theory, and $f$ is a long distance
function that is not calculable in perturbation theory. It is however
“universal” and can be measured experimentally once and for all. In essence,
the function $f$ describes the distribution (and dynamics) of partons in the
colliding hadrons. As such $f$ is termed the _parton distribution function_
(PDF)141414Considering final state hadrons, $f$ may be taken to be the
_fragmentation_ function (usually denoted $D$) [7]., and the subscripts $i$
and $j$ in Eq. (2.36) refer to the specific parton flavour (including gluons).
The symbols $Q$ and $\mu_{F}$ denote, respectively, the partonic hard and
factorisation scales. The latter marks the boundary between short distance and
long distance, or equivalently between perturbative and non–perturbative,
dynamics. Thus a parton emitted at a scale less than $\mu_{F}$ will be
absorbed into the parton distributions, while a parton emitted at a scale
larger than $\mu_{F}$ will be considered as part of $\hat{\sigma}$. The more
orders included in $\hat{\sigma}$ the less dependence of $\hat{\sigma}$ on the
scale $\mu_{F}$. The last term in Eq. (2.36), with $p$ being a real number, is
there to indicate that factorisation holds up to corrections that behave as
inverse powers of $Q$. It is worth noting that the factorisation formula
(2.36) has been proven for sufficiently inclusive observables in a variety of
processes, e.g., Drell–Yan process (hadron + hadron $\to\ mu^{+}\mu^{-}$ \+ X)
and scattering of colourless hadrons (hadron + hadron $\rightarrow$ hadron +
X), where X denotes “anything else” in the final state [62, 63, 64, 114].

Since physical quantities, such as $\sigma$, cannot depend on the (arbitrary)
factorisation scale $\mu_{F}$ then

$\frac{\partial\,\ln\sigma}{\partial\ln\mu_{F}}=0.$ (2.37)

Although parton distributions are not calculable from first principles of QCD,
as stated above, (2.37) allows for perturbative calculations of their
dependence on the scale $\mu_{F}$. Substituting Eq. (2.36) into Eq. (2.37), we
schematically have

$\frac{\partial\ln f}{\partial\ln\mu_{F}}\propto P,\qquad f=q_{i},g.$ (2.38)

where the function $P$ can only depend on the shared variables between
$\hat{\sigma}$ and $f$; the coupling and convolution variables. The solution
of Eq. (2.38) describes the “evolution” of the PDFs between two scales, say,
$\mu_{F}$ and $\mu^{\prime}_{F}$. Once measured experimentally at a scale
$\mu_{F}$, they can be perturbatively predicted at arbitrary high scales,
$\mu^{\prime}_{F}\gg\Lambda_{\text{QCD}}$, (provided the coupling remains
small over the evolution range). The function $P$ actually refers to the
Altarelli-Parisi splitting functions (or evolution kernels) and Eq. (2.38) is
the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equation
[115, 116, 117, 118]. It is one of the fundamental equations in pQCD
(analogous to the $\beta$ function equation (2.31)). Written in the standard
form, including explicit dependence on the partons’ (longitudinal) momentum
fractions, the latter evolution equation reads [7].

$\frac{\partial
f_{i}(x,\mu_{F})}{\partial\ln\mu_{F}}=\frac{\alpha_{s}\left(\mu_{F}\right)}{2\pi}\,\sum_{j}\int_{x}^{1}\frac{\textrm{\,d}z}{z}\,f_{j}(z,\mu_{F})\,P_{ij}\left(\frac{x}{z},\alpha_{s}(\mu_{F})\right).$
(2.39)

The splitting functions $P_{ij}$ essentially correspond to the rate of
production of parton $i$ from parton $j$ as the process evolves in
$\ln\mu_{F}$. They have the following perturbative expansion in the running
coupling:

$P_{ij}\left(z,\alpha_{s}\right)=P_{ij}^{(0)}(z)+\frac{\alpha_{s}}{2\pi}P_{ij}^{(1)}(z)+\cdots,$
(2.40)

where the leading splitting functions are

$\displaystyle P_{qq}^{(0)}(z)$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left[\frac{1+z^{2}}{(1-z)_{+}}+\frac{3}{2}\,\delta(1-z)\right],$
$\displaystyle P_{qg}^{(0)}(z)$ $\displaystyle=$
$\displaystyle\mathrm{T_{F}}\left[z^{2}+\left(1-z\right)^{2}\right],$
$\displaystyle P_{gq}^{(0)}(z)$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left[\frac{1+\left(1-z\right)^{2}}{z}\right],$
$\displaystyle P_{gg}^{(0)}(z)$ $\displaystyle=$ $\displaystyle
2\,\mathrm{C_{A}}\left[\frac{z}{(1-z)_{+}}+\frac{1-z}{z}+z(1-z)\right]+\left[\frac{11\,\mathrm{C_{A}}-4\,n_{f}\mathrm{T_{R}}}{6}\right]\delta\left(1-z\right).$

Additionally $P_{qq^{\prime}}^{(0)}=P_{q\bar{q}}^{(0)}=0$. The “plus”
distribution used in the above equations is defined by

$\int_{0}^{1}\textrm{\,d}x\;\frac{h(x)}{(1-x)_{+}}=\int_{0}^{1}\textrm{\,d}x\;\frac{h(x)-h(1)}{(1-x)},$
(2.42)

where $h(x)$ is an arbitrary smooth function. The state–of–the–art of the
splitting kernels is up to $\mathcal{O}(\alpha_{s}^{3})$ [119, 120].

Lastly, notice that the DGLAP equation (2.39) can be solved numerically either
in $x$-space or in its conjugate Mellin space:

$f(N,\mu)=\int_{0}^{1}\textrm{\,d}x\,x^{N-1}f(x,\mu).$ (2.43)

In the latter space it may be possible to solve the equation analytically.
Inverting back to the $x$–space is not, except in very special cases, however
analytically possible. Throughout this thesis we shall adopt the standard
choice $\mu_{F}=Q$ and thus do not worry about the evolution of parton
densities. There is a plethora of numerical programs that straightforwardly
handle such an evolution, see e.g., [121] (and references therein). Lastly
note that a conventional way of estimating PDF uncertainties is to choose the
renormalisation scale $\mu_{R}$, which has so far been implicitly taken to be
equal to $\mu_{F}$, to be $\mu_{R}^{2}=(x_{\mu}\mu_{F})^{2}$ and vary
$x_{\mu}$ in the range $1/2<x_{\mu}<2$ (often just takes the values
$x_{\mu}=1/2,1,2$) [122].

In the next chapter, we address more specific issues that are of direct
concern in the current thesis. Namely the phenomenology of “jets”.

