Soft-Collinear Effective Theory

Master Thesis

by

Esther Sabine Bonig

Supervisor & 1st corrector: Junior-Prof. Dr. Nicolas Borghini2nd corrector: Prof. Dr. York Schroder

University of BielefeldOctober 25, 2010

Wenn fur dich eine Sache schwer zu bewaltigen ist,

darfst du nicht gleich denken,

sie sei fur Menschen unmoglich;

du musst vielmehr glauben,

wenn uberhaupt etwas fur den Menschen moglich ist

und in seinem Bereich liegt,

dass es auch fur dich erreichbar sei.

Marcus Aurelius

3

4

Contents

1 Introduction 7

1.1 A Short Review of the Standard Model . . . . . . . . . . . . . . . . . . . . 71.2 Introduction to Quantum Chromodynamics . . . . . . . . . . . . . . . . . . 81.3 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 Quark Pair Production with Gluon Loop Correction . . . . . . . . . 12

1.4 Dimensional Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Renormalisation Group Equation . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Soft-Collinear Effective Theory 19

2.1 Effective Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Lightcone Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Motivation and Definition . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 The Lagrangian in Lightcone Coordinates . . . . . . . . . . . . . . . 212.2.3 The SCET Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.4 Power Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.5 Collinear Leading-order Lagrangian . . . . . . . . . . . . . . . . . . 242.2.6 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Wilson Lines in SCET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Decoupled SCET Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Jet and Soft Functions 33

3.1 Jet Angularity and Jet Algorithms . . . . . . . . . . . . . . . . . . . . . . . 333.2 Jet Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 Quark Jet Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Gluon Jet Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Soft Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Final Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.1 Renormalisation Group Equations . . . . . . . . . . . . . . . . . . . 48

4 Conclusion and Outlook 51

5 Appendix 53

5.1 Quark Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5

Contents

6

1 Introduction

In this work, we study specific aspects of the description of jet physics, which is relevantfor heavy ion collisions. For the calculation of a process including different energy scales,the soft-collinear effective theory (SCET) can be used. This effective theory, derived fromquantum chromodynamics (QCD), aims to control infrared divergencies, which occur fromparticles with a small energy. The main goal of this Master Thesis is to show how SCETworks and to calculate jet and soft functions occuring in the factorised cross section of aprocess. These functions include soft and collinear particles. Before we turn to specificcalculations, we will give an introduction to QCD and explain the ideas and procedure fora calculation in SCET.

1.1 A Short Review of the Standard Model

The basic framework of our knowledge on the behaviour of particles is the Standard Modelof particle physics. It includes quarks and leptons (fermions) as elementary particles aswell as gauge bosons, which are the mediators for the strong, weak and electromagneticforce. The fermions are arranged in three generations as shown in Table 1.1. The gaugeboson for the electromagnetic force (coupling strength α = 1/137) is the photon γ, whereasthe Z0 and the W± are mediating the weak force (coupling strength αw = 10−6). Thestrong force, in which we are interested in this work is mediated by the gluon g (couplingstrength g2

s = 4παs). Only the charged particles are exposed to the electromagnetic force(i.e. all fundamental particles except the neutrinos) and similarly only the quarks are ableto interact strongly, because they are the only fermions carrying colour charge. The weakforce acts between all particles, because it is possible for every fermion to couple to thegauge bosons of the weak force. All the physics of the standard model is encoded in theLagrangian, from which the Feynman rules can be derived. From these the cross sectionsand decay rates of physical processes can be calculated.[1]

Quarks

u c t ← charge +2/3d s b ← charge −1/3

Leptons

e µ τ ← charge −1νe νµ ντ ← charge 0

Bosons

γ g Z0 W±

Table 1.1: Overview of the elementary particles with their charge and gauge bosons. For everyparticle there exists also its antiparticle with opposite electric charge. Note, that each quark comesin three colours.

7

1 Introduction

q(g)

g(g, r)

q(r)

Figure 1.1: The straight lines represent quarks, the curled line denotes the mediating gluon.

1.2 Introduction to Quantum Chromodynamics

Quantum Field Theory (QFT) is the combination of special relativity and quantum me-chanics, which results in a field theory: All particles are described as fields, which can bequantised similar to position and momentum operators in quantum mechanics [2]. Thisdescription of particles and their interaction is essential for the understanding of their na-ture and behaviour. A special case of QFT is Quantum Chromodynamics [3, 4]. That isthe underlying theory for the formation of hadrons. Hadrons consist of quarks, which areglued together by gluons, the gauge bosons of the strong interaction. An important char-acteristic is the colour charge of each quark. The colour is a conserved quantum number,which can be red (r), green (g) or blue (b). The gluon is double colour charged [5], sothat the conservation of colour is ensured in any process. More precisely, the gluon ”carriesaway”and ”brings” another colour, which is carrying away an anticolour. One may visualisethis in a diagram (see Fig.1.1). The strength of the coupling depends on the momentumscale Q, known as the running coupling constant αs(Q

2). This has a special behaviour incontrast to quantum electrodynamics (QED) due to anti-screening. As a result αs is big, ifthe single quarks of a hadron are far away from each other but becomes weaker and weakerthe nearer the quarks get together. This leads to the so-called asymptotic freedom, thatis the behaviour of the quarks as quasi-free particles at small distances. In contrast, thequarks are forced together at larger distances, which results in the so-called confinement,that is the statement of the impossibility to observe a single quark. The reason for thisconfinement is the non-abelian structure of the QCD underlying gauge group SU(3), whichwe will discuss later in this chapter.

The QCD Lagrange density (further denoted as Lagrangian) is a straightforward exten-sion of the QED Lagrangian, which can be derived from local gauge invariance and is givenby

LQED = −1

4FµνF

µν + ψ(iγµ(∂µ + ieAµ︸ ︷︷ ︸

Dµ

)−m)ψ, (1.2.1)

where ψ is a Dirac spinor and ψ = ψ†γ0 is the adjoint Dirac spinor, Aµ is the photonfield, Fµν = ∂µAν − ∂νAµ is the electromagnetic field strength tensor and Dµ denotes thecovariant derivative, which we need to make the Lagrangian gauge invariant with respect tolocal U(1) gauge transformations considering the photon field. In contrast to QED, QCD

8

1.2 Introduction to Quantum Chromodynamics

is based on the SU(3) gauge symmetry [6]

ψ(x)→ S(x)ψ(x), ψ(x) =

ψr(x)ψg(x)ψb(x)

, (1.2.2)

which results in the local gauge transformation

S(x) = exp

[

i

8∑

i=1

αi(x)λi

2

]

, (1.2.3)

where αi(x) are space dependent phase factors. In analogy to the QED Lagrangian theQCD Lagrangian can stated to be

LQCD = −1

4Ga

µνGaµν +

∑

f

ψif

(i /D −mf

)

ijψj

f , (1.2.4)

with

Gaµν = ∂µA

aν − ∂νA

aµ − gsf

abcAbµA

cν , (1.2.5)

(Dµ)ij = δij∂µ − igs

∑

a

λaij

2Aa

µ, (1.2.6)

where Aµ = Aaµt

a are the gluon fields, the sum over f means the sum over all flavours

(which are the different quarks), a denotes the colour index (a=1,...,8) and fabc are theantisymmetric structure constants. The ta = λa

2 are the generators of the SU(3) gaugegroup and λa are called the Gell-Mann matrices [1]:

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0i 0 00 0 0

, λ3 =

1 0 00 −1 00 0 0

,

λ4 =

0 0 10 0 01 0 0

, λ5 =

0 0 −i0 0 0i 0 0

, λ6 =

0 0 00 0 10 1 0

,

λ7 =

0 0 00 0 −i0 i 0

, λ8 =1√3

1 0 00 1 00 0 −2

. (1.2.7)

Naively one would say there exist nine gluons, because there are three colours which canbe combined to one colour octet and one singlet state. But one has to be aware that thecolour singlet state does not exist in nature, because colourless gluons would not couple toquarks and single free gluons are not observed. For the matrices the following commutationrelations can be found, which define the Lie algebra of the SU(3) gauge group [7]:

[

λa, λb]

= 2ifabcλc. (1.2.8)

Because of the non-vanishing commutation relation SU(3) is called a non-abelian gaugegroup and therefore QCD is called a non-abelian theory. Hence, in comparison to abeliangauge theories like QED, there is an extra term in the field strength tensor (Eq.(1.2.5)),which corresponds to the self interaction of the gluons.

9

1 Introduction

1.3 Perturbation Theory

Perturbation theory is used for the description of a system at high energies where αs ≪ 1[8]. We will use perturbation theory for the calculations of QCD [10]. One has to be awarethat the application is only possible in the regime of a small coupling constant, that iswhen the distance between the quarks is small. In a collision this is the regime, wherethe momentum transfer is large. Richard Feynman invented a graphical representation forthe terms of the infinite perturbation series, called Feynman diagrams. These diagramsinclude different lines for quarks and gluons and vertices for their interaction and picturethe discrete terms of the perturbation series. Along with these diagrams there come rulesfor their calculation and hence for the calculation of the perturbation series, which can beread off from the Langrangian. For QCD we will now shortly state the rules and give thepropagators and vertex factors. Later on also the Feynman rules in SCET will be given.

1.3.1 Feynman Rules

We will shortly review the Feynman rules and calculate as an example a gluon loop correction[3]. For the derivation of the Feynman rules for QCD it is necessary to introduce a gauge-fixing term which gives rise to a so-called ghost term:

LQCD = −1

4Ga

µνGaµν + i

∑

f

ψif

(i /D −mf

)

ijψj

f + Lgauge−fixing + Lghost, (1.3.1)

where

Lgauge−fixing = − 1

2ξ

(∂µAa

µ

)2. (1.3.2)

For the gauge-fixing term it is most common to use Feynman gauge, which is ξ = 1. Becausethis term breaks the gauge invariance of the Lagrangian, a so-called ghost term is included.This absorbs the unphysical degrees of freedom and can be derived via the Fadeev-Popovtrick [3].

To expose the Feynman rules we will label in the following each line with a four-momentum p or q respectively and each vertex, which is the interaction point, with agreek letter. These are the Feynman rules for the calculation of a diagram [1]:

1. External Lines

For the external lines take p to be the four-momentum:

Incoming: u(p) Incoming: u(p)Electrons Quarks

Outgoing: u(p) Outgoing: u(p)

Incoming: v(p) Incoming: v(p)Positrons Antiquarks

Outgoing: v(p) Outgoing: v(p)

Incoming: ǫµ(p) Incoming: ǫµ(p)Photon Gluon

Outgoing: ǫµ∗(p) Outgoing: ǫ∗µ(p)

10

1.3 Perturbation Theory

with ǫ being the polarisation of the mediator.

2. Internal Lines

Each internal line is called a propagator and can be an electron/positron or quark/antiquarkas well as a mediator. Its momentum is labeled with q:

Electrons and Positronsi(/q+m)

q2−m2 Quarks and Antiquarksi(/q+m)

q2−m2

Photons−igµν

q2 Gluons−igµνδab

q2

3. Vertices

QED igγµ Quark-Gluon iαsγµta

with ta = λa

2 . There are also gluon-gluon interactions and therefore three- and four-gluon vertices, but they are not relevant for this work and can be looked up e.g. in[1].

4. Energy-Momentum Conservation

At each vertex and for the overall diagram energy and momentum are conserved.Therefore we have to write a δ−function, e.g. δ(p1 + p2 − q), as a constraint with afactor of (2π)4 for each vertex.

With each internal momentum there comes along a four-dimensional integration with afactor of 1/(2π)4. After the evaluation of these integrals the δ−function stating the overallenergy-momentum conservation will be left. If we factorise out this and one factor of (2π)4

the result corresponds to −iM, where M is the amplitude we need to compute the crosssection of a process and gives the relation between initial and final states [3]:

lout < ~pf|~pi >in= iM (pi → pf) (2π)4δ(4)

∑

i

pi −∑

f

pf

(1.3.3)

with ~pf being the final and ~pi being the initial momenta of the particles and the δ−functiondenotes momentum conservation. In a collision experiment with two initial particles (A andB with 4−momenta pA and pB) producing n final particles (with 4−momenta pj) the crosssection is given by

dσ =|M2|

4√

(pApB)2 −m2Am

2B

n∏

j=1

d3~pj

(2π)32E~pj

(2π)4δ

n∑

j=1

pj − pA − pB

. (1.3.4)

11

1 Introduction

p3p8

e−,p2

e+,p1

q,p7

q,p6

σ µ

p5 ρ

p4 ν

Figure 1.2: Quark pair production with a gluon loop correction.

The integrated cross section σ given in the above form is also called ”Fermi’s Golden Rule”[9] and is highly important for collision experiments as it is a measure for the probabilityof the process to happen.

1.3.2 Quark Pair Production with Gluon Loop Correction

As an example we will now calculate the quark pair production with a vertex correction, thegluon loop (see Fig.1.2). This process is called the ”Drell-Yan process” of QCD. We will findthis kind of loop correction also in the calculation of the jet functions in SCET. In particular,it will contribute to the anomalous dimension, which contains the loop corrections of thefunctions. Using the Feynman rules stated in section 1.3.1 we can determine the amplitude∗:

− iM = (2π)4∫

d4p3d4p4d

4p5d4p8u(p1) (−ieγσ) v(p2)

(−igσµ

p23

)

uq(p6) (−iαsγρta)

× i (γαpα5 +m5)

p25 −m2

5+ iǫ

(−ieγµ)i(

γβpβ4 +m4

)

p24 −m2

4 + iǫ

(

−iαsγνtb)

vq(p7)

( −igνρ

p28 − µ2 + iǫ

δab

)

×δ(p1 + p2 − p6 − p7)δ(p1 + p2 − p3)δ(p5 + p4 − p3)δ(p4 + p8 − p6)δ(p5 − p8 − p7)

= (2π)4∫

d4p8u(p1) (−ieγσ) v(p2)

( −igσµ

(p1 + p2)2

)

uq(p6) (−iαsγρ)

×i((6 p7+ 6 p8) +m2

5

)

(p7 + p8)2 −m5 + iǫ

(−ieγµ)i(( 6 p6− 6 p8) +m2

4

)

(p6 − p8)2 −m2

4 + iǫ(−iαsγ

ν)∑

a,b

tatbδabvq(p7)

×( −igνρ

p28 − µ2 + iǫ

)

δ (p1 + p2 − p6 − p7)

= −e2α2s

∫

d4p8 (2π)4 δ (p1 + p2 − p6 − p7) (1.3.5)

×u(p1)γµv(p2)uq(p6)γρ (i ( 6 p7+ 6 p8 +m5)) γ

µ (i ( 6 p6− 6 p8 +m4)) γρCF vq(p7)

(p1 + p2)2(

(p7 + p8)2 −m2

5 + iǫ)(

(p6 − p8)2 −m2

4 + iǫ) (p28 − µ2 + iǫ

) .

