Event shapes in electron-positron annihilation at NNLO accuracy

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MSc Physics

Track: GRAPPA

Event shapes in electron-positron

annihilation at NNLO accuracy

Author:

Rik Albers

10195343

Local Supervisor: Prof. Z. Tr´ocs´anyi

Assessor: Prof. Dr. E.L.M.P. Laenen Second assessor: Dr. B.W. Freivogel

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Abstract

Various event shape distributions are computed for two-jet production in electron-positron annihilation at NNLO accuracy using the CoLoRFulNNLO method. After adjusting the cutoff parameter ymin, we obtain agreement on the three-jet event shape predictions as well as the analytic result for the total cross section.

Subsequently, the distributions for thrust, heavy jet mass and total and wide jet broad-ening are resummed and matched with SCET predictions. For the thrust and heavy jet mass, the SCET prediction is available up to N3_{LL, while for the jet broadening it is} avail-able up to NNLL. Using the resummed predictions, one can apply a non-perturbative shift to account for hadronization effects and perform a fit on the magnitude of this shift as well as the value of αs(MZ).

We find reasonable agreement for the thrust, heavy jet mass and total jet broadening, but observe that the quality of the predictions is not enough for a precision fit on αs. The result for the wide jet broadening does not agree with the rest of the event shapes consid-ered, this might be explained by a lack of understanding of the SCET scale dependence.

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Acknowledgements

First of all, I would like to thank my supervisor, Eric Laenen, for introducing me to this project and the Particle Physics Research Group in Debrecen, and for his guidance throughout the project.

I would like to thank my local supervisor in Debrecen, Zoltan Tr´ocs´anyi, for his hospi-tality and giving me the opportunity to take part in the research conducted in the Particle Physics group.

Furthermore, I would like to thank the Research Group in Debrecen and in particular G´abor Somogyi, Adam Kardos and Zolt´an Sz˝or for their continuous help and feedback during the project. I greatly appreciate the way in which I was welcomed into the group and the great atmosphere throughout the year.

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Chapter 1 Introduction

Recent progress in the experimental precision of high energy physics by experiments such as LEP [1, 2] have produced precise measurements of various event shape distributions for electron-positron annihilation. This experimental data allows for a detailed compar-ison between theory and experiment, and places the main source of uncertainty on the theoretical prediction of the event shape distributions. Since the LEP experiments were performed, interest has arisen in next-to-next-to-leading order (NNLO) corrections to var-ious electron-positron annihilation observables. In recent years, a number of methods have been able to produce results for NNLO corrections to three-jet event shapes [3, 4]. With the availability of these calculations, precise predictions of for example the strong coupling constant αs should become possible.

More recently, the CoLoRFulNNLO method for NNLO corrections was incorporated into a general framework to produce high precision results for various QCD processes [5– 12]. This method was first applied to higgs boson decay [13] and has recently produced results for three jet production in electron positron annihilation [14].

In addition to the truncation of terms of higher order in αs, the theoretical precision is limited by the handling of logarithmic terms that spoil the order-by-order convergence. The first next-to-leading-log (NLL) resummation of various event shapes was achieved by Catani, Trentadue, Turnock and Webber in 1993 ref. [15]. With the calculation of NNLO corrections, there is a need to increase the logarithmic accuracy to N3_{LL, so that} all logarithmic terms that appear at _O(α3

s) are accounted for. Nonetheless, higher order calculations were not achieved until the development of SCET [16–18]. This framework has been shown to allow for resummation up to N3_{LL for thrust [19], heavy jet mass [20] and} up to NNLL for the total and wide jet broadening [21]. Recently the R-matching scheme has also been extended to N3_{LL [22].}

A third source of theoretical uncertainty is the hadronization process, since fixed order calculations only provide a prediction at the parton level, while detectors collect informa-tion on the hadrons that are produced. In order to account for this one can employ power corrections. The framework for power corrections was developed in 1996 [23] and has been applied to NLO distributions with NLL resummation [24] for multiple event shape distribu-tions. When the thrust distribution became available at NNLO with N3_{LL resummation,}

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the power corrections have been studies for this observable [19, 25]. The heavy jet mass was also studied [20] at NNLO and N3_{LL and seemed to rule out the framework of power} corrections as a consistent way of dealing with non-perturbative effects.

In this thesis, we will start with a description of some aspects of Quantum Chromo Dynamics (QCD) at NNLO and the process of two-jet production in electron-positron annihilation. The subtraction scheme on which the CoLoRFulNNLO method is based will be outlined in chapter 3. The results of an implementation of the two-jet process as well as a discussion on some ambiguities are presented in chapter 4.

Subsequently, for a number of the event shapes predicted at NNLO, we will apply resummation to N3_{LL (for thrust and heavy jet mass) or NNLL (for the total and wide jet} broadening). The process of resummation in Soft Collinear Effective Theory (SCET) will be outlined in chapter 5.

For the resummed event shapes, we will then employ power corrections, which are de-scribed in chapter 6. Finally, in chapter 7, fits to the parameter for the power correction and the strong coupling constant are presented as well as the resulting distributions for various event shapes. In doing so, we will be able to draw new conclusions on the consis-tency of power corrections as a framework for accounting for non-perturbative effects. We conclude the thesis in chapter 8.

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Chapter 2 QCD at NNLO

Quantum Chromo Dynamics (QCD) is the theory that describes the strong interaction and the behaviour of quarks and gluons as a result. The Lagrangian in QCD is given by [26]

L = −1₄Fa µνF

µν

a + q(i /D− m)q (2.0.1)

with /D = γµDµ, q is the quark field and Fµνa is the field strength tensor of the gluon. The field strength Fµν is given by

Fµνa = ∂µAaν − ∂νAaµ+ gf abc

AbµA c

ν. (2.0.2)

The covariant derivative is defined by

Dµ= ∂µ− igAaµta, (2.0.3)

where ta _{are the matrices that form a representation of the generators T}a _{of the group.} The generators, as well as the representation matrices, satisfy the SU (3) algebra in terms of the structure constants fabc_:

[Ta_{, T}b_{] = if}abc_Tc_. _(2.0.4)

This algebra is an example of a non-abelian gauge group SU (N ), whose elements do not commute. The non-abelian nature of the underlying gauge group of QCD is a key aspect of the theory, as for example it allows for interactions between gluons. One of the implications of gluon self-interaction is that the strong force becomes stronger with increasing distance. Individual partons (quarks and gluons) are forced to form constituent particles (hadrons) in order to be stable. The result is a theory in which we cannot observe free partons (at the energy of current detectors). In a detector, the quarks and gluons that are created as a result of a collision will hadronize and form jets of (relatively) stable particles. More details on jets will follow in section 2.2.

In this thesis only the lightest five quarks (d, u, s, c, b) will be taken into account, since these are relevant at the energy scale that is examined (Q = MZ ≈ 91.2GeV ). Further-more, their mass is neglected in all computations. Considering massless QCD allows for

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some simplification, but also introduces the necessity of corrections to adjust for this ap-proximation. The mass of the bottom quark, the heaviest of the relevant quarks in our calculations, is one of the reasons that we need to introduce power corrections as will be discussed in chapter 6. One of the goals of this work is to implement the process of electron-positron annihilation into two jets into the CoLoRFulNNLO framework. In sec-tion 2.3, some of the properties of this process will be outlined and we will discuss the steps necessary to perform the implementation.

2.1 Ultraviolet renormalization

Loop corrections to the vertices and propagators in QCD lead to results that do not make sense in d = 4 dimensions. As a result, it is necessary to replace the parameters as they appear in the Lagrangian by renormalized ones. To perform the UV renormalization, we use the conventions and notation as they appear in the CoLoRFulNNLO method. Therefore, in this thesis the ultraviolet (UV) renormalization is achieved with dimensional renormalization in d = 4_{− 2 dimensions and in the MS scheme. UV renormalization will} not be discussed in great detail, but merely to an extend that allows us to present the conventions and notation used.

The m-parton unrenormalized amplitudes are denoted by _|Ami and the renormalized amplitudes are denoted by _|Mmi. Thus, once we are using the notation |Mmi, we assume that UV renormalization has already been applied to all matrix elements.

To outline the conventions for UV renormalization, we start out with the loop expansion of the unrenormalized amplitude _|Ami [14]:

|Ami = (4παs)q/2 h |A(0) m i + αB s 4π |A (1) m i + αB s 4π 2 |A(2) m i + O(α 3 s) i , (2.1.1)

where _|A(n)m i is the n-loop correction to the amplitude, α_sB is the bare coupling (as it appears in the QCD Lagrangian) and q = m_{− 2 with m being the number of final state} partons at the Born level.