##  Chapter 3 Jet phenomenology

Ideally, theoretical predictions and/or interpretation of experimental data of
final states at colliders are to be performed in terms of short distance
colour–charged partons. However, what is observed instead at these colliders
are long distance colour–neutral hadrons (along with other particles such as
leptons), which are composite states of the latter partons. More precisely,
the final state hadrons, and particles, are observed in collimated sprays,
rather than large–angle separated individual particles, forming “jets” that to
some extent resemble the final state of the underlying partonic process. Fig.
3.1 is a schematic illustration of jet formation at hadron colliders111For
$e^{+}e^{-}$ machines, only the schematic view of the final state applies. The
initial–state is simply point–like electrons and positrons.. The whole
process, from initial colliding beam particles to final state detected
particle jets, may be thought of as, approximately, three disentangled stages
according to the energy scales (distance) probed: low (long), high (short) and
then low (long). Such a separation is a product of factorisation discussed in
the previous chapter (Subsec. 2.3.1).

Figure 3.1: A schematic view of jet production at hadron colliders. The
figure, modified from [1], shows a proton–proton collision at the LHC.

We summarise the three stages, bottom–up in Fig. 3.1, as follows222We have
already described hadronic collisions in Subsec. 2.3.1 in a broad sense. Here
we provide more details and set the scene for the subsequent sections.. As the
circulating bunches of protons in the LHC beam–pipe collide head–on, a single
“event” takes place between a proton–proton pair. A single parton from each
proton participates in the short–distance hard scattering process. The
distribution of partons within each proton is determined through PDFs which
depend, among other variables, on the factorisation scale $\mu_{F}$ (usually
set equal to the hard process scale) and the parton’s momentum fraction $x$
[123]. The insofar described stage is considered non–perturbative due to the
fact that PDFs are not calculable in pQCD but rather determined from global
fits to data collected from various experiments (primarily Deep Inelastic
Scattering (DIS), see e.g., [124, 125]). On their way to the hard interaction
point, the incident partons will radiate. Such a radiation is dubbed (for
obvious reasons) initial state radiation (ISR), and will contribute either to
the evolution of the PDFs (DGLAP equation (2.39)), if it is collinear
radiation, or to the final state, if it is wide–angle radiation333It is still
not entirely clear whether the ISR collinear radiation completely factorises
out from the final state for non-inclusive observables in hadron-hadron
collisions. See for instance Refs. [126, 127, 128, 129] for detailed
discussion on QCD coherence and factorisation breaking.. The spectators, the
other constituent partons of the incident protons, also interact with each
other and emit radiation, but at a much softer scale relative to the hard
scattering scale, contributing to the non–perturbative underlying event (UE).

At the interaction point the short–distance perturbative stage takes place. It
is characterised by large momentum transfers and may result in changing the
flavour of the scattering partons and/or producing more partons or other BSM
particles. The corresponding scattering cross section is calculable in pQCD.
While leaving the interaction point the final state coloured partons emit
radiation in the form of low energy and/or collinear (to the emitting parton)
gluons and quark–antiquark pairs. This is the final state radiation (FSR),
which forms the showering phase and leads to large logarithmic contributions
to the final cross section that may be “resummed” either analytically or
numerically via parton shower Monte Carlos (PSMCs), to a certain level of
accuracy (see for instance [32, 76]). This issue -resummation- stands at the
heart of this thesis and will be examined in detail in Sec. 3.2. It should be
emphasised, as stated previously, that the above picture, that ISR and FSR are
disentangled from each other, is strictly speaking a probabilistic
(approximate) picture, which is implemented in PSMCs. Physically, there will
always be “quantum interferences” between the two radiations leading to
contributions of the type investigated in [130] (“fifth” form factor).

As the process evolves away from the hard interaction point towards lower
energies (longer distances) the coloured partons resulting from the showering
as well as those coming from softer interactions, both from UE and PU444See
definitions in introductory chapter 1., undergo a non–perturbative
hadronisation phase yielding colour singlet hadrons. The latter (massive)
hadrons are produced either at ground or excited state. The excited state
hadrons, or “resonances”, decay to lighter ones, at ground state, with high
(low) longitudinal (transverse) momentum, with respect to the direction of the
outgoing resonance. At high centre–of–mass energies, $\sqrt{s}\gg$ EW scale
$\sim 90\text{ GeV}$, massive particles, both elementary and bound–state
resonances in SM and BSM, decay into lighter particles that are highly boosted
in the lab frame. The resultant final jets have a different topology to the
unboosted case. This field has seen substantial developments in the very
recent years (see e.g., the boost11 review [26]).

At leading–order, parton jets are infinitely narrow and only start to build
non–vanishing width at next–to–leading order. The showering and hadronisation
phases contribute towards the increase of the latter width, thus smearing out
the energy distribution within the jet (a “splash–out” effect [1]). The UE and
PU, on the other hand, give rise to a “splash–in” effect whereby extra (soft)
energy is added to the jets. The final hadron (jets) eventually pass through
the various layers of the detector where they undergo some interactions
resulting in -yet another level of complexity- “calorimeter” jets, shown at
the top of Fig. 3.1, which embodies the detector response. The latter
(detector-level jets) is beyond the scope of the present thesis and we refer
the interested reader to Ref. [1] for more information. Hitherto, we have
presented no formal definition of a jet that applies at all levels; parton,
shower, hadron and detector. We do so in Sec. 3.1. Once a jet is properly
defined, we then exploit its usage in understanding QCD through exploring its
“internal” and “external” properties in Sec. 3.2. We succinctly summarise the
main points in Sec. 3.3.

It is worth noting that the field of jets (and related topics) has seen a
substantial growth over the last few years, mainly due to the fact that the
final state at the LHC is predominantly jets. Different research groups
tackled the issue from different angles and with different aims (understanding
QCD, background–signal discrimination, $\cdots$). A sample of literature
reviews on the topic may be found in Refs. [1, 26, 27, 31, 80, 131, 132, 133,
134] (and references therein).

###  3.1 Jet clustering algorithms

The measured kinematics (four–vectors) of “jets” at the detector level (from
energy depositions into calorimeter cells and tracks of charged particles)
provide a measure of the kinematics of the underlying short–distance theory
partons and/or decaying massive (boosted) particles555After effects of FSR,
ISR, hadronisation, UE and PU are properly estimated and corrected for at a
sufficiently reliable precision level. [1, 135, 136, 137, 138]. Jets are not,
however, intrinsically well–defined objects, just as partons are not
well–defined due to their infinite branching probabilities in pQCD, and thus a
prescription for defining them is required. A wide variety of “jet
definitions” have been proposed in the literature (for a review see, for
instance, Refs. [139, 133]). A typical jet definition consists of a _jet
algorithm_ and a _recombination scheme_. A jet algorithm is essentially a set
of mathematical rules that details the procedure of merging (“daughter”)
objects into “protojets” (parent objects) based on a measure of their
“closeness”. The objects, described by their $4$–vectors, refer to partons in
perturbative final state, final state hadrons in the simulated Monte Carlo
events or calorimeter output at detectors. The recombination scheme specifies
how the kinematics of the parent protojets are constructed from those of the
daughters.

