"Stochastic exclusive modelling of NLO QCD evolution"

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Stochastic exclusive modelling of NLO QCD

evolution

Magdalena Sªawi«ska

Henryk Niewodnicza«ski Institute of Nuclear Physics

Polish Academy of Sciences

Thesis presented for the degree of Doctor of Philosophy

written under supervision of

Prof. Stanisªaw Jadach

November 2, 2011

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Chapter 1 Introduction

The experiments at the Large Hadron Collider (LHC) in CERN have recently started collecting data. They are at one of the most promising frontiers of the fundamental research, likely to widen our knowledge on the fundamental interactions governing the origin and structure of our Universe. Proton collisions at the new record energies at the LHC will enable the production of the famous missing particle of the Standard Model (SM), the Higgs boson, and possibly other massive particles, changing completely our understanding of fundamental interactions, matter constituents, space and time. The detectors for the LHC experiments feature the high resolution, never achieved before in hadron colliders experiments. Moreover, high luminosity of the LHC will provide the large number of recorded collision events, thus assuring the high statistical precision of the measurements. The ambitious research programme of the LHC can only be fully accomplished if theoretical predictions for the physical observables (for instance dierential cross sections) within and beyond the Standard Model match the precision of the experiments.

The centre of mass energies of the colliding protons at the LHC are expected to reach 14TeV. At this scale the structure of protons can be resolved. While approach-ing the interaction point they act as streams of gluons and quarks (partons). The theory explaining interactions among quarks and gluons, so-called strong interactions, is Quantum Chromodynamics (QCD).

The strength of strong interactions among partons are governed by the value of the coupling constant of QCD, αS. The coupling decreases signicantly with the energy

scale. Fig. 1.1 presents measured values of αS at dierent energies. It equals 0.35

at 1.2 GeV, which is about the energy of a proton at rest (0.938 GeV), hence the term strong interactions. At about 90GeV (mass of a Z boson) the coupling equals 0.1184(7) [1]. The phenomenon of vanishing coupling constant at high energies is called the asymptotic freedom of quarks and gluons [2, 3]. The fact that αS is small at large

energies enables doing calculations for collider experiments within the perturbative Quantum Chromodynamics (pQCD) [3]. This methodology allows to calculate the physical observables as power series in the strong coupling.

Theoretical calculations for the LHC face several dicult problems. Let us mention some of them, that are related to the subject of this dissertation.

At the LHC energies many quarks and gluons, carrying colour charges, can be produced and interact with other particles from proton beams, before they scatter in the so-called hard process. These interactions change the momenta distributions, as well as the colour and avour structure of the partons, hence aecting all physical

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6 CHAPTER 1. INTRODUCTION

Figure 1.1: Summary of the values of αS at dierent values of scale µ, at which they

are measured. The data are from τ width, Υ decays, deep inelastic scattering, e+_e−

event shapes at 22GeV, 58GeV, 135GeV and 189GeV and from Z width. Taken from Ref. [1], http://pdg.lbl.gov/2004/reviews/qcdrpp.pdf.

observables. In particular, all LHC processes will be accompanied by abundant low-energetic and collinear gluons carrying colour charge. This problem will be investigated in this dissertation.

The multi-gluon eects are crucial for many physical observables important for the LHC experiments. It has been proved [48] already in the early days of QCD that they inuence the region of low transverse momentum in the transverse momentum spectrum of the Drell-Yan processes [9, 10]. Since then, many eorts have been made to include the eects of soft gluon emissions to all orders, in analogy to QED [11]. These include the inclusive exponentiation of leading-order emissions [12] and continuations of works of ref. [7] in developing techniques of exponentiating gluons emissions in a way applicable to many processes [13]. There are noteworthy eorts in exponentiating gluon emissions in analogy to emissions of soft photons in QED [11] in the next-to-eikonal limit [1416].

Another diculty in predicting the outcomes of collider experiments is due to the connement of quarks and gluons [17]. They do not exist as isolated particles, but form hadrons objects of zero colour charge at the last stage of the collision process. It is these composite particles that are identied by detectors, hence the theoretical models for many observables should include hadronization of quarks and gluons into hadrons. These models are implemented using Monte Carlo methods, see for instance ref. [18,19].

Exploiting eciently all data recorded by the LHC experiments in order to dig out interesting new phenomena requires developing the calculation methods for these QCD eects and tools to evaluate physical observables to higher order in perturbative expansion. These theoretical calculations will be confronted with the data measured in experiments or used to analyse them. Therefore they must reach the unprecedented precision of the LHC experiments. The experiments at hadron colliders, however, are very challenging from the point of view of theoretical calculations. The hadronic col-lisions feature complicated nal states, because hadrons are composite objects.

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More-1.1. THE METHODOLOGY OF BEYOND LO CALCULATIONS 7 over, huge energy range of accelerated protons requires computations of interactions among partons to be done beyond the leading order, see next chapter. Another chal-lenge comes from experiments capabilities of resolving transverse momenta of multiple resolved partons. Therefore theoretical calculations for the LHC processes should give predictions not only as functions of longitudinal momenta, but should also provide insight into the transverse momentum plane.

Massive eort is required in developing new better methodologies of calculating the Quantum Chromodynamics eects in the measurements at the LHC. This thesis presents work being a part of this eort.

1.1 The methodology of beyond leading order

calcu-lations

The calculations of physical observables in high energy physics are carried out with the help of the so-called collinear factorisation theorems (CFT1_{) [2025]. They tell us that}

each complicated physical observable (for example inclusive dierential cross-section dσincl) can be schematically reduced to a convolution:

dσincl = FP DF ⊗ σhard⊗ Ff ragm, (1.1)

where the ⊗ symbol is understood as (f ⊗ g)(x)

Z

dzdyf (y)g(z)δ(x − yz), (1.2)

and x, y and z are fractions of longitudinal momenta along the proton beams or energetic partons emitted in the hard process.

All functions in (1.1) are evaluated at the so-called factorisation scale and dˆσhard

dx =

1

Q2C(x, Q

2₎ _(1.3)

is the dierential cross-section of the hard scattering process, C is the coecient function and Q is a typical energy scale of a given process, for example Q = MZ in

case of a Z boson production. Since a typical scale Q is large, this part of a hadronic process is called a hard process. If a factorisation scale is chosen to be equal to the hard process scale, the dependence of the coecient function C on Q enters only through the running coupling α:

C(x, Q2) = C(0)(x) + α(Q

2₎

2π C

(1)_{(x) + · · ·} _(1.4)

This choice of factorisation scale (xing it equal to the hard process scale) has been adopted in the calculations presented in this thesis.

Below a certain energy scale ∼2GeV, the perturbative expansion is no longer valid and these components must be parametrised using parton distribution functions and fragmentation functions, FP DF and Ff ragm, respectively. The rst describes the

long-distance interactions inside the incoming proton, the second the fragmentation of (out-going) jets into hadrons. These functions in equation (1.1) are evaluated at the same

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8 CHAPTER 1. INTRODUCTION scale as C, called the factorisation scale. The changes (evolution) of functions F due to variations of the factorisation scale Q are governed by the DGLAP [26] evolution equation: Q2 d dQ2F (x, Q 2 ) = α(Q 2₎ 2π P (x, α(Q 2 )) ⊗ F (x, Q2), (1.5)

which is related to the renormalisation group equation for the physical observable dσincl [27]. The function P (x, α(Q2)), governing the evolution of F in (1.5) is called the evolution kernel and is calculable order by order in perturbative expansion. It will be discussed in detail in the next chapter.

Beyond the leading order the decomposition (1.1) is not unique. Functions C, FP DF

and Ff ragm must be, however, calculated within the same scheme, such that scheme

dependence cancels for physical observables. The actual choice of factorisation scheme is a non-trivial aspect of every calculations using factorisation theorems. In the next chapter I shall elaborate on the factorisation scheme used in the calculations in this thesis.