To achieve renormalization, we should replace the bare coupling αB

s with the running coupling αs(µ), which depends on the renormalization scale µ. In the MS scheme this replacement is given by αB sµ 2 0 S MS = αsµ h 1₋αs 2π β0 + α_s 2π 2 β2 0 2 − β1 2 +_O(α3 s) i . (2.1.2)

In this equation, the factor SMS

is given by SMS

= (4π) _exp(

−γE), (2.1.3)

and the coefficients βm are the coefficients that appear in the renormalization group equa-tion for the coupling constant

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dαs(µ) d log µ2 =−αs(µ) X m αs(µ) 4π m βm, (2.1.4) β0 = 11 3 CA− 4 3nfTR, β1 = 34 3 C 2 A− 20 3 CATRnf − 4CFTRnf. (2.1.5) By substituting the bare coupling for the running coupling, we can define the renormalized amplitudes in terms of the unrenormalized ones [14],

In chapter 3, the cancellation of infrared (IR) divergent terms will be discussed. Once this is done, the regularization parameter can be set to = 0. The -dependent terms do, in principle, not contribute to the result of the calculation. It is however important that one takes care to use the same renormalization conventions throughout the calculation to ensure the cancellation of -dependent terms. For the rest of this thesis, we will assume all matrix elements are UV renormalized.

2.2 Jets

In a QCD event in a collider, the final state particles are quarks and gluons. Due to confinement, quarks and gluons can not be observed individually. When highly energetic quarks and gluons are produced in a collider they will produce jets, sprays of highly en-ergetic hadronic particles. After this hadronization process, the information regarding energy and momentum for the seperate partons need to be analysed in such a way that it can be compared to theoretical predictions, in which only a limited number of partons are considered.

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2.2.1 Jet algorithms

To analyze the jet events in which large numbers of hadrons are produced, jet algorithms are used to cluster particles into jets and to assign properties to each jet instead of to each individual particle. A number of different jet algorithms are used and each combines par-ticles through different conventions. An often used method is to define a distance measure in momentum space, dij, and minimum distance dcut [27, 28]. To prevent ambiguities from possibly overlapping jets, an iterative procedure is used. Firstly, one selects the smallest of all values dij. If this distance is larger than the minimum distance dcut, the momenta are considered to be ’apart’ enough to be defined as separate jets. If this distance is smaller than dcut, the momenta i and j are combined into a single momentum (so that they are in the same jet) and the procedure starts again. The value of dcut thus determines the definition of what one considers to be an n-jet or an n + 1-jet event.

Jet algorithms use different definitions for the distance measure dij. A commonly used scheme is the so called E-scheme [28], in which the invariant mass is used as the distance measure,

dij = Mij2/Q

2_. _(2.2.1)

The JADE scheme, which will be used in the code used in this thesis, uses a measure relevant for massless QCD [29],

dij = 2EiEj(1− cos θij), (2.2.2)

where θij is the angle between particles i and j. Another example is the Durham scheme, in which the distance measure is defined by

dij = 2min(Ei2, Ej2)(1− cos θij). (2.2.3) Not only the distance, but also the recombination scheme can differ between algorithms. For the JADE algorithm for example, it is possible to rescale the spatial part of the combined momentum:

pij =

Ei+ Ej |~pi+ ~pj|

(~pi+ ~pj), (2.2.4)

so that the resulting momentum is also massless.

2.2.2 Event shapes

Event shape variables are infrared safe observables that describe the energy and momentum of final state particles. Event shapes allow the analysis of the distribution of energy and momentum in a consistent way and allow for comparison between theoretical predictions and results obtained in an experiment. Event shapes are required to be infrared safe by definition; soft particle emission or collinear parton splitting do not change the value of an event shape for a particular final state.

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An event shape distribution dσ/dJ for some event shape J is computed using a delta function containing the jet function’s dependence on final state momenta:

dσ dJ = Z dσ dJdJ δ(J− J({p})) = Z dσδ(J _{− J({p})).} (2.2.5)

Such an event shape distribution can be compared between theoretical an experimental results, since it contains information about the energy and momenta of jets instead of individual partons.

In this thesis, distributions for a number of different event shape variables are presented: Thrust T : The thrust [30] is defined by the expression

T(m) _{= max} ~ n Pm i=1|~pi· ~n| Pm i=1|~pi| , (2.2.6)

where ~n is a unit vector, chosen in the direction for which the value of the thrust is maximal. For a two (massless) particle final state the value of the thrust is trivially T = 1, since then the thrust vector is aligned with both back-to-back momenta. For final states with more than two particles the thrust takes values in the range 0.5≤ T ≤ 1. A value T = 1 coincides with the limit of two jets of collinear particles, while T = 0.5 occurs in the limit of a spherically symmetric distribution. It is common to use the parameter τ = 1_{− T , so that the distribution is situated in the} region 0_{≤ τ ≤ 0.5.}

Scaled heavy jet mass ρ: The heavy jet mass [30] is the largest of the invariant masses in the two hemispheres Hi (split by the plane normal to the thrust vector),

MH = max(M1(~n), M2(~n)), (2.2.7) where M2 i = X k∈Hi pk 2 . (2.2.8)

Finally the parameter ρ, the scaled heavy jet mass, is defined as ρ = M

2 H

s . (2.2.9)

Note that in the case of two massless final state partons, the value of the parameter is trivial, ρ = 0.

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C parameter: The eigenvalues λi of the tensor Θab ₌ P ipaipbi/|~pi| P j|~pj| , (2.2.10)

where a, b = 1, 2, 3, define the value of the C parameter [31] by

C = 3(λ1λ2+ λ2λ3+ λ3λ1). (2.2.11) In the case of two final state partons with momenta along the z axis, the tensor has only one nonzero value and thus only one nonzero eigenvalue. Thus in that case, C = 0.

Wide and narrow Jet Broadening BW, BN: Values related to the transverse momen-tum measured with respect to the thrust axis. For each hemisphere Hi (split by the plane normal to the thrust vector) one defines [32]

Bi = P k∈Hi|~pk× ~n| 2P i|~pj| , (2.2.12)

which is trivially zero in the case of two final state partons. The wide jet broadening is then defined as the largest of the two

BW = max(B1, B2), (2.2.13)

while the narrow jet broadening is the smaller of the two

BN = min(B1, B2). (2.2.14)

Total Jet Broadening BT: From the values for both wide and narrow jet broadening, the total jet broadening is defined as the total of the two

BT = BW + BN. (2.2.15)

Rapidity y: A parameter defined as

y = 1 2ln E + pk E_{− p}k , (2.2.16)

where the momentum pk is the momentum parallel to an axis. This axis is often chosen to be the thrust axis, but can also be an arbitrary axis. In the case of two massless final state partons, if measured along an arbitrary axis, the rapidity is the only non-trivial eventshape listed here.

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Cont. e+_e− → X loop corrections Born q ¯q tree V q ¯q 1-loop R q ¯qg tree VV q ¯q 2-loop RV q ¯qg 1-loop RR q ¯qq ¯q, q ¯qgg tree Table 2.1: Contributions to the process e+_e−

→ q¯q at up to NNLO. The table shows the contribution, the final states produced and the number of loops to be included.

2.3 e

+

e

−

_{→ 2 jets at NNLO}

e+ e− q ¯ q p4 p5 p1 p2 γ

Figure 2.1: Tree level Feynman diagram for the process e+_e− → qq

One of the goals of this thesis is to produce up to next-to-next-to-leading order QCD corrections to the process e+_e−

→ q¯q, which is labeled as a ”2 jet” process since the born contribution includes two final state QCD partons that will produce two jets in a detector. The tree level process is shown diagrammatically in figure 2.1. The corrections at NLO are identified by a virtual and a real correction. The former describes the correction from a one-loop matrix element, which includes a virtual particle. The latter is a correction for the born matrix element that produces an extra gluon in the final state, which thus results in another jet (which might be collinear to one of the other jets and may thus not be detected as being another jet) in a detector. At the NNLO level one distinguishes three contributions. The first is the doubly-virtual correction, which is the born matrix element with a two-loop correction. The second is a real-virtual correction, which includes both a virtual (loop) correction and a real (additional final state parton) correction. The third is the doubly-real correction, which describes the contribution from the process producing an additional two final state partons. The processes to be considered in each of the contributions are shown in table 2.1.

