Soft-Collinear Effective Theory
Robin van der Leeuw
May 6, 2008
Supervisor: prof. dr. E.L.M.P. Laenen
Abstract
We consider interactions of initial hadrons with energy much larger than their mass. Collinear and soft
degrees of freedom give rise to infrared divergences, hence we study an the soft-collinear effective theory
(SCET) which separates these. We derive the lowest order collinear quark and gluon SCET Lagrangian
and study the matching procedure involving Wilson lines. This procedure is then performed to prove
factorization to all orders in αs of B → Xs γ decay, Deep Inelastic Scattering (DIS) and Higgs production
via gluon fusion. For this last process we also consider the resummation of large logarithms.
3
4
Acknowledgements
During the year I spent working on my thesis, I learned a lot about physics and doing research in general.
For this I would like to thank first and foremost my supervisor Eric Laenen, with whom I worked very
pleasantly. His door was always open for questions, and he motivated me when needed. I especially
appreciate that he gave me just the right amount of attention; he made me think first by myself, but was
there to help if it didn’t work out.
Furthermore I thank the Nikhef institute in general and the theory group in particular for providing a
very pleasant working environment. I would like to thank Gerben Stavenga and Chris White in particular
for the enjoyable discussions on physics, from which I learned a lot.
In my first half year of doing research, room ‘301 de gekste’ was great, good working atmosphere and
good fun. I have to mention Ivo as my ping-pong partner during those times; I still need revenge for that
last match!
I owe a lot to my family and friends; I thank my parents for supporting me during my study. Although
Utrecht can be somewhat far away (especially from Ireland), it was always great to come home, be it to
‘de olm’ or to Feakle.
Also thanks to Daan, as good friend from the QSSP and roommate on the Nikhef, who listened to all
my whining when I got stuck again. Cheers for putting up with me! With him I would like to thank my
friends for providing the necessary distractions during my last year, especially Froukje, Duveke, Floor,
Giel, Malou and all other people I’m forgetting.
My five and a half years in Utrecht would not have been the same without the QSSP, with whom even
the most boring lectures could turn out fun (especially after roaming the city the night before.) Bedankt
gasten en Dorien, het waren en zijn mooie tijden!
5
6
Contents
1 Introduction
11
2 The Drell-Yan process
13
2.1
2.2
Computing the cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
14
2.1.2
2.1.3
The invariant amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
18
The resulting cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3 Landau equations and power counting
3.1
3.2
3.3
21
Landau equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Coleman-Norton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Infrared power counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
23
25
3.3.1
3.3.2
Pinch surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The power counting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
26
4 Intro SCET
4.1 Momentum modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
4.2
4.3
Power Counting in SCET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The collinear SCET Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
32
4.3.1
4.3.2
The quark Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The label operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
35
4.3.3
4.3.4
The gluon field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Collinear gluon Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
37
4.4
4.5
Feynman rules for SCETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
41
4.6
Wilson lines in SCET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Finding the form of the Wilson line . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 The ultrasoft Wilson line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
44
46
4.7
The heavy-to-light decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Redefinition of the fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
48
7
8
Contents
4.8
4.9
SCETII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.8.1
Wilson lines in SCETII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.8.2
Gauge transformations in SCETII . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Using SCET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5 The inclusive B → Xs γ decay
5.1
5.2
Calculating the decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.1.1
The physics of the decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Matching QCD onto SCETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.2.1
The operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.2.2
Hard kernel and jet function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.2.3
The structure function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.2.4
The final factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6 Deep Inelastic Scattering
6.1
6.2
6.3
67
The kinematics of the process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6.1.1
The Breit frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6.1.2
Powercounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
The DIS cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.2.1
SCET operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6.2.2
Charge conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6.2.3
The parton distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6.2.4
The matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
7 Higgs production through gluon fusion
7.1
7.2
7.3
53
Kinematics of Higgs production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
79
80
Matching at Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
7.2.1
82
The matching coefficient CHgg . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Matching at Q (1 − z)
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
8 Conclusions
87
A Conventions
89
A.1 Dirac Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B Lightcone Coordinates
89
91
B.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
B.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
CONTENTS
9
C HQET
C.1 Constructing HQET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
95
C.1.1 Feynman rules for HQET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
D Properties of Wilson lines
D.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
99
D.2 Properties of Wilson lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
D.2.1 causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
D.2.2 Multiplication of Wilson lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
E Moments
103
10
Contents
Chapter 1
Introduction
Many processes in current particle accelerators have large regions of phase space where the energy of
incoming hadrons is very large, much larger than their mass, EH mH . Examples are decays of a B
meson (e.g. B → Xs γ, B → πeν, B → ππ) and hard scattering events like Deep Inelastic Scattering
(DIS) and Drell-Yan like processes. At these large energies quarks and gluons move typically along the
lightcone direction, and will form a collinear jet. A simple example of a collinear jet is shown on the
front cover of this thesis. Here we have shown a lighthouse which emits a light beam travelling in a
certain direction. This can be seen as a collinear jet, as the photons all move along the same direction.