The practical computations of C and F parts employ dierent methodologies. The hard process part C is calculated at xed order in perturbative expansion with a de-ned number of the on-shell initial and nal state particles. The dominant contribution to proton structure and fragmentation functions, on the other hand, comes from many emissions, which are computed approximately and resummed to innite order. The multi parton radiation consists of the emitted partons, which are low-energetic (soft) or collinear with the emitter. Therefore it is usually referred to as the soft part. It is typically simulated by means of Monte Carlo methods2 _{in the so-called Parton Shower}

(PS) algorithms [29, 30] and/or parametrised using the parton distribution functions (PDFs) [31,32].

The recent progress in calculating the hard process matrix elements for the the LHC-related processes has been signicant, see for example refs. [3336]. They are calculated as on-shell matrix elements, typically employing sophisticated algebraic techniques [3741]. For the most interesting hard processes the matrix elements computed at next-to-leading (NLO) or next-to-next-to-leading order (NNLO) in QCD are already available. This means that the simplest conguration of particles for a given process (the leading order) is supplemented with additional (one or two, respectively) emissions of partons.

The improvements of the PS Monte Carlos (PSMCs) by means of incorporating higher-order corrections to the implementations of soft parts are, on the contrary, is rather slow. The rst PSMCs in collider physics simulated the evolution of structure functions [29] and hadronic jets [30] as the Markov chain of subsequent emissions, according to the DGLAP [26] evolution kernels. Despite 30 years that have passed since the rst implementations, PS MC still remain at the leading order [4244]. On the other hand, the analytical calculations of evolution kernels of eq. (1.5) has been done at NLO, see for instance [21], and recently at NNLO [45, 46] using Mellin transform, see for instance [47, 48]. The results obtained with Mellin transform are integrated over the phase-space (transverse momenta) and hence are not suitable for any realistic Monte Carlo implementation. Another reason for PSs remaining at LO is that, while leading-order PSs can be easily simulated as Markov chains of independent emissions,

2_{A noteworthy example of a dierent approach is the QCDNUM program [28]. It solves the}

integro-dierential equation (1.5) as a set of linear equations on a grid in (x, ln Q2₎ _{variables, iterating over}

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1.2. THE AIM OF THIS WORK 9 the explicit inclusion of interferences inside the NLO kernels introduces correlations within pairs of partons, a new problem not encountered in the LO.

The progress in including incomplete higher-order (NLO) contributions into Par-ton Showers [49, 50] is worth mentioning. In Herwig++ [49] the authors use multiple splittings in a simplied kinematics and in this way include NLO amplitude-squared-type diagrams. The clever choice of the ordering variable (angle), incorporates eec-tively contributions of interferences, otherwise absent in this construction.3 _{The works}

of [50,51] proceed in a slightly dierent direction. The authors choose virtuality as the ordering variable and include only the leading terms in the 1/N2

C expansion, where NC2

is the number of colours in QCD.

Other important activities aimed at improving theoretical predictions for QCD observables focus on combining leading-order PDFs with NLO matrix elements for hard processes. This has been started already in the late 1970s with the NLO calculations of semi-inclusive distributions of the W/Z production processes, see [8,5254]. Combining the NLO-corrected hard process with LO parton shower MC is a non-trivial task. The notable examples of this ongoing eort are reported in refs. [55,56].

Another important issue concerns the dependence of physical observables calculated at a given order of perturbative expansion on the renormalisation scheme used in the calculations. This dependence is used as an estimate of the uncontrolled higher order eects. Since it decreases while going from LO to NLO, the NLO calculations are regarded as more reliable and precise than LO ones.

1.2 The aim of this work

The study presented in this thesis is a part of the eort of incorporating exact next-to-leading corrections into the Monte Carlo simulation of DGLAP [26] evolution of parton distributions in the exclusive (unintegrated) form, see refs. [57,58]. The DGLAP evolu-tion of the parton distribuevolu-tions at the NLO level will be modelled by the Monte Carlo program itself, as opposed to other approaches, where the evolution of PDFs is done using non-MC methods, see for instance [28]. The simulation is to be fully exclusive, which means that evolution is modelled in the MC within the unintegrated phase space. Therefore the evolution kernels considered here will not be one-dimensional functions of the lightcone variable x, but rather the fully dierential distributions depending on four-momenta of all real emitted partices.

So far, the eorts in developing the aforementioned new NLO PS are focused on constructing initial state radiation (ISR) the multiple emission of quarks and gluons from partons of the incoming protons, at the NLO level. The nal state radiation (FSR) radiation from (massive) particles after interaction is limited to LO in QCD (hadronization is not included). There are also preliminary studies on how to construct the matrix element for the hard process and combine it with the shower part, see [59]. The processes considered include deep inelastic scattering (DIS), Drell-Yan and W/Z boson production (and leptonic decay). The latter process features no QCD radiation in the nal state, hence the discussion in this thesis will be restricted to the initial state radiation (ISR) kinematics, of course having in mind that transition between DIS and

3_{Apart from the improvements of the evaluation of PDFs from interactions of single partons,}

Herwig++ incorporates also eects of multiple interactions of partons inside a proton. Although very relevant for the LHC simulations, they do not change the order of perturbative expansion included strictly in the code, which remains LO.

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10 CHAPTER 1. INTRODUCTION Drell-Yan is feasible, see ref. [21] for details. The entire programme of constructing NLO PS with NLO hard ME is quite ambitious, its range and time scale going beyond the scope of this thesis.

In simulating ISR by means of Monte Carlo methods, real and virtual Feynman diagrams contributing to NLO kernels are treated dierently. The emissions of real particles aects phase space for subsequent emissions, whereas virtual emissions enter only into overall normalisation. Therefore, while talking about exclusive modelling I will usually have in mind real emissions diagrams only.

From the point of view of the above project, the analysis of soft singularities and their cancellations is crucial for implementing the evolution of parton densities in the exclusive form. This is because the Monte Carlo program consists of two steps. Firstly, a shower kinematics is generated according to a simplied (crude) distribution. Sec-ondly, this distribution is corrected using a MC weight, which is the ratio of the exact NLO distribution and the crude:

MC weight = exact NLO distribution_{crude distribution} . (1.6) The variance of the distribution of the above MC weights should be as small as possible (with respect to the average weight), the weights themselves being positive.

In the Monte Carlo algorithm the collinear factorisation will play the leading role. The important point is that, while collinear singularities between the evolution kernels and between kernels and the coecient function are identied and taken care of, the internal soft singularities of Feynman diagrams are still present and may spoil the MC computations!

The crude distribution is usually constructed using LO distributions, but due to the above requirements on the MC weight, it should also contain as many as possible improvements borrowed from the NLO. The numerator in equation (1.6) represents a distribution from a single Feynman diagram. It is not necessarily gauge invariant and may contain infrared singularities. Unless the LO-improved crude distribution re-produces exactly these divergences, the MC weight distribution will be unacceptable (featuring for example unintegrable tails). Therefore a detailed, diagram-wise anal-ysis of infrared singularities and their cancellations among dierent diagrams are so important in constructing the Parton Shower MC.

The main aim of this dissertation is to give account of the studies on infrared sin-gularities in the two-real parton emissions Feynman diagrams contributing to the NLO kernel. The analysis goes beyond the well-known discussion of doubly logarithmic sin-gularities analysed in refs. [60, 61] and the construction of inclusive PS based upon coherent branching [62, 63]. I shall investigate the structure of singularities of uninte-grated distributions and examine carefully also singly-logarithmic singularities. I shall analyse the cancellations of these singularities among diagrams in physical gauge, giv-ing special care to the structure of sgiv-ingularities in the soft limit. It should be stressed that the cancellations of infra-red singularities analysed in this thesis are not of the usual KLN [64] nature, i.e. between integrated real and virtual Feynman diagrams at a given perturbative order, for two reasons. Firstly, the analysis is restricted to diagrams with two-real emissions only, the cancellations being governed by their colour and spin quantum numbers and not by the interplay among real and virtual diagrams, although soft limits are closely related upon integration to the virtual corrections. Secondly, all distributions will be analysed before integration, within the standard phase space.