The matrix elements used for the doubly-virtual correction are computed in ref. [33]. In this paper the real-virtual and doubly-real corrections are computed as well, but these were previously available in the code, as will be explained in section 2.3.2.

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2.3.1 Event shapes

The total cross section is known up to NNLO from direct calculations [34] and can be written as σ = σ0 1 +αs π C1+ α_s π 2 C2 , (2.3.1)

where σ0 is the total cross section at born level, and the factors are known to be C1 = 1 and C2 = 365 24 − 11ζ(3) − 11 12− 2ζ(3) 3 Nf, where ζ(x) is the Riemann Zeta function.

For the event shapes defined before in this chapter, most values are trivial at the born level of the 2-jet process and thus also trivial for the virtual and doubly-virtual corrections. When this is the case, the event shape distribution calculated from the leading order contribution to the 2-jet process is clustered in one bin, since it has only a single value, which means that in the rest of the distribution only the NLO and NNLO corrections to the 2-jet process contribute. The NLO contribution to the 2-jet process therefore gives the leading contribution to the event shape distribution. The first correction is coming from the NNLO contribution to the 2-jet process. For this reason it is chosen to consider the rapidity y on an arbitrary axis, so that the contribution from the LO 2-jet contribution is non-trivial and it becomes a true NNLO calculation. The parameters T, ρ, BT, BW and C are all trivial at leading order, as was illustrated in chapter 2. Take note that we will still write ’NLO contribution’ for the contribution to an event shape coming from the NLO correction to the two-jet event, even if this is the leading contribution to the event shape distribution.

2.3.2 Implementation

A part of the matrix elements used in the code for the process e+_e− _{→ 3jets, which was} implemented before e+_e−

→ 2jets, can be used in the code for the process e+_e−

→ 2jets. Table 2.2 illustrates which matrix elements can be used, and what their counterparts are. For the 3 jet process, an extra parton is emitted at the born level, which thus coincides to the real correction for the 2 jet process. Similarly, the 3 jet real and virtual corrections coincide with the 2 jet process’ doubly-real and real-virtual corrections, respectively.

Additionally, one must specify the patterns to be considered for each contribution. One needs to distinguish between up-type and down-type quarks, so that for example for the born process (and similarly for the virtual and doubly-virtual corrections) two patterns need to be considered:

1) e+_e−

→ u¯u, 2) e+_e−

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3 jets 2 jets

Born _→ Real correction

Real correction → Doubly-real correction Virtual correction → Real-virtual correction Table 2.2: The matrix elements for the process e+_e−

→ 3jets that are used in the code for the process e+_e− _{→ 2jets.}

For the real and real-virtual correction, the patterns are

1) e+_e−

→ u¯ug, 2) e+_e−

→ d ¯dg, while for the doubly-real correction we have seven patterns:

1) e+_e− → u¯ugg, 2) e+_e− → d ¯dgg, 3) e+_e− → u¯ud ¯d, 4) e+_e− → u¯uu¯u, 5) e+_e− → u¯uc¯c, 6) e+_e− → d ¯dd ¯d, 7) e+e−→ d ¯ds¯s.

Since the subtraction scheme used in the code needs a number of colour connected matrix elements, these need to be provided too. The colour connected matrix elements are defined by

Mm,(i,k) =hMm| Ti· Tk|Mmi . (2.3.2)

The coefficients for the connected matrix elements can be obtained from colour algebra [5],

Ti· Tj = Tj · Ti (i6= k), T2i = Ci, (2.3.3) where Ci = CF, CA in the fundamental and adjoint representation, respectively. Using colour conservation we can further write

X

i

Ti|Mi = 0. (2.3.4)

For the 2-jet NNLO subtraction scheme, the born level colour connected matrix element it needed, as well as the born level doubly correlated matrix element

|M(i,k),(j,l)|2 =hMm|{Ti· Tk, Tj· Tl}|Mmi , (2.3.5) where {A, B} = AB − BA is the anti-commutator.

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It is worth noting that the factor S that appears in the matrix elements in [33] differs from the ones used in the UV-renormalization in the CoLoRFulNNLO framework. In the literature the factor SMS¯

is often abbreviated to S, however in CoLoRFulNNLO there is a distinction [5]: SMS¯ = (4π) _exp( −γE), (2.3.6) S = (4π Γ(1− ). (2.3.7)

The normalization of this ambiguity is done at the level of the insertion operators, so that when implementing the matrix elements from [33] one should not change the normalization.

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Chapter 3 Kinematic singular behaviour

In a QCD jet event, the observed jet structure determines the process to which the event is assigned. The experimental analysis of an n-jet process, for which one would naively assume the n jets originate from n final state QCD particles, can contain contributions from processes with n + 1, n + 2, etc. final state particles. These contributions are the real corrections, where the n + 1 jet structure resembles that of an n jet event, and the event will be observed as an n particle event. Specific limits of this behavior give rise to singular behavior.

In a process involving n + 1 final state particles, it is possible for two particles to cluster together and to be observed as one jet. The properties of the jet will resemble that of a jet caused by one QCD particle. As a result, the event produced by an n + 1 particle event will be assigned to an n particle event. The collinear limit, in which two momenta become completely parallel, and its implications will be discussed in section 3.1.

Another possibility for an n + 1 particle event to be observed as an n particle event is when one particle has low energy. The soft particle will not be observed and the event will be assigned to an n particle process. The soft limit, in which a particle’s energy approaches zero, and its implications will be discussed in section 3.2.

For n+2 particle events, which become relevant at NNLO, more situations are possible. For example, three particles can become collinear [5], or a combination of two collinear particles and a soft one. More details on kinematic singular behavior at NNLO are given in section 3.4.

In an analytic computation, the Kinoshita-Lee-Nauenberg (KLN) theorem ensures the cancellation of infrared singular terms in the total cross section [35]. The KLN theorem states that, after summing over all initial and final states, the transition rate should be an infrared safe observable. For a numerical calculation (Monte Carlo integration) however, it is not possible to make use of the cancellation of divergent terms, since the matching of singular terms occurs after integration. When making use of Monte Carlo integration, several methods can be employed to assure regularization of the singular terms, including [36]:

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In the slicing method, which will not be explained to great detail since it is not the method used in this thesis, the part of the phase space in which the integration diverges is sliced out [37, 38]. Consider a cross section computed using a jet resolution variable J, which is singular in the region where J → 0. The singular part of the phase space can be sliced out by defining a parameter Jδ << Q

σ = Z 0 dJdσ dJ = Z Jδ 0 dJdσ dJ + Z Jδ dJdσ dJ, (3.0.1)

below which the non-singular part can be neglected, since its integral over the phase space below Jδ will be of O(Jδ/Q). Thus the integral over the differential cross section in the part of the phase space below Jδ can be approximated by the singular part, which can be integrated analytically

σ = Z Jδ 0 dJdσ sing dJ + Z Jδ dJdσ dJ +O(Jδ/Q) (3.0.2) = σsing |J =Jδ+ Z Jδ dJdσ dJ +O(Jδ/Q). (3.0.3)

The singular part of the cross section can be handled in the same way as in an analytic calculation, i.e. the singularities in the real and virtual contributions will cancel, while the rest of the phase space can safely be integrated over using Monte Carlo integration.

subtraction:

Another method to eliminate divergences from numerical calculations is the subtrac-tion method. As mensubtrac-tioned before, infrared divergences cancel from the total cross section after integration over the phase space, which is different for the real and virtual corrections. Thus the expression

σN LO ₌ Z m+1 dσR_J m+1+ Z m dσV_J m, (3.0.4)

is infrared safe, where the real correction dσR _{is integrated over the m + 1 parton} phase space, while the virtual correction dσV _{is integrated over the m parton phase} space. To regularise both integrals seperately, an approximate cross section dσA _is introduced, which is added and subtracted [33, 39]

σN LO = Z m+1 dσR Jm+1− dσR,AJm + Z m dσV + Z 1 dσR,A Jm, (3.0.5)

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such that the singular behavior is regularised in each of the integrals seperately. For this method to work properly, dσA_{is required to match the singular behavior of dσ}R_. Additionally, it must be possible to integrate dσA _{over the one-parton phase space} without the use of a jet function. The regularization of singular terms by subtraction will be discussed in detail in this chapter.