These collinear degrees of freedom, together with soft momenta, can be identified as the regions which
lead to infrared divergences using Landau equations [1] and the Coleman-Norton method [2]. Using this
method factorization formulas have been proved, which essentially separate the hard, collinear and soft
momenta. This factorization has proved successful yet difficult using this full QCD approach. The main
disadvantage of this conventional approach is that the powercounting is done only at the end of the
calculation, in stead of simplifying the calculation by doing this from the start. This can be solved using
an effective field theory.
For the regions where the large scale is Q ≈ EH ΛQCD , the scale at which non-perturbative effects
dominate, we would like to have such an effective field theory (EFT). Effective field theories treat only
those degrees of freedom which are relevant for the problem. The Lagrangian in an EFT therefore only
takes care of the appropriate degrees of freedom. Such an effective field theory can be used to separate
perturbative effects with a coupling constant αs (Q) from the non-perturbative effects (αs (ΛQCD )). This
factorization will result in a convolution of a hard kernel, a jet function and a soft function. The hard
part describes the perturbative short distance physics, the jet function the particle in collinear jets and
the soft function the non-perturbative long distance physics. The soft-collinear effective theory (SCET)
[3, 4, 5] can be used to decouple the soft interactions to all orders from the start of the calculation.
In this effective theory the relevant degrees of freedom, i.e. hard, collinear and soft, have momenta
with a well-defined scaling in the small parameter λ. The value of this parameter defines two distinct
p
theories. In SCETI it is defined by λ = ΛQCD /Q, while in SCETII we have λ = ΛQCD /Q. These
two theories differ in the invariant mass of the collinear particle; in SCET I we have p2C ∼ QΛQCD , while
11
12
Chapter 1. Introduction
for SCETII p2c ∼ Λ2QCD . In both theories the hard degrees of freedom are integrated out, and appear in
Wilson coefficients, while the long distance fluctuations will appear as effective fields in the final effective
theory.
The thesis is built up in the following way. Chapter 2 introduces the Drell-Yan process, in which
we first encounter the singularities coming from the soft and collinear degrees of freedom. Chapter 3
shows that we can use a powercounting procedure to keep track of the singularities. Then in 4 SCET is
introduced. This is the basis of this thesis, where we obtain the techniques used in SCET. This is a quite
technical chapter, but is of uttermost importance to understand the remainder of the thesis.
The last three chapters discuss well known and described processes, which show the advantages of using
SCET over the full theory. Chapter 5 shows how the cross-section of the B → X s γ decay is factorized
to all orders in αs using SCET. The same is done for Deep Inelastic Scattering (DIS) in chapter 6, in
which some useful steps emerge for chapter 7, where the large logarithms for Higgs production via gluon
fusion are calculated. We conclude with a brief summary and conclusion in chapter 8. The conventions
which are used are elaborated in appendix A. In appendix B the lightcone coordinates and corresponding
notation which are used throughout this thesis are introduced, while in appendix C we show how Heavy
Quark Effective Theory (HQET) [6, 7] is built up, which is used for the heavy-to-light current in the B
decay.
Chapter 2
The Drell-Yan process
The Drell-Yan process (Drell & Yan..) is a well understood and described process. It is an inclusive
hadron-hadron scattering, which is very closely linked to Deep Inelastic Scattering, which is discussed in
chapter 6.
The goal of this chapter is to compute the real and virtual next to leading (NLO) graphs for DrellYan, in which two hadrons interact and form two leptons, plus any additional remnants (figure 2.1). The
process is given by
A(k1 ) + B(k2 ) → l(p1 ) + l(p2 ) + X,
(2.1)
where hadrons and leptons are all massless, k12 = k22 = p21 = p22 = 0. Two quarks of hadrons A and
B interact, creating a photon which decays into a lepton-antilepton pair. The remnant of the hadrons
form the unspecified particles X. The leading term of the cross-section is given by the Born crosssection, σB = 4πα2 /(9Q2 s)Q2f [8]. In this leading term there are no gluon interactions, hence no QCD
contribution. The NLO terms for the Drell-Yan process are the one gluon emission and loop contributions,
shown in fig (2.2). These are the diagrams we will be interested in.
Figure 2.1: The Drell-Yan process. Two hadrons, A and B, interact which produces
a lepton pair and a remnant X. This remnant contains the remaining quarks from the
interacting hadrons and additional gluon emissions.
13
14
Chapter 2. The Drell-Yan process
Figure 2.2: The leading and next-to-leading (NLO) diagrams for the Drell-Yan process.
On the left the Born diagram is shown, while the NLO gluon emission and one-loop
diagram are shown in the middle and on the right respectively.