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1.2. THE AIM OF THIS WORK 11 In this thesis I shall also present tools for graphical (numerical) analysis of mul-tidimensional distributions developed during these studies, as well as the analytical analysis of singularities in the unintegrated form. To be precise, in the graphical anal-ysis the distributions will be in most cases averaged over the azimuthal angle, which is motivated by legibility of the plots. The angle will remain unintegrated in the analytical expressions and in the MC generation.

The next chapter presents the framework of the calculations. It will give an outline of the collinear factorisation of CFP [21], which provides quantum eld theoretical grounds for the systematic extraction of kernels beyond the leading order, followed by introducing the factorisation scheme more suitable for Monte Carlo, that is inspired by the above. The Monte Carlo scheme will be used throughout this thesis.

The main results of this dissertation are contributions to the exclusive NLO evolu-tion kernel. They include non-singlet and selected singlet diagrams. The contribuevolu-tions to the exclusive kernel are given diagram-wise and in a fully dierential form, which have hitherto not been done in the literature. The distributions given in this thesis are four-dimensional and are implemented in a computer Monte Carlo program.

Chapters 3 and 4 present the numerical and analytical analysis of exclusive two-real Feynman diagrams contributing to the NLO kernel. Special care will be given to the structure of infrared singularities appearing in these diagrams and their cancellations. Gauge cancellations of infrared singularities are investigated for the rst time on the level of unintegrated distributions. This will be done analytically and numerically in the soft limit.

The analytical distributions have been implemented in a computer program CCNAS and generated using a Monte Carlo event generator Foam [65]. The CCNAS program will be presented in chapter 5. The implementation of contributions from diagrams to the NLO kernel is an important step before applying these distributions to a program simulating the evolution of parton densities. The new contributions have been cross-checked with integrated formulae in the literature. In some cases I will also present results of analytical integration of diagrams.

Finally, chapter 6 provides conclusions.

A part of this dissertation covers collaborative work published in refs.: [58, 59, 6671]. Most of dissertation, however, presents calculations and analysis done by its author, often crosschecked by the collaborators. The most important part of this thesis elaborates on the unintegrated contributions to the singlet kernel, analysis of colour coherence eects of the non-singlet and singlet diagrams as well as collinear and soft limits of Vg distribution (in chapters 3 and 4) and summarises the individual research activity of the dissertation author. The same is true for the whole numerical and graphical calculation framework used in the analysis of the soft limits, presented in chapter 5.

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Chapter 2 Collinear factorisation

The factorisation theorem is the main tool used in the development of the exclusive NLO Monte Carlo for DGLAP evolution. In this chapter the collinear factorisation scheme of CurciFurmanskiPetronzio (CFP) [21] will be reviewed.

CFP is a particular implementation of the EGMPR [20] scheme. Both of them are using physical gauge, like in older QED papers [72].

The collinear factorisation, however, is not the only factorisation technique devel-oped and applied in the NLO computations. In the complementary approach, the soft factorisation, soft singularities are resumed rst, followed by the collinear ones [24] and the general gauge is employed. This factorisation was used for instance in calculat-ing the time-like radiation (jets) in the e+_e− _{annihilation process beyond the leading}

order [73,74]. A dierent approach has recently been adopted in refs. [7577].

The CFP scheme appears to be the most solid starting point for the MC applica-tions, yet not applicable directly. At the end of this chapter a new scheme named MC factorisation scheme will be outlined. Many of its technical ingredients are common with the CFP scheme.

In the following I shall dwell upon these technical ingredients of the CFP factorisa-tion scheme, that are common with the MC scheme. They determine the calculafactorisa-tion techniques used in computing Feynman diagrams in the next chapters. In particular, I shall discuss the role of collinear factorisation theorem in identifying and regulating collinear singularities. The remaining infrared divergences will be analysed in the next chapters.

2.1 The general CFP prescription

In the following it will be explained, how the formula (1.1) is obtained in the framework of CFP factorisation. This section is a short review of the CurciFurmanskiPetronzio (CFP) collinear factorisation of ref. [21].

In the CFP factorisation it is assumed that UV divergences associated with Feyn-man diagrams for a given process have already been regularised dimensionally (the number of dimensions n = 4 − 2ε, ε > 0) [78] and subtracted. After subtracting the UV poles the result is analytically continued to n = 4 + 2ε, ε > 0 and the only remain-ing sremain-ingularities are soft and collinear. The startremain-ing point is the EGMPR expansion [20] of a process cross-section σ, see Fig. 2.1:

σ = (1+K0+K0⊗K0+. . .)⊗C0⊗(1+K00+K 0 0⊗K 0 0+. . .) = 1 1 − K0 ⊗C0⊗ 1 1 − K₀0. (2.1) 13

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14 CHAPTER 2. COLLINEAR FACTORISATION

q

qq Z

q

Z

K K K K K’₀ K’₀ K’₀ K’₀

C

0 0 0 0 0

Figure 2.1: Diagrammatic representation of a factorised cross-section for a Z boson production process.

Eq. (2.1) is sometimes called the generalised ladder expansion. The rungs of the ladder (kernels), K0 and C0, are sums of two-particle-irreducible (2PI) diagrams, see Fig. 2.1.

The relation between QCD diagrams and functions F introduced in (1.1) depends on the type of the process. For calculating structure functions in DIS-type process, FP DF ≡ _1−K1

0 and Ff ragm. ≡

1

1−K₀0. In e

+_e− _{annihilation into quarks, only the}

frag-mentation functions are present, F(1) f ragm. ≡ 1 1−K0 and F (2) f ragm. ≡ 1 1−K0 0, whereas FP DF is

absent. In the calculation of cross-section for W/Z production with leptonic decays, depicted schematically in Fig. 2.1, F(1)

P DF ≡ 1 1−K0 and F (2) P DF ≡ 1 1−K0 0. The

generalisa-tion of (2.1) to processes with more external quark lines requires more independent ladders, one for each hadronic channel [20]. For the sake of simplicity, the factorisation procedure will be discussed on a process with only one ladder of emissions.

The functions F represent the momentum distributions of quarks and gluons inside hadrons. Below a certain scale q0 (typically ∼2GeV), where perturbative QCD no

longer can be used, they are tted from the experiments. It is understood that the evolution of parton distribution functions has boundary conditions at q0 (the initial

condition) and Q, being the scale of the hard process. In this thesis, proton wave function will be neglected for simplicity. All incoming partons' momenta will be on-shell, normalised to 1.1

All 2PI diagrams are free of collinear singularities, that originate from integration over momenta connecting the kernels. According to EGMPR the C part of equation (2.1) is already free of collinear singularities. The decomposition (2.1) is not a factorised formula of the type of eq. (1.1), because for instance C and F are still contracted by means of spinor indices and four-momenta integration.

In the following I shall recapitulate the classic factorisation procedure of CFP [21]. In this and next chapters some problems associated with the CFP factorisation will be discussed, in order to motivate the construction of a dierent factorisation procedure, more suitable from the point of view of its implementation in a Monte Carlo [57, 58]. The MC scheme will be employed in all analysis performed later in this thesis. The dierences between the two schemes will be discussed in section 2.4 and in chapter 3 on a practical example. Before that, the construction of the CFP factorisation will be

1_{To be precise, I will hence analyse the evolution of Green functions of the evolved PDFs rather}

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2.1. THE GENERAL CFP PRESCRIPTION 15 presented, as many of its calculation methods are inherited by the MC scheme.

The four-momenta integration and contraction of spinor indices between all rungs of the ladder get disentangled by introducing the projection operators P (see next page): K0 =PK + (1 − P)K0. The operator P acting on K0 extracts the singular part, while

(1 −P)K0 is the nite part.

