Resummation; global and non-global Logarithms

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U

NIVERSITY OF

A

MSTERDAM

B

ACHELOR

T

HESIS

Resummation; global and non-global

Logarithms

Author: T.M. VONK

Student ID: 10534806

research conducted between: 01-04-2016 and 07-07-2016

Supervisor: Prof. dr. Eric Laenen Second assessor: Wouter Waalewijn Institute: Institute for Theoretical Physics Amsterdam

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i

UNIVERSITY OF AMSTERDAM

Abstract

Resummation; global and non-global Logarithms by T.M. VONK

Event shapes are an important tool to extract data from jets in particle col-lider experiments. There are large logarithms in each term of the perturba-tive expansions of those event shapes, which slow the convergence of the series. In order to still make predictions when these logarithms are large, the event shapes can be resummed. In this thesis the resummation of a global event shape, thrust, is presented in detail. Resummation of event shapes that are not global present difficulties when resumming; techniques that work for global event shapes do not account for so-called non-global logarithms in the perturbative expansion. These logarithms originate from a miscancellation of real and virtual emissions of gluons.

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ii

Populair wetenschappelijke

samenvatting

De figuren hierboven zijn illustraties van twee metingen die gemaakt zijn door een experiment bij een deeltjesversneller. In het midden zijn twee deeltjes (een electron en een positron) op elkaar gebotst. Alle energie van die deeltjes is daarbij omgezet naar twee nieuwe deeltjes (in dit geval twee quarks). Deze quarks zijn vervolgens uiteen gevallen in heel veel andere deeltjes. De paden van al die deeltjes zijn gemeten met de detectors en dat zijn de sporen in de figuren. Zo’n spray van deeltjes heet een jet. De jets links en rechts zijn niet helemaal hetzelfde, de linker is breder dan de rechter. In de vorm van de jets zit informatie over de natuurkunde die gaat over quarks, gluonen en andere deeltjes. Om die natuurkundige wetten te testen moet je voorspellingen doen over de vorm van de jets en dan een experiment doen om te kijken of je voorspelling klopt, en mijn onderzoek ging over het maken van voorspellingen.

De ‘vorm’ van de jet is natuurlijk een beetje vaag. Daarom hebben natu-urkundigen grootheden bedacht die iets zeggen over de vorm. Die groothe-den heten ‘event shapes’ (botsingen als in de figuren heten events). Het voorspellen van zo’n event shape is heel moeilijk, maar het is mogelijk om dat op te splitsen in heel veel simpelere delen. Stel dat ik de event shape A wil uitrekenen, dan kan ik zeggen:

A = a1+ a2+ a3+ a4+ ...

Dit werkt goed als elke a die je uitrekent steeds kleiner wordt, bijvoorbeeld: A = 90 + 9 + 0.9 + 0.09 + ...

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iii Dan kan ik namelijk alleen de eerste twee, a1 = 90en a2 = 9, uitrekenen:

A ' 90 + 9 = 99

Het probleem wat ik onderzocht heb in mijn onderzoek is wat je nog kan doen als dat heel langzaam gaat:

A = 10 + 7 + 5 + 3 + 1 + .3 + .1 + ...

Als je dan alleen de eerste twee a’s uitrekent zit je er een flink stuk naast. Heel veel a’s uitrekenen is heel moeilijk en duurt heel lang. Mijn onderzoek gaat over een techniek die er voor zorgt dat je van alle a’s het grootste deel tegelijkertijd uitrekent en optelt, zodat je een goede schatting kan maken van de event shape. Als die schatting dan overeenkomt met de experi-menten dan weet je dat de natuurkundige wetten kloppen in die situatie.

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iv

Introduction

This thesis is about resummation of global and non-global logarithms. Re-summation is a mathematical technique that is used in jet physics to im-prove predictions from quantum chromodynamics, but is also used in other branches of physics. The basic problem that resummation techniques solve is the following: let’s look at a quantity R that has a perturbative expansion:

R =X

n=0

cnαnS, (1.1)

and the cnare not all the same size, but contain logarithms in a systematic

way. For instance, each cn could contain a term like log2n(τ ), where τ is

some variable. This means that the sum will only converge quick enough if αSlog2(τ )is much much smaller than one. But αS∼ 0.1 at energies that are

common in colliders, so that would exclude a large zone where τ is small. What is really wanted then is a way to reorder the sum in such a way that the large logarithms are identified and grouped together, so that the sum can be approximated by summing the contributions of these logarithms. This is what resummation does. In the example above, the variable R could after resummation take a form like:

R ∼ easL2_. _(1.2)

Here log(τ ) is abbreviated as L. Usually approximations are needed to per-form the resummation of a quantity. This means that in the example above that there might be extra, smaller terms apart from αsL2 in the exponent,

for instance αSL. These terms are called next-to-leading logarithmic, or NLL,

while the αSL2terms are called leading logarithmic, also abbreviated to

lead-ing log or LL.

In the next section I will make the connection between jet physics and resummation and summarize the history of resummation in jet physics. Af-ter that I will give a schematic overview of the structure of a resummation calculation.

1.1 Resummation and jet physics

One field of physics where resummation is often used is jet physics. This is also the focus of my thesis. A jet is a spray of particles that forms when a particle collider produces particles with a color charge. The particles will emit gluons (figure 1.1 who will then form hadrons (figure 2.2). A mea-surement apparatus will only measure the hadrons and we would like to relate these measurements to the fundamental process that produced the

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Chapter 1. Introduction 2

FIGURE1.1: A high energy quark emits gluons. Those glu-ons will later form hadrglu-ons and form a jet.

FIGURE 1.2: The emitted gluons form hadrons. These hadrons can be measured in particle detectors. Fig-ure adapted from http://www.gk-eichtheorien. physik.uni-mainz.de/Dateien/Zeppenfeld-3.

pdf

quarks. For that purpose new observables have been constructed, called event shapes. An ideal event shape can be quickly extracted from the data and easily predicted from theory. This prediction is not always easy to do, as there can be large logarithms in the perturbative expansion. That prob-lem can be solved by resummation.