Here we have introduced a term +iǫ in the denominator of the propagators and a smallnon-zero gluon mass µ which can be set to zero in the end to control the divergences in theintegral. In the last step the quadratic Casimir operator CF occurs, which is defined as thesum CF ≡

∑

a tata = N2−1

2N for a group SU(N). Let us first have a look at the numerator.

∗The calculation follows [3]

12

1.3 Perturbation Theory

We will rewrite it using

γργµ = 2gρµ − γµγρ , γµγρ = 2gµρ − γργ

µ,

γρ/pγρ = −2/p, (1.3.6)

/k = (6 p7 + /p8) , /k′ = ( /p6 − /p8).

Numerator = e2α2su(p1)γµv(p2)uq(p6)[−2/kγµ/k

′

+2m4 (/kγµ + γµ/k)︸ ︷︷ ︸

2kµ

+2m5

(/k′γµ + γµ/k

′)

︸ ︷︷ ︸

2k′µ

−2m5m4γµ]CF vq(p7)

= −2e2α2su(p1)γµv(p2)uq(p6)[/kγ

µ/k′+m2γµ − 2m(kµ + k

′µ)]CF vq(p7)

= −2e2α2su(p1)γµv(p2)uq(p6)[/lγ

µ/l +(

−y/p3+ z/p7

)

γµ(

(1− y)/p3+ z/p7

)

+m2γµ − 2m (2lµ + (1 − 2y)pµ3 + 2zpµ

7 )]CF vq(p7). (1.3.7)

For the last equality we have used that

l ≡ k + yp3 − zp7,

k′ = p3 − k, (1.3.8)

1 = x+ y + z,

and that the masses of the quark-antiquark pair are the same for both. The denominatoralso needs to be reexpressed. This is done using the Feynman-trick:

1

A1...An=

∫ 1

0dx1...dxnδ

(n∑

i=1

xi − 1

)

(n− 1)!

(x1A1 + ...+ xnAn)n. (1.3.9)

The factor 1(p1+p2)

2 can be written in front of the integral, because it does not depend on

p8. Then we get for the denominator for n = 3:

1(

(p7 + p8)2 −m2

5 + iǫ)(

(p6 − p8)2 −m2

4 + iǫ) (p28 − µ2 + iǫ

)

=

∫ 1

0dxdydzδ (x+ y + z − 1)

2

D3(1.3.10)

with

D = x(k2 −m2

)+ y

(k′2 −m2

)+ z

((k − p)2 − µ2

)+ iǫ

= k2(x+ y + z︸ ︷︷ ︸

1

) + 2k(yp3 − zp7) + yp23 + zp2

7 − xm2 − ym2 − zµ2 + iǫ

= l2 + xyp23 − (1− z)2m2 − zµ2 + iǫ. (1.3.11)

With the use of the anticommutation relation

/p, γµ = 2pµ (1.3.12)

13

1 Introduction

and the Dirac equation (/pv(p) = mv(p); u(p)/p = u(p)m) we get:

Numerator = −2e2α2su(p1)γµv(p2)uq(p6)

[

γµ

(

−1

2l2 + (1− x)(1 − y)p2

3

1

1+ (1− 2z − z2)m2

)

+ (pµ7 + pµ

6 )mz(z − 1) + pµ3m(z − 2)(x− y)

1

1+ 2l

]

CF vq(p7) . (1.3.13)

Since the denominator D only depends on l2, we can simplify the numerator further bysymmetry relations:

∫d4l

(2π)4lµ

D3= 0,

∫d4l

(2π)4lµlν

D3=

∫d4l

(2π)41/4gµν l2

D3.

(1.3.14)

Now we can put everything together and find:

M = 2ie2α2s

1

(2π)41

(p1 + p2)2

∫

d4l

∫ 1

0dxdydzδ (x+ y + z − 1)

2

D3u(p1)γµv(p2)

×uq(p6)[γµ

(

−1

2l2 + (1− x)(1 − y)p2

3 + (1− 2z − z2)m2

)

+(p′µ7 + pµ

)mz(z − 1) + pµ

3m(z − 2)(x− y)]CF vq(p7). (1.3.15)

To evaluate this integral one needs to apply Wick rotation. The further steps of the com-putation are not relevant for this thesis and can be looked up in [3]. We restrict ourselvesto the above steps as they already illustrate the technique to tackle loop diagrams.

1.4 Dimensional Regularisation

When calculating Feynman diagrams one often encounters divergent integrals over internalmomenta, which can not be carried out because of infinities, either due to a small energycorresponding to long distance phenomena (infrared divergences) or due to a large energycorresponding to small distance phenomena (ultraviolet divergences). The superficial degreeof divergence is determined via power counting. To get the true degree of divergence onehas to carry out the integral with applying regularisation. To control these undesirabledivergences one can use different regularisation methods, which introduces a regulator asa parametrisation of these divergences. The method commonly used for gauge theories isdimensional regularisation, which we will use in sections 3.2 and 3.3 to calculate the jet andsoft function. We introduce this technique on the example of a simple loop integral.[11, 12]Consider the integral over a scalar propagator, e.g. a meson, with momentum kµ and massm:

I =

∫ddk

(2π)di

k2 −m2 + iδ. (1.4.1)

14

1.4 Dimensional Regularisation

First of all it is very useful to apply Wick rotation t → iτ , k0 → ip0, which avoids thesingularities in the integrand, that is at k2 = m2 in our example:

∫

ddki

k2 −m2 + iδ= i

∫

ddki

(ip0)2 − (~p)2 −m2 + iδ

=

∫

ddp1

p2 +m2 − iδ , (1.4.2)

where we may now neglect the iδ term in the denominator because there is no pole left.By now the integral is not regularised and we would have to integrate in d = 4 dimensions,which leads to a quadratic divergence discovered by counting the powers of p in the integral†.Therefore we will introduce a small parameter ǫ and evaluate the integral in d = 4 − 2ǫdimensions. Additionally we have to introduce a renormalisation scale µ. As a prefactor ofµ2ǫ it assures the mass dimension of the final result to be independent of ǫ. In the followingspherical coordinates are used:

I =

∫ddp

(2π)di

p2 −m2 + iδ= µ2ǫ

∫d4−2ǫp

(2π)4−2ǫ

1

p2 +m2

= µ2ǫ 1

(2π)4−2ǫ

∫

dΩ3−2ǫ

∫

dpp3−2ǫ

p2 +m2, (1.4.3)

where the angular integral can be evaluated via: [3]

(√π)d

=

(∫

dxe−x2

)d

=

∫

ddxe−Pd

i=1 x2i =

∫

dΩd−1

∫ ∞

0dxxd−1e−x2

︸ ︷︷ ︸

12

R ∞0 dx2x2( d

2−1)e−x2

=1

2Γ (d/2)

∫

dΩd−1 (1.4.4)

⇒∫

dΩd−1 =2πd/2

Γ (d/2). (1.4.5)

Therefore we are left with

I = µ2ǫ 2π2−ǫ

(2π)4−2ǫΓ(2− ǫ)

∫

dpp3−2ǫ

p2 +m2

= µ2ǫ 2π2−ǫ

(2π)4−2ǫΓ(2− ǫ)1

2m2−2ǫ

∫

duu(2−ǫ)−1

u+ 1

= µ2ǫ 2π2−ǫ

(2π)4−2ǫΓ(2− ǫ)1

2m2−2ǫB(2− ǫ, ǫ− 1), (1.4.6)

where we applied the substitution u = p2/m2 in the second step and used the Beta functionB(2− ǫ, ǫ− 1) defined by

B(x, y) =

∫ ∞

0du

ux−1

(1 + u)x+y=

Γ(x)Γ(y)

Γ(x+ y). (1.4.7)

†ddp = d4p has dimension 4, 1/p2 has dimension 2, which leads, when multiplied, to a quadratic divergence.

15

1 Introduction

With x = 2− ǫ and x+ y = 1 the integral becomes

I = µ2ǫ 2π2−ǫ

(2π)4−2ǫ

1

2m2−2ǫ Γ(2− ǫ)Γ(ǫ− 1)

Γ(2− ǫ)Γ(1)

= µ2ǫ 2π2−ǫ

(2π)4−2ǫ

1

2m2−2ǫΓ(ǫ− 1)

=(m

4π

)2(

4πµ2

m2

)ǫ

Γ(ǫ− 1). (1.4.8)

We now expand I in ǫ around 0 and get to first order

I ≈(m

4π

)2(

−1

ǫ− 1 + γE − ln

4πµ2

m2

)

, (1.4.9)

where γE is the Euler constant. When comparing two experimental results at differentscales µ1 and µ2 and taking the difference everything cancels except of the logarithm of theratio of the energy scales. Hence the integral I is logarithmically divergent with respect tothe physical scales µ1 and µ2, which are then set by the experiment.

1.5 Renormalisation Group Equation

For the explanation of the renormalisation group [13] we first need to clarify the concept ofcounterterms. Counterterms are inserted into the Lagrangian to cancel the divergent partsof the Feynman graphs. We need specifications for choosing the particular structure of thecounterterms, at which we will arrive via renormalisation. A common and very popularrenormalisation scheme, which is also used when doing the calculation of the jet function, isthe modified minimal subtraction scheme (MS) [3]. In this scheme one has to remove thepoles (1/ǫ), which is done by neglecting them and all the finite parts in the integral, whichdo not include the mass. In the example given in section 1.4, the final and therefore finiteresult in the MS−scheme is given by:

I ≈(m

4π

)2(

−1

ǫ− 1 + γE − ln

4πµ2

m2

)

→(m

4π

)2(

− lnµ2

m2

)

. (1.5.1)

Every renormalisation scheme gives a slightly different finite result, from which follows, thatif one wants the physics to stay constant while changing the renormalisation scheme one hasto change a quantity with respect to µ. Hence this invariance of the physical result is thetheoretical reason for the need of a running coupling constant, which becomes a couplingconstant dependent on µ. The scale then has to be chosen such that in the end µ is notinvolved anymore, because this so-called renormalised coupling is measured in experimentsand therefore needs to be independent of µ. To calculate the running of a quantity oneneeds differential equations, called Renormalisation Group Equations (RGEs).In general, a bare, that is unrenormalised, function can be expressed through the renor-malised function (note that we have neglected the mass dependence) via‡:

FB(pi, αB , ǫ) = ZF (ǫ, µ)F (pi, α, µ), (1.5.2)

‡Follows [14]

16

1.5 Renormalisation Group Equation

whereB labels the bare quantities and Z is a renormalisation factor depending on the chosenrenormalisation scheme. When applying the derivative with respect to µ on equation (1.5.2)and multiplying with µ

ZF, we get the renormalisation group equation(

γF (µ) + µd

dµ

)

F (pi, α, µ) = 0 (1.5.3)

with the anomalous dimension being defined as

γF (µ) ≡ µ

ZF

dZF

dµ. (1.5.4)

The splitting of the total derivative leads to the definition of the β−function:

µd

dµ= µ

∂

∂µ+ µ

∂α

∂µ

∂

∂α,

β(α) ≡ µ∂α

∂µ∼= −β0α

2 − β1α3 − ... . (1.5.5)

In QCD, where α = αs, Wilczek, Gross and Politzer first calculated the coefficient of theβ−function to leading order. Here also the next-to-leading order is given [15]:

β0 =33− 2Nf

12π,

β1 =153− 19Nf

24π2, (1.5.6)

where the βi are calculated via perturbation theory and Nf is the number of flavours. Withthis the renormalisation group equation can be written as the well-known ”Callan-Symanzikequation”:

(

µ∂

∂µ+ β(α)

∂

∂α+ γF (µ)

)

F (pi, α, µ) = 0, (1.5.7)

which has the solution

F (pi, α, µ) = F (pi, α0, µ0)exp

[

−∫ α

α0

dα′ γF (µ)

β(α′)

]

. (1.5.8)

The function F (pi, α, µ) is now expressed in terms of a new scale µ0 and a new couplingα0. Therefore we can evolve a function at a scale µ0 to another scale µ. If one has severalfunctions at different energy scales they can be evolved to the same scale and it is thenpossible to resum the individual parts and give the cross section at one particular scale.