M = C0 ∞ X i=0 K₀i = C0{1 + ∞ X i=1 K₀i−1(P + (1 − P))K0}. (2.2)

After recursively factorising mass singularities in K0 (starting from the last one):

M (1 −PK0) = C0{1 + (1 −P)K0 + ∞ X i=1 K₀i−1(P + (1 − P))(K0(1 −P)K0)} M (1 −PK0−P(K0(1 −P)K0)) = C0{1 + (1 −P)K0+ (1 −P)(K0(1 −P)K0) + ∞ X i=1 K₀i(1 −P)(K0(1 −P)K0)}, (2.3) one nally obtains:

M 1 −P K0 1 1 − (1 −P)K0 = C0 1 1 − (1 −P)K0 . (2.4) I dene now K = K0 1 1 − (1 −P)K0 , (2.5)

and the expression for M takes the form:

M = C0 1 − (1 −P)K0 1 1 −PK = C Q2 µ2, αS(µ 2₎ ⊗ Γ(αS(µ2), ε). (2.6)

Here, the series on the left represent a coecient function C, and the one on the right parton densities Γ. The convolution symbol ⊗ has been dened in equation (1.2).

All collinear and soft singularities are located in Γ, leaving C nite. It is written explicitly in (2.6) that only C depends on the factorisation scale Q, equal to the hard process scale, like in eq. (1.4). On the other hand, Γ encapsulates all collinear singu-larities as poles in ε. Spinor indices are decoupled and the four-momentum convolution is now replaced by only one-dimensional (in lightcone plus variables, see next section). The CFTs, such as equation (2.6) are formulated for inclusive objects. This fact makes their direct implementation in Monte Carlo impossible. Due to the construction of projection operator P, CFTs violate conservation of transverse momenta (in an inclusive calculations integrated over anyway), whereas in a reliable Monte Carlo the conservation of four-momenta is absolutely necessary.

Another important point is the treatment of collinear and infrared singularities. The collinear ones are in CFTs extracted after the phase space integration, whereas in the MC they have to be separated before the phase space integration. In the inclusive approach it is sucient that the infrared singularities cancel after adding all diagrams entering the kernel, whereas, as advocated in equation (1.5), the separation of infrared singularities must be done diagram-wise in the MC approach.

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16 CHAPTER 2. COLLINEAR FACTORISATION Moreover, computing Γ directly from eq. (2.6) features large oversubtractions that should be avoided in numerical calculations. This issue has been addressed in ref. [57]. In this thesis I will show one aspect of these oversubtractions using a specic example in the next chapter.

All the above outlined problems concerning the use of the classical inclusive CFTs in the MC led to constructing a novel factorisation scheme better suited for this purpose. It uses many technical details introduced in the CFP scheme, that will be presented in the next section. Before the end of this chapter, this new MC factorisation scheme will be presented in some limited form. For more details see ref. [57]. The selected properties of this scheme, in particular treatment of infrared singularities will be discussed later, in chapters 3 and 4.

2.2 Technicalities of CFP scheme

In this section I am going to describethe essential steps necessary to extract kernels for Monte Carlo from Feynman diagrams in the CFP factorisation scheme. All particle's momenta in Feynman diagrams are calculated in n dimensions, n = 4+2ε, ε > 0. I shall characterise briey divergences/singularities associated with the diagrams and their regularisation in this approach before going to four dimensions, where the distributions for kernels will be implemented numerically.

2.2.1 The choice of gauge

The axial gauge is often used in the collinear factorisation theorems (CFT), because it enables the generalised ladder decomposition, in which the evolution kernels are free of collinear singularities [20,72,79].

The light-like axial gauge introduces an arbitrary light-like four-vector η, η2 _{= 0}_,

that together with the four-momentum of the incoming parton (usually denoted by p) are the basis for longitudinal components of all other four-momenta. The gluons' propagators and vertices in this gauge are given in Appendix B.

The relation of the axial gauge to Wilson loops can be found in refs. [8082].

2.2.2 Sudakov variables

All particles' momenta will be parametrised in terms of Sudakov variables [12]. In this section I will introduce notation used in the following chapters and give details on kinematical parametrisation of the two-particle phase space used in the calculation of NLO kernels with two real emissions. The momenta of the massless on-shell emitted particles are denoted by ki, i = 1, 2:

ki = αip + βiη + kTi; k

2

i = 0, (2.7)

where η is the gauge vector (η2 _{= 0}_{), p is the momentum of the incoming parton}

(p2 _{= 0}_{), and k}

Ti is the transverse momentum. The on-shellness of ki puts constraint

on the Sudakov variables αi and βi. They are not independent, βi = − k2

Ti

2αiη·p. Another

constraint on the lightcone variables originates from keeping constant the lightcone variable x ≡ η·q

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2.2. TECHNICALITIES OF CFP SCHEME 17 The distributions originating from Feynman diagrams will be parametrised using rapidity related variables:

ai=

kTi

αi

. (2.8)

Its modulus, |a| ≡ a will be called angular variable.2

The four-momentum of the most virtual parton is denoted by

q = p − k1− k2, (2.9)

and −q2 _{is the virtuality of the (o-shell) parton closest to the hard process (in the}

ISR the most virtual one) that equals −q2 _{= −}1 − α2 α1 k_T2 1 − 1 − α1 α2 k_T2 2− 2k 2 T1k 2 T2cos ϕ12 = −(1 − α1)v21− (1 − α2)v22− 2 √ α1α2v1v2cos ϕ12. (2.10) where v1 (i = 1, 2) are related to the lightcone minus variables and v1 = −

k_Ti √

αi. (The

latter form of equation (2.10) is equivalent to equations (3.24) and (3.25).) The invari-ant mass squared of the two emitted partons equals

k2 = (k1+ k2)2 = 1 − α2 α1 k2_T₁ + 1 − α1 α2 k2_T₂ − 2kT1kT2cos ϕ12. (2.11)

Additionally, I dene other useful variables related to q2 _{and k}2_{, but expressed in the}

angular variables: a2 = (a1− a2)2 = a21+ a 2 2− 2a1a2cos ϕ12= k2 α1α2 ˜ q2 = 1 − α2 α2 a2₁+ 1 − α1 α1 a2₂+ 2a1a2cos ϕ12 = q 2 α1α2 . (2.12) It is convenient to use the logarithms of ratios of the above Sudakov variables of the two emitted partons while investigating the structures of Feynman diagrams. The convenient two-dimensional projection of the two-particle phase space in the Sudakov variables is depicted in Fig. 2.2. On the horizontal axis is the logarithm of ratio of the two particle's angles a1, a2. The maximum angle is xed by the factorisation scale and

hence the ratio measures the relative values of both angles. Similarly, the condition α1 + α2 = 1 − x makes only one of the longitudinal variables independent and the

logarithm of their ratios on the vertical axis measures the relative hardness of the two emitted partons. The half-plane ln(α1/α2) < 0 corresponds to the phase-space

region where the particle 2 is harder. Fig. 2.2 presents also three characteristic lines on the Sudakov plane. The line ln(a1/a2) = 0 (in red) is the line of equal angles

of the emitted particles with respect to the direction of the incoming parton. The middle (violet) line marks equal minus lightcone variables v1 = v2, vi =

k_Ti √

αi. Finally,

the diagonal line (magenta) is the line of equal transverse momenta of the emitted particles, kT1 = kT2. All contributions from diagrams are normalised to the eikonal

phase-space dΦ, dened below. Hence, we multiply the traces of Feynman diagrams W by the relevant phase-space factors to obtain unintegrated distributions ˜W, that

2_{Traditional rapidity variable ζ is related with a through ζ =} ln(k+/k−)

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18 CHAPTER 2. COLLINEAR FACTORISATION T₂ T1 k = k v = v₁ ₂ log ( a / a )1 2 α₁ α₂ a₁ a₂ 1 log ( / ) 1 0 =

Figure 2.2: The logarithmic Sudakov plane used in the discussion of singularities. form the NLO kernel. The relation between two-particle phase-space dΨ and eikonal phase-space dΦ in four dimensions is given by:

dΨ ≡ dΨ(k1)dΨ(k2) = d4_k 1 (2π)4(2π)δ + (k2₁)d 4_k 2 (2π)4(2π)δ + (k₂2) = = 1 (2π)6 dα1 2α1 dα2 2α2 dkT1|kT1| dkT2|kT2|dϕ1dϕ12 =da1 a1 da2 a2 dα1 2α1 dα2 2α2 dϕ1 2π dϕ12 2π α2 1α22a21a22 (2π)4 ≡ dΦ α2 1α22a21a22 (2π)4 , (2.13)

where ϕ1 is the azimuthal angle of kT1 and ϕ12 is the angle between kT1 and kT2.