1.1.1 Event shapes

There are many event shapes, with different uses and strengths. They can be divided into two categories, global and non-global. In this and the fol-lowing chapter I will only use global event shapes and non-global event shapes are discussed in chapter 3. For now it is sufficient to say that global event shapes are quantities that depend on the momenta of all detected hadrons, while non-global event shapes only depend on a subsection of the momenta. For instance, if the event shape only depends on the momentum of particles in one hemisphere it is a non-global event shape.

A much used global event shape is the thrust, which is defined as fol-lows: T = max ˆ n P all particles|ˆn · ~pi| P |~pi| . (1.3)

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Chapter 1. Introduction 3 Here ˆnis a three dimensional unit vector and ~pi is the three momentum

of a hadron. The unit vector that maximises the sum lies along what is conventionally called the thrust axis. Another global event shape is the heavy jet mass, which is defined through the hemisphere jet masses. The left-hemisphere jet mass is defined as:

ρL=   X ~ pi·ˆn>0 pi   2 , X Ei 2 , (1.4)

with pi the four momentum of the hadrons, ~pithe three momentum and ˆn

a unit vector along the thrust axis. The right-hemisphere mass is similarly defined: ρR=   X ~ pi·ˆn<0 pi   2 , X Ei 2 , (1.5)

The left and right hemisphere jet masses are examples of non-global event shapes. They can be combined to form a global event shape, the heavy jet mass:

ρheavy= max{ρL, ρR} (1.6)

The difference between global and non-global logarithms is explained in more detail in section 3.1

1.1.2 History of resummation in jet physics

The field of jet physics and resummation started with the paper from Ster-man and Weinberg (1977) that predicted jets from QCD. This paper and the experimental discovery of jets posed the question of how to analyse these jets. This led to the invention of event shapes, with early examples being the thrust by Farhi (1977) and the spherocity by Georgi and Machacek (1977). These event shapes where then calculated at fixed order1and compared to data and simulations, which showed that the fixed order approach fails in certain situations, for example at high T , because of the large logarithms. In the early nineties Cantani,Turnock, Webber and Trentadue published two papers2 in which they resummed the thrust for the first time. They did this to next-to-leading logarithmic accuracy. Over the years this result has been expanded upon by resumming other event shapes and improving the accuracy: in recent years the thrust has been resummed to next-to-next-to-next-to-leading logarithmic (N3LL) accuracy (Abbate et al., 2011), and the effects of hadronization have been numerically analysed. The resummation of event shapes could even be automated (Banfi, Salam, and Zanderighi, 2005). Also resummation techniques for situations other than e+e− colli-sions have been developed, nowadays deep inelastic scattering (Antonelli, Dasgupta, and Salam, 2000) and proton-proton experiments (Banfi, Salam, and Zanderighi, 2004) can be analysed with resummed event shapes.

However that was not the end; in 2001, Dasgupta and Salam showed that there exist previously unknown logarithms in non-global event shapes

1_{only contributions from α}n S, α

n−1

S , ...for a certain n. 2_{(Catani et al., 1991; Catani et al., 1993)}

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Chapter 1. Introduction 4 (Dasgupta and Salam, 2001). This is important because sometimes experi-mental restrictions mean that it is only possible to measure non-global event shapes; the experiment might not be able to measure particles in some ar-eas of the phase space. In 2015 two papers came out that presented tech-niques for the resummation of non-global logarithms: Caron-Huot (2015) and Larkoski, Moult, and Neill (2015). A short review of these papers can be found in section 3.3 of this thesis. During the writing of this thesis, an-other paper was published, with again a different method for resummation (Becher et al., 2016).

Other methods for resummation exist. Soft-collinear effective field the-ory, or SCET, is a theoretical framework made to deal with interacting par-ticles of different energies and can be used to resum event shapes. For an overview of soft-collinear effective field theory I refer to Lee (2015). Monte Carlo simulations are also very useful and are often used in tandem with analytical methods.

1.2 Rough sketch of the resummation of a global event

shape

In this section I will give a schematic overview of resummation, so that in the next chapter I can discuss the resummation of thrust in detail.

The physical setup is as follows: in an electron-positron collision two quarks with opposite colour and momentum are created. The quarks will then emit gluons, as illustrated in figure 1.1. These gluons eventually com-bine into measurable hadrons. These three processes occur on different timescales and can therefore be treated separately.

Resummation calculations are mainly concerned with the stage where gluons are emitted, because corrections from the hadronization process are suppressed by inverse powers of the total center-of-mass energy (Luisoni and Marzani, 2015). Because the hadronization process is non-perturbative, the corrections must be calculated from simulations.

The goal of resummation is to improve the accuracy of predictions of event shapes. These event shapes carry characteristics from the initial pro-cess, so that when compared to measurements information about that ini-tial process can be extracted from the data. Because it is not possible to predict the thrust of a single event shape, it makes more sense to look at the distribution of an event shape,

1 σ

dσ dτ.

σ is the total cross section. It turns out that the cumulative distribution is more easily calculated. It is defined as:

Σ(τ ) = Z τ 0 1 σ dσ dτ, (1.7)

and sums all the contribution to the cross section up to a certain value for that event shape. The lower limit, in this example τ = 0, is the limit where the large logarithms dominate the pertubative expansion. The expansion

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Chapter 1. Introduction 5 for the cumulative cross section takes the form:

Σ(τ ) = ∞ X n=0 Z dPS|M |2 nΘ(τ − ˆτ (n)) (1.8) |M |2

n is the matrix element squared for splitting n gluons of the original

particle, PS stands for phase space, Θ is the Heaviside step function and ˆτ the definition of the event shape. The step function is what makes this the cumulative cross section: it collects all contributions up to a certain τ .