Different jet definitions implement different jet algorithms and/or
recombination schemes with different strengths and weaknesses. Some important
general properties that should be fulfilled by a jet definition for it to be
practically useful were set out by the “Snowmass accord” [140]. They include,
along others, the crucial requirement of infrared and collinear (IRC) safety
of the jet algorithm. The presently available jet algorithms may be divided
into two main categories, depending on the manner they operate: cone
algorithms and sequential recombination algorithms. In what follows below, we
first provide a brief description of the first type and take it as an example
to address the issue of IRC safety. We then consider the second type of jet
algorithms. Notice that our calculations in this thesis are confined to the
latter type of algorithms and they are thus discussed in more detail (than
cone algorithms).

####  3.1.1 Cone algorithms

The first appearance of a jet algorithm was in the pioneering work of Sterman
and Weinberg [141], wherein they calculated the (partial) cross section for
$e^{+}e^{-}$ hadronic events, in which all but a fraction $\epsilon\ll 1$ of
the total $e^{+}e^{-}$ energy is emitted within some pair of oppositely
directed _cones_ of half–angle $\delta\ll 1$. This was the first basic
definition of a cone algorithm in terms of two parameters; energy and angle.

A modern cone algorithm operates in two distinct steps. In the first step it
searches for _stable_ cones666A stable cone is a cone whose axis coincides
with the axis of the sum of all three–momenta of particles it contains.. In
the second step it runs a _split–merge_ procedure to resolve any overlapping
stable cones. The outcome of this procedure is that each object, in the
initial list, is assigned to a single jet in the final list. Insofar the only
IRC safe cone algorithm available is the seedless infrared safe cone (SISCone)
algorithm developed in Ref. [142]. A detailed description of various cone
algorithms may be found in the review [133].

####  3.1.2 IRC safety of jet algorithms

As mentioned in the outset, a generic jet algorithm clusters objects that are
close to each other (in phase space). This feature guarantees that divergent
perturbative contributions from virtual and soft and collinear real emissions
cancel out. This implies that the number of hard jets found by the algorithm
must remain intact when an object undergoes a collinear splitting or a soft
emission is added to the ensemble of the initial objects. It is only through
this real–virtual cancellation that the algorithm can serve to define finite
quantities, such as cross section. Furthermore, being insensitive to collinear
splittings means that the jet algorithm will not significantly be affected by
non–perturbative processes such as fragmentation and decay of hadrons
(collinear splittings). Also, being insensitive to soft radiation helps in
minimising exposure to contaminations from UE and PU [133].

Fig. 3.2 schematically illustrates IRC unsafe jet algorithms of cone type,
where stable cones are found by iterating from _all_ objects (seeds) and then
running a split–merge procedure, as described above. The unsafety is
manifested in the change of the number of final jets found by the algorithm.
In Fig. 3.2 $(a_{1})$, the leftmost and middle objects are within a distance
$R$ while the rightmost is more than a distance $R$ away from the middle
object. The said cone algorithm, with radius $R$, finds two stable cones with
no overlapping and hence two final jets. Adding a soft object to the ensemble,
as in Fig. 3.2 $(a_{2})$, leads to three overlapping stable cones. The
split–merge procedure may then easily lead to the identification of three
final jets. This is IR unsafety.

Now consider a cone algorithm that iterates from the hardest object first. In
Fig. 3.2 $(b_{1})$, the middle object is the hardest and hence sets the first
seed. Given that the other two objects are within a distance $R$, the
algorithm finds a single stable cone and thus one final jet. If the middle
object undergoes a collinear splitting, as in Fig. 3.2 $(b_{2})$, then the
hardest object becomes the leftmost and hence sets the first seed. Iteration
leads to a jet containing the leftmost and middle objects only. The remaining
rightmost object provides a new seed and goes on to form a separate second
jet. This is collinear unsafety.

Figure 3.2: Figures $(a_{1}),(a_{2})$ illustrate the output of a an IR unsafe
cone algorithm and figures $(b_{1}),(b_{2})$ illustrate the output of a
collinear unsafe cone algorithm. The plane $(\mathbf{p_{t}},y)$ is the 2D
projection of the 3D phase space $(\mathbf{p}_{t},y,\phi)$.

It is crucial to note that perturbative calculations based on IRC unsafe jet
algorithms yield infinite answers at some order, say $\alpha_{s}^{n}$, in the
perturbation expansion. In our example, Fig. 3.2 $(b_{1})$ and $(b_{2})$, the
algorithm fails real-virtual cancellation at the first order, $\alpha_{s}$.
Thus, at and beyond this particular order ($n$) the perturbative series is not
meaningful since all terms are as large as the $(n-1)^{\rm th}$ term and
cannot be ignored. Therefore we shall exclusively work with IRC safe
algorithms when performing any calculation throughout this thesis. Moreover,
all the jet algorithms that we shall be discussing and many others are
implemented in the FastJet package [143].

Next we discuss the second type of jet algorithms, which are both much simpler
and IRC safe.

####  3.1.3 Sequential recombination algorithms

The JADE algorithm [144] was the first practically useful777As mentioned in
[133] earlier original, and somewhat complicated, work on sequential
recombination algorithms and related properties appeared in e.g., Refs. [145,
146]. version of a sequential recombination (SR) algorithm applied on
$e^{+}e^{-}$ hadronic multijet final states. SR algorithms proceed onto
building final jets by recursively recombining pairs of objects in a
$2\rightarrow 1$ sequence. At the perturbative level they work in a “reverse”
fashion to parton branching ($1\rightarrow 2$ sequence). This
-clustering–branching- analogy is fostered further by the fact that the
clustering procedure, of an SR algorithm, takes on an approximate
probabilistic picture, just as the branching process does. These and other
discernible characteristics have made SR algorithms a powerful tool not only
to find jets but to carry out detailed analyses on the internal structure of
jets. The latter “jet substructure” analyses, based on smart utilisations of
SR algorithms along with some other methods, have grown sufficiently large and
deserve a separate section (Subsec. 3.1.4).