3.1 Collinear limit

In the case of three final state partons, at NLO there are three cases of parton splitting that can lead to collinear behaviour [40], as indicated in figure 3.1. Factorization is used to write the n + 1 parton matrix element as an n parton matrix element combined with a parton splitting function. This can be used to generalize the collinear limit for any process involving the same final state particles, using the Altarelli-Parisi splitting functions defined by the expression [5] Cir M (0) m+1 2 = 8παsµ2 1 sir hM (0) m | ˆP (0) qg |M (0) m i , (3.1.1)

where ˆPqg(0) is the Altarelli-Parisi splitting function and Cir is an operator that takes the collinear limit of particles i and r and keeps only the leading singular terms.

M g q q (a) q_{→ q + g} M q ¯ q g (b) g_{→ q + ¯q} M g g g (c) g_{→ g + g} Figure 3.1: Diagrams for parton splitting processes at NLO. The collinear limit can be parametrized by the Sudakov parametrization [40].

iµ_{= z} ipµ+ kµ⊥− k2 ⊥ zi nµ spn (3.1.2) rµ_{= z} rpµ− kµ⊥− k2 ⊥ zr nµ spn (3.1.3) where zi+zr = 1 are the momentum fractions and the momenta iµand rµbecome collinear in the direction pµ_{. In this parametrization, the transverse momentum k}µ

⊥ obeys k⊥· p = k⊥· n = 0 and in the collinear limit k⊥ → 0. Note that all momenta, with the exception of k⊥, are lightlike (p2 = i2 = r2 = n2 = 0).

The splitting functions will be evaluated in the Feynman gauge, so that dµν(p) = −gµν + (pµnν + nµpν)/p· n where nµ is an arbitrary reference momentum. The Feynman

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gauge is a physical gauge, in which the leading collinear singularities arise as a result of external partons splitting in a collinear fashion [8]. This property allows the use of cut graphs, which only account for splitting of external partons.

3.1.1 Collinear

q

_{→ q + g splitting}

pi pir

pr

Figure 3.2: Cut diagram for the process q → q + g

In the case of a quark splitting to a quark and a gluon, the cut diagram in figure 3.2 applies. From this diagram we derive the partial amplitude

Fqqg = CFg2sµ 2/i + /r sir γµ_/iγν_d µν(r, n) /i + /r sir = CF4παsµ2 /i + /r sir γµ/iγν −gµν+ rµnν + nµrν r· n /i + /r sir = CF4παsµ2 /i + /r sir −γµ_/iγ µ+ /r/i/n + /n/i/r r_{· n} /i + /r sir , (3.1.4)

where we have included the propagator of the quark with momentum pir = pi + pr, since this is necessary to compute the total amplitude shown in figure 3.1. We use the identities given in appendix A.2 to write

(/i + /r)(−γµ_/iγ

µ)(/i + /r) = (d− 2)(/i + /r)/i(/i + /r)

= (d_{− 2)/rs}ir, (3.1.5)

(/i + /r)(/r/i/n + /n/i/r)(/i + /r) = (/i + /r)(sirn/− srn/i + sin/r)(/i + /r)

= sir(sni/i + snr/r + sin/i − srn/r + /i/n/r + /r/n/i)

= sir(2sin/i + sin/r + srn/i − sirn),/ (3.1.6) since i2 _{= r}2 _{= 0. We can then rewrite equation 3.1.4 as}

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Fqqg = CF4παsµ2 1 sir (d− 2)/r + 2 srn (2sin/i + sin/r + srn/i − sirn)/ . (3.1.7)

Inserting the Sudakov parametrization and using the relations from appendix A.2 we obtain

Fqqg = CF4παsµ2 1 sir (d_{− 2)z}r+ 1 zr (2z2 i + zizr+ zrzi) /p +O(1/k⊥) = CF4παsµ2 2/p sir −zr+ 1 + z2 i zr +_O(1/k⊥). (3.1.8)

We have kept only the terms that contribute to the leading singular part (_O(1/k2

⊥)), which is the part that appears in the Altarelli-Parisi splitting kernel. The splitting kernel is, in this case, a matrix in the spin state of the unresolved parton,

Cir M (0) m+1 2 = 8παsµ2 1 sir hM (0) m |si hs| ˆP (0) qg |s 0 i hs0 |M(0) m i . (3.1.9)

The m + 1 parton matrix element is factorized to an m parton matrix element and the Altarelli-Parisi splitting kernel, which takes into the account the collinear splitting of an external (in this case) quark. Comparing this equation to 3.1.8, we see that the kernel takes the form

hs| ˆPqg(0)(zi, zr, k⊥; )|s0i = CFδss0 −zr+ 1 + z2 i zr , (3.1.10)

which is diagonal in the spin state of the unresolved parton. The factor /p in eq. 3.1.8 comes from the two external ¯u,u that were taken into account by including the entire propagators for the external quarks. Since the ¯u,u should be associated with the m parton matrix element, the factor /p should not be included in the splitting kernel.

3.1.2 Collinear

g

_{→ q + q splitting}

µ ν

pir pr

pi

Figure 3.3: Cut diagram for the process g _{→ q + ¯q}

For the case of a gluon splitting into two quarks, the amplitude is derived from the diagram shown in figure 3.3. The methods used in this calculation are relatively simple (contracting

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indices and rewriting in terms of sij) and thus most of it will not be discussed in great detail. The amplitude is given by

Fgqq,µν =TR4παsµ2 dµρ(i + r) sir Tr[γρ/iγσ /r]dσν(i + r) sir =TR16παsµ 2 s2 ir −gµρ+ 2 (i + r)µnρ+ nµ(i + r)ρ sir,n iρrσ+ rρiσ− 1 2g ρσ sir × −gσν + 2 (i + r)σnν + nσ(i + r)ν sir,n =TR16παsµ 2 s2 ir [F1,µν+ F2,µν+ F2,νµ+ F3,µν] , (3.1.11)

where we have defined the functions Fi,µν according to the powers of sir,n = sin+ srn that occur: F1,µν = iµrν+ rµiν − 1 2gµνsir, (3.1.12) F2,µν = 1 sir,n h iµrσ + rµiσ− 1 2g σ µ sir ih 2(i + r)σnν + 2nσ(i + r)ν i = 1 sir,n h − iµ(i + r)νsrn− rµ(i + r)νsin+ nµ(i + r)νsir i , (3.1.13) F3,µν = 1 s2 ir,n h 2(i + r)µ(i + r)νsinsrn+ nµ(i + r)νsirsrn+ nµ(i + r)νsirsin − nµ(i + r)νsir,nsir i = 1 s2 ir,n h 2(i + r)µ(i + r)νsinsrn i . (3.1.14)

Inserting the Sudakov parametrization, we write the amplitude in powers of sir ∝ k2⊥, so that we can neglect any terms that will not contribute to the leading singular part. We write, using sir =− k2 ⊥ zizr, iµ = zipµ+ kµ⊥+ zr sir spn nµ, (3.1.15) rµ = zrpµ− kµ⊥+ zi sir spn nµ, (3.1.16) (i + r)µ = pµ+ sir spn nµ, (3.1.17)

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and substitute these into equations 3.1.12 - 3.1.14 using the relations given in appendix A.2, F1,µν =2zizrpµpν + (zr− zi)(kµ⊥pν + pµk⊥ν)− 2k ⊥ µk ⊥ ν + sir spn (zi2+ z 2 r)(pµnν + nµpν)− 1 2gµνsir+O(k 3 ⊥), (3.1.18) F2,µν =− zriµ(i + r)ν − zirµ(i + r)ν + sir spn nµ(i + r)ν =_{− 2z}izrpµpν + (zi− zr)k⊥µpν + sir spn h 2zizrpµnν + (1− zi2− zr2)nµpν i +O(k3 ⊥), F2,µν + F2,νµ=− 4zizrpµpν + (zi− zr)(k⊥µpν+ pµkν⊥) +O(k 3 ⊥), (3.1.19) F3,µν =2zizr(i + r)µ(i + r)ν =2zizrpµpν + 2zizr sir spn (pµnν + nµpν) +O(k⊥3). (3.1.20) Summing the three contributions, which cancels the _O(1/k4

⊥) terms, and plugging them back into eq. 3.1.11 gives

Fgqq,µν = TR16παsµ2 1 s2 ir −1 2gµνsir− 2k ⊥ µk ⊥ ν + sir spn (pµnν + nµpν) +O(1/k⊥) = TR8παsµ2 1 sir −g µν+ 4zzr k⊥ µk ⊥ ν k2 ⊥ +pµnν+ nµpν p_{· n} ! +_O(1/k⊥). (3.1.21) This expression contains a gauge dependent term, involving nµ. By gauge invariance this term must vanish when contracting Fgqq,µν with the matrix element, so we can drop it when writing down the expression for the Altarelli-Parisi splitting kernel. In this case the partial amplitude is a matrix in the spin state of the unresolved gluon:

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As an extra note, the external µ_,ν∗ _{can, in contrast to the case of q} _{→ qg splitting, not} be found explicitly in equation 3.1.21. However, it can be shown that

Fgqq,µν = dµσ(i + s)Fgqqσλdνλ(i + s),

which shows that the factors for the external gluons are actually present, but hidden, in the expression (recall dµν(p) = µ(p)∗ν(p)).