The next steps have the goal of factorizing the integrals. Schematically, this means that

∞ X n=0 Z dPS|M |2 nΘ(τ − ˆτ ) = ∞ X n=0 1 n! Z (one-gluon process) n .

which can be recognized as the series expansion of an exponential. Σ(τ ) = exp

Z

(one-gluon process)

Factorization requires both the matrix element and the step function to split up into parts that only depend one emitted gluon. It is possible to split up the matrix element if the emissions of the gluons occur independently, or if they can be approximated as independent emissions. The usual method is to approximate the emissions as independent and add some corrections to improve the accuracy. The step function can be split by transforming it with an integral transformation, for instance a Laplace or Mellin transformation. Whether this is possible or not depends on the event shape.

It turns out that for some regions of the phase space the logarithms in the one-gluon integral become dominant and can be approximated by

Z

(one-gluon process) = λg1(λ) + g2(λ).

Here, λ = αSlog τ, and g1 and g2 are functions. The first function collects

the leading logarithms and g2 collects the next-to-leading logarithms. This

form is consistent with the expansion in eq. (1.1) if g1can be approximated

by log τ , because then:

Σ(τ ) =X n=0 1 n!α n Slog2nτ

The origin of the logarithms can be found in the integral of the one-gluon process. The propagator of a gluon is

1 2Eg(1 − cos θ)

,

with Egthe energy of the gluon and θ the angle between the gluon and the

quark. Both of these result in an logarithmic divergence when integrated over.

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Chapter 1. Introduction 6 be found. This approximation is only valid in a region where the loga-rithms are large. For this reason resummation techniques are always used in tandem with fixed order calculations, where the perturbative expansion is calculated up to a certain order of αS. In the next chapter I shall discuss

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7

Chapter 2

Resummation of thrust in e

+

e

−

In this section I will derive the resummed cumulative distribution of the thrust, to leading logarithmic order. This derivation closely follows the one in Luisoni and Marzani, 2015, but I aim to give more details. In the first section I discuss the kinematics of thrust in order to give an expression for the thrust which factorizes easily. In the next section I use this expression for the thrust in order to formulate the one-gluon integral. After that I will evaluate this integral and comment on the result.

2.1 Kinematics

As in the introduction, the thrust is defined as: T = max ˆ n Pn i=1|~pi· ˆn| P |~pi| , (2.1)

with ˆna unit vector and ~pithe momentum of a hadron in the jet. For highly

relativistic hadrons this is the same as: T = max ˆ n Pn i=1|~pi· ˆn| Q . (2.2)

Q is the center-of-mass energy of the e+e− collision. This is the same be-cause of the conservation of energy:

|~pi| = Ei→

X |~pi| =

X

Ei = Q. (2.3)

The vector ˆnthat maximises the fraction in definition 2.1 is called the thrust vector and it has some useful properties. For instance, it defines the two hemispheresHLandHR, with the properties that for every vector inHL

the inner product with ˆnis positive. Every vector in HR has a negative

inner product with ˆn. Another property of ˆnis that the total momentum of both hemispheres is parallel to ˆn. The total transverse momentum of a hemisphere defined by the thrust vector is therefore zero.

I shall now express the total momenta in light-cone variables. As I ex-plain in appendix B, a general four-vector can be expressed as

pµ= z P + ¯z ¯P + kt. (2.4)

Where P and ¯P are the following four-vectors: P = Q 2 1 ˆ n , P =¯ Q 2 1 −ˆn .

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Chapter 2. Resummation of thrust in e+_e− ₈

If I pick the axis in the definition of the light-cone variables to be the thrust axis, the expression for the total momenta simplifies as the transverse total momentum is equal to zero.

q1 = zP + q₁2 zQ2P ,¯ (2.5) q2 = ¯z ¯P + q₁2 ¯ zQ2P. (2.6)

q1and q2are the total momentum of all the hadrons inHLandHR

where ˆnis the thrust axis. Because all inner products are positive in the first hemisphere and negative in the second hemisphere I can bring the summa-tion inside the absolute value.

T = 1 Q  |ˆn · X i∈HL ~ pi| + |ˆn · X j∈HR ~ pj|  = 1 Q(|ˆn · ~q1| + |ˆn · ~q2|) (2.8) This can then be approximated by1

T = s 1 + 2 q 2 1 Q2 + q2 2 Q2 , (2.9)

which I can approximate with a Taylor series: T = 1 + q 2 1 Q2 + q2₂ Q2. (2.10)

I assumed q₁2 Q2_{, because the hadrons are relativistic particles. My next}

goal is to derive an expression for the thrust that is suitable for factorization, that is, an expression that is of the form

1 − T = X

all particles

V (ki). (2.11)

To do that I use that

q1= X i∈HL pi, as well as = zP. q2= X i∈HL pi, as well as = ¯z ¯P .

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Chapter 2. Resummation of thrust in e+_e− ₉ If I introduce τ = 1 − T then τ = zP · P i∈HLpi+ ¯z ¯P · P j∈HRpj Q2 . (2.12)

This looks like it has the required form, but I am not quite there yet. I still need find P · pi in terms of ktiand z, the splitting ratio. This is required to

perform the one-gluon integral. The momentum of a quark or gluon inHL

can be written as:

pi = xiP + k⊥+ x−i P .¯ (2.13)

I assume they are highly relativistic, which implies pi· pi = 0. In light-cone

variables this means

2xix−i P ¯P − k 2 ⊥= 0, so, x−_i = k 2 ⊥ 2xiP ¯P . (2.14)

For particles in the second hemisphere: xi=

k_⊥2

2x−_i P ¯P. (2.15)

Now, I can compute the inner products in equation 2.12. zP pi = zP x−_i P =¯ zk_⊥2 xi for i ∈HL (2.16) ¯ z ¯P pi = ¯z ¯P xi P =¯ ¯ zk2_⊥ x−_i for i ∈HR (2.17) The fraction xi

z is the splitting ratio zi for particles in HL and x−_i

¯

z is the

splitting ratio (also zi) for particles inHR. Substituting the splitting ratios

in equations 2.16 and 2.17 results in an expression for V (ki).