Since the differential cross-section is given by
dσ
1
|M |2 dP S, or 2 =
dσ =
72s
dQ
Z
dP S|M |2
(2.2)
we need to compute the squared of the invariant amplitude |M |2 and the phase space, dP S. In the above
s is the center of momentum energy, s = (k1 + k2 )2 = Q2 . Let us start by calculating the invariant
amplitude for the various contributions.
2.1
Computing the cross-section
The cross-section is found by computing the appropriate diagrams. For this we clearly need the Feynman
rules for the QCD theory.
2.1.1
Feynman rules
As the Feynman rules can be found in every Quantum Field Theory book, we do not derive them here,
yet for completeness we do state them. For the massless photons and gluons we take the Feynman gauge,
in which the propagators have a simple form. The quark, photon and gluon propagators are given by
=
−i/
k
+ m2
k2
(2.3)
=
1 µν
η
p2
(2.4)
=
1
δab
q2
(2.5)
2.1 Computing the cross-section
15
while the gluon-quark-quark and photon-quark-quark vertices are given by
=
−ige γ µ δ(p1 + p2 + k)
(2.6)
=
−igs T a γ α δ(p1 + p2 + q)
(2.7)
(2.8)
We will not care about the non-abelian structure of QCD at the moment. Therefore we will ignore the
group generators T a in this chapter. However, these will be important in the next chapters.
The NLO terms have virtual and real contributions. In this section we will concentrate on the virtual
graphs; for a detailed calculation of the real part see for instance Pötter [8].
From a simple calculation, which is done further on, it is seen that only the exchange of a virtual
gluon between the incoming quarks (fig. 2.3) contributes, the two virtual graphs where a quark exchanges
a gluon with itself will just cancel each other. Therefore we only consider the former graphs from here
on.
The process is depicted as a cut graph in this figure. Cut diagrams are very useful graphical representations for calculating amplitudes. When calculating an amplitude, we take a diagram times the complex
conjugate, both with outgoing lines taken on-shell. Here the cut graphs come in. These represent just
the above; on the left of the cut, the amplitude can be calculated via the normal Feynman rules, yet
on the right the complex conjugate has to be taken. The lines which are cut are all on-shell, as these
represent outgoing particles. This representation will be used throughout this thesis where we calculate
amplitudes.
2.1.2
The invariant amplitude
For convenience, we have left out the outgoing leptons in figure 2.3. This can be done as they would yield
2N
the same overall factor to all orders. This factor can be calculated to be L(s, ) ≡ 8π2sαe 1−
[8]. This
overall factor should finally be reinserted in the result to obtain the final Drell-Yan cross-section.
To calculate the invariant amplitude squared for the virtual diagrams, |M (v) |2 , we use the Feynman
16
Chapter 2. The Drell-Yan process
k1
q
l
k2
Figure 2.3: The cut graph of the computed virtual gluon interaction. This is a NLO
term of the Drell-Yan process. The leptons which only contribute to a factor of L(s, )
have been omitted for convenience. The quark propagators between the gluon and photon
vertex have momentum k1 + l for the upper and k2 − l for the lower quark line.
rules for this theory, given above. We have:
1
1
dn l 1
T r[/
k 1 γα (/
k 1 + /l)γµ (/
k 2 − /l)γ α k/2 γ µ ].
n
2
2
(2π) l (k2 − l) (k1 + l)2
(2.9)
2
Using FORM [9] the trace can be calculated. This gives, up to order terms,
X
|M (v) |2 = −2igs2 µ2 e2 N CF L(s, )
Z
Tr[/
k 1 γα (/
k 1 + /l)γµ (/
k 2 − /l)γ α k/2 γ µ ]
=
+
+
8 s2 − 32(l · k1 )(l · k2 ) + 16(l · k1 )s − 16(l · k2 )s
8 −s2 + l2 s + 4(l · k1 )(l · k2 ) − 2(l · k1 )s + 2(l · k2 )s
82 −l2 s .
(2.10)
The integration over the gluon momentum is a thorough job. It will be done using Feynman parametrization and the identities
Z
dn l [l2 (l + k1 )2 (l − k2 )2 ]−1 = C0 (1, 2, 3)
(2.11)
Z
dn l [(l + k1 )2 (l − k2 )2 ]−1 = B0 (2, 3)
(2.12)
Z
s
dn l (k2 · l) [(l + k1 )2 (l − k2 )2 ]−1 = − B0 (2, 3),
(2.13)
4
where
C0 (1, 2, 3)
=
B0 (2, 3)
=
4π
Γ(1 − ) 1
i 1
+ ζ(2)
16π 2 s −Q2
Γ(1 − 2) 2
4π
Γ(1 − ) 1
i
+2
16π 2 −Q2
Γ(1 − 2) (2.14)
(2.15)
are standard integrals [?]. First let us rewrite the denominators as
l2 = D1 , (k1 + l)2 = D2 , (k2 − l)2 = D3 .