QCD Running Coupling

To calculate the running coupling constant of QCD one has to define the coupling at a largemomentum scale and then evolve the formula to a lower momentum scale by renormalisationgroup evolution. Therefore we use the Callan-Symanzik equation (Eq.(1.5.7))and take F tobe α(µ):

0 = µ∂α(µ)

∂µ+ β(α)

∂α(µ)

∂α+ γα(µ)α

⇒ µ∂α(µ)

∂µ= −β(α) (1.5.9)

17

1 Introduction

with the anomalous dimension γα(µ) being zero. Integrating this one gets:

−∫ µ

µ0

1

µ′dµ′ =

∫ αs(µ)

αs(µ0)

1

β(α)dα

⇒ lnµ

µ0=

1

β0

(1

αs(µ)− 1

αs(µ0)

)

, (1.5.10)

where we have used only the first order term of the expansion of the β−function given inEq.(1.5.5). Therefore the coupling constant is in lowest order given by

αs(µ) =αs(µ0)

1 + αs(µ0)β0 lnµ/µ0. (1.5.11)

With the definition of the scale Λ

Λ = µ0 exp

[

− 1

β0αs(µ0)

]

, (1.5.12)

which sets the scale for the coupling being large, we get an even simpler expression [15, 16]:

αs(µ) =1

β0 lnµ/Λ. (1.5.13)

18

2 Soft-Collinear Effective Theory

2.1 Effective Field Theories

Effective field theories (EFTs) are important if the underlying theory can not be treatedrigorously due to its complexity. The calculation is much more easily done in the frameworkof an EFT than applying a full theory, because one does not have to understand all thedifferent phenomena at distinct scales as EFTs may only be used in particular regimes. Thefirst step in the application of an EFT is to set the scales much larger or much shorter thanthe scale of the quantity under consideration to infinity or zero respectively. Therefore theobtained result is just an approximation to the full theory and the effects at other scales canbe thought of as small perturbations. If we for example look at a particle with a particularmass, we eliminate the heavier particles from the theory and get an EFT, which is valid onlyin an energy regime with E < mass of the heavy particles. This procedure can be iterated tolower masses, so that we evolve the EFT to a low energy regime and all the heavy particlesare successively freezed out. With each iteration, that is every time when approaching asmaller mass scale than where we have started from, we have to adopt another effectivetheory. This change has to be continuous at the boundary, which is assured by the so-calledmatching condition. In lowest order this is simply the equality of the coupling constantsin both regimes. The matching is done via the renormalisation group, already discussed insection 1.5. [17]

The theory we will use in this work is the soft-collinear effective theory (SCET), whichdescribes the interaction between energetic and non-energetic particles, e.g. collinear quarksand (ultra-)soft gluons. A particle is characterised via the scaling of its momentum p. Itis for example called soft if p ∼ λ, where λ is a process dependent parameter smaller than1. The cross section will be factorised, so that it is split up into three functions eachvalid in a different energy regime. Before doing so we have to make sure that the powercorrections in λ are small. In particular there are two types of power corrections. Oneis due to the particular jet algorithm∗, because whether a special soft particle contributesto one jet or another depends on the chosen algorithm. The other correction comes fromthe approximation of the jet axis. The difference between the actual parton direction andthe jet axis must not be too large. If this is assured, the factorised cross section containsmatrix elements which are identified with functions describing short distance physics (hardfunction), the propagation of energetic particles (jet function) and long distance physics (softfunction), which all depend on a different factorisation scale µi. The soft function cannotbe calculated in perturbative QCD (pQCD), because as it is dedicated to the long distancephysics it involves a small momentum transfer and therefore a large coupling constant. Inthis regime pQCD is no longer valid and we need an effective field theory like SCET tocalculate the soft function. Evaluating the hard function is not the focus of this work, wewill rather elaborate on the evaluation of the jet and the soft functions. [18, 19, 20, 21]

∗see chapter 3.1.

19

2 Soft-Collinear Effective Theory

Figure 2.1: Sketch of the different validities of SCETI and SCETII .

Because of the different scales of the energetic and non-energetic particles different de-scriptions, that is different effective theories, of the physics are needed. Hence there existtwo versions of soft-collinear effective theories valid in different energy regimes (see Fig.2.1).One is called SCETI and can be applied if the collinear momentum of the particle in thefinal state is p2

c ∝ ΛQCD, where ΛQCD is the energy scale at which the coupling is of order1. Then the ultrasoft, that is p ∼ λ2, and collinear fields can be decoupled. If the collinearmomentum is p2

c ∝ Λ2QCD, SCETI is only used as an intermediate step to get to lower

energies and the description is done via SCETII . To arrive at this theory one has to derivethe new matching coefficients and replace the ultrasoft gluon fields arising in SCETI withthe soft gluon fields. Hence the soft and collinear fields can be decoupled. [22, 23]

2.2 Lightcone Coordinates

2.2.1 Motivation and Definition

In a collision experiment at very high energies the colliding particles and most of the emittedparticles are ultrarelativistic and therefore move approximately along the lightcone. Hencewe will use lightcone coordinates for the mathematical fomulation of SCET. We define thelight-like directions with the conditions n2 = n2 = 0 and n · n = 2 as

nµ = (1, 0, 0, 1) , nµ = (1, 0, 0,−1). (2.2.1)

The particles with momenta collinear to these coordinates move along the ±z−direction.There are three different directions, ”+”, ”−” and ”⊥”. Hence a four-vector can be writtenas

pµ = n · pnµ

2+ n · pn

µ

2+ pµ

⊥, (2.2.2)

and

p ≡ (nµpµ, nµp

µ, n⊥µ pµ) = (p+, p−, p⊥). (2.2.3)

The momentum scale of collinear particles moving along the n−direction is given by pc =E(λ2, 1, λ), whereas for soft respectively ultrasoft particles all momentum components scale

20

2.2 Lightcone Coordinates

as ps = E(λ, λ, λ) respectively pus = E(λ2, λ2, λ2). Here E is a large energy scale and the

parameter λ is given as λ =ΛQCD

E ≪ 1 when working in SCETII†.

In our calculations we will use dimensional regularisation in terms of lightcone coordinates.A momentum integral in d = 4− 2ǫ dimensions can be written in the following way:

∫

d4−2ǫq =

∫

dq+dq−d2−2ǫq⊥ =

∫

dq+dq−dq⊥(

q⊥)1−2ǫ

dΩ2−2ǫ, (2.2.4)

where q+ and q− are always positive as they are lightcone coordinates.

2.2.2 The Lagrangian in Lightcone Coordinates

Let us now compute the collinear QCD Lagrangian in lightcone coordinates [20]. We areonly looking at processes where light quarks are involved and therefore neglect the masses.We start with the QCD-Lagrangian for massless particles, which is given by

LQCD = LquarkQCD + Lgluon

QCD , (2.2.5)

where the quark and gluon Lagrangian read

LquarkQCD = Ψi /DΨ, (2.2.6)

LgluonQCD = −1

2Tr(Ga

µνGµνa

)= −1

4

(Ga

µν

)2, (2.2.7)

with the trace being the sum over the colour index a and the Lorentz indices µ and ν. Towrite the Lagrangian in lightcone coordinates, we have to decompose the quark field ψ inton−collinear and n−collinear components. Notice, that in the following we will focus onn−collinear particles‡. To achieve the decomposition we use the projection operators Pn

and Pn:

ψ =

(/n/n

4+/n/n

4

)

︸ ︷︷ ︸

1

ψ = (Pn + Pn)ψ ≡ ξn + ξn. (2.2.8)

With the expression of the covariant derivative in terms of its lightcone components we getfor the Lagrangian:

Lquark =(ξn + ξn

)i

(/n

2n ·D +

/n

2n ·D + /D⊥

)

(ξn + ξn) . (2.2.9)

When calculating each term one notices, that a lot of them vanish. See Appendix 5.1 fordetails. The fermionic Lagrangian becomes

Lquark = ξn

(

i/n

2n ·Dξn + i /D⊥ξn

)

+ ξn

(

i /D⊥ξn + i/n

2n ·Dξn

)

. (2.2.10)

†Note that for SCETI the parameter is given by λ =

q

ΛQCD

E≪ 1.

‡If one wants to look at n−collinear particles one has to make the change n ↔ n.

21

2 Soft-Collinear Effective Theory

With these simplifications and the equation of motion

0 = ∂µ∂L

∂(∂µξn)− ∂L∂ξn

⇒ 0 = i /D⊥ξn +/n

2(in ·D)ξn | · /n

2

⇒ 0 =/n

2i /D⊥ξn + in ·Dξn

⇒ ξn = − 1

in ·Di/n

2/D⊥ξn =

1

in ·Di/D⊥

/n

2ξn, (2.2.11)

the fermionic Lagrangian can be written as

Lquark = ξn

(

in ·D + i /D⊥1

in ·Di/D⊥

)/n

2ξn + ξn

(

i /D⊥ +/n

2i /D⊥

/n

2

)

ξn, (2.2.12)

where the second term involving the ξn field can be neglected. [24]

2.2.3 The SCET Lagrangian

Now that we have already written the QCD Lagrangian in lightcone coordinates, we willdiscuss the Lagrangian for SCET [20]. The full Lagrangian is given by the sum of a partcontaining only soft, a part containing only ultrasoft and a part containing collinear fields:

LSCET = Ls + Lus + Lc. (2.2.13)

In the following we will focus on the collinear Lagrangian as we consider only collinear quarkfields. For the SCET Lagrangian the QCD covariant derivative needs to be replaced by theSCET covariant derivative

Dµ → DµSCET = ∂µ − igsA

µSCET , (2.2.14)

where AµSCET is the gluon field in SCET. Since the high energy gluons are no subject to

SCET, it is written as

AµSCET = Aµ

us +Aµn,q, (2.2.15)

where Aµn,q is a collinear gluon field with label momentum q and Aµ

us is an ultrasoft gluonfield. Note that in SCETII the ultrasoft gluon field becomes the soft gluon field. Thefermionic part and the gluonic part of the collinear SCET Lagrangian are nearly thesame as in QCD, the only difference pertains to the definition of the covariant derivative(Eq.(2.2.14)):

LquarkSCET = ξn

(

in ·DSCET + i /D⊥,SCET1

in ·DSCETi /D⊥,SCET

)/n

2ξn, (2.2.16)

LgluonSCET =

1

2g2s

Tr([Dµ,SCET ,Dν,SCET ]2

). (2.2.17)

The quark Lagrangian does not only describe the quark-quark interactions, but also theinteractions between quarks and gluons as the covariant derivative includes the gluon field.

22

2.2 Lightcone Coordinates

incoming quark incoming gluon outgoing quark

collinear: Q(λ2, 1, λ) collinear: Q(λ2, 1, λ) collinear: Q(λ2, 1, λ)collinear: Q(λ2, 1, λ) usoft: Q(λ2, λ2, λ2) collinear: Q(λ2, 1, λ)usoft: Q(λ2, λ2, λ2) collinear: Q(λ2, 1, λ) collinear: Q(λ2, 1, λ) (off-shell)usoft: Q(λ2, λ2, λ2) usoft: Q(λ2, λ2, λ2) usoft: Q(λ2, λ2, λ2)

Table 2.1: Change of momentum in different interactions. Here the outgoing quark momentum isgiven as the sum of the incoming quark and gluon momentum with the notation p = (p+, p−, p⊥).

In the following we will drop the index SCET . The soft and the ultrasoft part of theLagrangian are given for completeness§ by [25]:

Ls = fs,p′(/P + gs /As,q

)fs,p −

1

2Tr(Ga

µνGµνa

), (2.2.18)

Lus = fusi /Dfus −1

2Tr(Ga

µνGµνa

), (2.2.19)

with f being a soft respectively ultraosft field and Gµν = igs

[Dµ,Dν ].

2.2.4 Power Counting

The procedure of power counting is a useful tool to expand expressions into a perturbativeseries. Before we can have a look at the scaling of the different terms in the Lagrangian,we have to think about the possible interactions between quarks and gluons and how theychange the momentum of the incoming quark, which can be seen in Table 2.1. The mo-mentum p of a quark can be split into a large part (label momentum q) and a small part(residual momentum k ∝ λ2). We can therefore define a label operator P, which acts onthe label momenta of a field and extracts them:

PΦ(pi) = qiΦ(pi). (2.2.20)

We write the quark field as ξn =∑

q e−iqxξn,p, where the exponential function imposes label

momentum conservation. With this we see that

∂µ∑

q

e−iqxξn,p =∑

q

e−iqx (−iPµ + ∂µ) ξn,p. (2.2.21)

We will now consider the scaling of the different parts of the quark Lagrangian. We splitthe label operator into

Pµ =nµ

2(n · P)

︸ ︷︷ ︸

”−”direction

+Pµ⊥ ∝ O(λ0) +O(λ). (2.2.22)

Therefore the partial derivative acting on a field only gives contributions of O(λ2). Thescaling of the gluon fields can already be seen in Table 2.1, but we will write it down

§As our attention lies on the collinear quark with the attachment of soft gluons, we do not need these partsof the Lagrangian for further calculations.

23

2 Soft-Collinear Effective Theory

explicitly [26]:

Aµus ∝ λ2,

Aµs ∝ λ,

n ·An,q ∝ λ2,

A⊥n,q ∝ λ,

n ·An,q ∝ λ0. (2.2.23)

With the redefinition ∂µ → ∂µ− iPµ we get the scaling for the terms included in the quarkand gluon Lagrangian:

inµDµ = inµ∂

µ + nµPµ

︸ ︷︷ ︸

0

+gs nµAµ

︸ ︷︷ ︸

A+n,q+nAus

∝ O(λ2),

inµDµ = inµ∂

µ + nµPµ

︸ ︷︷ ︸

P

+gs (nAn,q + nAus)︸ ︷︷ ︸

A−n,q+nAus

∝ P + gsA−n,q

︸ ︷︷ ︸

inµDµc

+O(λ),

i /D⊥ = i/∂⊥ + /P⊥ + gs

(/Aus,⊥ + /An,q,⊥

)∝ /P⊥ + /An,q,⊥︸ ︷︷ ︸

i /Dc⊥

+O(λ2). (2.2.24)

For the quark Lagrangian no other terms have to be taken into account. But for the gluonLagrangian, to see which terms can be neglected at leading order, one has to evaluate indetail the trace over the square of the commutator, which leads to a large number of terms.We will therefore just give the final result from power counting for the SCET Lagrangianin the next section.