Therefore, the contributions from diagrams to the kernel, ˜W are their Dirac traces W multiplied by a factor Aa =

α2 1α22a21a22

(2π)4 .

Moreover, the n-dimensional phase space used in the CFP scheme is denoted by: dΨ(n) ≡ dΨ(n)_(k 1)dΨ(n)(k2) = dnk1 (2π)n−1δ +_(k2 1) dnk2 (2π)n−1δ +_(k2 2) (2.14)

In spherical variables related to angular variables it reads: dΨ(n) = 1 (2π)6+4εda1a 1+2ε 1 α2+2ε1 da2a1+2ε2 α2+2ε2 dα1 2α1 dα2 2α2 sin2εϕ12dϕ12 sin2εϕ1dϕ1 =dΦ(n)(α1α2a1a2) 2+2ε (2π)4+4ε , (2.15)

where ϕ12 ≡ ϕ1 − ϕ2 is the angle between the particles' kT's and dΦ(n) is the

n-dimensional eikonal phase space. In n dimensions A(n)

a = (α1α2a1a2)

2+2ε

(2π)4+4ε .

2.3 Inclusive CFP kernels from Feynman diagrams

In this section the extraction of inclusive MS kernels according to the CFP prescrip-tion will be reviewed. Despite not being the main interest of this thesis, the detailed description of extracting the NLO CFP inclusive kernels will be useful for introducing the new factorisation scheme. The latter factorisation, suitable for exclusive approach, will be used to calculate dierential distributions, presented in chapters 3 and 4 of this thesis, to be implemented in the MC computer program.

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2.3. INCLUSIVE CFP KERNELS FROM FEYNMAN DIAGRAMS 19 The new scheme will often be referred to as the MC factorisation scheme, see [58, 59] for reference. The discussion of technical dierences between the two schemes will be made on a concrete example in chapter 3. In this thesis the discussion of the MC factorisation scheme will be restricted to the emissions of the two real partons.

The evolution kernel P (x, αS(Q2)), introduced in (1.5) is also given by the

pertur-bative series: Pij(x, αS(Q2)) = αS 2πP 0 ij(x) + α_S 2π 2 P_ij1(x) + · · · , (2.16) where P1

ij(x) is the NLO kernel for splitting a parton j into a parton i. A kernel

is a matrix in avour space, which was written explicitly in the above, the indices i, j ∈ {q, ¯q, g}.

It is obtained in the CFP factorisation from the parton density Γ by means of taking a residue in the ε pole:

Pij(x) = 2Resε=0(Γij(x, ε)) (2.17)

In the following I will give formulae for calculating contributions to the CFP kernel from selected Feynman diagrams.

The parton density Γ, dened in (2.6), is a power series in K. To obtain the parton densities at NLO, the series is truncated to second order in αS:

K = K0(1 + (1 −P)K0) + O(α3S),

Γ = 1

1 −PK = 1 +PK + (PK)(PK) + · · ·

= 1 +PK0+P(K0(1 −P)K0) + (PK0)(PK0) + O(α3S)

(2.18)

The action of projection operator P on a kernel K0 is the following. The spin part

Pspin decouples spinor indices between neighbouring kernels, Pkin sets the incoming

momentum of the kernel to its left on-shell, and the pole part P P extracts a residue in ε after the phase space integration.

Γij(x, αS, 1 ε) = Zj δ(1−z)δij+P P Z dnk (2π)nxδ x − η · (p − k1− k2) η · p UiK 1 1 −PKLj , (2.19) where Zj, j ∈ {q, ¯q, g}, are renormalisation constants, calculable from virtual diagrams.

The Pspin consists of two operators: Ui and Li, the upper and lower parts, dened

depending on the type of partons connecting the kernels: Uq= / η 4η · q; Lq= /p; Ug = −gµν; Lg = 1 2 + 2ε −gµν+η µ_pν _{+ p}µ_ην η · p . (2.20)

For the action of spin operators on antiquark lines I follow the convention of ref. [21], that Uq¯= −Uq, in order to obtain a non-negative inclusive kernel Pq ¯q. The projection

operators of (2.20) are diagrammatically depicted in Figs. 2.3 and 2.4.

The contribution to a bare PDF, Γ(X)_{(x, ε)}_{, at NLO is obtained from a 2PI diagram}

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20 CHAPTER 2. COLLINEAR FACTORISATION ε (the pole part denoted by P P below) upon integration:3

ΓX(x, ε) = P P Z

dΨ(n)WX(k1, k2, ε)δα1+α2=1−xΘs(k1,k2)<Q. (2.21)

In the above, n-dimensional phase space element has been dened in (2.14), and the distribution

WX = Cw

X

q4

comes from a trace of the Dirac matrices. The list of Feynman rules for cut and uncut propagators and QCD vertices can be found in Appendix B. Detailed formulae for calculating W will be given in Appendix C. In the above C is the colour coecient. The following convention:

CF = N2− 1 2N = 4 3, CA = N = 3, Tf = 1 2f = 3 (2.22)

is adopted in this thesis, where N is the number of colours and f the number of avours in QCD. The wX _{function is dimensionless in the energy units (transverse momenta),}

q2 _{is quadratic, and s is typically linear.}

In equation (2.21) δα1+α2=1−x denotes the Dirac delta function δ(α1+ α2− 1 − x).

Similarly, Θs(k1,k2)<Qdenotes the Heaviside function Θ(Q−s(k1, k2)), which denes the

upper phase space boundaries. The choice of the s function is an important point. I will consider s(k1, k2) = max(a1, a2) instead of the original choice of CFP, which was

the maximal virtuality −q2. The denition of the phase space limit in terms of the

angular variable, which in the LO parton showers is related to the ordering variable, is a popular choice in the Monte Carlo programs, see for example [49]. At the leading order it enables including some NLO interference eects [30]. In the inclusive MS kernel of CFP the choice of the upper boundary has been proved to play no role, see also [70]. The question arises, how the choice of ordering variable will inuence results of the MC kernels. This issue has been addressed in ref [71], and it will not be further discussed in the present work. However, the analysis of singularity structure of dierential distributions contributing to the MC kernels in chapters 3 and 4 will conrm that the use of the angular variable is feasible. This choice will only be justied by the actual implementation in the Monte Carlo program.

In the following I will use a term contribution to inclusive evolution kernel from a Feynman diagram X in CFP scheme, meaning P(X)

ij (x) = 2Res0(ΓXij(x, ε)).

The parton distribution functions Γ (and the NLO kernel) do not depend on the hard process scale Q2, as indicated explicitly in eq (3.8). This property was investigated

in ref. [70], where we have examined mechanisms ensuring this property in the Pqq

kernel by explicit calculations. More precisely, we have checked that the inclusive MS kernel is independent of the choice of function s(k1, k2)dening the upper phase space

boundaries in the calculations of inclusive kernels.

The decomposition of a cross-section into a nite part representing a hard process and a divergent one, encapsulating all singularities interpreted as parton distribution functions of incoming or outgoing hadrons is realising the ideas of the old parton model of the early days of the theory of strong interactions. The decomposition (2.6) beyond

3_{In this notation the spin indices of Γ are dropped with respect to eq. (2.19), because they are}

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2.4. KERNELS FOR MONTE CARLO 21 the LO, however, is not unique. One can construct dierent factorisation schemes by means of redening the projection operator P, see ref. [83]. The choice of MS by CFP [21] was motivated by simplicity of practical calculations and a close connection to the operator product expansion technique [84,85] and parton model [86].