(2.16)
2.1 Computing the cross-section
17
Now the amplitude (2.9) can be written as
X
|M (v) |2
=
+
=
+
dn l
1
8 s2 + (D3 − D1 )(D2 − D1 ) − s(D3 + D2 )
n
(2π) D1 D2 D3
2
−s + D1 s − (D3 − D − 1)(D2 − D1 ) + s(D3 + D2 ) + 2 (−sD1 )
(2.17)
Z
n
2
d l
s
s
i32π 2 αs αe µ2 N CF L(s, )8
(1 − ) +
( − 1)
(2π)n D1 D2 D3
D1 D2
2
s
1
1
(l · k2 − l · k2 + 2) +
( − 1) +
(1 − ) +
( − 1) .
(2.18)
D2 D3
D1 D3
D1
D2
−2igs2 µ2 e2 N CF L(s, )
Z
Using the identities (2.11), (2.12), (2.13) and the fact that massless tadpoles like
contribution, thus
Z
dn l
1
(2π)n D1 D2
=
=
=
=
=
and in the same way
X
|M (v) |2
R
R
d4 k k12 have a vanishing
1
dn l 1
(2π)n l2 (k1 + l)2
Z 1 Z
1
dn l
dx
n (x(k + l)2 + (1 − x)l 2 )2
(2π)
1
0
Z 1 Z
1
dn l
dx
n (l2 + 2xl · k )2
(2π)
1
0
Z 1 Z
dn l
1
dx
(2π)n (l2 )2
0
0
Z
dn l D11D3 = 0, while at the same time
R
dn l D11 =
R
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
dn l D12 , we finally obtain
3
1
i32π 2 αs αe µ2 N CF L(s, )8 C0 (1, 2, 3)(s2 − s2 ) + B0 (2, 3)( s + s − 2 s) (2.24)
2
2
2 1
2
Γ(1 − )
31
4πµ
CF
s
+
+ 4 + ζ(2) .
(2.25)
= 2αs αe
−Q2
3
Γ(1 − 2) 2
2
=
It is important to note that we have not averaged over the spin, flux and colour yet. These averages yield
an overall factor of 1/8sN 2 with which |M (v) |2 has to be multiplied,
X
|M (v) |2 = −
1
αs
2π 2
4πµ2
−Q2
CF
Γ(1 − )
1
3
s 2+
+ 4 + ζ(2) ,
Γ(1 − 2) 2
(2.26)
where we have reinserted L(s, ).
For gluon emission the invariant amplitude can be computed likewise [8],
X
|M
(r) 2
2
| = 4αs µ CF
t
u
2sQ2
(1 − )( + ) +
− 2 .
u
t
tu
(2.27)
with u, t, s Mandelstam variables, defined as u = (k1 − l)2 , t = (k2 − l)2 and s = (k1 + k2 )2 like before.
18
Chapter 2. The Drell-Yan process
2.1.3
The Phase Space
R
2π
2
The next step is to compute the phase space. This phase space is just dP S(1) = 2π
s δ(1−Q /s) = s δ(1−
z) for the gluon loop [10]. For the gluon emission the phase space is less trivial, as we have now to consider
tow outgoing particles. In the center of momentum frame, the momentum n−vectors are give by k 1 =
√
√
√
√
( 2s , 0, . . . , 0, 2s ), k2 = ( 2s , 0, . . . , 0, − 2s ) for the incoming quarks, and l = (l 0 , 0, . . . , 0, |l| sin θ, |l| cos θ),
q = (q 0 , 0, . . . , 0, −|l| sin θ, −|l| cos θ) for the gluon and photon respectively. Here we have chosen the
quarks to propagate in the xn direction. For θ ∼ 0 and θ ∼ π the gluons propagate in this same
direction, and are hence collinear to the quarks.
For the case of gluon emission, the two particle phase space in n = 4 − 2 dimensions is given by
P S(2)
dn q dn l
2πδ(q 2 − Q2 ) 2πδ(l2 ) (2π)n δ (n) (l1 + l2 − l − q)
(2π)n (2π)n
Z ∞
Z 1
−
√
1 (4π)
d|l||l|1−2
d cos θ 1 − cos2 θ
δ(s − Q2 − 2 s|l|)
4π Γ(1 − ) 0
−1
Z
=
=
2
Introducing now the new variables z = Qs , y = 12 (1 + cos θ) we arrive at the following for the two particle
phase space:
Z
Z 1
1
4π
1
1−2
dP S =
z (1 − z)
dy[y(1 − y)]− .