2.2.5 Collinear Leading-order Lagrangian

The leading order Lagrange density L has to be of order λ4 because S =∫d4xL is of order

1 and d4x ∝ O(λ−4) as x+ ∝ λ−2, x− ∝ λ0 and x⊥ ∝ λ−1. We see that this is satisfied forthe already obtained leading order quark Lagrangian as the quark fields are of order λ: [25]

Lquark,0 = ξn,p′︸︷︷︸

O(λ)

in ·D︸ ︷︷ ︸

O(λ2)

+ i /Dc⊥

︸︷︷︸

O(λ)

1

in ·Dc︸ ︷︷ ︸

O( 11)

i /Dc⊥

︸︷︷︸

O(λ)

/n

2ξn,p︸︷︷︸

O(λ)

(2.2.25)

with

in ·Dc = P + gsA−n,q,

i /Dc⊥ = /P⊥ + /An,q,⊥. (2.2.26)

After power counting, the leading order gluon Lagrangian is given by

Lgluon,0 =1

2g2s

Tr([iDµ + gsA

µn,q, iDν + gsA

νn,q′]2)

(2.2.27)

with

iDµ ≡ nµ

2P + Pµ

⊥ +nµ

2in ·Dus (2.2.28)

24

2.2 Lightcone Coordinates

where Dus is the covariant derivative including only an ultrasoft gluon field. To derive theFeynman rules from the Lagrangian we need a gauge-fixing term. Along with this comesa ghost term, which cancels the longitudinal degrees of freedom of the gluon fields. Afterpower counting the leading order terms are given by [25]

Lgauge−fixing =1

ξTr([iDµ, A

µn,q

]2)

, (2.2.29)

Lghost = 2Tr(cn,p

[iDµ,

[iDµ + gsA

µn,q, cn,p

]])(2.2.30)

with ξ being the gauge fixing parameter and cn,p is the collinear ghost field. Therefore thefinal leading order collinear SCET Lagrangian is [25]

Lcollinear,(0)SCET = ξn,p′

(

in ·D + i /Dc⊥

1

in ·Dci /D

c⊥

)/n

2ξn,p +

1

2g2s

Tr([iDµ + gsA

µn,q, iDν + gsA

νn,q′]2)

+1

ξTr([iDµ, A

µn,q

]2)

+ 2Tr(cn,p

[iDµ,

[iDµ + gsA

µn,q, cn,p

]]). (2.2.31)

This is true for the regime where SCETI can be used. For the SCETII Lagrangian theultrasoft gluon field appearing in the derivatives (Eq.(2.2.28)) has to be replaced with thesoft gluon field.

2.2.6 Feynman Rules

The massless quark propagator can be obtained from the QCD result (see chapter 1.3.1) byrewriting the momenta in lightcone coordinates:

Dξ(q) =i/q

q2=i/n

2

n · q(n · q)(n · k) + q2⊥

, (2.2.32)

where we have rewritten the denominator with nininjnj = 2δij ninj. The gluon propagatoris gained in the same way from the QCD gluon propagator. In Feynman gauge it is

DA(q) =−igµν

q2δab =

−igµν

(n · q)(n · k) + q2⊥δab. (2.2.33)

From the leading order fermionic Lagrangian we can derive the Feynman rules for the (ultra-)soft and the collinear gluon vertex by looking at the different interaction terms. From thefirst term of Eq.(2.2.25)

ξn,p′ (in ·D)/n

2ξn,p = ξn,p′

(in · ∂ + gsn · A(u)s + gsn · An,q

) /n

2ξn,p (2.2.34)

we can read off the vertex for the coupling of an (ultra-)soft gluon to a collinear quark bylooking at the term in the middle, throwing away the gluon and quark field:

Vξ,A(u)s= igsn

µ /n

2ta. (2.2.35)

For the vertex of the coupling of a collinear gluon to a collinear quark we need the secondterm of Eq.(2.2.25) and rewrite it

i /Dc⊥

1

in ·Dci /D

c⊥ = i /D

c⊥

1

P1

1 + gsA−n,q/P

i /Dc⊥. (2.2.36)

25

2 Soft-Collinear Effective Theory

⇒

Figure 2.2: Integrating out the off-shell propagators leads to the introduction of a Wilson line. Ifthe gluons are collinear as in this figure, they are summed up into a collinear Wilson line.

Expansion ingsA−

n,q

P leads to

i /Dc⊥

1

in ·Dci /D

c⊥ = i /D

c⊥

1

P

(

1−gsA

−n,q

P

)

i /Dc⊥. (2.2.37)

Therefore the second term reads

ξn,p′

(

i /Dc⊥

1

in ·Dci /D

c⊥

)/n

2ξn,p

= ξn,p′(/P⊥ + gs /An,q,⊥

)(

1

P − gsn · An,q

P2

)(/P⊥ + gs /An,q,⊥

) /n

2ξn,p (2.2.38)

with Dc given in Eq.(2.2.26). By writing out every term explicitly one can identify threeterms being responsible for the leading order collinear quark-collinear gluon vertex as theyinclude only one collinear gluon field and are of order gs. To get the right vertex factorone needs to include the last term of Eq.(2.2.34) as it also corresponds to the leading ordercollinear quark-collinear gluon vertex. Again it can be read off:

Vξ,An,q = igs

(

nµ +/p′⊥γ

µ⊥

n · p +γµ⊥/p⊥n · p′ −

/p′⊥/p⊥nµ

(n · p′)(n · p)

)

/n

2ta. (2.2.39)

2.3 Wilson Lines in SCET

In this section we give a description of the different Wilson lines and explain why they areneeded in SCET. We will shortly present the collinear Wilson line and give the calculation forfinding the (ultra-)soft Wilson line. In general Wilson lines are a mathematical constructto build gauge invariant operators like the current. Consider the radiation of (ultra-)softgluons in scattering processes. Theoretically an infinite large number of (ultra-)soft gluonscan be emitted and so the radiation corrections would be too large to use perturbationtheory for calculating the cross section. Handling all this (ultra-)soft radiation requiresthe absorption into an expression, which is called an (ultra-)soft Wilson line. To be more

26

2.3 Wilson Lines in SCET

⇒

Figure 2.3: Combination of collinear gluons into a collinear Wilson line Wn.

explicit, when attaching gluons to a collinear quark the quark can go off-shell. These off-shell propagators need to be integrated out as they lead to divergences. This is done byshrinking them into one point (Fig.2.2), which means summing up the gluons into a Wilsonline (Fig.2.3). Generally it is given by [27]

Φy[z1, z2;A] = Pexp

[

igs

∫ z2

z1

dzµAµ(z)

]

, (2.3.1)

where Φy[z1, z2;A] denotes the phase factor of the gluon field and P is the path ordering.To motivate the collinear Wilson line, we notice that the fermionic part of the Lagrangian(Eq.(2.2.25)) is not gauge invariant under collinear gauge transformations. Collinear gaugetransformations Uc are a subset of the QCD gauge transformations which obey ∂µUc(x) ∝(λ2, 1, λ). A possible choice in analogy to the QCD gauge transformations is [28]

Aµn,q → UcA

µn,qU

†c −

i

gsUc∂

µU †c , A

µ(u)s → UcA

µ(u)sU

†c . (2.3.2)

In order to make the Lagangian invariant under this gauge transformation, we have tointroduce an Operator, which is the collinear Wilson line:

Wn =∑

perm

exp[

− gs

n · P n · An,qTa]

. (2.3.3)

By Fourier transformation in position space it can be written as

Wn(x) = P exp

[

igs

∫ x

−∞dsn ·Aa

n,q(n · s)T a

]

. (2.3.4)

The path ordering P denotes the ordering of the gluon fields, where the fields closer to thepoint x are moved to the left. For the Wilson line the following identities hold [25]:

WW † = 1 = W †W, (2.3.5)

f(P + gsn · An,q) = Wf(P)W †, (2.3.6)

27

2 Soft-Collinear Effective Theory

p

k1 k2

ν µ

Figure 2.4: Attachment of two (ultra-)soft gluons to a collinear quark.

with f being an arbitrary function. This and Eq.(2.2.38) can now be used to write theleading order collinear Lagrangian in a collinear gauge invariant way. In SCETI this is

Lquarkc = ξn,p′

(

in ·Dus + gn · An,q + (/P⊥ + gs /A⊥n,q)Wn

1

PW†n(/P⊥ + gs /A

⊥n,q′)

)/n

2ξn,p.

(2.3.7)

If we also want to rewrite the gluonic part of the Lagrangian we see, that the Wilson linesall cancel each other. This is due to Eq.(2.3.5) and the cyclicity of the trace. This formof the Lagrangian still includes coupled collinear and (ultra-)soft fields. To see in SCETI

that the ultrasoft gluon fields decouple from the collinear quark fields, we need to redefinethe collinear fields by splitting them into a bare field without any soft gluon contribution,denoted with the superscript 0, and an ultrasoft Wilson line Yn(x) [25, 29]:

ξn,p(x)→ Yn(x)ξ0n,p(x) , An,q(x)→ Yn(x)A0n,q(x)Y

†n (x),

W (x) → Yn(x)W 0(x)Y †n (x). (2.3.8)

Before inserting this into the Lagrangian, we will give an explanation of what the (ultra-)softWilson lines are and how they are computed [25].

Consider two (ultra-)soft gluons (momenta kµ1 and kµ

2 ), represented by the fields A(u)s,k1

and A(u)s,k2, coupling to a collinear quark (momentum pµ = mvµ + pµ) represented by the

field qc, where v is the velocity and p is the residual momentum of the particle (Fig.2.4).To explicitly show that the quark goes off the mass shell, we introduce a quark mass mwhich will drop out in the end of the calculation automatically. After the interaction withthe first gluon, the quark has the momentum

pµ + kµ1 = mvµ + pµ +

nµ

2n · k1 +

nµ

2n · k1 + kµ

1,⊥ ∝ mvµ +nµ

2n · k1 (2.3.9)

and has gone off-shell due to the radiative correction. After the attachment of the secondsoft gluon, the momentum changes to

pµ + kµ1 + kµ

2 ∝ mvµ +nµ

2n · (k1 + k2) , (2.3.10)

which is again an on-shell quark as it corresponds to an outgoing state. Let us first look atthe amplitude of this process. It is given by

M = qci(/p + /k1 + /k2 +m)

(p + k1 + k2)2 −m2igsT

bγµ i(/p+ /k1 +m)

(p + k1)2 −m2igsT

aγνAaµ,(u)s,k1

Abν,(u)s,k2

qc.

(2.3.11)

28

2.3 Wilson Lines in SCET

After some replacements:

γµAaµ,(u)s,k1

= /Aa(u)s,k1

∝ /n

2n ·Aa

(u)s,k1,

/p+ /k1 + /k2 +m = m/v +/n

2n · (k1 + k2) +m,

/p+ /k1 +m = m/v +/n

2n · k1 +m,

(p+ k1 + k2)2 = m2 v2

︸︷︷︸

1

+m(n · v)n · (k1 + k2) +nnnn

4(k1 + k2)

2

︸ ︷︷ ︸

→0

,

(p + k1)2 = m2 +m(n · v)(n · k1), (2.3.12)

the amplitude becomes:

M = qcg2s

m(v + 1) + /n2n · (k1 + k2)

m(n · v)n · (k1 + k2)

/n

2n ·Aa

(u)s,k1

×m(v + 1) + /n2n · k1

m(n · v)(n · k1)

/n

2n · Ab

(u)s,k2T bT aqc. (2.3.13)

With some reordering and the further simplifications (note that /vqc = qc):

/n/n = 0,

m(/v + 1)/n

2mn · v qc =1

2n · v (v /n+ n)qc =1

2n · v (2gµν − γνγµ)vµnν

=1

2n · v (2v · n− /n/v + /n)qc (2.3.14)

= qc

the final result for the amplitude is:

M = qcg2s

n · Aas,k1

n ·Abs,k2

n · (k1 + k2)n · k1T bT aqc. (2.3.15)

Of course we also have to take the crossed Feynman diagram into account, where the gluonsare cross-attached to the quark. To find the amplitude we have to interchange k1 and k2.By relabeling a↔ b we get

Mcrossed = qcg2s

n ·Aa(u)s,k1

n ·Ab(u)s,k2

n · (k1 + k2)n · k2

(

T bT a + fabcT c)

qc. (2.3.16)

To read off the (ultra-)soft Wilson line for infinitely many attached gluons we have togeneralise the expression for the amplitude by realising that a series is involved:

M =

∞∑

n=0

∑

perm

qc(−gs)n

n ·Aa1

(u)s,k1· · · n · Aan

(u)s,kn

(n · k1)(n · (k1 + k2)) · · · (n ·∑n

i=1 ki)T an · · · T a1qc

= qc∑

perm

exp

[−gsn ·Aa

(u)s,kTa

n · k

]

qc = qc∑

perm

exp

[−gsn · Aa

(u)s,kTa

n · P

]

qc,

(2.3.17)

29

2 Soft-Collinear Effective Theory

where the permutations over ai include the crossed graphs. We have now integrated out theoff-shell quark propagators. For the ultrasoft gluons, as they arise in SCETI , the collinearquark field qc is simply ξn,p and the ultrasoft Wilson line can be read off:

Yn = exp

[−gsn ·Aus,kTa

n · P

]

,

Yn(x) = P exp

[

igs

∫ x

−∞dsn ·Aa

us(n · s)T a

]

. (2.3.18)

Because SCETII is the EFT of SCETI , the ultrasoft and collinear fields need to be decou-pled before reading off the soft Wilson line. Fortunately the redefinition (Eq.(2.3.8)) onlyaffects the collinear fields, which are outside the exponential in Eq.(2.3.17), so that the softWilson line is the same as the ultrasoft one with the replacement Aus,k → As,k:

Sn = exp

[−gsn ·As,kTa

n · P

]

,

Sn(x) = P exp

[

igs

∫ x

−∞dsn · Aa

s(n · s)T a

]

. (2.3.19)

It is seen, that these Wilson lines describe incoming gluons coupled to an incoming quark(from −∞ to x) or an outgoing antiquark in opposite direction. If we want to describeoutgoing gluons attached to an incoming quark (from −∞ to x) or an outgoing antiquarkin opposite direction, the Wilson lines are given by

Zn =∑

perm

exp

[

(−gsn ·A(u)s)1

−n · P†

]

,

Zn(x) = P exp

[

igs

∫ x

−∞dsn ·A(u)s(n · s)

]

, (2.3.20)

where Zn may be replaced by Yn or Sn, depending on which gluon energy we look at. TheWilson line for an outgoing quark (from x to ∞) emitting (ultra-)soft gluons is written as

Z†n =

∑

perm

exp

[

(−gn ·A(u)s)1

n · P†

]

,

Z†n(x) = P exp

[

ig

∫ ∞

xdsnA(u)s(n · s)

]

. (2.3.21)

This is done in a similar way for all other combinations. But one has to take care that forthe cases of an outgoing antiquark (from x to ∞) and an incoming antiquark (from −∞ tox) we need an anti path ordering. By writing down all the possible ways of the quark onecan see, which soft Wilson line is needed for the redefinition of the collinear fields. This canbe seen in detail in [30].