Modern SR algorithms come in a rich spectrum. They share a common strategy
though. In Algorithm 1 we present the details of an _inclusive_ variant of a
typical SR algorithm that depends on a radius $R$ and a continuous parameter
$\rm p$. Such algorithm is referred to in [143] as “generalised $k_{t}$”
algorithm. The _exclusive_ variant of Algorithm 1 is similar except that:

  1. 1.

After step $5$ (and before step $6$): If $d_{\min}<d_{\rm cut}$, where $d_{\rm
cut}$ is a resolution cut–off parameter, then continue to step $6$. Otherwise,
declare all remaining objects to be final jets and terminate the iteration.

  2. 2.

Step $9$: if $d_{\min}$ is a $d_{iB}$ then object $i$ is considered to be part
of the “beam jet” and is thus discarded. Return to step $1$.

Algorithm 1 Inclusive sequential recombination algorithm (generalised $k_{\rm
T}$ [143])

0: list of objects $\\{p_{i}^{\mu}\\}$, jet-radius $R$, real parameter $\rm
p$.

0: $p_{i}^{\mu}$ is in terms of ($p_{ti},y_{i},\phi_{i}$), all measured with
respect to the beam direction.

1: Put the set of current objects equals to the set of all objects in the
event.

2: repeat

3: For each object compute its distance to the beam $B$ $d_{iB}=p_{ti}^{2\rm
p}.$ (3.1)

4: For each pair of objects $i,j$ compute the pairwise distance metric
$d_{ij}=\min\left(p_{ti}^{2\rm p},p_{tj}^{2\rm p}\right)\frac{\Delta
R^{2}_{ij}}{R^{2}},$ (3.2) where $\Delta
R_{ij}^{2}=\left(y_{i}-y_{j}\right)^{2}+\left(\phi_{i}-\phi_{j}\right)^{2}$.

5: Find the minimum $d_{\min}$ of all $d_{ij}$ and $d_{iB}$.

6: if $d_{\min}$ is a $d_{ij}$ then

7: merge objects $i$ and $j$ into a single protojet using a recombination
scheme and add (remove) the protojet (objects $i$ and $j$) to (from) the
current list of objects.

8: else

9: if $d_{\min}$ is a $d_{iB}$ then

10: declare object $i$ to be a final jet and remove it from the list of
current objects.

11: end if

12: end if

13: until No more objects left.

14: return list of final jets.

Note that the radius parameter $R$ is usually taken to be of order $1$ in the
exclusive variant.

The inclusive SR algorithms that will be considered in this thesis correspond
to the following special choices of the parameter $\rm p$:

  * •

$\rm p=-1$: anti-kT algorithm [73].

  * •

$\rm p=\;\;\;0$: Cambridge/Aachen (C/A) algorithm [147, 148].

  * •

$\rm p=\;\;\;1$: kT algorithm [149].

It should be noted that unlike the kT (and C/A), which proceeds by clustering
soft objects first (or via energy–independent clustering in C/A), the anti-kT
proceeds by first clustering soft objects to the harder ones in the
neighbourhood. As such the anti-kT algorithm is similar in spirit to a cone
algorithm with the exception that it maintains IRC safety due to its
hierarchical nature. While offering a valuable replacement for the cone
algorithm888Note that calculations with cone algorithms are far simpler than
with typical SR algorithms. The anti-kT retains the simplicity of the cone
algorithms while dispelling their IRC culprit., the substructure defined by
the anti-kT algorithm cannot be mapped onto that of QCD jets resulting from
parton branching. As a consequence of this, care should be exercised when
carrying out substructure studies using the anti-kT algorithm.

Each of the above listed algorithms has some strengths and weaknesses, which
are highlighted in [142], and using a variety of them for physics analyses is
highly desirable as concluded in [1]. Lastly it is worth mentioning that a
flurry of variants of SR algorithms999Some with extra parameters that need, in
most cases, to be tuned to the specific physics analysis being carried out.
have been developed over the past few years. These include, for instance,
flavoured $k_{t}$ algorithm [150], variable–$R$ algorithm [151] and arclus
($3\rightarrow 2$ clustering sequence) algorithm [152].

####  3.1.4 Jet grooming techniques

Jet grooming algorithms, which may be regarded as variants of SR algorithms,
were largely motivated by UE and PU investigations. Soft radiation coming from
the latter non–perturbative processes and which end up in jets may be
“groomed” (mitigated) away through modifying the substructure of the
respective jets. Each grooming algorithm operates on a single jet and produces
a “groomed” jet. The latter contains less junk–radiation and is thus of
improved resolution, leading to more accurate extractions of its properties.
Popular jet grooming algorithms include: filtering [14, 153], trimming [30]
and pruning [29]. Although all three methods aid in better identifying the
origin of a particular jet, trimming was originally proposed for use on jets
from QCD light quarks while filtering and pruning have typically been used on
jets from decay of (boosted) heavy particles [26].

As mentioned earlier, jet grooming is one of the main jet substructure
techniques devised to boost new physics searches in a jetty environment such
as that experienced at LHC. Other substructure tools and methods, aside from
jet shapes which we address below, include: Y–splitter [20], various taggers
[37, 154, 155, 156, 157, 158], shower deconstruction [159], three–body
kinematic variables [160] and many others (see Refs. [26, 27]101010See also
the boost12 page at “http://ific.uv.es/~boost2012/”.). Most of these
substructure tools have been incorporated into the FastJet
package111111SpartyJet [161] is another software/package built around FastJet
and provides a variety of other jet–oriented tools including measurements of
some event and jet observables.. This package, written in a “jet–oriented”
manner, has served as a framework for the development of a whole branch of
study that is primarily concerned with jets: “jetography”. The reader is
referred to Ref. [133] for more details.

###  3.2 Jet shapes

Event shape observables (or simply event shapes), such as thrust, characterise
the geometrical properties of the energy flow in QCD final state events. For
example, a pencil–like event produces a value of the (thrust) event shape that
is different from that produced by a sphere–like event. Event shapes are
constructed by taking a weighted sum of the four–momenta, transverse momenta,
energies etc of final state particles121212Particles here refer, as previously
discussed in Sec. 3.1, to: calorimeter towers, clusters or charged tracks at
the detector level, particles at hadronic level, many partons at the shower
level and few partons at perturbative level.. While imparting information on
the _global_ shape of an event, hence dubbed _global (event) shapes_ , event
shapes convey no information regarding the jet content of the event, such as
energy distribution within individual jets.