τ = X all particles V (ki) (2.18) V (ki) = k2_⊥i Q2_z i (2.19)

2.2 Factorization

Now we turn to an essential step in the resummation calculation: factoriza-tion of the matrix element and the step funcfactoriza-tion. As seen in the introduc-tion, the general formula for a cumulative distribution is

Σ(τ ) = Z

dPS|M |2_{Θ(τ − ˆ}_{τ ).} _(2.20)

Here, PS stands for phase space, |M |2 _{is the matrix element, Θ is the unit}

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Chapter 2. Resummation of thrust in e+_e− ₁₀

final particles. In our case it turns out that is is useful to express ˆτ as ˆ

τ =XV (ki),

as derived above. The factorized matrix element |M |2 _{can be derived via}

the coherent branching mechanism, which is done in chapter 3 of Luisoni and Marzani, 2015. The matrix element is

|Mn|2= 1 n! n Y i=1 2CF αS(k2t,i) 2π pgq(zi)

This is derived with the assumption that the emissions are independent. The n! accounts for the different orderings of the gluons and the Altarelli-Parisi splitting function pgqis

pgq =

2 − 2z + z2 z

CF, the color factor, accounts for the different colors of gluons a quark can

emit. CF = 4₃ (Ellis, Stirling, and Webber, 1996). The coupling strenght

αS(k2t)depends on the energy of the interaction. If I combine all these

ele-ments, it turns out that the integral for the real contribution to the cumula-tive distribution is WR= ∞ X n=0 1 n! Z n Y i=1 dzi dk_ti2 k2 ti dφ 2π2CF αS(kti) 2π pgq(zi) Θ τ − n X i=1 V (ki) ! . (2.21) It is important to recognise that the phase space and matrix element part of the integral only depend on kti and zi, the momentum of one gluon, and

not on the momentum of other gluons. This means the matrix element is factorized. To get a fully factorized expression, I need to split up the step function. This can be done by a Laplace transformation:

Θτ −XV (ki) = Z dν 2πiνe ν(τ −P V (ki)) ₌ Z dν 2πiνe ντY e−νV (ki)_. _(2.22)

To factorize WR I substitute above in equation 2.21.

WR= Z dν 2πiνe ντ ∞ X n=0 1 n! n Y i=1 Z dzi dk_ti2 k2 ti dφ 2π2CF αS(kti) 2π pgq(zi) e −νV (ki) (2.23) The integrals that are multiplied are all the same, so the product of all these integrals is the same as one integral raised to the power n.

WR= Z dν 2πiνe ντ ∞ X n=0 1 n! Z dzi dk_ti2 k2 ti dφ 2π2CF αS(kti) 2π pgq(zi) e −νV (ki) n (2.24) = Z _dν 2πiνe ντ_exp Z dzi dk2_ti k2_ti dφ 2π2CF αS(kti) 2π pgq(zi) e −νV (ki) (2.25)

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Chapter 2. Resummation of thrust in e+_e− ₁₁

In the second line I used that the series expansion of an exponent isP xn_/n!.

WRis only the real part of the cross section, the virtual part has to be expo-nentiated in a similar way. I have done this in appendix A. As can be seen in that appendix the virtual part contributes a −1 to the exponent.

Σ(τ ) = WR+WV = Z dν 2πiνe ντ_expZ _dz i dk2_ti k2_ti dφ 2π2CF αS(kti) 2π pgq(zi) e−νV (ki)_{− 1} . (2.26)

The only job left to do is evaluating the one-gluon integral. In order to do so three approximations are made. First, the running coupling is evaluated with the one-loop QCD β function,

αS(k2t) = 1 1 + αS(Q)β0log k 2 t Q2 ,

which relates αS(kt2)to αSat the center of mass energy Q. β0 is a constant.

Second, the (e...− 1) is replaced by (Catani and Trentadue, 1989) Θ V (ki) − eγE ν .

The last approximation is that I only use the divergent part of the splitting function pgq, which is z/2. The other part of the splitting function, −2 + z is

only responsible for higher order (next-to-leading logarithmic) corrections. All of these approximations leave us with the following integrals, which I shall evaluate in the next section:

I = −4CF 2π Z dz z Z dk2_t k2 t 1 1 + αSβ log k 2 t Q2 Θ k_t2 zQ2 − eγE ν . (2.27) Σ(τ ) = Z _dν 2πiνe ντ_eI

2.3 One-gluon integral

2.3.1 Limits on the integral

The upper limit on k2

t is determined from the conservation of momentum,

while the lower limits come from the step function. To determine the upper limit I express kµas a four-vector:

kµ= (zQ 2 , ~kt, k

3₎

Now I can determine k3 from k0 and ktbecause I assume that the emitted

gluons are on-shell:

k2= 0 = z 2_Q2 4 − k 2 t − (k3)2 means that k3 = ± r z2_Q2 4 − k 2 t

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Chapter 2. Resummation of thrust in e+_e− ₁₂

This is only real if:

k_t2< z

2_Q2

4 . (2.28)

The lower limit comes from the theta function, which dictates: k_t2

zQ2 >

1 ˜ ν, with ˜ν = eγE_ν._{This implies}

k2_t > zQ

2

˜

ν . (2.29)

The upper limit on k2

t must be bigger then the lower limit in equation 2.28,

z2Q2 4 >

zQ2 ˜ ν , and this gives a lower limit on z:

z > 4 ˜

ν. (2.30)

2.3.2 Evaluating the integral

Now that we now the limits of the integral in equation 2.27, I can evaluate this integral. I = −4CF 2π Z 1 4/˜ν dz z Z Q2z2/4 Q2_z/˜_ν dk2_t k2 t 1 1 + αSβ0log k 2 t Q2

Here, αS = αS(Q2)and Q is the center of mass energy of the e+e−collision.