3.1.3 Collinear

g

→ g + g splitting

µ ν

pir pr

pi

Figure 3.4: Cut diagram for the process g_{→ g + ¯g}

In the case of a gluon splitting into two gluons, we derive the the amplitude Fggg from the cut graph shown in figure 3.4,

Fggg,µν = CA4παsµ2

dµρ(i + r) sir

Vρ,α,β(−i − r, i, r)dαγ(i)Vσ,γ,δ(−i − r, i, r)dβδ

dσν(i + r) sir = CA4παsµ 2 s2 ir h F1,µν + F1,νµ+ F2,µν+ F2,νµ+ F3,µν i , (3.1.24)

where we have defined functions Fi,µν in a similar fashion as in the g→ q¯q case: F1,µν =(− 1)(i + r)µ(i + r)ν − 4iµrν + 2iµiν srn sin + 2rµrν sin srn + 4sir sinsrn (srnrµnν + siniµnν− sirnµnν) + 2 srn sin + sin srn (iµrν + (i + r)µnν − gµνsir), (3.1.25) F2,µν = 2 sir,n h iµ(i + r)ν(1− 2)(sin− srn) + (i + r)µ(i + r)ν(1− )(srn− sin) + nµ(i + r)νsir+ s2 in srn (i + r)µ(i + r)ν − srn sin + sin srn (iµ(i + r)νsin+ nµ(i + r)νsir) i , (3.1.26)

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F3,µν = 2 s2 ir,n h 2(_{− 1)(i + r)}µ(i + r)ν(sin+ srn)2 i . (3.1.27)

Similarly to the g → q¯q case, straightforward algebra was used to rewrite these expressions and will not be shown in detail. The Sudakov parametrization is introduced and the functions Fi,µν are combined to give a relatively compact equation. Inserting equations A.2.11 and A.2.12 and adding the contributions gives

Fggg,µν =CA4παsµ2 4sir zzr gµν(1− 2zzr)− 8(1 − )(zrµ− zriµ)(zrν + zriν) + sir spn 4 zzr h (iµnν+ nµiν)(zr− z) + ((i + r)µnν + nµ(i + r)ν)((3− 2z)z − 2) i +_O(1/k⊥). (3.1.28) Finally, we insert equations 3.1.15 - 3.1.17 to obtain

Fggg,µν = CA4παsµ2 s2 ir h 8(1_{− )k}⊥_µk_ν⊥_{− 4} zi zr +zr zi gµνsir − 8sir spn zi zr − zr zi (nµpν + pµnν) i +O(1/k⊥) =CA8παsµ 2 s2 ir h − 4(1 − )zizr k⊥ µk ⊥ ν k2 ⊥ − 2 z_zi r +zr zi gµν − 2 zi zr +zr zi nµpν + pµnν p_{· n} i +O(1/k⊥). (3.1.29)

From this expression, the splitting kernel is derived using similar arguments to the g _{→ q¯q} case: hµ| ˆP(0) gg (zi, zr, k⊥; )|νi = CA − 4(1 − )zizr k⊥ µkν⊥ k2 ⊥ − 2 zi zr + zr zi gµν . (3.1.30)

Note that the factors for the external gluons in the m parton matrix element are, similary to the case of g _{→ q¯q splitting, not explicitly present in expression 3.1.29.}

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3.1.4 Spin averaged splitting kernels

The spin averaged form of the splitting kernels are also required. They can be derived from the expressions we derived by contraction with δss0/2 in the case of an unresolved fermion

and dµν(p)/(d− 2) in the case of an unresolved gluon [40]. Thus the spin averaged kernels are given by h ˆP(0) qg (zi, zr, k⊥; )i = CF h − zr+ 1 + z2 i zr i (3.1.31) h ˆPq ¯(0)q (zi, zr, k⊥; )i = TR d− 2 h − dµνgµν − 4zizr i = TR h 1₋ 2zizr 1− i (3.1.32) h ˆPgg(0)(zi, zr, k⊥; )i = CA d− 2 h 4(1− )zizr− 2 zi zr +zr zi dµνgµν i = 2CA h zizr+ zi zr +zr zi i , (3.1.33) where we have, as usual, removed the gauge terms and used

dµνgµν =−gµνgµν+

p_{· n + n · p}

p· n =−d + 2. (3.1.34)

3.2 Soft limit

At NLO accuracy, the emission of a soft gluon is the only situation one needs to consider to account for the singularities in the soft limit [40]. The momentum of the soft gluon will be parameterized as sµ= λqµ, such that in the soft limit λ→ 0.

In this case it is not possible to use cut graphs, since the soft gluon can be emitted from any of the external legs. Therefore, in the soft limit one must take the square of the matrix element summed over the external legs. An operator Ss is defined [5],

Sshcs|Mm+1i = gsµ(s)Jµ(s)|Mmi , (3.2.1) which takes the soft limit and keeps the leading singular terms. The factor Jµ is the soft current, which among others contains the colour factor for the soft gluon and sums the soft emission over all possible external legs. cs represents the colour for the soft gluon. For the squared matrix element, the soft limit takes the form

Ss|M (0) m+1| 2 _{= 4πα} sµ2µ(s)ν∗(s)hMm| Jµ(s)Jν(s)|Mmi =_−8παsµ2 X i,k 1 2Sik(s)hM (0) m | Ti· Tk|M(0)m i , (3.2.2) where we used µ_(s)ν∗_{(s) = d}

µν(s) =−gµν+ (sµnν+ nµsν)/s· n. Note that in the last line the eikonal factor Sik does not contain the gauge terms coming from dµν; we ignore such

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terms due to gauge invariance. The factor 1_/₂ _{is written in the sum to emphasize that} _S ik is symmetric in its indices, which will become apparent when we derive an expression for it. The eikonal factor is obtained from the equation

X

i,k

Sik(s)Ti· Tk = gµνJµ(s)Jν(s). (3.2.3)

3.2.1 Soft gluon emission from quark

M

µ

pis ps

pi

Figure 3.5: Diagram for soft gluon emission from a quark

For the emission of a soft gluon from a quark, the diagram shown in figure 3.5 is used. For this diagram (excluding M) we find,

Fqqg = gsTisu(i)γ¯ µ /i + /s

sis

µ(s). (3.2.4)

where Ts

i is the colour charge operator associated with the soft gluon. Inserting the parametrization sµ= λqµ and taking only the leading singular terms, we get

Fqqg = gsTis

sis ¯

u(i)γµ/iµ(s) +O(λ0) =₋gsT s i sis ¯ u(i)/iγµµ(s) + 2 gsTis sis ¯ u(i)iµµ(s) +O(λ0) =gsTis iµ i_{· s}u(i)¯ µ (s) +O(λ0 ), (3.2.5)

where we have used the (massless) Dirac equation, ¯u(i)/i = 0. From this expression we can read of the soft current,

Jµ(s) = X

k

T_ks kµ

k_{· s}, (3.2.6)

where k is summed over the external legs. From equation 3.2.3 the eikonal factor is ob-tained:

Sik(s) = 2sik sissks

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3.2.2 Soft gluon emission from gluon

M µ ν λ pis ps pi

Figure 3.6: Diagram for soft gluon emission from a gluon

For the soft emission of a gluon from a gluon, the diagram shown in figure 3.6 is used. For this diagram, the partial matrix element is found to be

Fggg = gsTbs µ_(s)d λσ_{(i + s)} sis Vµ,σ,ν(−s, i + s, −i)ν(i), (3.2.8) where Ts

a is the colour charge operator of the soft gluon. We derive

dλσ(i + s)Vµ,σ,ν(−s, i + s, −i) = h − gλσ + 2(i + s) λ_nσ_{+ n}λ_{(i + s)}σ sis,n i ×h− (i + 2s)νgσµ+ (−i + s)σgµν + (s + 2i)µgσν i =h_{− g}λσ_{+ 2}iλnσ + nλiσ si,n ih − iνgσµ− iσgµν + 2iµgσν i +_O(λ), where in the second line we inserted the parametrization for the soft limit, sµ_{= λq}µ_{. Using} the identities (i)· i = (i) · n = 0, we arrive to the expression

Fggg = gsTas sis iλµ_(s)ν_(i) − 2iµµ(s)λ(i) + 2 sin h − (i · n)µ(s)µ(i)iλ i +_O(λ0₎ =− gsTas iµ i_{· s} µ (s)λ(i) +O(λ0 ).