30

2.4 Decoupled SCET Lagrangian

2.4 Decoupled SCET Lagrangian

Inserting the redefinitions from Eq.(2.3.8) into the quark Lagrangian leads to the decoupledSCETI Lagrangian:

Lquarkc = ξ0n,p′Y

†n

(

in ·Dus + gYnn ·A0n,qY

†n + (/P⊥ + gsYn /A

⊥,0n,q Y

†n )

× YnW0nY

†n

1

P YnW†,0n Y †

n (/P⊥ + gsYn /A⊥,0n,q′Y

†n )

)

Y †n

/n

2Ynξ

0n,p. (2.4.1)

Because /P⊥ commutes with the Wilson line and the relation [25]

Y †n in ·DusYn = in · ∂ (2.4.2)

holds, the quark Lagrangian is given as

Lquarkc = ξ0n,p′

(

in · ∂ + gn ·A0n,q + (/P⊥ + gs /A

⊥,0n,q )W 0

n

1

PW†,0n (/P⊥ + gs /A

⊥,0n,q′)

)/n

2ξ0n,p.

(2.4.3)

Doing this for the gluonic, gauge-fixing and ghost term of the SCET-Lagrangian leads tothe cancellation of all ultrasoft Wilson lines due to the cyclicity of the trace [25]. Thereforethese parts of the Lagrangian look the same as in chapter 2.2.5 with the replacement of thecollinear gluon and the ghost fields by the bare collinear gluon and bare ghost fields. Toachieve the decoupling of soft and collinear fields in SCETII one has to match the SCETI

result onto SCETII by integrating out the ultrasoft degrees of freedom. See therefore forexample [31].

31

2 Soft-Collinear Effective Theory

32

3 Jet and Soft Functions

When describing a collision experiment one combines different generated particles into oneobject, called a jet. The shape of the jet, as reconstructed from jet algorithms, gives insightto the substructure of a jet which leads to information about QCD and its dynamics.The three most important jet shapes are jet broadening, thrust and angularity. We willconcentrate on the calculation of the jet and soft function for a scattering process withrespect to the angularity. To make the calculation of the cross section feasible, factorisationis applied. By factorisation we mean the splitting of the cross section in functions valid indifferent energy regimes. Each part can be calculated independently so that one does nothave to consider the physics at different energies. Note that the guideline and the givenresults for the calculations come from [32].

3.1 Jet Angularity and Jet Algorithms

The angularity of a jet is given by [32]

τa =1

2EJ

∑

i∈J

∣∣pi

T

∣∣ e−ηi(1−a), (3.1.1)

where EJ is the jet’s energy, i represents a particle in the jet, pT is the transverse momen-tum of the particle relative to the jet axis, η is the pseudorapidity and a is an arbitraryparameter. To assure infrared-safety of the angularity, a has to be between −∞ and 2. Thepseudorapidity η is given by η = − ln (tan(Θ/2)), where Θ is the angle between a particleand the jet direction. Setting a to 1 corresponds to jet broadening [33]. The angularityvanishes for a big exponent, that is large η. The pseudorapidity takes large values for asmall angle Θ, that is collinear particles. The jet axis is obtained from the used jet algo-rithm which combines different particles into a jet. There are many of these algorithmsavailable, each characterised by certain properties. We will explain the two most importantjet algorithms below.

Cone Algorithm There are many different types of cone algorithms, but they are all basedon one principle. First a cone in the plane of rapidity and azimuth with radius R (R has tobe chosen appropriately) is formed. Then all particles within this radius are grouped into ajet. In the fixed cone algorithm the hardest particle of all particles, that is the particle withthe highest momentum, is taken as the jet axis. In the iterative cone algorithm the first stepis to align the hardest particle along the cone axis, which is then redefined after ”drawing”the cone by the collective direction of the three-momenta of the particles inside the jet. Af-ter a few iterations of drawing a cone and redefining the jet axis the cone should be stableand is then called a jet. While finding the stable cones, there will be an overlap of some ofthese cones which lead to the general problem that the jets would not be well separated. To

33

3 Jet and Soft Functions

X X

ql

Figure 3.1: The momentum labeling of the Feynman diagrams used for the calculations.

solve this one can for example use a split-merge method, which we will not discuss here. [34]

Sequential Recombination Algorithm The underlying principle of a sequential recombi-nation algorithm is in contrast to the cone algorithm to first look at the created particlesand their distance to each other before determining the jet axis. One type of sequentialrecombination jet algorithm is the kt algorithm which searches for the smallest distancebetween all particles and recombines the two particles, which are nearest to one anotherto a new particle (in detail, for a recombination the distance has to be smaller than a setparameter ycut). The new particle is then given by adding the four-momenta of the oldparticles. If there is no other particle in the neighbourhood it is called a jet. This procedureis done until no single particle is left. [34]

It is important to note, that these two jet algorithms are infrared and collinear safeat least to NLO. If one adds a soft parton to an observable and it remains the same, theobservable is called infrared safe. If the replacement of a massless parton with two collinearmassless partons has no effect on the observable, it is called collinear safe. When lookingat these jet algorithms, we see that all the collinear particles are recombined into a jet andthe soft particles do not affect the procedure of jet reconstruction because of their smallfour-momenta.

3.2 Jet Function

When performing the calculation of the jet function one has to make use of the zero-binsubtraction to avoid double counting of the soft modes: Because the jet function can beextended to any energy scale of the loop momenta, the soft regime would be consideredtwice, once in the jet and then again in the soft function. Hence we have to subtract the”soft jet function” by zero-bin subtraction. Therefore one has to set the loop momentumproportional to λ2 in the jet function, which is taking the soft limit, and subtract it fromthe naive contribution. [35]

Before we turn to the actual calculation, we have to clarify the labeling of the differentmomenta (see Fig.3.1). We take the momentum going in and out of the graph as l and splitit up in a label momentum l− = ω, which is directed in the n−direction and a residual partl+, which is directed in the n−direction. The loop momentum is q.

In order to keep the calculation as easy as possible, we impose certain cuts throughthe Feynman diagrams (see Fig.3.2), which lead to phase space restricitons. There are twopossibilities where to cut such a diagram. On the one hand we cut a propagator. This leadsto a virtual and a tree level diagram with one particle in the final state. These integrals donot contribute to the overall amplitude, because the integrals are scaleless and therefore can

34

3.2 Jet Function

Figure 3.2: Cut through a propagator (left) and a loop (right).

Restriction for any part of the jet function

Cone algorithm Θcone = Θ(tan2(R/2)− q+/q−)Θ(tan2(R/2)− l+−q+

ω−q−)

kT− algorithm ΘkT= Θ(tan2(R/2) − q+ω2

q−(ω−q−)2)

Zero-bin subtraction Θ0alg = Θ0

cone = Θ0kT

= Θ(tan2(R/2)− q+/q−)

Additional restriction for the measured jet function

Restriction δR = δ(τa − 1/ω(ω − q−)a/2(l+ − q+)1−a/2 − 1/ω(q−)a/2(q+)1−a/2)

Zero-bin subtraction δ0R = δ(τa − 1/ω(q−)a/2(q+)1−a/2)

Table 3.1: Restrictions for both particles being inside the jet.

be set to zero when working in dimensional regularisation. On the other hand we considera cut through a loop. The diagram is then divided into two diagrams with two particles inthe final state, which correspond to real emission diagrams. These can contribute in threedifferent ways: It is possible that both particles are inside the jet or that one of the particlesexits the jet with an energy E smaller than the cutoff Λ∗ (in this last case the remainingparticle is inside the jet with an energy E > Λ). These restrictions can be written for thedifferent algorithms in a mathematical way (see Table 3.1). For the measured jet functionthere is in contrast to the unmeasured jet function, which is independent of the observable,an additional restriction as they may contribute to the jet’s angularity. The amplitude isthen just the sum of all the cut diagrams.

3.2.1 Quark Jet Function

In this section we will calculate in detail the part of the quark jet function where a gluonis outside a measured jet and has an energy smaller than the cutoff Λ to NLO and give theresults for the measured and unmeasured quark jet function. We present the results for thecone algorithm and accomplish the calculations with these restrictions.

Gluon Outside a Measured Quark Jet

Naive Contribution

The naive contribution of a measured jet function with a gluon being outside the jet in

∗Note that λ =q

Λω

for SCETI where ω is the label momentum and therefore a large energy scale.

35

3 Jet and Soft Functions

dimensional regularisation for the cone algorithm is given by

Jq,outω (τa) = g2

sµ2ǫCF

∫dl+

2π

1

(l+)2

∫d4−2ǫq

(2π)4−2ǫ

(

4l+

q−+ (2− 2ǫ)

l+ − q+ω − q−

)

×2πδ(

q−q+ − (q⊥)2)

Θ(q−)Θ(q+)2πδ

(

l+ − q+ − (q⊥)2

ω − q−)

×Θ(ω − q−)Θ(l+ − q+)Θ(2Λ − q−)Θ

(q+

q−− tan2 (R/2)

)

δ(τa)

= g2sµ

2ǫCF

∫dl+

2π

1

(l+)2

∫

dq+dq−dq⊥dΩd′

(

q⊥)1−2ǫ 1

(2π)4−2ǫ

×(

4l+

q−+ (2− 2ǫ)

l+ − q+ω − q−

)

2πδ(

q−q+ − (q⊥)2)

Θ(q−)Θ(q+)

×2πδ

(

l+ − q+ − (q⊥)2

ω − q−)

Θ(ω − q−)Θ(l+ − q+)

×Θ(2Λ− q−)Θ

(q+

q−− tan2 (R/2)

)

δ(τa). (3.2.1)

The last Θ− and δ−functions come out of the phase space restrictions (Table(3.1)). Note,that the argument of the Θ−function is turned around, as it otherwise would restrict thegluon to be inside the jet. The second Θ−function of Θcone in Table(3.1) does not haveto appear in the calculation, as the cone axis points directly in the direction of the quarkif the gluon is outside the jet. The other functions come out of the Feynman diagrams.Furthermore the angular integral is in lightcone coordinates written as an integration ind′ = d − 2 dimensions as one angle is already fixed and d = 4 − 2ǫ. Therefore the angularintegral is given as (Eq.(1.4.4))

∫

dΩd′ =2πd′/2

Γ (d′/2)=

2π1−ǫ

Γ(1− ǫ) . (3.2.2)

Throughout the calculation we make use of the relation for evaluating the integral over adelta function:

δ (f(x)) =∑

i

δ(x− xi)

|f ′(xi)|, (3.2.3)

where the xi denote the roots of f(x) and the sum runs over all roots. Note that because allthe momenta are positive as we work in lightcone coordinates, we do not need to considernegative roots. First we calculate the integral over the transverse component q⊥

∫

dq⊥(

q⊥)1−2ǫ

δ

q

−q+ − (q⊥)2︸ ︷︷ ︸

f(q⊥)

δ

(

l+ − q+ − (q⊥)2

ω − q−)

. (3.2.4)

The only root is q⊥0 =√

q−q+ such that f ′(q⊥0 ) = −2√

q−q+ and the q⊥−integral becomes

1

2(q−q+)−ǫδ

(

l+ − q+ − q−q+

ω − q−)

. (3.2.5)

36

3.2 Jet Function

Calculation of the q+−integral

∫

dq+(q−q+)−ǫ

(

4l+

q−+ (2− 2ǫ)

l+ − q+ω − q−

)

δ

l+ − q+

(

1 +q−

ω − q−)

︸ ︷︷ ︸

f(q+)

Θ(q+)

×Θ(l+ − q+)Θ

(q+

q−− tan2(R/2)

)

(3.2.6)

with q+0 = l+

ω (ω − q−) and f ′(q+0 ) = −(

1 + q−

ω−q−

)

leads to

(q−l+)−ǫ

(ω − q−ω

)1−ǫ(

4l+

q−+ (2− 2ǫ)

l+q−

ω(ω − q−)

)

Θ(l+)Θ

(l+(ω − q−)

q−ω− tan2 (R/2)

)

.