First I evaluate the ktintegral:

I =−4CF 2π Z 1 4/˜ν dz z log 1 + αSβ0log k2_t Q2 Q2_z2_/4 Q2_z/˜_ν =−4CF 2π Z 1 4/˜ν dz z h log 1 + 2αSβ0log z 2 − log1 + αSβ0log z ˜ ν i

To be able to do the z integral I have to separate the log(z) terms: I = −4CF

2π Z 1

4/˜ν

dz

z [log ((1 − 2αSβ0log 2) + 2αSβ0log z) − log ((1 − αSβ0log ˜ν) + αSβ0log z)] For brevity:

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Chapter 2. Resummation of thrust in e+_e− ₁₃

Then I can perform the z integral: I =−4CF

2π Z 1

4/˜ν

dz

z [log (A + 2αSβ0log z) − log (B + αSβ0log z)] =−4CF

2π

1 2αSβ0

(A + 2αSβ0log z) log (A + 2αSβ0log z) − log z

− 1 αSβ0

(B + αSβ0log z) log (B + αSβ0log z) + log z

1 4/˜ν =−4CF 2π A 2αSβ0 log A − B αSβ0 log B − 1 2αSβ0 A + αSβ0log 4 ˜ ν log A + αSβ0log 4 ˜ ν + 1 αSβ0 B + log 4 ˜ ν log B + log4 ˜ ν

It is now helpful to unpack A and B again, so that I can group together the log ˜νterms. I will use λ˜ν = αSβ0log ˜ν.

I =−4CF 2π 1 − 2αSβ0log 2 2αSβ0 log (1 − 2αSβ0log 2) − 1 − λν˜ αSβ0 log (1 − λ˜ν) − 1 2αSβ0 (1 − 2λν˜) log (1 − 2λν˜) + 1 αSβ0

(1 − 2λν˜+ 2αSβ0log 2) log (1 − 2λν˜+ 2αSβ0log 2)

The first term does not contain ν so that can be discarded, as I am only in-terested in the logarithmic behaviour. Now I will separate the log(2) terms from last term, because they turn out to be of next-to-leading logarithmic order.

1 αSβ0

(1 − 2λ˜ν+ 2αSβ0log 2) log (1 − 2λ˜ν+ 2αSβ0log 2)

= 1 αSβ0

(1 − 2λ˜ν) log (1 − 2λ˜ν+ 2αSβ0log 2) + 2 log(2) log (1 − 2λν˜+ αSβ0log 2)

The last term is already next-to-leading log, and the first term can be sepa-rated by using an expansion around x = 1 − 2λν˜

log(x + ) ' log(x) +

x + ... (2.31)

At large ν only the first term is not negligible. This leaves us with I =−4CF 2π −1 − λν˜ αSβ0 log (1 − λν˜) − 1 2αSβ0 (1 − 2λν˜) log (1 − 2λν˜) + 1 αSβ0 (1 − 2λ˜ν) log (1 − 2λ˜ν) = CF παSβ0 [2(1 − λν˜) log (1 − λ˜ν) − (1 − 2λν˜) log (1 − 2λν˜)] . (2.32)

Earlier I defined ˜ν = eγE_{ν, but it turns out that the γ}

E term is only

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Chapter 2. Resummation of thrust in e+_e− ₁₄

γE terms from the leading logarithmic terms.

I = CF παSβ0 [2(1 − λν − αSβ0γE) log (1 − λν− αSβ0γE) − (1 − 2λ_ν − 2α_Sβ0γE) log (1 − 2λν− 2αSβ0γE)] = CF παSβ0 [2(1 − λν) log (1 − λν− αSβ0γE) − αSβ0γElog (1 − λν− αSβ0γE) − (1 − 2λ_ν) log (1 − 2λν − 2αSβ0γE) − 2αSβ0γElog (1 − 2λν− 2αSβ0γE)]

The γE inside the logs can be separated out with the expansion in equation

2.31. This results in I = CF

παSβ0

[2(1 − λν) log (1 − λν) − (1 − 2λν) log (1 − 2λν)

+αSβ0γE(2 log (1 − 2λν) − log (1 − λν))] . (2.33)

The last term is next-to-leading log. Now we have the result of the integral in 2.27 to leading logarithmic order, which is conventionally known as g1.

I = CF παSβ0 [2(1 − λν) log (1 − λν) − (1 − 2λν) log (1 − 2λν)] ≡ g1(λν) αS (2.34) Now I can go back to the expression for the cumulative cross section in equation 2.26.

Σ(τ ) =

Z _dν 2πiνe

ντ_eg1(λν)/αS _(2.35)

To be able to do this integral I expand g1around log ν = log τ ,

g1(log ν) = g1(log τ ) +

∂g1(log τ )

∂ log τ (log τ + log ν)

Σ(τ ) = eg1(λτ)/αS Z _dν 2πiνexp  ντ + ∂g1(log τ )

∂ log τ (log τ + log ν)

αS





Which is (Luisoni and Marzani, 2015)

Σ(τ ) = exp h g1(λν) αS(Q) i Γ 1 − α−1_S ∂log τg1(λτ) . (2.36)

2.4 Matching and possible improvements

The result in equation 2.36 is valid up to leading logarithmic order. A next-to-leading logarithmic calculation can be done by using a higher order beta function (two-loop instead of the one-loop used here), including the non-divergent part of the splitting function and keeping the next-to-leading log terms that I neglected in the calculation above, such as the γE term in

equa-tion 2.33 and the log(2) term in 2.32. Figure 2.1 compares the different loga-rithmic orders. It is also possible to improve the prediction using computer simulations of the hadronization process. From those simulations correc-tions to the cross section can be calculated and added to the prediction from

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Chapter 2. Resummation of thrust in e+_e− ₁₅

FIGURE 2.1: A comparison between different logarithmic

orders of accuracy. Plot taken from Monni, Gehrmann, and Luisoni (2011).

FIGURE2.2: The addition of hadronization corrections im-proves the prediction of the differential cross section. In this graph the data from OPAL is plotted against predic-tions with and without hadronization. Figure from Salam

(2005)

fixed order + resummation. This is illustrated with a different event shape in figure 2.2.