Consequently we find that the soft charge and eikonal factor are equal to the previous case, apart from an overal minus sign in the soft charge.

3.2.3 Soft collinear overlap

There is a region of phase space where the soft and collinear limits overlap. The soft gluon can also be collinear to the parton that emitted it. For this we need to consider the collinear limit of the operator Sr on the squared matrix element. When the gluon (r)

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becomes collinear to its parent parton (i), sik = zispk and skr = zrspk. The eikonal factor becomes Sik = 2sik sirskr → 2zi sirzr . (3.2.9)

So that the collinear limit of equation 3.2.2 can be written as

CirSr|M (0) m+1| 2 =−8παsµ2 X k6=i 2zi sirzr hM (0) m | Ti· Tk|M(0)m i .

We have seemingly picked up a factor 2, but this comes from the fact that, as mentioned before, the eikonal factor is symmetric in its indices. So when we explicitly choose particle i (and remove the sum over index i) we need to include both_Sik and Ski. One should also note that the sum over index k does not include i: in equation 3.2.2 this contribution would be zero (Sii = 0), so we should not include it here. Finally, we use the colour summation identity, X k Ti· Tk= 0, (3.2.10) to write CirSr|M (0) m+1| 2 _{= 8πα} sµ2Ti2 2zi sirzr|M (0) m | 2_. _(3.2.11)

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3.3 NLO subtraction scheme

To illustrate the methods used, the subtraction scheme will be explained in greater detail at NLO than at NNLO. The NLO method was presented in Ref. [39] and uses an approximate cross section to regularize kinematic singularities, such as was described at the beginning of this section in equation 3.0.5. As was mentioned there, the approximate cross section must match the singular behavior of dσR _{and it must be integrable over the one-parton} phase space of the unresolved particle. The goal is therefore to find an expression that satisfies these two requirements.

Firstly, we define an expression that matches the singular behavior of the squared matrix element in the unresolved regions of the phase space,

A_|M(0)_m+1_|2 =X r h X i6=r 1 2Cir + Sr− X i6=r CirSr i |M(0)m+1| 2_. _(3.3.1)

In this expression, Cir and Sr are operators that take the collinear and soft limit of the squared matrix element, respectively, as seen in the sections 3.1 and 3.2. The factor one-half is required to prevent double counting of the combinations of partons. The last term on the right hand side is necessary to cancel the overlapping of the soft and collinear regions of phase space [5]. In this region there is a double subtraction, since we are subtracting both the soft limit of Cir and the collinear limit of Sr.

Equation 3.3.1 is only defined in the strict limit, since the operators used in this expres-sion take the strict limit of the squared matrix element. For the subtraction to work, we need an expression that is valid across the whole phase space. It is necessary to factorize the m + 1-parton phase space into an m-parton one times a one-parton one,

dφm+1({p}) = dφm({˜p})[dp1], (3.3.2)

where one should note that the notation [dp1] denotes a yet to be determined function that includes the one parton phase space. The original m + 1 parton phase space momenta {p} are mapped to the momenta {˜p}, which enter the m parton phase space. This way, the unresolved parton is factorized out of the original phase space (such that the two can be integrated over seperately), while momentum conservation is still valid in the new m parton phase space. In the soft and/or collinear limits, the mapped momenta ˜p should mimick the regular momenta in the corresponding limit. In this way, the counterterms can be defined across the entire phase space, will match the point-wise singular behaviour of the squared matrix element in the unresolved regions and can be integrated seperately over the unresolved parton’s phase space.

Following the conventions of Ref. [39], we then write the approximate cross section using the notation A to denote the extension of A over the whole phase space,

dσm+1R,A = dφm[dp1]A|M (0)

m+1|2, (3.3.3)

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resembles that of eq. 3.3.1, however it is not defined in the sense of operators that take scrict limits. Instead, it is a regular expression,

A_|M(0)_m+1_|2 ₌X r h X i6=r 1 2Cir+Sr− X i6=r CirSr i , (3.3.4)

twhere the seperate term _Cir is given by

Cir({p}) = 8παsµ2 1 sir hM

m({˜p}(ir))| ˆPfi,fr(zi,r, zr,i, k⊥,i,r; )|Mm({˜p}

(ir)₎

i , (3.3.5) The labels fi,frdenote the flavours of the collinear partons as they appear in the definition of Cir. Note that the m parton matrix element now takes the mapped momenta as its argument, while _Cir is a function of the original momenta. Similarly, the other two terms are given by Sr({p}) = −8παsµ2 X i,k 1 2Sik(r)|M (0) m,(i,k)({˜p} (r) )|2 , (3.3.6) CirSr({p}) = 8παsµ2 1 sir 2zi,r zr,i T_i2_|M(0)_m (_{˜p}(r)₎ |2_, _(3.3.7)

where we have introduced the notation _Mm,(i,k) = hMm| Ti · Tk|Mmi for the colour con-nected matrix element.

So far the mapped momenta ˜p, the transverse momentum k⊥ and the momentum fractions zi and zr have not been made explicit. In ref. [39] these quantities are made explicit, in such a fashion that they fulfill all the requirements stated in this section so far. In the collinear counterterm, the m + 1 momenta are mapped to m momenta so that

˜ pµ_ir = 1 1− αir (pµ_ipµ r − αirQµ) (3.3.8) ˜ pµn = 1 1− αir pµn, (3.3.9) where αir is defined to be αir = 1 2 y(ir)Q− q y2 (ir)Q− 4yir . (3.3.10)

Note that we mapped the m + 1 momenta _{{p} = {p}1, .., pi, pr, ..pm+1} to the m momenta {˜p}(ir) ₌_{{ ˜}_p

1, .., ˜pir, .., ˜pm+1}, so that the collinear momenta pi and pr are combined explic-itly and all other momenta are mapped to ensure momentum conservation.

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k_⊥,i,rµ = ζi,rpµr − ζr,ipµi + ζirp˜µir, (3.3.11) ζi,r= zi,r− yir αiry(ir)Q , (3.3.12) ζr,i = zr,i− yir αiry(ir)Q , (3.3.13) ζir = yir αiry( ˜ir)Q (zr,i− zi,r), (3.3.14)

while momentum fractions used in these equations are given by

zi,r = yiQ y(ir)Q , zr,i = yrQ y(ir)Q . (3.3.15)

Note that in these equations the momentum fractions carry two indices, since there can be more than two final state partons and it needs to made explicit which ones are involved, as compared to the two final state partons in the calculations in section 3.1.

Considering that in the collinear limit sir → 0 (and thus also yir → 0), one can see that the transverse momentum has the correct behaviour (i.e. that of the one used in the Sudakov parametrization) when squared,

k2

⊥,i,r =ζi,rpr,µ− ζr,ipi,µ+ ζirp˜ir,µζi,rpµr − ζr,ipµi + ζirp˜µir

=−ζi,rζr,isir+ ζi,rs( ˜ir)rζir− ζr,is( ˜ir)iζir

=−zr,izi,rsir+O(s2ir), (3.3.16)

where we used

ζir =O(sir), ζi,r= zi,r+O(sir), ζr,i = zr,i+O(sir), (3.3.17) and

ζi,rs( ˜ir)r− ζr,is( ˜ir)i = siQ s(ir)Q 1 1_{− α}ir (sir− αirsrQ)− srQ s(ir)Q 1 1_{− α}ir (sir− αirsiQ) = sir 1− αir (siQ− srQ) = O(sir). (3.3.18)

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˜ pµ n = Λ µ ν[Q, (Q− pr)/λr](pνn/λr), (3.3.19) λr =p1 − yrQ, (3.3.20) where Λµ

ν is a Lorentz transformation, given explicitly in ref. [39]. In this case, the m+1 momenta _{{p} = {p}1, .., pr, ..pm+1} were mapped to the m momenta {˜p}(r) ={ ˜p1, .., pm+1}, where in the last case pr was ’mapped away’ and is thus absent.