(3.2.7)

Therefore the jet function remains to be

Jq,outω (τa) = g2

sµ2ǫCF

1

Γ(1− ǫ)1

22−2ǫ

1

π2−ǫ

∫

dl+1

(l+)2

∫

dq−(q−l+)−ǫ

×(ω − q−ω

)1−ǫ(

2l+

q−+ (1− ǫ) l+q−

ω(ω − q−)

)

Θ(l+)Θ(q−)

×Θ

(l+(ω − q−)

q−ω− tan2 (R/2)

)

Θ(ω − q−)Θ(2Λ− q−)δ(τa). (3.2.8)

The first Θ−function in the third line of Eq.(3.2.8) gives a lower bound for the l+−integration,which is always larger than zero. Therefore we are left with

∫∞ωq− tan2 (R/2)

ω−q−

dl+1

(l+)2(q−l+)−ǫ

(ω − q−ω

)1−ǫ(

2l+

q−+ (1− ǫ) l+q−

ω(ω − q−)

)

=1

ǫ

(1− ǫ)(q−)2 − 2ω(q− − ω)

(q−)2ǫ+1ω2 tan2ǫ (R/2). (3.2.9)

As a last step we need to compute the q−−integral. Note that the restriction for q− beingsmaller than 2Λ is stronger than q− being smaller than ω. Therefore the integration reads

Jq,outω (τa) = g2

sµ2ǫCF

1

Γ(1− ǫ)1

22−2ǫ

1

π2−ǫ

∫

dq−Θ(q−)Θ(2Λ− q−)δ(τa)

×(

1

ǫ

(1− ǫ)(q−)2 − 2ω(q− − ω)

(q−)2ǫ+1ω2 tan2ǫ (R/2)

)

= −g2sµ

2ǫCF1

Γ(1− ǫ)1

22−2ǫπ2−ǫδ(τa)

1

(2Λ tan(R/2))2ǫ

×

1

ǫ2− 2Λ2

ω2ǫ− 4Λ

2ωǫ2 − ωǫ︸ ︷︷ ︸

4Λǫω

12ǫ−1

. (3.2.10)

37

3 Jet and Soft Functions

With Taylor expansion around ǫ = 0 in the last term of Eq.(3.2.10)

1

2ǫ− 1≈ −1− 2ǫ (3.2.11)

a finite term can be extracted and the naive contribution of the jet function for a gluonbeing emitted outside the jet is finally

Jq,outω (τa) = −g2

sµ2ǫCF

1

Γ(1− ǫ)1

22−2ǫπ2−ǫ

1

(2Λ tan(R/2))2ǫ

(1

ǫ2− 1

ǫ

(2Λ2

ω2− 4Λ

ω

)

+8Λ

ω

)

δ(τa).

(3.2.12)

Zero-bin Contribution

The zero-bin contribution in dimensional regularisation of the jet function with the gluonbeing outside the jet is given by taking the soft limit of Eq.(3.2.1)

Jq,out,(0)ω (τa) = g2

sµ2ǫCF

∫dl+

2π

1

(l+)2

∫d4−2ǫq

(2π)4−2ǫ

(

4l+

q−+ (2− 2ǫ)

l+ − q+ω − q−

)

×2πδ(

q−q+ − (q⊥)2)

Θ(q−)Θ(q+)

×2πδ(l+ − q+)Θ

(q+

q−− tan2 (R/2)

)

Θ(2Λ− q−)δ(τa)

= g2sµ

2ǫCF

∫dl+

2π

1

(l+)2

∫

dq+dq−dq⊥dΩ2−2ǫ

(

q⊥)1−2ǫ 1

(2π)4−2ǫ

×(

4l+

q−+ (2− 2ǫ)

l+ − q+ω − q−

)

2πδ(

q−q+ − (q⊥)2)

Θ(q−)Θ(q+)2πδ(l+ − q+)

×Θ

(q+

q−− tan2 (R/2)

)

Θ(2Λ− q−)δ(τa).

(3.2.13)

The angular integral is evaluated in the same way as in Eq.(3.2.2). The q⊥−integral gives

∫

dq⊥(

q⊥)1−2ǫ

δ(q−q+ − q⊥2

︸ ︷︷ ︸

f(q⊥)

) =

∫

dq⊥(

q⊥)1−2ǫ δ(q⊥ − q⊥0 )

|f ′(q⊥0 )|

=

∫

dq⊥(

q⊥)1−2ǫ δ(q⊥ −

√

q−q+)

2√

q−q+

=1

2

(q+q−

)−ǫ, (3.2.14)

and the zero-bin jet function becomes

Jq,out,(0)ω (τa) = g2

sµ2ǫCF

1

22−2ǫπ2−ǫ

1

Γ(1− ǫ)

∫

dl+1

(l+)2

∫

dq+dq−(q−q+)−ǫ

×(

2l+

q−+ (1− ǫ) l

+ − q+ω − q−

)

δ(l+ − q+)Θ(q−)Θ(q+)

×Θ

(q+

q− − tan2 (R/2)

)

Θ(2Λ− q−)δ(τa). (3.2.15)

38

3.2 Jet Function

The q+−integral

∫dq+

1

(q−q+)ǫ

(

2l+

q−+ (1− ǫ) l

+ − q+ω − q−

)

Θ(q+)δ(l+ − q+)Θ

(q+

q− − tan2 (R/2)

)

= 2(l+)1−ǫ

(q−)1+ǫΘ(l+)Θ

(l+

q− − tan2 (R/2)

)

(3.2.16)

leads to a lower bound for the l+−integral:∫ ∞

q− tan2(R/2)dl+

2

(l+)2(l+)1−ǫ

(q−)1+ǫ= 2(q−)−1−ǫ

(q− tan2 (R/2)

)−ǫ

ǫ. (3.2.17)

This result holds for q− tan2R/2 and ǫ being larger than zero, which is satisfied. Thereforewe have only the q−−integral left to do:

Jq,out,(0)ω (τa) = g2

sµ2ǫCF

1

22−2ǫπ2−ǫ

1

Γ(1− ǫ)

∫

dq−2

ǫ

Θ (q−)Θ (2Λ− q−)

(q−)1+ǫ (q− tan2 (R/2))ǫ . (3.2.18)

This can be evaluated as∫ 2Λ

0dq−(q−)−1−2ǫ =

[1

−2ǫ(q−)−2ǫ

]2Λ

c

=1

2ǫ

(c−2ǫ − (2Λ)−2ǫ

), (3.2.19)

where c denotes a constant term, which we will neglect. The final result for the zero-bincontribution therefore becomes

Jq,out,(0)ω (τa) = −g2

sµ2ǫCF

1

22−2ǫπ2−ǫ

1

Γ(1− ǫ)1

(2Λ tan (R/2))2ǫ

1

ǫ2δ(τa). (3.2.20)

To get the total gluon outside measured quark jet function, we have to add the leadingorder term, which is δ(τa), to the next-to-leading-order naive contribution and subtract thezero-bin term:

Jq,out,(0)ω (τa) = δ(τa) + Jq,out

ω (τa)− Jq,out,(0)ω (τa)

= δ(τa)

(

1− g2sµ

2ǫCF1

22−2ǫπ2−ǫ

1

Γ(1− ǫ)1

(2Λ tan (R/2))2ǫ

×(

−1

ǫ

(2Λ2

ω2− 4Λ

ω

)

+8Λ

ω

))

. (3.2.21)

We see that this contribution to the jet function is power suppressed by O(λ2) as the1/ǫ2−term drops out and Λ

ω scales as λ2. The same holds for every other function with oneparticle being outside a jet.

Measured Quark Jet

The measured quark jet function for both particles being inside the jet is given by

Jqω(τa) = g2

sµ2ǫCF

∫dl+

(l+)21

2π

∫

ddq1

(2π)d

(

4l+

q−+ (d− 2)

l+ − q+ω − q−

)

2πδ

(

q−q+ −(

q⊥)2)

×Θ(q−)Θ(q+)2πδ

(

l+ − q+ −(q⊥)2

ω − q−

)

Θ(ω − q−

)Θ(l+ − q+

)

×Θ(tan2(R/2)− q+/q−

)δR. (3.2.22)

39

3 Jet and Soft Functions

The zero-bin contribution reads

⇒ Jq,0ω (τa) = 4g2

sµ2ǫCF

∫dl+

2π

1

l+

∫ddq

(2π)d1

q−2πδ(q−q+ − q⊥2)Θ(q−)Θ(q+)

×2πδ(l+ − q+)Θ(l+ − q+)Θ(tan2(R/2)− q+/q−)

×δ(

τa −1

ω(q−)a/2(q+)1−a/2

)

= 4g2sµ

2ǫCF1

Γ(1− ǫ)2π1−ǫ

(2π)4−2ǫ

∫

dl+1

l+

∫

dq+dq−dq⊥1

q−

(

q⊥)1−2ǫ

×δ(

q−q+ − q⊥2)

Θ(q−)Θ(q+)δ(l+ − q+)Θ(l+ − q+)

×Θ(tan2(R/2)− q+/q−

)δ

(

τa −1

ω(q−)a/2(q+)1−a/2

)

, (3.2.23)

where the q⊥ integral is the same as in Eq.(3.2.14) and gives a factor 12 (q+q−)

−ǫ. As a next

step we evaluate the integral over the momentum in ”− ”direction:

∫

dq−(

1

q−

)1+ǫ

δ

τa −

1

ω(q−)a/2(q+)1−a/2

︸ ︷︷ ︸

f(q−)

Θ(q−)Θ

(

tan2(R/2) >q+

q−

)

.

(3.2.24)

The Θ−function restricting q− to be positive is fulfilled automatically by the second Θ−function,because q+ and tan2(R/2) are always positive. Therefore the second Θ−function gives usthe lower bound for the integral:

∫ ∞

q+

tan2(R/2)

dq−(

1

q−

)1+ǫ δ(q− − q−0 )

|f ′(q−0 )| =

(1

q−0

)1+ǫ 1

|f ′(q−0 )|Θ(

q−0 −q+

tan2(R/2)

)

=2ω

a(τaω)−1−2/aǫ (q+

)(2/a−1)ǫΘ

(

(τaω)2/a (q+)1−2/a − q+

tan2(R/2)

)

, (3.2.25)

with

q−0 = (τaω)2/a(q+)1−2/a,

f ′(q−0 ) = − a

2ω(τaω)1−2/a (q+

)2/a−1. (3.2.26)

By performing these calculations the zero-bin jet function becomes:

Jq,0ω (τa) = 4g2µ2ǫCF

1

Γ(1− ǫ)π2−ǫ

(2π)4−2ǫ

2ω

a(τaω)−1−2/aǫ

∫ ∞

0dl+

1

l+

∫ ∞

0dq+

(q+)(2/a−2)ǫ

δ(l+ − q+

)

×Θ(l+ − q+

)Θ(q+)Θ

(

(τaω)2/a(q+)1−2/a − q+

tan2(R/2)

)

. (3.2.27)

40

3.2 Jet Function

The last Θ−function gives the upper bound of the q+−integral:

Jq,0ω (τa) =

4ǫ

π2−ǫg2µ2ǫCF

1

Γ(1− ǫ)

∫ ∞

0dl+

1

l+

∫ tana(R/2)τaω

0dq+

2

aτaδ(l+ − q+

)

×Θ(l+ − q+

)Θ(q+). (3.2.28)

The integration of the l+−integral gives 1q+ as the function δ (l+ − q+) is involved. Therefore

the remaining q+−integral gives

∫ tana(R/2)τaω

0dq+

(q+)2ǫ(1/a−1)−1

Θ(q+)

=1

2ǫ(1/a− 1)(τaω)2ǫ(1/a−1) tan2ǫ(1−a)(R/2)Θ (τaω tana(R/2))

(q+)2ǫ(1/a−1)

(3.2.29)

and the zero-bin contribution of the measured jet function with both particles being insidethe jet is

Jq,0ω (τa) = g2µ2ǫCF

(1

4π

)1−ǫ 1

π

1

Γ(1− ǫ)1

τ1+2ǫa ǫ(1− a)

(

tan2(1−a)(R/2)

ω2

)ǫ

. (3.2.30)

For the naive contribution we only give the integrated result as the detailed calculation isquite extensive:

Jqω(τa) = g2µ2ǫCF

∫dl+

2π

1

(l+)2

∫ddq

(2π)d

(

4l+

q−+ (d− 2)

l+ − q+ω − q−

)

2πδ(

q−q+ − (q⊥)2)

×Θ(q−)Θ(q+)2πδ

(

l+ − q+ − (q⊥)2

ω − q−)

Θ(ω − q−)Θ(l+ − q+)

×δ(

τa −1

ω(ω − q−)a/2(l+ − q+)1−a/2 − 1

ω(q−)a/2(q+)1−a/2

)

Θalg (3.2.31)

=αsCF

2π

1

Γ(1− ǫ)

(4πµ2

ω2 tan2(R/2)

)ǫ(1

ǫ2+

3

2ǫ

)

δ(τa) +αs

2πJq

alg(τa). (3.2.32)

The naive contribution depends on the choice of the used jet algorithm. But this differenti-ation only affects the final part of the jet function and can therefore be summed up in thefinite contribution Jq

alg(τa). [32]The naive and the zero-bin contribution can then be put together to get the final ex-

pression for the measured quark jet function with both particles being inside the jet. Whiledoing this the modified minimal subtraction (MS) scheme is applied:

Jqω(τa) = δ(τa)

(

1 +αsCF

π

(1− a/2(1− a)ǫ2 +

1− a/2(1− a)ǫ ln

µ2

ω2+

3

4ǫ

))

− αsCF

π

[Θ(τa)

(1− a)τaǫ

]

+

+αs

2πJq

alg(τa). (3.2.33)

The plus distribution at the end of the first line is defined as

[Θ(τa)g(τa)]+ = limǫ→0

d

dτa(Θ(τa − ǫ)G(τa)) (3.2.34)

41

3 Jet and Soft Functions

with G(τa) =∫ τa

1 dτ ′ag(τ′a) and the fulfilled boundary condition

∫ 10 dτa [Θ(τa)g(τa)]+ = 0.

Here g(τa) = 1(1−a)ǫτa

and hence

[Θ(τa)

(1− a)τaǫ

]

+

= limǫ→0

d

dτa

(

Θ(τa − ǫ)ln τa

(1− a)ǫ

)

= limǫ→0

1

(1− a)ǫ

(Θ(τa − ǫ)

τa+ δ(τa − ǫ) ln τa

)

.