(2.28)
8π Q2
Γ(1 − )
0
2.2
The resulting cross-section
Having computed the phase space, the cross-sections can be computed and added up. The virtual NLO
cross-section is easily seen to be
dσ ( v)
dQ2
=
=
=
Z
dP S|M ( v)|2
(2.29)
2 1
4πµ
2π
Γ(1 − )
δ(1 − z) 2 αs
s
CF
s
2π
−Q2
Γ(1 − 2)
αs CF 4πµ2
Γ(1 − ) 1
3
−
+
+4−
π
Q2
Γ(1 − 2) 2
2
−
1
3
+ 4 + ζ(2)
+
2
2
π2
δ(1 − z),
3
(2.30)
(2.31)
where we have used (−) = exp(iπ) = 1 + iπ − 12 π 2 2 + O(3 ), while the cross-section should be real,
thus the imaginary part has to be omitted.
Because of the remaining integral over y in the phase space, the NLO real cross-section is not so trivial.
2
The Mandelstam variables s, u and t can be expressed in terms of z and y by s = Qz , t = (k1 − l)2 =
2
2
2
Q
Q
Q
1
2
2 (1 − z )(1 − cos θ) = − z (1 − z)(1 − y) and u = (k2 − l) = − z (1 − z)y. These y’s will be integrated
2.2 The resulting cross-section
19
over when computing the cross-section:
dσ (r)
dQ2
=
=
Z
X
|M (r) |2
dP S
Z 1
αs CF 4πµ2
1
1−2
z (1 − z)
dy[y(1 − y)]− ×
2π
Q2
Γ(1 − )
0
1−y
y
−2 −1
−1
+ 2z(1 − z) y (1 − y) − 2 .
+
(1 − )
1−y
y
The y integral can be taken using the Beta function
dσ (r)
αs CF
=
dQ2
π
4πµ2
Q2
Γ(1 − )
Γ(1 − 2)
−
1
1
(2.32)
(2.33)
and yields
z (1 − z)1−2 + 2z 1+ (1 − z)−1−2 .
(2.34)
Now we have completed the calculation of the real and virtual NLO cross-section separately. We have
to add them up to hopefully get rid of the poles. In the real cross-section the powers of z can be rewritten
so that we get poles in which may cancel against those in equation (2.31). The 1 − z terms in equation
(2.34) can be rewritten in terms of the delta and plus distributions,
1
1
= − δ(1 − z) +
1+2
(1 − z)
2
1
1−z
+
− 2
ln(1 − z)
1−z
+
+ O(2 ).
(2.35)
Using this we see that the terms in square brackets in (2.34) become
1 z (1 − z)1−2 + 2z 1+ (1 − z)−1−2
(1 − z)2
2z
1 +
− z
(1 − z)1+2
(1 − z)1+2
1
1
− z (1 + z 2 )
(1 − z)1+2
"
#
1
1
ln(1 − z)
1
2
− 2
− (1 + ln z)(1 + z ) − δ(1 − z) +
2
1−z +
1−z
+
"
#
1
1 (1 + z)2
ln(1 − z)
1 + z2
2
ln z .
δ(1 − z) −
+ 2(1 + z )
−
2
(1 − z)+
1−z
1−z
+
−
=
=
=
=
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
The cross-section finally yields
dσ (r)
dQ2
=
αs CF 4πµ2
Γ(1 − )
×
π
Q2
Γ(1 − 2)
"
#
1
1 (1 + z)2
ln(1 − z)
1 + z2
2
ln z .
δ(1 − z) −
+ 2(1 + z )
−
2
(1 − z)+
1−z
1−z
+
(2.41)
In this form of the cross-section, double and single poles have appeared. These poles have a physical
1 The
Beta function is defined by B(α, β) =
R1
0
dyy α−1 (1 − y)1−β =
Γ(α)Γ(β)
Γ(α+β)
20
Chapter 2. The Drell-Yan process
explanation. When the outgoing gluon is collinear (θ → π or 0, thus y → 0 resp. 1), the integral over y
diverges. This divergence results in a 1/ pole after integration. In the same way, when the gluon is soft,
thus s → Q2 , and z → 1, another 1/ singularity appears. From the first term in eq. (2.35) we see that
this singularity results in a single pole in . The 1/2 double pole can be traced back to the integral C0 .
From this integral, eq. (2.11), it is clear that one of the ’s is the result of a soft pole, while the other
comes from a collinear one. The singularities in the cross-section come from specific momentum modes
of the gluons! This is an important observation, on which more will follow.
Using eq. (2.41) we can add the virtual and real terms and arrive at the following final form of the
cross-section:
dσN LO
dQ2
=
=
−
dσ (r)
dσ (v)
+
2
dQ
dQ2
Γ(1 − )
π2
3
αs CF 4πµ2
−
4
+
−
δ(1 − z)
π
Q2
Γ(1 − 2)
2
3
#
1 (1 + z)2
ln(1 − z)
1 + z2
2
+ 2(1 + z )
ln z .
−
(1 − z)+
1−z
1−z
+
(2.42)
(2.43)
The terms with double poles 1/2 cancel each other, and we only have to deal with single poles when
computing the cross-sections.
After grouping the 1/ terms, the above NLO cross-section can be written in a more convenient way
as
dσN LO
αs
=
dQ2
π
4πµ2
Q2
Γ(1 − )
Γ(1 − 2)
1
R(z) − Pqq .