In the MS scheme inclusive parton density (and fragmentation) functions are uni-versal, the whole dependence on the type of the hard process being encapsulated in the coecient functions. This property enables applying PDFs measured in one process to obtain predictions for other processes in the collider experiments involving hadrons. For example, PDFs measured in the DIS process at HERA can be used in the data analysis of the W/Z boson production process at hadron colliders, such as Tevatron and LHC. The evolution kernels satisfy the so-called sum rules of the parton model:

(i) the fermion number conservation: Z 1

0

dx(Pqq(x) − Pq ¯q(x)) = 0,

(ii) momentum conservation, which for the space-like kernels reads: X

p0

Z 1

0

dx xPp0_p(x) = 0.

I follow the convention, in which the Pp0_p kernel describes the p → p0 splitting (P_p0_←p).

2.4 Kernels for Monte Carlo

In previous sections I have reviewed, how one obtains the contributions to the NLO ker-nels from Feynman diagrams according to the CFP factorisation. This is a convenient starting point for introducing Monte Carlo exclusive kernels, which will be done in this chapter. The practical calculation of MC kernels will be postponed to the following chapters.

In the CFP factorisation scheme, the contribution to the inclusive NLO kernel is identied as the coecient in front of 1

ε term of Γ, see equation (3.8). For most of NLO

diagrams after factorising o this pole, the rest of the calculations is done essentially in four dimensions. Diagrams with internal singularities that give rise to the additional

1

ε pole require special care. More precisely, for such diagrams terms of order O(ε 1₎

from Dirac traces and phase space are kept and together with 1

ε2 contribute to the

NLO kernel. Moreover, in CFP scheme there are oversubtractions of the two-real contributions ∼ 1

ε2 in P(K(1 − PK) compensated by (PK)(PK).4 Introducing the

MC scheme is motivated by the necessity of avoiding this kind of oversubtractions in the kernels used in the MC simulation, see [57]. At the same time, it is aimed to remain as close as possible to the CFP scheme at the technical level and preserve universality of PDFs in the MC scheme as it is done in the MS.

4_{Double poles do not contribute to the NLO kernel, yet enable to see that the real emissions in CFP}

build PDFs from the apparent geometrical series: Γ ' e+1

ε = 1

1−(1−e− 1ε)= 1 + (1 − e −1

ε) + (1 − e−1ε)2, instead of obtaining it directly from the exponent: Γ ' e+1

ε = 1 + 1 ε+ 1 2! 1 ε2 + . . ., as it is organised in the MC scheme. The CFP scheme hence features huge oversubtractions, that are unacceptable in the MC, see discussion in ref. [57]. Consequently, (PK)(PK) terms are absent in the MC scheme.

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22 CHAPTER 2. COLLINEAR FACTORISATION The scheme implemented in the MC employs the correspondence:

Z Q 0 ˜ Q4ε−1d ˜Q = Q4ε 1 4ε ↔ Z Q q0 d ˜Q ˜ Q = ln(Q/q0) (2.23)

between n- and four-dimensional integrals.5 _{In the above, the details of regulating the}

four-dimensional integral over scale ˜Q(here with a small cut-o parameter q0) play no

role, because the NLO kernel the coecients multiplying singular 1

2ε or ln(Q 2_/q2

0) are

the same. The inclusive MC kernel P(X)

M C(x)is dened as P_{M C}(X)(x) = ∂ ∂ ln Q2G (X) (Q, x), G(X) = Z dΨ(k1)dΨ(k2) δ x − q · η p · η WX(k1, k2, ε = 0) ΘQ>s(k1,k2)>q0, (2.24) where G(X) denotes a contribution from a diagram X to the NLO kernel calculated in

four dimensions from the beginning. The exclusive MC kernels are dened as integrands of P(X)

M C(x). Identifying the singular parts on the level of integrands and correctly

subtracting o internal singularities of WX (see next sections) is a highly non-trivial

task yet crucial for constructing MC. Therefore, the the analysis of the singularity structure of underlying Feynman diagrams described in chapters 3 and 4 in this thesis will play an important role in obtaining MC kernels.

I will postpone further discussion of the dierences between the two factorisation schemes until the example of the bremsstrahlung diagram, calculated both in n dimen-sions in the CFP scheme and in four dimendimen-sions in the MC scheme will be presented. The implementation of diagrams with a collinear singularity in the Monte Carlo will also be partly addressed.

The denition of the MC scheme is less formal than CFP and so far it is dened case by case for dierent classes of diagrams, see [57, 58]. In this section it will be dened explicitly for two dierent cases: ladder-type diagrams with collinear countert-erms and genuinely 2PI diagrams. Example Feynman diagrams corresponding to these contributions are displayed in Figs. 2.3 and 2.4. The gures present explicitly spin parts of projection operators; later in this thesis gures will show Feynman diagrams without (external) projection operators.

2.4.1 Diagrams requiring counterterms

The crucial dierence between evolution kernels in the CFP and MC schemes arises for the diagrams featuring the internal collinear singularity, thus requiring collinear counterterms.

The subtracted diagrams originate from terms P(K0(1 −P)K0) in eq. (2.18). In

both factorisation procedures they require counterterms P(K0PK0) to be subtracted

from P(K0K0), where K0 is truncated at the leading order. Due to dierent ways of

applying projection operators in MC and CFP, however, the resulting evolution kernels will be dierent. Also, in the MC scheme P(K0)(PK0)contributions are absent. These

dierences will be studied in the case of non-singlet bremsstrahlung diagrams in order

5_{When mentioning dimensionality in this thesis, I will refer to 4- or n-momenta, not the}

distri-butions. A convenient phrase n-dimensional contribution to the kernel, for example will mean a distribution depending on dimensionally regularised momenta (n-momenta).

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2.4. KERNELS FOR MONTE CARLO 23 to analyse the dierences in the structure of infrared singularities between the two schemes. In the following I will dene the MC scheme.

The spin parts of projection operators, dened in (2.20), are the same in both schemes. Fig. 2.3 displays the Feynman diagrams with the spin parts of projection operators applied corresponding to the counterterms discussed in this thesis. The orientation of diagrams is such, that the incoming proton is at the bottom and a hard process on top. U_q U_q L_g L_g L_g U_g Lq U_q L_q U_q Lq U_q

Figure 2.3: The illustration of the usage of spin projection operators, that were dened in (2.20), in the calculations of non-singlet (left) and singlet counterterms (middle and right) discussed in this thesis.

Formally, the Monte Carlo exclusive kernels are given by the integrand of GX_a(Q, x) = Z dΨ(k1)dΨ(k2) δ1−x=α1+α2 x hP0 (K0K0)Θq0<s(k1,k2)<Q−P 0 (K0P0(K0))Θq0<s(k1)<s(k2)<Q i = Z dΨ(k1)dΨ(k2) δ1−x=α1+α2 h Wladder(k1, k2, 0)ΘQ>s(k1,k2)>q0 − W ct (k1, k2)ΘQ>s(k1)>(k2)>q0 i , (2.25)

The cancellations of double poles are executed before taking the derivative in (2.24). Hence, P0_{operator acts on the integrands and extracts singular parts in four dimensions}

unlike the projection operator of CFP that extracts poles from n-dimensional integrals after phase space integration.

In eq. (2.25) the function s determining the upper phase-space boundaries is de-ned as s(k1, k2) = max(a1, a2), and q0 is again a technical cuto. The counterterms

feature an ordering of emissions. The (additional) ordering Θ function comes from the insertion of P0 _{operator between two LO kernels. It also changes kinematics, setting}

the momentum q1 = (p − k1) on its right on-shell.

The exact analytical forms of dierential distributions of counterterms Wct

i will be

given in sections 3.1.2 and 4.1.1, because they are discussed together with diagrams requiring subtractions. I will also show explicitly the integration of a subtracted dia-gram in section 3.1.2, while analysing the example of the double-gluon bremsstrahlung diagram. Since counterterms are subtracting the leading-order behaviour from ladder diagrams, their dierential distributions will be typically expressed using leading order

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24 CHAPTER 2. COLLINEAR FACTORISATION Altarelli-Parisi [26] kernels: Pqq(z) = 1 + z2 1 − z , Pqg(z) = 1 2(z 2 + (1 − z)2), Pgq(z) = 1 + (1 − z)2 z , Pgg(z) = 1 1 − z + 1 z − 2 + z(1 − z). (2.26)

2.4.2 Genuinely 2PI diagrams

All contributions to the NLO kernel from genuine 2PI kernels in P0_K

0 terms in (2.18)

are calculated in the following way in the MC scheme: G2P I(Q, x) = Z dΨ(k1)dΨ(k2) δ1−x=α1+α2 h W₂2P I(k1, k2, 0)ΘQ>s(k1,k2)>q0 i (2.27) Here, only one projector operator P0 _{is applied. It puts on-shell only the incoming}

parton. The function Θ does not impose the ordering of emissions, contributions from both a1 > a2 and a1 < a2 are present. The example 2PI NLO diagrams are presented

in Fig. 2.4, with spin projection operators shown explicitly.