(3.2.35)

Unmeasured Quark Jet

In the same manner as the measured quark jet the unmeasured quark jet can be obtained.The only difference is the dependence on the angularity. To be more explicit, one has todrop the δ−function depending on τa in Eq.(3.2.31). The zero-bin contribution vanishesand the full unmeasured quark jet function is

Jqω = 1 +

αsCF

2π

(1

ǫ2+

3

2ǫ+

1

ǫln

µ2

ω2 tan2R/2

)

+αs

2πJq

alg. (3.2.36)

3.2.2 Gluon Jet Function

In addition to the quark jet function there is also a gluon jet function which contributesto the whole jet function. Gluon jet functions are characterised by having gluons in theoutgoing states. The results for the measured and unmeasured gluon jet functions are:

Jgω(τa) =

(

1 +αsCA

π

(1− a/21− a

(1

ǫ2+

1

ǫlnµ2

ω2

)

+11

12ǫ

)

− αs

3πTRNf

1

ǫ

)

δ(τa)

−αsCA

π

1

ǫ

1

1− a

[Θ(τa)

τa

]

+

+αs

2πJg

alg(τa), (3.2.37)

Jgω = 1 +

αs

2π

(

CA

(1

ǫ2+

11

6ǫ+

1

ǫln

µ2

ω2 tan2R/2

)

− 2

3ǫTRNf

)

+αs

2πJg

alg.

(3.2.38)

3.3 Soft Function

After having found the mathematical description of the jet function, we will have a look atthe soft function. The soft function to NLO can be characterised as the sum over all softgluon emissions from all Wilson lines which interfere. Note that such a soft gluon could beemitted into the final state which could lead to the creation of an extra jet and change theresult. The soft function would not be infrared safe. Therefore we need some constraintsfor the calculation, which are given in Table 3.2. These restrictions make sure, that a softparticle is emitted only into an already existing jet or has a small energy (E < Λ) whenbeing outside a jet. The sum over the soft gluon emissions can be divided into three differentparts:

1. a) gluon ∈ measured jet and contributes to the angularity,

b) gluon /∈ jet with E < Λ,

42

3.3 Soft Function

Gluon outside a jet Θ(k0 < Λ)

Gluon inside jet i Θ(

k+

k− < tan2R/2)

Gluon inside measured jet δ(

τa − 1ω (k−)

a/2(k+)

1−a/2)

Table 3.2: Restrictions for the calculation of the soft function. At NLO they are the same for coneand kt algorithms.

c) gluon ∈ unmeasured jet with any energy.

This classification is simple but turns out to be disadvantageous when actually performingthe calculation as for the calculation of a particle being not in a jet one has to subtractall the jets from the whole space, which is rather difficult. Therefore we translate thissectioning into a better calculable one:

1. gluon ∈ measured jet and contributes to angularity Smeasij (τk

a ),

2. gluon ∈ measured jet with E > Λ and does not contribute to angularity Skij,

3. gluon ∈ measured jet with E < Λ and does not contribute to angularity Skij,

4. gluon is anywhere with E < Λ and does not contribute to any angularity Sinclij .

Therefore the soft function in NLO is given by the sum

S(

τ la

)

= a+ (b+ c) = 1 + (4− 3 + 2)

=∑

i6=j

Smeasij (τk

a )

M∏

l 6=k

δ(τ la) +

∑

i6=j

(

Sinclij −

∑

k∈meas

Skij +

∑

k/∈meas

Skij

)M∏

l

δ(τ la)

=∑

i6=j

Smeasij (τk

a )M∏

l 6=k

δ(τ la) +

∑

i6=j

(

Sinclij +

N∑

k=1

Skij

)M∏

l

δ(τ la). (3.3.1)

The last line is a simplification due to the fact, that the contribution without an energyrestriction (c) involves a scaleless integral when using dimensional regularisation. Thereforethe equation Sk

ij + Skij = 0 holds.

Inclusive Soft Function

In the following the inclusive soft function will be calculated. This is the function wherethe gluon may be anywhere with an energy smaller than Λ and does not contribute to theangularity. It is given by

Sinclij = −g2

sµ2ǫ ~Ti

~Tjni · nj

(2π)4−2ǫ

∫

d4−2ǫk1

(ni · k)(nj · k)2πδ(k2)Θ(k0)Θ(k0 < Λ). (3.3.2)

First of all the integral needs to be rewritten in spherical coordinates:∫

d4−2ǫkδ(k2)Θ(k0)Θ(k0 < Λ) =

∫

dk0d3−2ǫkδ(

(k0)2 − ~k2)

Θ(k0)Θ(k0 < Λ)

=

∫ Λ

0dkdϑ1dϑ2

π1/2−ǫ

Γ(1/2− ǫ)1

2k1−2ǫ sin1−2ǫ ϑ2 sin−2ǫ ϑ1. (3.3.3)

43

3 Jet and Soft Functions

Further we define ~ni lying in z−direction and ~nj being in the xz−plane. Therefore theproducts in the denominator of Eq.(3.3.2) do not contain any y−components:

ni · k = n0k0 − ~n · ~k = |k|(1− cos ϑ2),

nj · k = |k|(1 − njx cos ϑ1 sinϑ2 − njz cos ϑ2). (3.3.4)

With this the inclusive soft function reads

Sinclij = −g2

sµ2ǫ ~Ti

~Tjni · nj

(2π)4−2ǫ2π

∫ Λ

0dk

∫ π

0dϑ1

∫ 2π

0dϑ2

π1/2−ǫ

Γ(1/2 − ǫ)1

2k1−2ǫ

× sin−2ǫ ϑ1 sin1−2ǫ ϑ2

k2(1− cosϑ2)(1− njx cos ϑ1 sinϑ2 − njz cos ϑ2)

= −αs

2π

(4π2µ2

4

)ǫ4ǫ

π~Ti~Tjni · nj

π1/2−ǫ

Γ(1/2 − ǫ)1

2

[k−2ǫ

2ǫ

]Λ

0

∫ π

0dϑ1

×2

∫ π

0dϑ2

sin−2ǫ ϑ1 sin1−2ǫ ϑ2

(1− cos ϑ2)(1 − njx cos ϑ1 sinϑ2 − njz cos ϑ2)

=1

ǫ

1

Γ(1− ǫ)αs

2π

(4πµ2

4Λ2

)ǫ

~Ti~Tjni · nj

4ǫ

2

Γ(1− ǫ)√πΓ(1/2 − ǫ)

∫ π

0dϑ1

×∫ π

0dϑ2

sin−2ǫ ϑ1 sin1−2ǫ ϑ2

(1− cos ϑ2)(1− njx cos ϑ1 sinϑ2 − njz cos ϑ2), (3.3.5)

where we have neglected the lower bound of the k−integral in the last step. For the cal-

culation of the ϑ1−integral the conditions 2Re(ǫ) < 1 and Re(

njz cos ϑ2−1njx sinϑ2

)

≤ −1 or ≥ +1

need to be fulfilled, which is achieved. The integral can then be carried out:

Sinclij =

1

ǫ

1

Γ(1− ǫ)αs

2π

(4πµ2

4Λ2

)ǫ

~Ti~Tjni · nj

4ǫ

2

Γ(1− ǫ)√πΓ(1/2 − ǫ)

(

−4−ǫ Γ(1/2 − ǫ)2Γ(1− 2ǫ)

)

︸ ︷︷ ︸

−4−ǫ Γ(1/2−ǫ)Γ(1−ǫ)

4ǫ√

π

×∫ π

0dϑ2

sin1−2ǫ ϑ2

(1− cos ϑ2)(−1− njx sinϑ2 + njz cos ϑ2)

×2F1

(

1, 1/2 − ǫ, 1− 2ǫ,2njx sinϑ2

1− njz cos ϑ2 + njx sinϑ2

)

= −1

ǫ

1

Γ(1− ǫ)αs

2π

(4πµ2

4Λ2

)ǫ

~Ti~Tjni · nj

4ǫ

2

×∫ π

0dϑ2

sin1−2ǫ ϑ2

(1− cos ϑ2)(−1− njx sinϑ2 + njz cos ϑ2)

×2F1

(

1, 1/2 − ǫ, 1− 2ǫ,2njx sinϑ2

1− njz cos ϑ2 + njx sinϑ2

)

, (3.3.6)

44

3.3 Soft Function

with 2F1 being the hypergeometric function. To make the last integral a little simpler, wemake the substitution cosϑ2 = u:

Sinclij = −1

ǫ

1

Γ(1− ǫ)αs

2π

(4πµ2

4Λ2

)ǫ

~Ti~Tjni · nj

4ǫ

2

∫ 1

−1du

(1 − u2)−ǫ

1− u

× 1

1− njzu+ njx

√1− u2

· 2F1

1, 1/2 − ǫ, 1− 2ǫ,2njx

√1− u2

1− njzu+ njx

√1− u2

︸ ︷︷ ︸

=:z′

.

(3.3.7)

This integral clearly has some singularities. One is at u1 = 1. The other can be computedby setting the denominator of the second fraction to zero and solve for u. This gives anothertwo singularities for

u2,3 =1

n2jx

+ n2jz

(

njz ± njx

√

n2jx

+ n2jz− 1)

. (3.3.8)

Because ~nj lies in the xz−plane, the equation√

n2jx

+ n2jz

= 1 holds and Eq.(3.3.8) can

be rewritten as u2,3 = u2 = njz =: n. These singularities correspond to z′1 = 0 andz′2 = 1. Because they lie inside the range of integration, the integral needs to be splittedin an integral from −1 to δ (we call this contribution Sincl

1 ) and from δ to 1 (we call thiscontribution Sincl

2 ) with n < δ < 1. For Sincl1 we rewrite the hypergeometric function with

the formula [36]

2F1(a, b, 2b, z) = (1− 1/2z)−a2F1

(a/2, 1/2 + 1/2a, b + 1/2, z2(2− z)−2

)(3.3.9)

⇒ 2F1

(

1, 1/2 − ǫ, 1− 2ǫ,2√

(1− n2)(1− u2)

1− nu+√

(1− n2)(1 − u2)

)

=1− un+

√

(1− n2)(1− u2)

1− un · 2F1

1/2, 1, 1 − ǫ, (1− n2)(1− u2)

(1− un)2︸ ︷︷ ︸

=:z

.

(3.3.10)

The contribution Sincl1 therefore reads

Sincl1 = −1

ǫ

1

Γ(1− ǫ)αs

2π

(4πµ2

4Λ2

)ǫ

~Ti~Tjni · nj

4ǫ

2

×∫ δ

−1du

(1− u2)−ǫ

(1− u)(un− 1)2F1

(

1/2, 1, 1 − ǫ, (1− n2)(1− u2)

(1− un)2

)

. (3.3.11)

45

3 Jet and Soft Functions

To calculate this integral, the hypergeometric function needs another rewriting [37]:

1

Γ(c)2F1 (a, b, c, z) =

[

−(1− z)c−a−b

Γ(a)Γ(b)2F1 (c− a, c− b, c− a− b+ 1, 1 − z) 1

Γ(c− a− b+ 1)

+1

Γ(c− a)Γ(c− b)2F1 (a, b, a+ b− c+ 1, 1− z) 1

Γ(a+ b− c+ 1)

]

× π

sin (π(c− a− b)) (3.3.12)

⇒ 2F1 (1/2, 1, 1 − ǫ, z) =

√π

cos ǫπ

(1− un|u− n|

)1+2ǫ

· 2F1 (1/2− ǫ,−ǫ, 1/2 − ǫ, 1− z) Γ(1− ǫ)Γ(1/2− ǫ)

+π

cos ǫπ

1

Γ(1/2− ǫ)1

Γ(−ǫ)︸ ︷︷ ︸

−ǫΓ(1−ǫ)

Γ(1− ǫ)Γ(3/2 + ǫ)

· 2F1 (1/2, 1, 3/2 + ǫ, 1− z)

= fa(z) + fb(z). (3.3.13)

The calculation of the part containing fb(z) is quite easy as it is of order ǫ. Therefore thewhole integrand containing fb(z) can be expanded around ǫ = 0:

∫ δ

−1du

π

cos ǫπ

−ǫΓ(1/2 − ǫ)Γ(3/2 + ǫ)

(1− u2)−ǫ

(1− u)(un− 1)· 2F1 (1/2, 1, 3/2 + ǫ, 1− z)

≃ − 2ǫ

(u− 1)(u− n)tanh−1

(u− n1− nu

)

= − 2ǫ

|n− u|(u− 1)tanh−1

( |n− u|1− nu

)

,

(3.3.14)

where we have used the fact that n is always larger than u = cos ϑ2 and z = (1−n2)(1−u2)(1−un)2

.

For the integration of the part containing fa(z) a trick has to be applied. In general anintegral over a product of a function f(u, ǫ) being divergent at u = u0 and a non-singularfunction g(u, ǫ) can be written as

∫

duf(u, ǫ)g(u, ǫ) =

∫

duf(u, ǫ)g(u0, ǫ) +

∫

duf(u, ǫ) (g(u, ǫ) − g(u0, ǫ)) . (3.3.15)

In the case of the integral from −1 to δ the covered singularity is at u = n. Therefore thedivergent and non-singular functions read

f(u, ǫ) =1

|u− n|1+2ǫ, (3.3.16)

g(u, ǫ) =

√π

cos ǫπ

(1− un)2ǫ

(1− u2)ǫ(u− 1)

Γ(1− ǫ)Γ(1/2− ǫ)2F1

(

1/2− ǫ,−ǫ, 1/2 − ǫ, 1− (1− n2)(1− u2)

(1− un)2

)

,

(3.3.17)

46

3.3 Soft Function

with g(n, ǫ) =√

πcos ǫπ

(1−n2)ǫ

n−1Γ(1−ǫ)

Γ(1/2−ǫ) . Applying Eq.(3.3.15) we get for the integral containing

fa(z)∫ δ

−1du

√π

cos ǫπ

(1− u2)−ǫ

(u− 1)

(1− un)2ǫ

|u− n|1+2ǫ

Γ(1− ǫ)Γ(1/2− ǫ) · 2F1

(

1/2− ǫ,−ǫ, 1/2 − ǫ, 1− (1− n2)(1− u2)

(1− un)2

)

=

∫ δ

−1du

1

|u− n|1+2ǫ

√π

cos ǫπ

(1− n2)ǫ

n− 1

Γ(1− ǫ)Γ(1/2 − ǫ) +

∫ δ

−1du

1

|u− n|1+2ǫ(g(u, ǫ) − g(n, ǫ)) .