2.4.1 Matching

Results like the one above are only valid for small τ , while fixed order cal-culations are valid for larger τ . These two can be combined to form an approximation with a larger range of validity then the two separate ap-proaches have. This is called matching. In figure 2.3 an example is given of a fixed order calculation that is matched to a NNLL resummation. Further details on matching resummation with fixed order can be found in Monni, Gehrmann, and Luisoni (2011).

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Chapter 2. Resummation of thrust in e+_e− ₁₆

FIGURE 2.3: The bands indicate theoretical uncertainty. This graph compares a matched calculation (NNLL +NNLO) to a fixed order calculation (NNLO). Because of resummation the matched calculation has higher precision at low τ = 1 − T . Plot from Monni, Gehrmann, and Luisoni

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17

Chapter 3

Non-global event shapes

Many event shapes have been resummed, some even up to next-to-next-to-next-to-leading log (N3 _{LL). All of these event shapes have one thing in}

common: they are global event shapes. In this chapter we will look at event shapes that are not global and why those quantities lead to problems when resumming. I shall review recent progress on the resummation these non-global event shapes in the last section of this chapter.

3.1 Definition of global and non-global

Global event shapes are event shapes that depend on all momenta (of all particles) in an equal way. The thrust is an example of a global event shape. An example of a non-global event shape is the jet mass of a single hemi-sphere: ρL/R= P i∈HL/Rki 2 P all particlesEi .

With HL/R the left/right hemisphere. Usually this hemisphere is defined

by the thrust axis1. This event shape is a clear example of a non-global event shape; ρLdoes not depend on momenta of particles emitted in to the right

hemisphere.

3.1.1 Subtleties in non-globalness

The distinction between global and non-global can be quite subtle. Take for instance the heavy jet mass: ρ_heavy = max{ρL, ρR}. This is a global

event shape: although the result is always the mass of a single hemisphere, momenta in the other hemisphere still have influence over the outcome; on cannot ignore particles in a calculation, because each particle contributes to the mass of its hemisphere and those hemispheres have an equal chances of becoming the more massive one.

There is also the case of indirectly global event shapes: the event shape does not explicitly depend on all momenta, but the event shape is indirectly sensitive to the other momenta, usually via conservation of momentum. Examples of indirectly global event shapes are the event shapes TzQ and

BzEthat are used to analyze deep inelastic scattering experiments.

A third subtlety lies in event shapes that do depend on all momenta, but not in an equal way. For instance, one could have a event shape that

1_{The thrust axis is the axis that maximises the sum in the definition of thrust, see section} 2.1

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Chapter 3. Non-global event shapes 18 goes like E2

t/Q2 in one hemisphere but like Et/Qin the other. Those event

shapes are not global and are called discontinuously non-global.

The last case is that of the dynamically discontinuously global observ-ables, where the event shape appears global but involves non-linearity’s such that not all regions of emission are treated equally. An example is the broadening BzEin deep inelastic scattering.

Why are these distinctions important? It turns out that non-global event shapes have logarithms in their expansions that do not appear in global event shapes. Those logarithms are called non-global logarithms. The next section concerns two questions: why does non-globalness cause logarithms and can we show that they actually exist in experiments.

3.2 Non-global logarithms

In 2001 Dasgupta and Salam discovered that there are previously unno-ticed logarithms in the expansion of non-global event shapes (Dasgupta and Salam, 2001). That meant that you cannot treat non-global event shapes in the same way as in chapter 2. In this section I will describe the origin of these non-global logarithms.

Before the discovery of these logarithms, physicists would resum a non-global event shape by excluding emissions in the region of phase space that does not contribute to the event shape. For instance, if one would calculate the thrust of a single hemisphere, one would follow the same approach as in chapter 2, while discounting all particles in the excluded hemisphere. If I choose the left hemisphere as the measurement hemisphere, equation 2.12 changes to τ = zP · P i∈HLpi+ ¯z ¯P · P j∈HRpj Q2 → zP ·P i∈HLpi Q2 . (3.1)

Which means that

τ = X i∈HL V (ki) = X i∈HL k2 t,i zQ2.

Which is almost the same as equation 2.19. The matrix element is similar, so the rest of the resummation calculation will be the same. However, this approach discards all gluons emitted in the right hemisphere, while those gluons could emit other gluons that go to the left hemisphere, as is illus-trated in figure 3.1. In global resummation the secondary emission of a real

FIGURE 3.1: Initially, the gluons on the right do not con-tribute to the event shape. However they can emit other

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Chapter 3. Non-global event shapes 19 gluon is canceled against the secondary emission of a virtual gluon: the ef-fect of the virtual gluon on the thrust is the same as the efef-fect of a real gluon. However, in the non-global case the effect of the gluons is not the same: the virtual gluon influences particles in a region of phase space that has no in-fluence on the event shape, while some real gluons travel to a hemisphere where they do contribute to the event shape. That means that the cancel-lation does not work. This creates a logarithm because the propagator of a gluon is proportional to

1 Eg(1 − cos θ)

,

with Eg the energy of the gluon and θ the angle with the emitting gluon.

The logarithms caused by these gluons are of next-to-leading logarithmic order because there is no divergence in θ; the emitted gluon can not be aligned with the emitting gluon, otherwise the emitted gluon will not reach the other hemisphere.

FIGURE3.2: The virtual gluon on the right only influences gluon in the hemisphere that does not contribute to the

event shape.

3.3 Resummation and observation of non-global

loga-rithms

In 2015, a lot of progress was made in the study of non-global logarithms. In this section I will review three papers that I found to be interesting and important contributions to that progress. The first two present resumma-tion techniques for non-global logarithms while the third presents a new non-global observable that features non-global logarithms that should be easier to measure in experiments.