U

_q

L

_g

U

_g

L

_q

U

_q

L

_g

Figure 2.4: Three examples of NLO cut diagrams. The rst two are not discussed in this thesis because they feature virtual emission (left) or belong to the gluon-gluon kernel (middle). The third one contributes to the singlet Pqg kernel, see chapter 4.

Since the 2PI diagrams do not feature any additional singularities,6 _their

contribu-tions to the evolution kernel are identical in the CFP and MC schemes.

2.4.3 Regularisation of infrared divergences

While collinear singularities are instrumental in the denition and extraction of evolu-tion kernels in the CFP and MC factorisaevolu-tion procedures, the infrared singularities of 2PI Feynman diagrams are still present and play an important role, especially in the MC implementation. They originate from quark and gluon propagators. The infrared divergences will be of the main interest in this thesis.

In the CFP scheme the infrared divergences are regularised using the principal value (PV) prescription, dened as:

P V Z 1 0 dα α = Z 1 0 dα α α2_{+ δ}2; δ 2 _{→ 0.} _(2.28)

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2.5. NON-SINGLET AND SINGLET EVOLUTION KERNELS 25 The prescription (2.28) is not unique. There are other possible ways of regulating divergent denominators discussed in the literature, see [80,81,8789]. The Mandelstam-Leibbrandt (ML) prescription of refs. [8789] has been employed in [90, 91] for calcu-lating the two loops splitting functions in CFP factorisation. In spite of having a few pleasant properties (such as Lorentz invariance), it is ill-suited for practical compu-tations, because it leads to a proliferation of diagrams (by roughly a factor of 2) due to additional ghost contributions. Therefore the MC scheme follows the CFP regu-larisation of longitudinal variables. Adopting the notation of ref. [21], the infra-red divergences are parametrised in the following way:

I0 ≡ Z 1 0 dα α = Z 1 δ dα α α2_{+ δ}2 = − ln δ + O(δ 2_), I1 ≡ Z 1 0 dα α ln α α2 _{+ δ}2 = Z 1 δ dα ln α α = − 1 2ln 2 δ + O(δ2), (2.29) where δ 1 is a cuto parameter.

The infrared divergences will be the main interest in this thesis. Let us remark that I0, due to its simple form, must originate from _α1

i factors in the Dirac traces and

a simple conguration of transverse momenta. The singular denominators are due to quark's and gluon's propagators. As will be shown, not all diagrams exhibit the I0

singularity. In some cases the spin structure of the numerators cancels the divergence. Such a detailed analysis is beyond the scope of this thesis, because the squared spin-summed Feynman diagrams do not expose their spin structure explicitly. Some work in this direction has been made using spin amplitudes, see for example [92] and [93], but such studies are beyond the scope of this thesis. Another simple observation is that the Dirac traces cannot produce logarithms. Hence, I1must originate from two-dimensional

integrations. Indeed, integrations over azimuthal angle and transverse momentum give rise to ln α1 or ln α2. If these terms have additional _α1_i factors, they combine to ln α_α_ii,

giving rise to I1 upon integration. Hence, I1 divergence is usually not a purely infrared,

but rather a combination of collinear-infrared divergences. These general remarks will be illustrated by the analytical forms of relevant diagrams in chapters 3 and 4.

For the purpose of this work it was useful to nd a parametrisation of multidi-mensional distributions that would make their singularities transparent on a three-dimensional plots. The logarithmic Sudakov variables depicted in Fig. 2.2 are par-ticularly suitable for graphical presentation and analysis of the infra-red singularities on the level of unintegrated distributions. The singly-logarithmic singularities appear on the Sudakov plane as innite narrow linear structures and doubly-logarithmic as innite two-dimensional structures (plateaux) stretching to innity in two directions.

2.5 Non-singlet and singlet evolution kernels

Before proceeding further with the detailed analysis of contributions to the NLO kernels in the MC scheme, I will dene the terms singlet and non-singlet evolution kernels used in chapters 3 and 4. The avour indices, for simplicity dropped in the previous section, will be now written out explicitly.

In general, the evolution equation for parton densities reads: Q2 d dQ2p(x, Q 2 ) = αS 2π X p0 (Ppp0 ⊗ p0)(x, Q2), (2.30)

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26 CHAPTER 2. COLLINEAR FACTORISATION where p and Ppp0 are the parton densities and evolution kernel for either space- or

time-like processes. The evolution kernel P is a matrix in the avour space and p(x, Q2₎ _is

a vector in avour space, p = {qi, ¯qi, G}. The order of indices in the Ppp0 kernel follows

the convention Pp←p0.

The explicit decomposition of the evolution equation (2.30) into quarks and gluons sector now reads:

Q2 d dQ2qi(x, Q 2_{) =} α(Q2) 2π " X k Pqiqk⊗ qk+ X k Pqiq¯k ⊗ ¯qk+ PqiG⊗ G # (x, Q2) Q2 d dQ2q¯i(x, Q 2 ) = α(Q 2₎ 2π " X k Pq¯iqk⊗ qk+ X k Pq¯iq¯k ⊗ ¯qk+ Pq¯iG⊗ G # (x, Q2) Q2 d dQ2G(x, Q 2_{) =} α(Q 2₎ 2π " X k PGqk ⊗ qk+ X k PG¯qk ⊗ ¯qk+ PGG⊗ G # (x, Q2) (2.31)

In practical calculations linear combinations of parton densities are used: q_i(+)= qi+ ¯qi ; q(+)= X i q_i(+) (2.32) q_i(−)= qi − ¯qi ; q(−)= X i q_i(−) (2.33)

together with the decomposition of P into valence (V) and sea (S) contributions: P(±)= PqqV ± P V q ¯q PF F = P(+)+ 2f PqqS PF G= 2f PqG PGF = PGq (2.34)

leading to the decomposition of matrix equation (2.30) into a singlet and non-singlet parts. This results in the non-singlet parts q(−)

i and q (+) i −f1q

(+) _{having their evolution}

equations decoupled from other avour combinations [83]: Q2 d dQ2q (+) (x, Q2) = α(Q 2₎ 2π [PF F ⊗ q (+) + PF G⊗ G](x, Q2) Q2 d dQ2G(x, Q 2 ) = α(Q 2₎ 2π [PGF ⊗ q (+) + PGG⊗ G](x, Q2) Q2 d dQ2q (−) i (x, Q 2_{) =} α(Q 2₎ 2π [P(−)⊗ q (−) i ](x, Q 2₎ Q2 d dQ2 q(+)_i − 1 fq (+) (x, Q2) = α(Q 2₎ 2π P(+)⊗ q_i(+)− 1 fq (+) (x, Q2) (2.35)

Possible means of implementing non-singlet and singlet DGLAP evolutions in a Monte Carlo program have been explored, see for instance [9496]. In ref. [95] evolution equations were implemented in the form (2.31), without diagonalising the kernel. Ref. [94] presents a solution, in which partons with dened avours are generated according to (2.35). The former approach adopted in ref. [96] is opposite: the singlet/non-singlet densities are generated throughout the evolution and the actual avours are associated with each event only at the nal step of generating each MC event.

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2.5. NON-SINGLET AND SINGLET EVOLUTION KERNELS 27 Summarising, this chapter presented the classic collinear factorisation methodology at the level of inclusive (integrated) evolution kernels depending only on x and Q. The main interest in this dissertation, however, will be the exclusive kernels dened in section 2.4. The decomposition into singlet and non-singlet is still valid for them.