(3.3.18)

With the identity 2F1 (1/2 − ǫ,−ǫ, 1/2 − ǫ, 1− z) = zǫ we can rewrite the hypergeometricfunction in the second integral of Eq.(3.3.18). It therefore reads

∫ δ

−1du

1

|u− n|1+2ǫ(g(u, ǫ) − g(n, ǫ))

=

√π

cos (ǫπ)|u− n|1+2ǫ

Γ(1− ǫ)Γ(1/2 − ǫ)

((1− n2)ǫ

1− u − (1− n2)ǫ

1− n

)

= −√π

(1

|u− n| (1− 2ǫ ln (|u− n|)))

1√π

(1− ǫ ln 4)

×(

1

1− u(1 + ǫ ln (1− n2)

)− 1

1− n(1 + ǫ ln (1− n2)

))

= − sgn(u− n)

(1− u)(1− n)(1− ǫ ln(4))

(

1− ǫ ln( |u− n|2

1− n2

))

= − sgn(u− n)

(1− u)(1− n)

(

1− ǫ ln(

4|u− n|21− n2

)

+O(ǫ2)

)

, (3.3.19)

where we have expanded the whole integrand around ǫ = 0 in the third line, which is possibleas the integrand vanishes for u = n. Therefore Eq.(3.3.18) becomes

∫ δ

−1du

1

n− 1

[ √π(1− n2)ǫ

cos (ǫπ)|u− n|1+2ǫ

Γ(1− ǫ)Γ(1/2 − ǫ) −

sgn(u− n)

(1− u)(1 − n)

(

1− ǫ ln(

4|u− n|21− n2

))]

(3.3.20)

and Sincl1 is

Sincl1 = −1

ǫ

1

Γ(1− ǫ)αs

2π

(4πµ2

4Λ2

)ǫ

~Ti~Tjni · nj

4ǫ

2

∫ δ

−1du

[

− 2ǫ

|n− u|(u− 1)tanh−1 |n− u|

1− un

+1

n− 1

[ √π(1− n2)ǫ

cos (ǫπ)|u− n|1+2ǫ

Γ(1− ǫ)Γ(1/2 − ǫ) −

sgn(n − u)1− u

+ǫ · sgn(n− u)1− u ln

(4|n− u|21− n2

)]]

.

(3.3.21)

This result differs from the calculation in [32] by a prefactor 11−n for the tanh−1 term. Due

to this disagreement we may not take the results for the other parts of the soft functionout of [32]. Calculating them by ourselves is unfortunately not doable in the frameworkof a Master Thesis. Therefore we will in the next section derive the renormalisation groupequations which are needed after one has obtained the different parts of the soft function.

47

3 Jet and Soft Functions

3.4 Final Summation

Because all the different parts of the functions need to be put together we need them allat one scale, which is done by renormalisation group evolution. As there are two types offunctions, measured and unmeasured ones, we also need two types of renormalisation groupequations where the RGE for the unmeasured functions is obtained in a similar way as inQCD. In the following we will have a closer look on the RGEs needed for the evolution andstate the proof that factorisation is valid.

3.4.1 Renormalisation Group Equations

The function being independent of an observable, in our case the angularity τa, will becalled F (µ) and the one being dependent on τa will be called F (µ, τa). The general ansatzfor the bare (unrenormalised) function, denoted with a B, is [32, 38]

FB(τa) =

∫

dτ ′aZF (τa − τ ′a, µ)F (τ ′a, µ). (3.4.1)

If F and the renormalisation factor Z are independent of τa one has the simplified ansatz

FB = ZF (µ)F (µ) (3.4.2)

from which the RGE follows by taking the derivative with respect to µ and multiplyingwith µ

ZF (µ) :

µd

dµF (µ) = γF (µ)F (µ), (3.4.3)

with

γF (µ) = − µ

ZF (µ)

dZF (µ)

dµ

= ΓF (αs) lnµ2

ω2+ γF (αs) (3.4.4)

being the anomalous dimension. In the last equality we split up the anomalous dimensionin a term being dependent on the renormalisation scale µ with a factor called the cuspanomalous dimension ΓF (αs) and a term only being dependent on αs.

If F depends on τa, it is a little more complicated. We first of all rewrite Eq.(3.4.1) inFourier space:

FB(ta) = ZF (ta, µ)F (ta, µ), (3.4.5)

where ta is the Fourier transform of τa. Acting with the derivative with respect to µ onEq.(3.4.5) and multiplying the whole equation with µ

ZF (ta,µ)one gets the RGE

0 =µ

ZF (ta, µ)

((

dZF (ta, µ)

dµ

)

F (ta, µ) + ZF (ta, µ)dF (ta, µ)

dµ

)

= −γF (ta, µ) +µ

F (ta, µ)

dF (ta, µ)

dµ(3.4.6)

48

3.4 Final Summation

with the anomalous dimension being

γF (ta, µ) = − µ

ZF (ta, µ)

dZF (ta, µ)

dµ. (3.4.7)

Now we leave the Fourier space and the anomalous dimension with respect to τa can bewritten as:

γF (τa, µ) = −∫

dτ ′aµ

ZF (τa − τ ′a, µ)

d

dµZF (τa, µ)

= −ΓF (α)

(2

jF

[Θ(τa)

τa

]

+

− lnµ2

ω2δ(τa)

)

+ γF (α)δ(τa), (3.4.8)

where the anomalous dimension has been split in a cusp and a non-cusp part. The “+” onthe inner bracket denotes the plus distribution defined in Eq.(3.2.34). The solution of thetwo RGEs (Eq.(3.4.3) and (3.4.6)) is given by [32]

F (µ) = eKF (µ,µ0)(µ0

ω

)ωF (µ,µ0)

︸ ︷︷ ︸

UF (µ,µ0)

F (µ0),

F (µ, τa) =

∫

dτ ′aeKF (µ,µ0)+γEωF (µ,µ0)

Γ(−ωF )

(µ0

ω

)jF ωF[

Θ(τa − τ ′a)(τa)1+ωF (µ,µ0)

]

+︸ ︷︷ ︸

UF (τa,µ,µ0)

F (µ0, τ′a),

(3.4.9)

where γE is the Euler constant and UF are the evolution kernels. In leading order KF (µ, µ0)and ωF (µ, µ0) are given as [32]

ωF (µ, µ0) = − Γ0F

jFβ0ln

αs(µ)

αs(µ0),

KF (µ, µ0) = − γ0F

2β0ln

αs(µ)

αs(µ0)−−2πΓ0

F

β20

αs(µ)αs(µ0) − 1− αs(µ)

αs(µ0) ln αs(µ)αs(µ0)

αs(µ), (3.4.10)

with Γ0F and γ0

F being the leading order coefficients of the expanded (cusp) anomalousdimension: [39]

Γ0F = 2CF , (3.4.11)

γ0F =

32παsCF for quark jetαs6π (11CA − 2Nf ) for gluon jet

. (3.4.12)

49

3 Jet and Soft Functions

Therefore the evolution kernels are given in leading order as: [32]

UF (µ, µ0) = exp

− γ0F

2β0ln

αs(µ)

αs(µ0)− 4πCF

β20

αs(µ)αs(µ0) − 1− αs(µ)

αs(µ0) ln αs(µ)αs(µ0)

αs(µ)

×(µ0

ω

)− 2CFjF β0

ln αs(µ)αs(µ0)

, (3.4.13)

UF (τa, µ, µ0) =1

Γ(

2CFjF β0

ln αs(µ)αs(µ0)

)

× exp

− γ0F

2β0ln

αs(µ)

αs(µ0)− 4πCF

β20

αs(µ)αs(µ0) − 1− αs(µ)

αs(µ0) ln αs(µ)αs(µ0)

αs(µ)

+γE

(

− 2CF

jFβ0ln

αs(µ)

αs(µ0)

)](µ0

ω

)− jF 2CFjF β0

ln αs(µ)αs(µ0)

×

Θ(τa)

(τa)1− 2CF

jF β0ln

αs(µ)αs(µ0)

+

(3.4.14)

By applying the RGEs to the soft function the different parts can be evolved to one scaleand put together. By doing so one could express a factorised cross section, for example forthe process e+e− → N jets, in terms of only the three scales, the hard, the soft and thescale of the jet function, which can then be chosen appropriately.

The anomalous dimensions can be used to examine the validity of the EFT by the so-called consistency relation. That is the sum of all anomalous dimensions for the differentenergy regimes has to be zero:

δ(τ ia)

γH(µ) +∑

i/∈meas

γJi(µ) + γunmeasS (µ)

+∑

i∈meas

(γJi(µ, τ

ia) + γmeas

S (µ, τ ia))

= 0.

(3.4.15)

The anomalous dimension of the hard function is obtained from the matching coefficients[40]. γJi and γS can be read off the jet respectively soft function and are given as the infiniteparts.

50

4 Conclusion and Outlook

In this work we gave an introduction to the soft-collinear effective theory and presentedthe main ideas. We introduced Wilson lines to ensure gauge invariance and decouple the(ultra-)soft modes from the collinear ones. The derivation of the collinear Lagrangian inSCET led to the Feynman rules. After understanding these basics of SCET we looked at thejet and soft function of a factorised cross section. First the derivation of the different partsof the jet function has been made before the ideas for the calculation of the soft functionwere exposed. The calculation of the inclusive soft function showed a different prefactorfor one term in comparison with [32]. Because the same method was applied it can onlybe due to an analytic error in one of the two works. As a next step the second part of theinclusive soft function has to be calculated before deriving the remaining two parts of the softfunction, that are the functions with the gluon being inside a measured or an unmeasuredjet k. The differences for these calculations in comparison to the inclusive soft function liein the phase space restrictions, namely the δ− and Θ−functions and the splitting into twodifferent cases: a) The jet k equals the direction of one of the Wilson lines or b) the jet klies in any other direction. This partition is due to an arising singularity for the case a),as the product of the normal vector ni with k, which appears in the denominator of theintegral (see Eq.(3.3.2)), is zero for k = i, j. The resulting total soft function depends on twodifferent scales, one for the measured part and one for the unmeasured part. Furthermorethe hard and the jet function are also sensitive to another scale. Because in the end wehave to put the three functions together, we need them all at one scale. To achieve this, thefunctions have to be evolved by renormalisation group equations. By applying the RGEsfor measured and unmeasured functions, the hard, jet and soft function are evolved to ascale µ0. An important improvement coming along with the evolution is the minimisationof logarithms, which tend to become very large and therefore induce a divergent crosssection. By choosing the initial scales appropriately the final cross section will be finite.By looking at a particular process like e+e− scattering, the cross section can be computedand then compared to experimental or simulated data. With the consistency relation of theanomalous dimensions the validity of the effective field theory can be proved.

As an improvement of the calculation for the jet and soft function one may take higherorders of λ into account. The parts of the jet function with a particle being outside a jetare then no longer suppressed and therefore have to be taken into account. Also the RGEsneed to be computed to higher orders.

The calculation of the cross section for e+e− to N jets is in comparison to other processeswell understood, as the interactions in this process are known. In pp collisions for examplethere exists a whole bunch of interactions in the initial quark states due to QCD backgroundreactions, which makes the calculation of the cross section very difficult. The hope is tobe able to derive the cross section for pp collisions as the produced jets are the same asfor the e+e− collision. Therefore one may be able to translate these calculations onto morecomplicated processes like proton proton collisions.

51

4 Conclusion and Outlook

52

5 Appendix

5.1 Quark Lagrangian

In this Appendix a detailed list of the vanishing terms arising in the SCET quark Lagrangianis given. First notice the following identities for the quark fields:

ξn = ψ†P†nγ

0 = ψ†Pnγ0 = ψ†γ0Pn,

ξn = ψ†P†nγ

0 = ψ†Pnγ0 = ψ†γ0Pn (5.1.1)

and

Pn/n = /n , Pn /n = /n,

Pn /n = 0 = /nPn , Pn/n = 0 = /nPn. (5.1.2)

Therefore the vanishing terms are:

ξn/n

2in ·Dξn = ψ†γ0Pni

/n

2n ·DPnψ = 0,

ξni /D⊥ξn = ψ†γ0i /D⊥PnPnψ = 0,

ξn/n

2in ·Dξn = ψ†γ0Pni

/n

2n ·DPnψ = 0,

ξn/n

2in ·Dξn = ψ†γ0in ·DPn

/n

2Pnψ = 0,

ξn/n

2in ·Dξn = ψ†γ0Pni

/n

2n ·DPnψ = 0,

ξn/n

2in ·Dξn = ψ†γ0Pn

/n

2in ·DPnψ = 0,

ξn/n

2in ·Dξn = ψ†γ0Pni

/n

2n ·DPnψ = 0,

ξni /D⊥ξn = ψ†γ0Pni /D⊥Pnψ = 0.

53

5 Appendix

54

List of Figures

1.1 Colour conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Quark pair production with a gluon loop correction . . . . . . . . . . . . . . 12

2.1 Sketch of the different validities of SCETI and SCETII . . . . . . . . . . . 202.2 Visualisation of integrating out off-shell propagators . . . . . . . . . . . . . 262.3 Combination of collinear gluons into a collinear Wilson line . . . . . . . . . 272.4 Attachment of (ultra-)soft gluons to a collinear quark . . . . . . . . . . . . 28

3.1 Momenutm labeling of the Feynman diagrams . . . . . . . . . . . . . . . . . 343.2 Cut diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

55

List of Figures

56

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Erklarung

Hiermit versichere ich, die vorliegende Masterarbeit selbststandig und nur unter Benutzungder im Literaturverzeichnis angegebenen Quellen erstellt zu haben.

Bielefeld, den 25.10.2010