(2.44)
Here Pqq are the Altarelli-Parisi splitting functions [11] which denote the possibility of finding a quark
inside a quark after gluon emission,
Pqq = CF
(1 + z)2
(1 − z)+
3
+ δ(1 − z)
2
(2.45)
and R(z) is defined as
R(z) = CF
"
π2
−4 +
3
2
δ(1 − z) + 2(1 + z )
ln(1 − z)
1−z
+
−
1 + z2
1−z
#
ln z .
(2.46)
In (2.44) the final form of the NLO Drell-Yan cross section is given. This cross-section still has to be
renormalized. This can be done, yet it is not in the scope of this thesis.
An important feature we have seen arise is that the singularities which we encounter when calculating
the NLO Drell-Yan cross-section, are the result of very specific momentum modes. These soft and collinear
modes will be in the center of this thesis.
We will return to this process at the end of the thesis, where we study the production of a Higgs
boson via gluon fusion. This process is very similar to Drell-Yan. We will see what the difference is when
using SCET in stead of calculating in full QCD, as we did in this chapter.
Chapter 3
Landau equations and power
counting
In this chapter we will look at the soft and collinear singularities in more physical way than we did before.
First we will look at what is needed for singularities to occur in the calculation of graphs and then we
will consider how to keep track of these singularities. Again we consider the Drell-Yan process, as we
have already established some theory for this. This chapter will roughly follow the reasoning of Eynck
[12].
3.1
Landau equations
For an arbitrary Feynman diagram, the amplitude G is given by
G=
L Z
Y
dd qi N (li , ps )
i=1
I
Y
1
2 − m2 + i .
l
j
j=1 j
(3.1)
Here N just stands for numerator, L is the number of loops, I the number of internal lines, p s the external
momenta and qr the momentum in loop r. The line momenta of the internal line j is denoted by l j .
Prefactors, like coupling constants, have been omitted, as these are not relevant for this discussion.
Using Feynman parameters1 the amplitude can be written in a more convenient way as
G = (I − 1)!
with D the denominator D =
PI
I Z
Y
i=1
j=1
1
i
0
dα δ(1 −
I
X
i=1
αi )
L Z
Y
j=1
d d lj
N (kj , ps )
DI
(3.2)
αj (kj2 (li , ps ) − m2 ) + i. Obviously, when D → 0 we will get
singularities, thus this is a necessary condition for them to occur. The denominator vanishes exactly
1 We can rewrite an integral containing several propagators with the same momentum using the method of Feynman
R1
R
R n
δ(1−α1 −α2 )
1
1
parameters. For three propagators this is done as dn l l12 (l+k)
d l (l+α (l+k)2 +α
2 (1−q)2 = 0 dα1 dα2
(l−q)2 )3
1
21
2
22
Chapter 3. Landau equations and power counting
when either αj = 0 or when the particle is on-shell, kj2 = m2j . Yet as the above integral is defined in
the complex αj and kjµ plane, we can use Cauchy’s theorem to deform the contour and thereby move far
enough away from the poles. Thus not all of these values yield actual singularities.
The denominator is quadratic in the momentum and can therefore result in at most two poles for the
kj2
= m2 singularity. Cauchy’s theorem now allows us to move the contour integral in the complex plane
away from the poles as long as the contour does not cross any singularities. This leads to a finite result,
as long as the two poles are well enough separated. However, in the case that the two poles coalesce,
with one pole in the upper and one in the lower half-plane, the contour cannot be deformed without
crossing either one of the poles. As the values of the momenta obviously have real values, the limit where
the poles approach the real axis “pinch” the contour integral. This leads to a singularity and is called a
pinch of the contour. For the single αj = 0 pole, we see that at the (fixed) endpoints of the contour, the
contour integral cannot be deformed away from the pole and therefore can result in a singularity as well.
As mentioned, the denominator D is a quadratic function of k. Therefore it can be seen as a parabol
which crosses the real axis (D = 0) at two points, represented by the two poles. For two poles which
coalesce and both approach the real line, this resembles approaching the minimum of the parabol which
can be found from ∂kµ D = 0. The necessary conditions for a singularity therefore are
D=0
and
∂D
=0
∂liµ
∀ i, µ.
(3.3)
These equations can be written in a more useful way as the so called Landau equations [1]
kj2 = m2j
X
or
αj
=
0,
αj kjµ jr
=
0
j∈ loop r
(3.4)
∀ j, r
(3.5)
where jr is the incidence matrix which ensures that the direction of the vector k jµ corresponds to the
direction of the contour,
jr


 +1
=
−1


0
when kjµ flows in the same direction as lr
when kjµ flows in the opposite direction of lr
(3.6)
otherwise.
These Landau equations hold for any diagram at any order and thus are a strong requirement for occurrence of pinches.