The phase space mappings described above generate the phase space factorization as required, where [dp1] is found to be

[dp1] =Jm(pr,{˜p}; Q) dpr

(2π)d−1δ+(p 2

r). (3.3.21)

Jm is a Jacobian factor, for which the formulae can once again be found explicitly in ref. [39].

As shown in equation 3.0.5, the subtraction termsCir,SrandCirSrneed to be integrated over the one parton phase space to determine the subtraction for the virtual correction. It is not very instructive to perform the integrals explicitly, as is done in ref. [39], however it is useful to understand the structure of the integrated approximate cross section that is to be achieved after integration, which is an m parton Born cross section into which the operator I is inserted,

Z

1

dσ_m+1R,A = dσB

m⊗ I(m − 1, ), (3.3.22)

where the insertion operator acts in the colour space of the m partons and contains all the integrated counterterms. It is worth noting that during the integration the colour charge dependence, the Born matrix element and some standard factors are taken out of the counterterms, so that the integrated counterterms are only functions of some kinematic parameters. The integrated collinear counterterm is defined by

Z [dp(ir)_1;m(pr, ˜p; Q)]Cir({p}) = αs 2πS µ2 Q2 Cir(y( ˜ir)Q; m− 1, )T 2 ir|M (0) m ({˜p} (ir)₎ |2_{. (3.3.23)} The integrated soft counterterm is defined in a similar way, except that the summation over the flavour indices of emitting partons is also taken out of the definition,

Z [dp(r)_1;m(pr; Q)]Sr({p}) = αs 2πS µ2 Q2 _X i X i6=k Sik(y˜i˜k, y˜iQ, ykQ˜ ; m−1, )T 2 i |M (0) m,(i,k)({˜p} (r)₎ |2_, (3.3.24) and lastly the soft-collinear counterterm is similarly defined by

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Z [dp(r)_1;m(pr; Q)]CirSr({p}) = αs 2πS µ2 Q2 CS(m_{− 1, )T}2 i |M (0) m ({˜p} (r)₎ |2_. _(3.3.25) In the insertion operator the counterterms are structured in a different way than before, as they are written in terms of the flavour of the parent parton,

Cq = Cqg− CS, Cg =

1

2Cgg+ nfCq ¯q− CS, (3.3.26) and the collinear-soft counterterm is absorbed into the newly defined collinear counterterms Ci. This redefinition allows the insertion operator in equation 3.3.22 to finally be written as I(_{{p}; m−1, ) =} αs 2πS µ2 Q2 X i h Ci(yiQ; m−1, )Ti2+ X k6=i Sik(yik, yiQ, ykQ; m−1, )TiTk i . (3.3.27)

3.4 NNLO subtraction scheme

At NNLO, multiple combinations of soft and collinear particles need to be considered to regularize all kinematic singularities. The cases in which one of these combinations occur in a QCD process are the doubly unresolved limits [5], for which the possibilities are

1. A collinear parton triplet, 2. Two pairs of collinear partons,

3. A pair of collinear partons and a soft gluon, 4. A soft quark-antiquark pair,

5. A pair of soft gluons.

To regularize the singularities encountered in the listed situations, a subtraction scheme will be used, similar to what was described at the beginning of this chapter. The NNLO contribution to the cross section is written as

σN N LO ₌ Z m+2 dσRR m+2Jm+2+ Z m+1 dσRV m+1Jm+1+ Z m dσV V m Jm. (3.4.1)

Here the integral over m+2 parton phase space contains all contributions from the doubly-real corrections, i.e. all processes that involve the emission of two extra external partons. The integral over m + 1 parton phase space contains all real-virtual contributions, which

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involve the emission of one extra external parton and a virtual correction (such as a loop diagram). Finally, the intregral over m parton phase space contains all doubly-virtual con-tributions, involving the same number of external partons as the leading order contribution and a double virtual correction. Each phase space integral is weighed by the appropriate jet function, which equals 1 for the total cross section.

Similarly to the NLO cross section, each of the integrals over a seperate phase space is kinematically divergent, but the final result is free of any kinematic divergences. Thus it is possible to consistently add and subtract terms that regularize each of the integrals seperately. Following the procedure of Ref. [6], we start by accounting for the doubly-unresolved limits, which occur in the doubly-real contribution:

σN N LO ₌ Z m+2 h dσRR m+2Jm+2 − dσm+2RR,A2Jm i + Z m+1 dσRVm+1Jm+1+ Z m h dσV Vm + Z 2 dσRR,A2 m+2 i Jm. (3.4.2)

At this point it is entirely unclear what the form of dσRR,A2

m+2 is, since the notation is symbolic and very inexplicit. For now, it should be clear that dσRR,A2

m+2 must match the doubly-unresolved singular behaviour of dσRR

m+2 and must be integrable over the phase space of the two unresolved partons without the use of a jet function. Since dσRR,A2

m+2 is weighed by an m parton jet function, it is added back in the integral over m parton phase space.

As mentioned above, dσRR,A2

m+2 only accounts for the doubly-unresolved regions of phase space. We must also account for the singly-uresolved limits of dσRR

m+2by subtracting dσ RR,A1

m+2 and adding it back in the integral over m + 1 parton phase space. Thus the cross section is written as σN N LO ₌ Z m+2 h dσRR m+2Jm+2 − dσm+2RR,A2Jm− dσm+2RR,A1Jm+1 i + Z m+1 h dσm+1RV + Z 1 dσRR,A1 m+2 i Jm+1 + Z m h dσV V m + Z 2 dσRR,A2 m+2 i Jm. (3.4.3) Additionally, dσRR,A2

m+2 contains singly-unresolved singularities that do not match those of dσRR

m+2 or dσ RR,A1

m+2 . To account for this, we add dσ RR,A12

m+2 , which matches the singular behaviour of dσRR,A2

m+2 in the singly-unresolved regions. The combination dσ RR,A1

m+2 −dσ

RR,A12

m+2 must not contain any additional singularities in the doubly-unresolved regions of phase space. The resulting expression reads

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σN N LO = Z

m+2 h

dσm+2RR Jm+2− dσm+2RR,A2Jm− dσm+2RR,A1Jm+1+ dσRR,Am+2 12Jm i + Z m+1 h dσRV m+1+ Z 1 dσRR,A1 m+2 i Jm+1 + Z m h dσV V m + Z 2 dσRR,A2 m+2 − Z 2 dσRR,A12 m+2 i Jm, (3.4.4)

where each approximate cross section is added back in the integral that matches the jet function it is weighed by. In equation 3.4.4 the integral over m + 2 parton phace space is now finite, simply because we required all singularities to be canceled.

Subsequently, we consider the real-virtual contribution. dσRV

m+1 has singly-unresolved singularities, which will be regularized by subtracting the term dσRV,A1

m+1 . Additionally, the integrated countertermR

1dσ RR,A1

m+2 can show singular behaviour in the singly unresolved re-gion of the m+1 parton phase space. To regularize this behaviour, the termR

1dσ RR,A1

m+2 A1

is subtracted, so that we arrive to the expression

σN N LO ₌ Z

m+2

dσRR

m+2Jm+2− dσRR,Am+2 2Jm− dσm+2RR,A1Jm+1+ dσm+2RR,A12Jm + Z m+1 dσRV m+1 + Z 1 dσRR,A1 m+2 Jm+1 − dσRV,A1 m+1 + hZ 1 dσRR,A1 m+2 iA1 Jm + Z m dσV V m + Z 2 dσRR,A2 m+2 − dσ RR,A12 m+2 +dσRV,A1 m+1 + hZ 1 dσRR,A1 m+2 iA1 Jm. (3.4.5) The integral over the m+1 parton phase space is now finite as well, since it was constructed as such. By the KLN theorem, the integral over the m parton phase space must therefore be free of kinematic singularities as well. Thus it is now safe to consider each integral seperately and in d = 4 dimensions. The final result is written in a similar fashion as 3.4.1

σN N LO ₌ Z m+2 dσN N LO m+2 + Z m+1 dσN N LO m+1 + Z m dσN N LO m , (3.4.6)

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dσN N LO

m+2 =

dσRR

m+2Jm+2− dσRR,Am+2 2Jm− dσm+2RR,A1Jm+1+ dσm+2RR,A12Jm ε=0 (3.4.7) dσN N LO m+1 = dσRV m+1+ Z 1 dσRR,A1 m+2 Jm+1− dσRV,A1 m+1 + hZ 1 dσRR,A1 m+2 iA1 Jm ε=0 (3.4.8) dσmN N LO = dσmV V + Z 2 dσRR,A2 m+2 − dσ RR,A12 m+2 +dσRV,A1 m+1 + hZ 1 dσRR,A1 m+2 iA1 ε=0 Jm. (3.4.9) The details of the structure of these approximate cross sections is beyond the scope of this thesis. The principles of phase space factorization and integration are similar to that in the NLO case, but the expressions for the counterterms and integrated counterterms become much larger in the NNLO case. In ref. [5], the operators that take the doubly unresolved limits (which at NLO were Cir and Sr) are constructed as well as the matching of overlapping regions of phase space (which at NLO lead to including the term CirSr in the operator A). The momentum mappings necessary for the phase space factorization to produce the actual counterterms are discussed in ref. [6] and ref. [7] for the doubly-real and real-virtual contributions, respectively. Subsequently, refs. [9–12, 41] give a detailed account of the procedures employed in the integration of the counterterms, so that finally the counterterms can be written in terms of insertion operators such as equation 3.3.27.