3.3.1 Resummation using color matrices

A resummation technique for non-global event shapes is presented in the paper ‘Resummation of non-global logarithms and the BFKL equation ’ by Caron-Huot (2015). A detailed review of this technique goes beyond the scope of this thesis but the central idea is that of color density matrices that ‘keep track’ of the color exchanges. A general expression for an event shape is R = ∞ X n=0 Z dP S|Mn|2u({pi}),

where PS stands for phase space, Mn the matrix element for emitting n

gluons and u some measurement function that depends on the momenta of the gluons. An example of a measurement function is the step function used

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Chapter 3. Non-global event shapes 20 in the thrust resummation of chapter 2. The paper introduces the following functional: R[U ] = ∞ X n=0 Z dP S [Ma1,...,an n ] ∗ Ua1b1_(θ 1)...Uanbn(θn)hMnb1,...,bn i u({pi}).

Ris a functional of the hermitian color-density matrices U . These matrices can be seen as color rotations according to Caron-Huot. This functional has divergences at low energies, and this paper demonstrates that these diver-gences exponentiate and can be canceled. For this one needs to introduce a renormalization energy scale µ. Caron-Huot shows that the renormalized functional depends on µ in the following way:

µ d dµ+ β d dαS Rren[U ; µ] = K U, δ δU, αS(µ), Rren[U, µ].

This differential equation can be used to resum the non-global logarithms; the hamiltonian contains the soft, wide-angle divergences responsible for the non-global logarithms and the differential equation specifies how these evolve at higher order. If one starts with a detailed description of the non-global logarithms at low order, the evolution equation tells you how these translate to non-global logarithms at higher order. There might still be global logarithms in R, but these can be resummed using conventional tech-niques.

3.3.2 Resummation with subjets

FIGURE3.3: The majority of the radiation that causes non-global logarithms comes from gluons near the boundary. These gluons can be viewed as a soft subjet and resummed,

according to Larkoski, Moult, and Neill (2015).

The approach of Larkoski, Moult, and Neill (2015) is different, in this pa-per they recognize that the majority of the secundary emissions that cause these non-global logarithms (as pictured in figure 3.1) come from gluons near the boundary. They can then find a subjet, a substructure of a jet, that is responsible for the radiation into the part of the phase space that con-tributes to the event shape cross section. The idea is then to resum this soft subjet in order to account for the radiation that causes non-global loga-rithms. A sketch of this idea is presented in figure 3.3. It is not easy to find these substructures; in this paper Larkoski et al. present observables called energy correlations that can be used to identify subjets.

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Chapter 3. Non-global event shapes 21

3.3.3 Observing non-global logarithms

Non-global logarithms can be hard to observe; they are often of next-to-leading logarithmic order and have to compete with other NLL and non-perturbative effects. In the paper ‘ Non-global correlations in collider physics’ by Larkoski and Moult (2016), a new observable is constructed that pro-vides a more ‘clean’ opportunity to measure non-global logarithms.

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22

Chapter 4

Conclusion and outlook

With this thesis I hope to show that resummation is a useful technique to improve predictions from QCD. In chapter two an important event shape, the thrust, is resummed up to leading logarithmic order. While the tech-niques used in chapter two are more broadly applicable to other event shapes, they do not work with a class of event shape called non-global event shapes. There are non-global logarithms in the perturbative expan-sion of these event shapes, that prevent resummation with the techniques used in chapter two. This is because the cancellation between real and vir-tual gluons that occurs in global resummation does not always happen in the non-global case. In recent years, new techniques have been developed to resum and observe these non-global logarithms.

4.1 Outlook

Non-global logarithms are a quickly developing subfield, and it will be in-teresting to see what can be done with the methods of Caron-Huot (2015) and Larkoski, Moult, and Neill (2015). Resummation is not an easy subject and it does not help that there are few comprehensive or detailed review articles. The article by Luisoni and Marzani (2015) is a great start, but my hope is that more articles will be written on resummation that take a more pedagogical approach, and I hope that this thesis can help people under-stand some of the details of resummation.

I would like to thank all the people who have helped me: Eric, Jordy and Jonathan. Thanks for introducing me to this interesting subject!

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23

Appendix A

Cancellation real and virtual

emissions

Because the variable is infrared and collinear safe the real emissions should cancel against the virtual emissions if there are no cuts by theta functions (Luisoni and Marzani, 2015). This means that:

Σ = 0 = ∞ X n=0 Z dW_nR+ Z dW_nV (A.1)

With dWnR/V the contribution of n gluons to the real/virtual cumulative

cross section. These contribution have to cancel term by term, so dWnR =

− dWV

n . However there is no constraint on the virtual part, only the real

gluons contribute to the thrust. Σ(τ ) = ∞ X n=0 Z dW_nR Θ (τ − ˆτ ) + Z dW_nV (A.2) It is now practical to multiply the virtual contribution with Θ(τ ). Because τ is always positive, the step function is always 1.

Σ(τ ) = ∞ X n=0 Z dW_nR Θ (τ − ˆτ ) + Z dW_nV Θ(τ ) (A.3)

Now I use the transformation in equation 2.22 and the fact that dWR n =

− dWV

n to factorize the expression:

Σ(τ ) = ∞ X n=0 1 n! Z " n Y i=0 dW₁R # Θ (τ − ˆτ ) − 1 n! Z " n Y i=0 dW₁R # Θ(τ ) ! = ∞ X n=0 1 n! Z " n Y i=0 dW₁R # (Θ (τ − ˆτ ) − Θ(τ )) ! = ∞ X n=0 1 n! Z " n Y i=0 dW₁R # Z _dν 2πiνe ντ −ν ˆτ ₋ Z dν 2πiνe ντ !

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Appendix A. Cancellation real and virtual emissions 24 Now I use that ˆτ =P V (ki)to factorize the exponential.