Event shape–like observables that are sensitive to the properties of single
jets in an event are known as _jet shapes_ 131313The term “jet shapes”
originally devised for the observable $\psi(r;R)$ which measures the fraction
of energy confined in an annulus of width $dr=R-r$ [162].. They can serve to
quantify a variety of topologies in the final state, from pure QCD–like to EW
bosons’ two–pronged decays to top quark (and other proposed new particles)
three–pronged decays141414More information on two– and three–pronged decays
may be found in [133].. Each of the latter mentioned topologies yields a
different value for the jet shape. Jet shapes are by construction of
_non–global_ nature, as they only measure specific parts, rather than the
whole, of the the event. Such a non–global property, besides making jet shapes
more delicate, opens up the door for a better scrutiny of QCD dynamics.

A trend of jet observables, under the broader class of jet substructure, have
been devised in the recent years. These include, for instance: N–subjettiness
[163], jet width and jet eccentricity [164], jet pull [165] and many others,
e.g., [166, 167, 168, 169, 170, 171, 172] (see Refs. [26, 27] for a review).
To facilitate later discussions, we concentrate in this introductory review on
the jet shape named _angularities_ [45, 167] (originally introduced in [38,
39] for two-jets in $e^{+}e^{-}$ annihilation), which embodies a spectrum of
other jet shapes through a continuous parameter $a$ and has the functional
form:

$\displaystyle\tau_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2E_{\mathrm{J}}}\sum_{i\in\,\rm
J}k_{ti}\,e^{-\eta_{i}(1-a)},$ (3.3)

where $k_{t}=\left\lvert\mathbf{k}_{t}\right\rvert$ is the transverse momentum
relative to the jet direction and $\eta=-\ln\tan(\theta/2)$ is the
(pseudo-)rapidity measured from the jet axis. $E_{\mathrm{J}}\sim Q/2$ is the
energy of the jet $\rm J$ (which will be replaced by the transverse momentum
of the jet in hadron–hadron scattering) and $Q$ is the hard scale. The
parameter $a$ runs over the range $-\infty<a<2$ due to IRC safety (see below).
For $a=0$ the angularities jet shape (3.3) reduces to the jet mass fraction,
defined by

$\rho=\frac{M_{\mathrm{J}}^{2}}{Q^{2}}=\frac{1}{Q^{2}}\left(\sum_{i\in\,\rm
J}k_{i}\right)^{2},$ (3.4)

where $k_{i}$ is the four–momentum of particle $i$. The case $a=1$ corresponds
to jet broadening ($B$) [173]. It has been shown in [174] that recoil of the
primary quarks (in $e^{+}e^{-}$ two-jet events)151515Recall that the quark
transverse momentum induced via recoil is (assuming the quark to be in the
right hemisphere):
$\mathbf{p}_{t}=-\sum_{i\in\mathcal{H}_{R}}\mathbf{k}_{ti}/Q$, where
$\mathcal{H}_{R}$ is the right hemisphere ($\eta_{i}>0$). against soft gluons
contributes to the $B$-distribution at single logarithmic (SL) accuracy. As
such for $a=1$ in $\tau_{a}$ (3.3) recoil effects cannot be neglected should
we aim to compute the $\tau_{a}$ distribution at the said accuracy. In fact,
for this value of $a$ ($a=1$) $\tau_{a}$ is independent of $\eta$ and
contributions of hard collinear radiation to the angularities jet shape are
equal to contributions from soft wide-angle radiation (both contribute at SL
accuracy). For $a>1$ the jet shape $\tau_{a}$, which becomes $\propto
e^{+\eta_{i}}$, receives _larger_ contributions from hard collinear radiation
than from soft wide-angle radiation. Since a full treatment of recoil effects
is beyond the scope of this thesis161616We refer the reader to eg., [31, 174]
for details on the treatment of recoil effects in event/jet shape studies., we
restrict our analyses to the case $a<1$ throughout.

A somewhat related general form of jet shapes was introduced in [31]:

$V\left(\\{\widetilde{p}\\},k\right)=d_{\ell}\left(\frac{k_{t}^{(\ell)}}{Q}\right)^{a_{\ell}}e^{-b_{\ell}\eta^{(\ell)}}g_{\ell}\left(\phi^{(\ell)}\right),$
(3.5)

where $\\{\widetilde{p}\\}$ represents the Born momenta after recoil against
emission $k^{\mu}$, with transverse momentum $k_{t}^{(\ell)}$,
(pseudo-)rapidity $\eta^{(\ell)}$ and azimuthal angle $\phi^{(\ell)}$ measured
with respect to the emitting parton leg $\ell$. IRC safety implies that
$a_{\ell}>0$ and $a_{\ell}+b_{\ell}>0$ [31, 38, 175]171717Clearly if
$a_{\ell}<0$ then $V\propto Q/k_{t}^{(\ell)}$ and hence the jet shape is IR
unsafe. Moreover, the contribution of any particle to the jet shape behaves as
$k_{t}^{(\ell)}e^{-b_{\ell}\eta^{(\ell)}}=\omega\sin\theta^{(\ell)}e^{-b_{\ell}\eta^{(\ell)}}\sim\omega
e^{-(a_{\ell}+b_{\ell})\eta^{(\ell)}}$ in the collinear limit. Collinear
safety ($V$ should stay finite as $\eta^{(\ell)}\rightarrow+\infty$) then
implies that $a_{\ell}+b_{\ell}>0$.. The angularities (3.3) then corresponds
to the specific choice: $a_{\ell}=d_{\ell}=g_{\ell}(\phi)=1,Q=2E_{\mathrm{J}}$
and $b_{\ell}=1-a$ with emissions restricted to be within the jet reach. The
said IRC conditions on $V$ then translate into $a<2$ for $\tau_{a}$
eqrefeq:Jets:Angularities.

##### IRC and recursive IRC safety

In a similar fashion to IRC safety of a jet algorithm, discussed in Subsec.
3.1.2, IRC safety of an observable means that [31] given an ensemble of
partons then the addition of a relatively much softer and/or collinear
splitting of one (or more) of those partons should not significantly change
the value of the observable. Precisely, the change should not be more than a
positive power of the softness/collinearity of the splitting(s), normalised to
the hard scale. Recursive IRC (rIRC) safety conditions slightly, but
crucially, extend those of “plain” IRC and may be formulated as in Eqs. (3.3),
(3.4) and (3.5) of Ref. [31]181818Instead of rewriting the full expressions
here we rather use them to carry out an explicit example below.. rIRC safety
not only ensures the validity of perturbative expansion of the corresponding
jet shape distribution but also ensures the “resummability”, or equivalently
exponentiation, of the latter perturbative series (see below, Subsec. 3.2.4).