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Chapter 4 NNLO: Results and discussion

4.1 Results for event shapes

To run the code the cutoff parameter ymin was set to be ymin = 10−6 for NLO processes and ymin = 10−8 for NNLO processes, a choice which will be discussed in detail in section 4.2. Furthermore, the results were obtained for a center of mass energy √s = 91.2GeV and with a strong coupling constant αs(Q) = 0.118. The bin size in all distributions is ∆X = 0.01, where X is the event shape variable considered in a given distribution. To compare the theory to experiment, we use the data from ALEPH [1].

Figures 4.1a, 4.1b show the distributions for thrust and the ρ parameter, respectively. For the process in consideration, both distributions are trivial at leading order, where the entire leading contribution is condensed in the first bin (at T = 1 and ρ = 0) and thus not relevant since both distributions use a weight on the y-axis that dampens these contributions. For this reason, only the NLO and NNLO contributions are shown. It is worth noting that the two distributions are identical in the case of the NLO contribution, where we have three massless partons in the final state. As expected for these distributions, the edge of phase space is reached at 1− T = 1/3, ρ = 1/3 for the NLO contribution.

The predictions for the jet broadening parameters BW and BT are shown in figures 4.2a and 4.2b, respectively. For these distributions, as mentioned before, the leading order result is trivial and thus condensed in a single bin (Bi = 0).

The prediction for the C parameter is shown in figure 4.3a. Some misbinning can be seen in the NNLO distribution, which occurs when large cancellations are placed in adjacent bins instead of the same bins. This effect occurs more often when smaller bin sizes are used.

The distribution for the rapidity is shown in figure 4.3b. The distribution shows the rapidity of the most energetic jet, with the axis (to which the momentum pk is parallel) chosen to be the z-axis. For this parameter the JADE jet algorithm is used with the momentum scaling mentioned in chapter 2,

~pij =

Ei+ Ej |~pi+ ~pj|

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0 5 10 15 20 (1 − T ) d σ d(1 − T ) (pb) 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 (1− T ) NLO NNLO cont. NNLO

(a) Thrust distribution for e+_e− _{→ 2jets at} NLO and NNLO accuracy. The thickness of the bar in each bin represents the error margin of the calculated value in that bin, while the actual result is always in the middle of the bar.

0 5 10 15 20 ρ d σ (pb)_dρ 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ρ NLO NNLO cont. NNLO

(b) Distribution for the rho parameter for e+_e−

→ 2jets at NLO and NNLO accuracy. The presentation has the same meaning as in figure 4.1a Figure 4.1 -5 0 5 10 15 20 25 30 35 40 BW d σ d BW (pb) 0.0 0.05 0.1 0.15 0.2 0.25 0.3 BW NLO NNLO cont. NNLO

(a) Distribution for the wide jet broadening for e+_e− _{→ 2jets at NLO and NNLO accuracy.} The presentation has the same meaning as in figure 4.1a -5 0 5 10 15 20 25 30 35 40 BT d σ d BT (pb) 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 BT NLO NNLO cont. NNLO

(b) Distribution for the total jet broadening for e+_e− _{→ 2jets at NLO and NNLO} accu-racy. The presentation has the same meaning as in figure 4.1a

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-5 0 5 10 15 20 C d σ d C (pb) 0.0 0.2 0.4 0.6 0.8 1.0 C NLO NNLO cont. NNLO

(a) Distribution for the C parameter for e+_e−

→ 2jets at NLO and NNLO accuracy. The presentation has the same meaning as in figure 4.1a 0 5 10 15 20 25 30 35 d σ d| y| (pb) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 |y| LO NLO cont. NNLO cont. NNLO

(b) Distribution for the rapidity y of the most energetic jet relative to the z-axis for e+_e−

→ 2jets at LO, NLO and NNLO accuracy. Jets have been clustered using the JADE algorithm with the resolution parameter ycut = 0.05. The presentation has the same meaning as in figure 4.1a

Figure 4.3

This algorithm uses the distance measure dij = yij, with the jet resolution parameter ycut = 0.05. In the figure we show the distribution for the absolute value of y, which prevents misbinning between negative and positive values of y. This misbinning occurs when two jets have near equal energy, such that numerics decide which jet is used to calculate rapidity. The difference between the results of the two jets will be a sign in the rapidity. Large cancellations might then be assigned to bins with opposite signs.

For all event shapes, except rapidity, one can compare the results of the two-jet code to the results of the three-jet code, which was already available. There is agreement between the results, but this only confirms the correctness of the sub-processes that are also present in the three-jet production. A stronger confirmation for the validity of the code is the correctness of the total cross section, which will be discussed in detail in the next section.

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4.2 Choice for the cutoff parameter

y

min

In the code framework, a cutoff value ymin for the parameters yij is used to ensure numer-ical stability of the code and reduce the estimated uncertainty of the results. The regions of phase space where the parameters yij are small include the regions where subtraction terms are canceling large, divergent contributions. In theory this region should be rendered safe by the subtraction scheme, but in the practice of dealing with numerical calculations, the large cancellations might cause instability. Despite the subtraction terms being valid, infinities and other errors (such as ’Not a Number’ (NaN)) might occur. The code frame-work handles these infinities properly, but as the number of these errors gets larger, the prediction made by the framework becomes less reliable.

As an analysis of the numerical stability of the code for different values for ymin, one could consider the number of infinities that occur in a certain number of phase space points. For a value of ymin = 10−5, a particular run of 1.6· 106 points leads to 4 occurrences of an infinite number. A value of ymin = 10−8 for the same number of points leads to a total number of 467 infinite numbers. Thus, even though the code does not take these occurences into account for the final prediction, it is clear the the numerical stability decreases with the value of ymin. The number of infinite numbers, even for ymin = 10−8, is still relatively small in comparison to the total number of phase space points and the calculation remains stable enough to be used safely.

In the process discussed in this paper a naive choice of ymin = 10−5, which appears sufficient for most processes, leads to an incorrect prediction for the total cross section. Both the real-virtual and doubly-real contributions show significant dependence on a choice for the value of ymin between ymin = 10−5 and ymin = 10−8 , as shown in figure 4.4b. In figure 4.4a it is shown that the prediction for the total cross section converges to the analytic prediction, σN N LO _{= 0.0817555 pb [34], as the value of y}

min decreases.

The main source of the relatively strong dependence on the cutoff value appears after an analysis of the separate sub-processes in the doubly-real contribution. At ymin = 10−5 the contribution of the process e+_e− _{→ b¯bgg to the total cross section seems to vanish,} since the integrated value is small (_O(10−4_{)) and the estimated error is always greater than} the value itself. This behaviour disappears when the value of ymin becomes smaller. From ymin = 10−6 and smaller, the contribution from this sub-process converges to a non-zero value with an estimated error that does not allow for a vanishing contribution. It thus seems that, especially for this sub-process, a relevant part of the physics is hidden below the value ymin = 10−5.

For the prediction of various event shape variable distributions, the cutoff parameter has a smaller relative impact than in the case of the total cross section. For the sub-process e+_e− _{→ b¯bgg the thrust and total jet broadening distributions are shown in figure 4.5, for} two values of the cutoff parameter. As can be seen in figure 4.5a, the thrust distribution is largely unaffected by an adjustment in the cutoff parameter. For this distribution one expects the distribution to only be affected significantly for small values of 1− T , since the events that contain small values for yij have a large thrust (a small yij coincides with xk → 1 (k 6= i, j) for which the thrust is large).

Event shapes in electron-positron annihilation at NNLO accuracy

MSc Physics

Track: GRAPPA