Let’s give a simple example of the use of these Landau equations. Consider the decay of a photon into
a quark-antiquark pair, as shown in figure 3.1. This should be interpreted as the decay of a virtual photon
into two partonic jets, which consist of a set of partons (e.g. quarks, gluons) moving nearly parallel to
each other. The one-loop correction to this is physically seen as the exchange of a gluon between the two
3.2 The Coleman-Norton method
23
(a)
(b)
(c)
(d)
Figure 3.1: The process of photon decay into two quarks and its one-loop corrections.
partonic jets and is proportional to [12]
G=
Z
d
d l
Z Y
i
dαi
1
δ(1 − α1 − α2 − α3 )
D3
(3.7)
where D = α1 l2 + α2 (p1 + l)2 + α3 (p2 − l)2 . From the Landau equations it follows that D = 0 and
α1 lµ + α2 (p1 + l)µ − α3 (p2 − l)µ = 0.
(3.8)
There are three solutions to the above, which can be categorized into one soft solution where the gluon
momentum vanishes and two collinear solutions for which the gluon momentum is proportional to p 1 or
p2 :
lµ
=
0
lµ
=
lµ
−ζpµ1 ,
=
and
ζ 0 pµ2 ,
α3
α2
=
=0
α1
α1
α1 ζ = α2 (1 − ζ),
α1 ζ 0 = α3 (1 − ζ 0 ),
(3.9a)
α3 = 0
(3.9b)
α2 = 0.
(3.9c)
If in the collinear limit, eqs. (3.9b) and (3.9c), the proportionality constant vanishes, ζ → 0, there is
both a soft and collinear singularity. This corresponds to the −2 term in the Drell-Yan process, which
we already established in the previous chapter, (2.41).
3.2
The Coleman-Norton method
For the above case of a three-point vertex this is rather simple, yet to do this for more complicated
diagrams would be rather laborious. Therefore we would like to have a more general method for finding
solutions to the Landau equations. The searching for pinches is simplified significantly by Coleman and
Norton [2], who designed a graphical representation.
Let’s first define
∆xµi = αi liµ
(3.10)
where αi liµ is identified with the spacetime vector ∆xµi . Here liµ is the on-shell line momentum like before.
24
Chapter 3. Landau equations and power counting
Figure 3.2: The reduced diagram of the collinear singularity in the first order QCD
correction to the decay of a photon into two quarks, of which the normal graph was shown
in fig. 3.1(c). The off-shell quark line with momentum p2 − k is contracted to a point.
There is a similar diagram where the line with momentum p1 + k is contracted to a point.
If we also identify the Feynman parameter αi as
αi =
∆x0i
li0
(3.11)
we can write this vector as ∆xµi = ∆x0i viµ , with viµ = (1, ~li /li0 ). This can be interpreted as follows:
on-shell particles are particles propagating freely along classical trajectories with velocity v iµ , and the
vertices are spacetime points separated by ∆xµi .
Looking at the collinear diagrams, for which (3.9b) and (3.9c) hold, we find for the two on-shell quark
and gluon lines, with momenta and l1 = p1 − k and l2 = k respectively, that
∆xµp1 −k = α2 (p1 − k)µ = α1 k µ = ∆xµk .
(3.12)
This shows that in the collinear limit the on-shell partons travel the same distance. We note that for
off-shell lines the Landau equations yield αi = 0. These off-shell lines are thus contracted to points in the
Coleman-Norton picture. This leads to reduced diagrams (figure 3.2), which describe a physical process:
in the above example two massless particles with momenta p1 − k and k are created at the first vertex
and then propagate freely to the second vertex where they recombine into the outgoing particle with
momentum p1 . The method described above can be generalized to arbitrary diagrams, by considering a
graph with a loop consisting of n lines. For physical reasons, the spacetime separation of one line (e.g.
the vector between vertex 1 and 2, defined as ∆x1,2 ) must equal that of all other n − 1 lines,
∆x1,2 + ∆x2,3 + . . . + ∆xn,1 = 0.
(3.13)
By the identification (3.10), this is identical to the second Landau equation.
We have shown that we can use the Coleman-Norton representation for general diagrams. This
representation is straightforward in use, as you can find pinches by just contracting off-shell lines to
points. However, one must be careful with the result, as it may not represent a physical process. This is
illustrated below for the soft pinch.
When applying the Coleman-Norton method to the soft limit of the one loop diagram, the diagrams
in figure 3.3 are obtained. Diagram 3.3(b) is excluded as the two jets created at the first vertex propagate
3.3 Infrared power counting
(a)
25
(b)
(c)
Figure 3.3: (a) The trivial diagram of the first QCD correction to the decay of a photon
into two quarks. (b) Reduced diagram of this same process, where the off-shell gluon line
is contracted to a point. This is a disallowed reduced diagram, because the quark jets are
freely propagating and thus cannot rejoin at some other point in spacetime. (c) Reduced
diagram where the gluon has tachyonic properties, thus do not represent a physical process.
freely and thus can never rejoin at a second vertex. In the third diagram a particle is created at the vertex,
follows a curve in spacetime and then returns to the same spacetime point. As mentioned, the vertices
in the Coleman-Norton picture represent spacetime points. Physical particles cannot follow spacelike
trajectories, therefore this diagram is excluded as well. The only reduced diagram which represents a
physical process is the trivial one, figure 3.3(a).