Let us verify, for instance, that the angularities $\tau_{a}$ jet shape (3.3)
is an rIRC safe observable. If we introduce the variable

$\widetilde{\tau}_{a}\left(\\{\widetilde{p}\\},k\right)=\frac{k_{t}}{Q}e^{-\eta(1-a)},$
(3.6)

and use Eq. (3.3) of [31], which states that
$\widetilde{\tau}_{a}(\\{\widetilde{p}\\},k_{i}(\lambda_{i}))=\lambda_{i}$
where $k_{i}(\lambda_{i})$ are momentum functions that depend on the
parameters $\lambda_{i}$, then the angularities $\tau_{a}$ in (3.3) reads:

$\tau_{a}\equiv\bar{\tau}_{a}\left(\\{\widetilde{p}\\},k_{1}(\epsilon\lambda_{1}),\cdots,k_{m}(\epsilon\lambda_{m})\right)=\sum_{i\in\rm
J}^{m}\widetilde{\tau}_{a}\left(\\{\widetilde{p}\\},k_{i}(\epsilon\lambda_{i})\right)=\epsilon\sum_{i\in\rm
J}^{m}\lambda_{i},$ (3.7)

where $\epsilon\ll 1$ is a small parameter. Applying the conditions of rIRC
safety given in Eqs. (3.4), (3.5a) and (3.5b) in [31] we find: (a) for the
first condition given in (3.4) we have

$\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon}\,\bar{\tau}_{a}\left(\\{\widetilde{p}\\},k_{1}(\epsilon\lambda_{1}),\cdots,k_{m}(\epsilon\lambda_{m})\right)=\sum_{i\in\rm
J}^{m}\lambda_{i},$ (3.8)

(b) for the second condition given in (3.5a) we have

$\lim_{\lambda_{m+1}\rightarrow 0}\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon}\,\bar{\tau}_{a}\left(\\{\widetilde{p}\\},k_{1}(\epsilon\lambda_{1}),\cdots,k_{m+1}(\epsilon\lambda_{m+1})\right)=\lim_{\lambda_{m+1}\rightarrow
0}\sum_{i\in\rm J}^{m+1}\lambda_{i}=\sum_{i\in\rm J}^{m}\lambda_{i}.$ (3.9)

and finally (c) for the third condition given in (3.5b) we have

$\lim_{\mu\rightarrow 0}\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon}\,\bar{\tau}_{a}\left(\\{\widetilde{p}\\},k_{1}(\epsilon\lambda_{1}),\cdots,\\{k_{a_{\ell}},k_{b_{\ell}}\\}(\epsilon\lambda_{\ell},\mu),\cdots,k_{m}(\epsilon\lambda_{m})\right)\\\
=\lim_{\mu\rightarrow
0}\Bigg{[}\sum_{i\in\mathrm{J},i\neq\ell}\lambda_{i}+\mu\lambda_{\ell}+(1-\mu)\lambda_{\ell}\Bigg{]}=\sum_{i\in\mathrm{J}}\lambda_{i},$
(3.10)

where $\mu^{2}=\left(k_{a_{\ell}}+k_{b_{\ell}}\right)^{2}/k_{t\ell}^{2}$ and
$\lim_{\mu\rightarrow 0}k_{a_{\ell}}+k_{b_{\ell}}=k_{\ell}$. Eqs. (3.8),
(3.9), and (3.10) proves that $\tau_{a}$ is an rIRC safe observable.
Consequently the $\tau_{a}$ _perturbative distribution_ can be resummed into
the form (1.3).

In the next section we first define the (integrated) jet shape distribution
and present an explicit calculation at one-gluon level. Then we elaborate on
some of the important new contributions that crop up at two-gluon level and
are related to the non-global nature of $\tau_{a}$.

####  3.2.1 Jet shape distributions

We define the _integrated jet shape distribution_ , or simply the _shape
fraction_ 191919Also called jet shape cross section., $f(\tau_{a})$, as the
fraction of events for which the observable is smaller than a given value
$\tau_{a}$ [7]:

$f\left(\tau_{a}\right)=\frac{1}{\sigma^{(0)}}\int^{1}_{0}\textrm{\,d}\sigma\,\Theta\left[\tau_{a}-\bar{\tau}_{a}\left(\\{\widetilde{p}\\},k_{1},\cdots,k_{n}\right)\right]\Xi^{\rm
JA}(k_{1},\cdots,k_{n}),$ (3.11)

where $\sigma^{(0)}$ the is Born cross section and $\textrm{\,d}\sigma$ is the
total differential cross section for producing $n$ partons in the final state.
The function $\Xi^{\rm JA}$ represents the resultant phase space constraint
due to applying the jet algorithm ($\rm JA$) to cluster final state particles
in the event. To study some of the features of the shape fraction
$f(\tau_{a})$, it is instructive to consider a specific example. For
simplicity we consider two-jet production in $e^{+}e^{-}$ annihilation
process, schematically depicted in Fig. 3.3, where the angularities shape of
only one jet is measured ($\rm J_{1}$). Final state particles are clustered
using the anti-kT jet algorithm.

Figure 3.3: Sources of global (configuration 1) and non–global (configuration
2) logarithms in dijet events at fixed–order. For configuration 2, we show a
real secondary emission (left) and its virtual correction (right).

There are two sources of large logarithmic contributions to the shape fraction
in the region of small $\tau_{a}$ ($\tau_{a}\ll 1$): “primary” (or
independent) emissions, such as gluon $1$ in Fig. 3.3, which are radiated off
the hard parton (initiating the jet) into the jet vicinity. For such
emissions, the corresponding Born $+$ first order correction to the shape
fraction (3.11) reads

$f(\tau_{a}^{\ell})=1+\int\textrm{\,d}\Phi_{1}\left\lvert\mathcal{M}(k_{1})\right\rvert^{2}\,\left[\Theta\left(\tau_{a}^{\ell}-k_{t1}e^{-\eta_{1}(1-a)}/Q\right)-1\right]\,\Xi^{\text{anti--$\rm
k_{T}$}}\left(k_{1}\right),$ (3.12)

where $\tau_{a}^{\ell}$ represents the angularities shape of the $\ell^{\rm
th}$ jet in the final state and $\textrm{\,d}\Phi_{1}$ is the one–gluon phase
space (B.9). The anti-kT function $\Xi^{\text{anti--$\rm k_{T}$}}\\!$, which
ensures that gluon $1$ is clustered with $\mathrm{J}_{\ell}$, is given by:

$\Xi^{\text{anti--$\rm k_{T}$}}\left(k_{1}\right)\equiv\Theta_{k_{1}\in\rm
J_{\ell}}=\Theta\left(R^{2}-\Delta R_{\ell i}^{2}\right),$ (3.13)

where $\Delta R_{ij}^{2}$ is defined in Algorithm 1 with $y\equiv\eta$. The
$-1$ term in (3.12) stands for the virtual contribution, whose matrix–element
is identical (modulo a minus sign) to that of the real contribution (B.5) in
the eikonal limit, as discussed in Appendix B.2 (Subsec. B.2.3). Specialising
to the centre-of-mass frame202020In which
$\Theta_{k_{1}\in\mathrm{J}_{\ell}}=\Theta\left(\eta_{1}\pm\ln\tan(2/R)\right)$,
where the pseudorapidity $\eta$ is defined in Eq. (B.10) and $+(-)$ is for the
right (left) jet., it is a straightforward exercise to show that at LL
accuracy and assuming small-$R$ jets (full details in the next chapter 4), Eq.
(3.12) yields

$f(\tau_{a}^{\ell})\propto
1-\mathrm{C_{F}}\frac{\alpha_{s}}{\pi}\,\ln^{2}\left(\frac{1}{\tau_{a}^{\ell}}\right).$
(3.14)

Therefore primary emissions give rise to double (soft and collinear)
logarithms (DL) contributions to the shape fraction. At the $n^{\rm th}$ order
of the perturbative _expansion_ of $f(\tau_{a}^{\ell})$, the contribution is
of the form $\alpha_{s}^{n}L^{2n},L=\ln(1/\tau_{a}^{\ell})$. i.e., up to two
logarithms per coupling $\alpha_{s}$. These logarithms thus exponentiate and
may be resummed to all–orders in a rather simple fashion, as shall be
elaborated on in Subsec. 3.2.4.

####  3.2.2 Non–global logarithms

The second source of large logarithms is due to “secondary” (or correlated)
emissions originating from the complementary region of phase space, $\Omega$
(out–of–jets region), as illustrated by configuration $2$ in Fig. 3.3. While
real emissions contribute to the value of the jet shape, $\tau_{a}^{\ell}$,
their corresponding virtual corrections do not. This creates a mismatch
between real–virtual divergent contributions leaving behind non–vanishing
large logarithms in the ratio of the energy scales of $\mathrm{J}_{\ell}$ and
$\Omega$. Such logarithms, dubbed _non–global_ logarithms (NGLs) and appearing
at $\mathcal{O}(\alpha_{s}^{n},n\geq 2)$, were first pointed out in [2]. For
our specific angularities example, we have the leading NGLs contribution to
the shape fraction at $\mathcal{O}(\alpha_{s}^{2})$, in the soft limit and
including the virtual corrections, given by

$f^{\mathrm{NG}}(\tau_{a}^{\ell})=-\int\textrm{\,d}\Phi_{1}\textrm{\,d}\Phi_{2}\,\left\lvert\mathcal{M}(k_{1},k_{2})\right\rvert^{2}\,\Theta\left(k_{t2}e^{-\eta_{2}(1-a)}-Q\,\tau_{a}^{\ell}\right)\Xi^{\text{anti--$\rm
k_{T}$}}\left(k_{1},k_{2}\right),$ (3.15)

where $\Xi^{\text{anti--$\rm
k_{T}$}}\left(k_{1},k_{2}\right)=\Theta_{k_{1}\notin\rm
J_{\ell}}\Theta_{k_{2}\in\rm J_{\ell}}$ and the soft part of the
matrix–element squared for the emission of two gluons is given by the term
$2\,S$ in Eq. (B.65). The minus sign in (3.15) is due to the fact real
emissions, which contribute to the shape fraction only in the region
$\bar{\tau}_{a}(\\{\widetilde{p}\\},k_{2})<\tau_{a}^{\ell}$, completely cancel
against virtual corrections, thus leaving only virtual corrections to
contribute in the complementary region
($\bar{\tau}_{a}(\\{\widetilde{p}\\},k_{2})>\tau_{a}^{\ell}$). The detailed
evaluation of the phase space integral in Eq. (3.15) will be postponed to
Chapter 4. For the time being, we simply state the final result. In the case
where only one jet, say $\mathrm{J}_{\ell}$, is measured and the other is left
unmeasured, we have for small-$R$ jets and assuming $k_{t1}^{2}\gg k_{t2}^{2}$
(which suffices to extract the leading non–global logarithms) the leading NGLs
contribution reads:

$f^{\mathrm{NG}}(\tau_{a}^{\ell})=-S_{2}\,\left[\frac{\alpha_{s}}{2\pi}\,\ln\left(\frac{1}{\tau_{a}^{\ell}}\right)\right]^{2},$
(3.16)

where we have denoted the NGLs coefficient at $\alpha_{s}^{2}$ by $S_{2}$,
which is given by $S_{2}=\mathrm{C_{F}}\mathrm{C_{A}}\,\pi^{2}/3$, i.e.,
independent of the jet radius (in the small-$R$ limit). Eq. (3.16) shows that
NGLs (which originate from soft emissions at large angles) are single
logarithms (SL), contributing at next–to–leading log (NLL) level to the
angularities shape variable. Due to the minus sign in (3.16), NGLs impact on
the shape fraction is thus to reduce the primary emission contribution.

Figure 3.4: An into-jet $\rm J_{1}$ coherent radiation from an ensemble of
soft large-angle gluons (that are outside $\rm J_{1}$). Such configurations
contribute to NGLs at higher-orders.

At the $n^{\rm th}$ order in the perturbative expansion of
$f(\tau_{a}^{\ell})$ the non-global contribution
$f^{\mathrm{NG}}(\tau_{a}^{\ell})$ corresponds to configurations where an
ensemble of $(n-1)$ soft, wide-angle particles212121To extract the leading
NGLs it suffices to consider these particles to be energy-ordered;
$\omega_{1}\gg\omega_{2}\gg\cdots\gg\omega_{n-1}$. outside the jet coherently
emit a single softest gluon into the jet, as depicted in Fig. 3.4. The
ensemble may assume any complicated geometry. As such it has not been possible
[2, 176] to _analytically_ resum these logs to all–orders, as has the case
been with primary (global) logs. However, employing the large-$\mathrm{N_{c}}$
limit one can either write an evolution equation [71] or resort to a dipole
evolution code [2] to _numerically_ compute the resummation. The solution to
the said all–orders evolution equation adheres a _factorised_ form [71, 177]:

$f(\tau_{a}^{\ell})=\Sigma_{P}(\tau_{a}^{\ell})\,S(t)$ (3.17)

where the first term $\Sigma_{P}$ is the Sudakov form factor given by
bremsstrahlung emission from primary hard partons and is discussed below. The
second factor arises from successive soft secondary branchings (as described
above). The parameter $t$ is known as the evolution parameter and is given by