Σ(τ ) = ∞ X n=0 1 n! Z " n Y i=0 dW₁R # Z _dν 2πiνe ντ −νP iV (ki)₋ Z dν 2πiνe ντ ! = ∞ X n=0 1 n! Z " n Y i=0 dW₁R # Z dν 2πiνe ντ n Y i=0 e−νV (ki)₋ Z dν 2πiνe ντ !! = Z _dν 2πiνe ντ ∞ X n=0 1 n! n Y i=0 Z dW₁Re−V (ki)_{− 1}

This we can recognize as an exponential: Σ(τ ) = Z dν 2πiνexp Z dW₁R e−V (ki)_{− 1} (A.4)

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25

Appendix B

Light-cone variables

In special relativity the position or momentum of particles is represented with a four-vector, xµ _{= (x}0_{, x}1_{, x}2_{, x}3_{). It turns out that for some}

appli-cations another representation can be more useful. This is called the light-cone representation. The four quantities that form that representation are:

X = x 0_{+ x}3 √ 2 , ¯ X = x 0_{− x}3 √ 2 , ~xt= x1 x2 (B.1) These are also called the Sudakov variables.

The traditional space coordinates are not used equally, x3 _{plays a more}

central role. Although you can use light-cone variables in any problem, they are equivalent to ‘traditional’ coordinates, they are really usefull in problems that have a central axis that is different from the other two. Ex-amples of this are hadron colliders, where the interacting particles have a high boost in the beam direction, and jet physics, where the two original quarks are higly boosted along one axis.

A highly relativistic particle going in the z direction has the following momenta: ‘traditional coordinates’ pµ= p     1 0 0 1     ‘light-cone coordinates’ pµ= √2p 2     1 0 0 0    

A boost in the z direction looks like this:

pµ=   p+ p− pt  → pµ=   eβp+ e−βp− pt  

where β depends on the velocity v: β = 1

2log 1 + v 1 − v.

The inner product of two vectors have to be independent of coordinates. The inner product in light-cone variables is defined as:

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Appendix B. Light-cone variables 26 because this results in the same answer as the inner product in traditional coordinates: vµwµ= v0w0+ v3w0− v0_w3_{− v}3_w3 2 + v0w0− v3_w0_{+ v}0_w3_{− v}3_w3 2 − v 1_w1_{− v}2_w2 = v0w0− v1w1− v2w2− v3w3 Finally, it is conventional to write:

kµ= k = zP + kt+ ¯z ¯P (B.3)

With k the momentum of an emitted particle and P, ¯P the momentum of the emitting particle in lightcone variables. An expression for ¯zis:

¯ z = k

2 t + k2

zP · ¯P

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27

Appendix C

Expression of thrust in

light-cone variables

In this appendix I will explain why 1 Q(|ˆn · ~q1| + |ˆn · ~q2|) = s 1 − 2 q 2 1 Q2 + q₂2 Q2 (C.1) The right-hand side is in terms of four-vectors instead of the three-vectors on the left-hand side. I start therefore with writing the total momenta in four-vectors. In light-cone coordinates the total momentum of the hemi-spheres can be expressed as:

q1= zP + q⊥+ w zP¯ q2= ¯z ¯P + ¯q⊥+ ¯ w ¯ zP Here, w = q12 Q2 and ¯w = q2 2

Q2. This expression is then the same as equation

B.3, because qt= 0. To find z in terms of w and ¯w, I use the conservation of

momentum.

q1+ q2 = P + ¯P

P and ¯P are the momenta of the initial quarks. This means that: z +w¯ ¯ z = 1 ¯ z +w z = 1 From this I can express z and ¯zin w and ¯w.

z = 1 2 h 1 − ¯w + w +p1 − 2(w + ¯w) + (w − ¯w)2i ≡ 1 2[1 − ¯w + w + t] ¯ z = 1 2[1 + ¯w − w + t]

twill turn out to be equal to the thrust. To see that I will express t in terms of z.

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Appendix C. Expression of thrust in light-cone variables 28 To express z in terms of three vectors I use that:

z = 2 ¯P

Q2 · q1and (C.3)

¯ z = 2P

Q2 · q2. (C.4)

This can be derived as follows, I start with: 2 ¯P

Q2 · q1=

2 ¯P

Q2 · (zP + ¯z ¯P ),

and use the inner product of the light-cone variables (formula B.2) and the fact that P · ¯P = 1₂Q2to get:

2 ¯P

Q2 · zP = z

To show that t is equal to the expression C.1, I express P and qi in

three-vectors: P = Q 2 Q 2nˆ ! , P =¯ Q 2 −Q₂nˆ ! and q1 = q0 1 ~ q1 ! , q2= q0 2 ~ q2 !

tis can now be expressed in three-vectors by combining C.2, C.3, C.4 and the vectors above.

t = 2 Q2 Qq0 1 2 + Q 2n · ~ˆ q1+ Qq₂0 2 − Q 2n · ~ˆ q2 − 1.

Conservation of energy dictates that q₁0 + q₂0 = Q, which means that the expression simplifies to:

t = 1

Q( ~q1· ˆn − ~q2· ˆn) = 1

Q(| ~q1· ˆn| + | ~q2· ˆn|) (C.5) Which is the same as equation 2.8, while t also equals:

t =p1 − 2(w + ¯w) + (w − ¯w)2 ₌ s 1 − 2 q 2 1 Q2 + q₁2 Q2 + q 2 2 Q2 − q2₂ Q2 2 (C.6) ' s 1 − 2 q 2 1 Q2 + q2 1 Q2 (C.7) Which is the same as equation 2.9. I neglected the third term in the square root because I assume that the hadrons are relativistic.

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29

Bibliography

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Resummation; global and non-global Logarithms

U

NIVERSITY OF

A

MSTERDAM

B

T

Resummation; global and non-global

Logarithms

Abstract

Populair wetenschappelijke

samenvatting

Contents

Chapter 1

Introduction

1.1

Resummation and jet physics

1.2

Rough sketch of the resummation of a global event

shape

Chapter 2

Resummation of thrust in e

+

e

−

2.1

Kinematics

2.2

Factorization

2.3

One-gluon integral

2.4

Matching and possible improvements