The main aim of this thesis will be presenting the structure of infrared singulari-ties of Feynman diagrams entering the quark-quark, quark-antiquark and gluon-quark kernels. The Dirac traces W of the two-loop Feynman diagrams, listed in Appendix C, have been calculated using FeynCalc7_{, the open-source package for Mathematica.}

The technique of calculating colour traces described in ref. [97] has been employed to obtain the colour factors for all topologies.

The diagrams will be divided into contributions to the non-singlet, discussed in chapter 3 and singlet kernel in chapter 4. All diagrams will be analysed at the level of unintegrated distributions. The numerical analysis performed using the CCNAS framework will be followed by numerical analysis of the most singular contributions in the soft Sudakov limit dened in eq. (3.1). The eects of colour coherence resulting from gauge cancellations among two-real Feynman diagrams will be examined for the rst time within the unintegrated distributions, thus extending the studies of [6063]. The important aspect will be a numerical analysis of these collinear and soft singu-larities and their graphical presentation. For this purpose, a set of numerical programs has been developed under a common name CCNAS. The functionality of these pro-grams include generating unintegrated distributions from gauge-invariant groups of Feynman diagrams, single diagrams as well as their parts (typically the most singular ones) and calculating integrals of these diagrams. The entire programming framework will be presented in chapter 5.

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Chapter 3 Non-singlet kernel

In the following the MC factorisation scheme, dened for two real emissions in chap-ter 2, will be employed to obtain exclusive NLO non-singlet evolution kernels. The contributions to the kernels will be given diagramwise. The analytical expressions will be supplemented by numerical results, obtained from the CCNAS framework. The infra-red singularities of diagrams and groups of diagrams will be analysed in the fully dierential form.

The Feynman diagrams considered in this chapter originate from the leading order amplitude:

by means of adding a second gluon emission to all lines .

Contributions to the non-singlet NLO kernel (depicted as crossed diagrams) origi-nate from the squared sum of the above. It is convenient to discuss the contributions into the following subsets originating from the following squares of amplitudes:

(1) The cross-section for emitting two gluons, given by 2 ,

includes genuinely non-singlet diagrams that will be discussed in this chapter. They will be further divided into two subgroups depending on their colour factors. (2) The process of emitting a q¯q pair is displayed below

2 .

It consists of diagrams belonging to both non-singlet and singlet kernels. This subset refers to quarkanti-quark kernel and will be considered in section 3.4. (3) The amplitude squared describing quarkanti-quark transitions (with emitting

two quarks): 2 , 29

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30 CHAPTER 3. NON-SINGLET KERNEL governed by the Pqq¯ kernel, that will be discussed in section 3.5.

In the following, analytical, fully dierential distributions will be presented diagram-by-diagram followed by their graphical analysis involving partial phase space integra-tion. The multidimensional distributions will be analysed numerically in the logarith-mic Sudakov variables dened in section 2.2.2. The discussion of the numerical results will be supplemented by extraction of the relevant analytical formulae describing their singularity structure. They will be derived in the soft limit:

(

αi, kTi → 0,

k_Ti

αi = ai =const.

(3.1) The grouping in the above subsets (1)-(3) will be used. The subset (1) will be analysed in sections 3.1 and 3.2, followed by the subset (2) in section 3.4 and nally the subset (3) in section 3.5.

The technicalities associated with phase space integration of all diagrams have been already presented in detail in ref. [58]. The aim of this thesis is to extract all singularities in the unintegrated form. All distributions implemented in the computer program are regularised by cutos imposed on the variables: α1, α2 > δ, the ratio of

angles y > ∆, and a2 _{> κ}_{. For suciently small δ, ∆ and κ, the IR structures in the}

plots are independent on the values of cutos. In the following δ = ∆ = κ = 10−10_was

used.

The contributions from non-singlet diagrams will be further divided into subsets on the basis of their colour coecients: C2

F and CACF, see ref. [97].

While presenting dierential distributions in this and next chapter, I will use nota-tion introduced in secnota-tion 2.2.2. The Feynman rules for cut and un-cut propagators in the axial gauge are listed in Appendix B. The calculations of Dirac traces and contrac-tions ( ˜W (k1, k2, ε)) are performed using Mathematica package FeynCalc. They are done

in n dimensions, n = 4+2ε, ε > 0. For the purpose of the MC scheme, four-dimensional distributions will be given in most cases and the notation ˜W (k1, k2) ≡ ˜W (k1, k2, ε = 0)

will be adopted.

3.1 Bremsstrahlung diagrams. Soft counterterm.

Dis-cussion of the evolution variable.

The diagrams considered in this section are shown in Fig. 3.1. They consist of the subtracted amplitude-squared diagram (left) and the interference (right). While the latter is a 2PI diagram, the former enters the NLO kernel from the P(K0· K0) term

together with a counterterm (−P(K0PK0)).

I will start the analysis by presenting dierential distributions from the interference diagram (Bx), followed by the ladder diagram (Br). Then, the singular structure of Br will be analysed and its counterterm introduced in Section 3.1.2. This is the only place in this thesis, in which I will focus on obtaining integrated kernels. The phase space integration will be done both in n dimensions, following CFP method and in four dimensions, introducing the extraction of kernels in the Monte Carlo scheme, with the aim of comparing the two schemes. The analytical formula will be also helpful for understanding the sources of infrared divergences. In sections 3.1.3 and 3.1.4 the infrared singularities of the diagrams in Fig. 3.1 will be analysed numerically and analytically.

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3.1. BREMSSTRAHLUNG DIAGRAMS 31

P

Figure 3.1: Bremsstrahlung diagrams considered in this section: the Br diagram (with the collinear counterterm) and the Bx interference.

3.1.1 Dierential distributions

In this section the dierential distributions, normalised to the eikonal phase-space are given. In the Dirac trace of the Br diagram terms of order O(ε) are kept for the purpose of discussion of dierences between MC and CFP schemes in the next section. For the use in Monte Carlo, only four-dimensional contributions are important.

The Dirac trace of the interference diagram Bx is given by eq. (C.2.4) in Appendix C. Calculating the trace leads to the dierential distribution:

˜ WBx(k1, k2) = 4 CF2 − 1 2CACF αS 2π2 2 a2 1a22 ˜ q4 T₀Bx+ T₁Bxa1· a2 a2 1 + T₂Bxa1· a2 a2 2 + T₃Bx(a1 · a2) 2 a2 1a22 , (3.2) where T₀Bx = 2x1 + x 2 1 − x 1 α1 + 1 α2 − 2x, T₁Bx = 1 + 2x 2 α1 − 1 + x − x2_, T₂Bx = 1 + 2x 2 α2 − 1 + x − x2, T₃Bx = 2(1 + x2). (3.3) In this and next chapters the dierential distributions are parametrised in terms of lightcone variables α1, α2 variables and angles, see section 2.2.2. The x variable, x =

1 − α1 − α2 is a constant.

As follows from (3.2), in the axial gauge this diagram contributes in both C2 F and

CACF subsets. In this section I take under consideration only the part of it proportional

to C2 F.

The distribution for the ladder (unsubtracted) diagram will be written out in n dimensions (as a function of ε) in order to discuss the calculation of contributions to NLO kernels in the CFP scheme. A brief outline of the integration procedure will be given in the next section, whereas more details (featuring all steps of integration) the reader can nd in Appendix D.

The Feynman diagram Br1 gives rise to the following trace: WBr1(k1, k2, ε) = C2 Fα2S µ4ε x T r[/pγν(/p − /k1)γβ/q/η/qγα(/p − /k1)γµ] 4q · η q4_{(p − k} 1)4 dµν(k1)dαβ(k2), (3.4)

where the dimensional parameter µ comes from dimensional regularisation. If we now normalise the above to the eikonal phase space of eq. (2.15) we obtain the ˜WBr1 dis-tribution:

˜

WBr1(k1, k2, ε) = A(n)a W Br1_(k