$t=\int_{e^{-L}}^{1}\frac{\textrm{\,d}x}{x}\frac{\alpha_{s}(xQ)}{2\pi},$
(3.18)

where $x=2\,k_{t}/Q$ and the logarithm $L\propto\ln(1/\tau_{a}^{\ell})$ (with
the exact formula of $L$ depending on the configuration under study). A widely
used parametrisation of $S(t)$ is given in [2]

$S(t)=\exp\left[-\frac{\mathrm{C_{A}^{2}}}{2}\,S_{2}\,\left(\frac{1+(at)^{2}}{1+(bt)^{c}}\right)t^{2}+\mathcal{O}(1/\mathrm{N_{c}^{2}})\right],$
(3.19)

where $S_{2}$ is the $\mathcal{O}(\alpha_{s}^{2})$ coefficient and the
parameters $a,b$ and $c$ are determined from fitting (3.19) to the output of
the MC program of Ref. [2]. Actually, for use in phenomenology one replaces
the large-$\mathrm{N_{c}}$ colour factor $\mathrm{C_{A}}/2$ by the
corresponding full colour factor such that when expanding (3.19) one recovers
the full $\mathcal{O}(\alpha_{s}^{2})$ result. Note that the neglected
$1/\mathrm{N_{c}^{2}}$ non-planar part contribute at the $10\%$ level.

In–depth discussion of this and other related features of NGLs is presented in
the subsequent chapters. It is in fact one of the central aims of this thesis
to compute, as accurately as possible, NGLs at fixed–order and quantify their
all–orders impact on the total resummed jet shape distribution.

####  3.2.3 Clustering logarithms

So far we have only dealt with jet shape distributions where final state jets
are defined with the anti-kT algorithm. As emphasised earlier (see Sec. 3.1)
the anti-kT algorithm works, particularly in the soft limit, like a rigid
cone. Soft partons are clustered independently to their nearest hard
neighbours. This simple picture is spoiled by other SR algorithms such as kT,
C/A as well as the SISCone algorithm, where soft partons are iteratively
clustered together, based on their closeness, and may well form separate jets
from hard partons’ jets. The impact of this on the final form of the jet shape
distribution is two fold:

  * •

Significant reduction in the NGLs contribution due to tighter phase space
constraints. This was first shown in [75] for gaps–between–jets distribution.

  * •

Breakdown of the naive single–gluon exponentiation (at NLL) associated with
primary emissions due to the appearance of a tower of large single logarithms.
This was first pointed out in [3, 74] for the same distribution
(gaps–between–jets). We refer to these logarithms as _clustering logarithms_
(CLs). They are a type of non–global logarithm as they only show–up for
non–global observables. As a result of this some authors, e.g., of Ref. [178],
refer to them as _abelian_ NGLs.

Postponing the discussion of both effects at fixed–order and all–orders to
later chapters, we note that the leading CLs contribution to angularities
shape fraction at $\mathcal{O}(\alpha_{s}^{2})$ is, for the case of one
measured jet $\mathrm{J}_{\ell}$, of the form

$f^{\rm JA}(\tau_{a}^{\ell})=C_{2}^{P,\rm
JA}\,\mathrm{C_{F}^{2}}\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\,L^{2},$
(3.20)

where $L=\ln(1/\tau_{a}^{\ell})$ and $C_{2}^{P,\rm JA}$ is the outcome of the
phase space integral. For small jet radii, $R\ll 1$, $C_{2}^{P,\rm JA}$ only
depends on the geometry of the final state (and neither $R$ nor the jet
shape). As $R$ increases $C_{2}^{P,\rm JA}$ dependence on $R$ is restored as
shall be explicitly shown in Chapter 5. For the kT and C/A algorithms:
$C_{2}^{P,\text{$\rm
k_{T}$}}=C_{2}^{P,\text{C/A}}\simeq\pi^{2}/27=S_{2}/\mathrm{N_{c}^{2}}$ as
$R\rightarrow 0$. This relation between $C_{2}^{P,\text{$\rm
k_{T}$}(\text{C/A})}$ and $S_{2}$ further fosters the non-global nature of
CLs.

Just like NGLs, the SL clustering logs persist at every order in the
perturbative expansion of the non–global jet shape distribution and must be
resummed for a given phenomenological calculation to be meaningful. Although
the all–orders result can currently only be obtained via the MC program of
[2], the situation is slightly more satisfactory than with NGLs. It has been
possible, for both gaps–between–jets and jet mass as discussed in Chapter 6,
to analytically compute the first few orders in the corresponding $g_{2}$ (see
Eq. (1.3)) CLs–tower and evidently show, by explicit comparison to the output
of the said MC program, that they capture the full behaviour of the all–orders
result. Full details are presented in Chapter 6.

Below we outline the necessary steps for a typical resummation calculation of
primary global logarithms (GLs).

####  3.2.4 Resummation

Fixed–order perturbative calculations usually involve a small number of
additional partons to the Born configuration and can produce accurate results
at the large–$\tau_{a}$ tail region of the jet shape distribution
($\tau_{a}\lesssim 1$). Such regions of phase space describe large departures
from the Born–event energy flow pattern due to the extra partons being
energetic and at large angles. Their contributions are free from large
logarithms, suppressed by powers of the coupling $\alpha_{s}$ and comply with
the convergence of the perturbative series. Fixed–order calculations are
usually carried out with the help of Fixed–order Monte Carlos (FOMCs) such as
event2 [179], eerad3 [180, 181], for $e^{+}e^{-}$ annihilation, and mcfm
[182], nlojet++ [58], for hadron-hadron scattering, to mention just a few.
These programs are fully exclusive over the final state and can produce
predictions for arbitrary IRC observables. The prevalent accuracy across the
majority of the available FOMCs is next–to–leading order (NLO), with the
exception of eerad3 which reaches next–to–next–to–leading order (NNLO) for
observables in $e^{+}e^{-}$ annihilation events.

Figure 3.5: Fixed–order vs (matched) resummed calculations and comparison to
experimental data for the total broadening shape variable (left) without
hadronisation effects and (right) with hadronisation effects (figure from
http://www.lpthe.jussieu.fr/~salam/talks/).

In the small $\tau_{a}$ region, $\tau_{a}\ll 1$, in which departures from the
Born energy–flow configuration are small due to the extra partons being soft
and/or collinear to the initial hard Born partons, and which constitutes the
bulk of final state configurations, fixed–order perturbative calculations are
unreliable, as illustrated in Fig. 3.5 (left). This is due to the fact that
each power of the coupling $\alpha_{s}$ is accompanied by up to two powers of
large logarithms associated with soft and collinear divergences, as
exemplified in the previous subsections. The all–orders perturbative series of
the shape fraction $f(\tau_{a})$ (3.11) may be cast in the form