3.3
Infrared power counting
In the previous section we discussed the necessary conditions for singularities which come in the form
of the Landau equations. Yet to find sufficient conditions for the occurrence of singular behaviour in
Feynman diagrams, infrared power counting is used. As can be expected, this is similar to the ultraviolet
power counting used in the analysis of ultraviolet divergences (see for instance [13, 10]). Using this
technique we may both identify regions in the momentum space which result in infrared divergences and
at the same time use it to identify infrared safe regions by putting bounds on the integrals.
3.3.1
Pinch surfaces
Let’s first introduce so-called pinch surfaces. The denominator D(ki , αi ) vanishes on a set of points in
(k, α) space. These points construct a surface in this space, yet not all of these surfaces will lead to
a singularity. Pinch surfaces are the surfaces in the (k, α) plane which do lead to such a singularity.
These pinch surfaces are defined in terms of normal and internal variables. The internal variables lie
in the surface and can be manipulated (e.g. by scaling) without changing the pinch. For instance, for
a collinear pinch rotation around the direction of propagation is an internal variable. By this rotation
the gluon remains close to the parton jet, thus remaining collinear. On the other hand, changing the
normal variables, which point out of the surface, will lead to leaving the pinch surface. The integral is
only singular in the normal variables κi , the internal variables just parameterize the pinch surface.
26
Chapter 3. Landau equations and power counting
To illustrate this, consider again the collinear pinch of the by now well known three-point vertex
(figure 3.2) where the gluon momentum k µ is collinear with pµ1 (3.9b). We will work in the rest frame of
the photon and with the outgoing parton jets in the ±z direction,
Q
Q
pµ1 = ( √ , 0, 0, √ )
2
2
Q
Q
pµ2 = ( √ , 0, 0, − √ )
2
2
(3.14)
When scaling the collinear gluon with momentum k + along the + direction, or rotating it by the azimuthal
angle φ of k ⊥ about the z−axis, it will remain collinear. k + and φ are therefore the internal variables.
The perpendicular component of the momentum, (k ⊥ )2 is obviously a normal component as it points
away from the collinear direction, but what about k − ? The denominator for the diagram, where p1 ∼ p+
1
and p2 ∼ p−
,
is
2
D = (p1 − k)2
=
=
=
−
−
⊥
⊥ 2
+
2(p+
1 − k )(p1 − k ) − (p1 − k )
+
−
⊥ 2
2(p+
1 − k )(−k ) − (k )
Q
−2 √ k − ,
2
(3.15)
where the internal variables are small compared to Q. So we see that k − is a normal variable as well.
3.3.2
The power counting procedure
Introducing a scaling variable λ as κi = λκ0i , where the ratio κi /κ0i is hold fixed, the pinch surface is
approached by taking λ → 0. Just like in ultraviolet power counting this results in having a factor of λ ns
in the integrand, which produces the singularity. Only for a non-positive n s , a singularity will occur.
This exponent ns consists of several contributions. The terms in the integral (3.1) which contribute
to ns are the line momenta li (κi , λ) in the denominator and numerator, and the measure dqi . Let us
call aj the contribution of the measure dqj , bj the contribution of the denominator D(kj , mj ) and c the
power of λ from the momenta in the numerator. The integral then becomes the following homogeneous
integral
YZ
YZ
n(κ0j , mj ) P (aj −bj )+c
n(κj , mj )
→
λ j
,
(3.16)
dd κ0j
dd κj
D(κj , mj )
D(κ0j , mj )
j
j
where the degree of divergence is ns =
behaves:
P


 >0
ns
=0


<0
j (aj ) − bj ) + c.
This degree of divergence shows how the integral
: finite integral
: logarithmic divergent integral
(3.17)
: power divergent integral.
We can now use the above to see whether an integral is divergent. This will be illustrated again by
the example of a decaying photon. Scaling the normal coordinates as described by the power counting
prescription, k − = λk 0− and (k ⊥ )2 = λ(k 0⊥ )2 , the degree of divergence can be found. The denominator
3.3 Infrared power counting
27
(3.15) is composed of the terms
(p1 + k)2
(p2 − k)
k2
2
=
=
=
−
+ −
⊥ 2
2p+
1 k + 2k k − (k )
+
−2p−
2k
+ −
+ −
+ 2k k − (k⊥)
2k k − (k ⊥ )2
2
∼
∼
∼
λ
(p1 + k 0 )2
(p2 − k 0 )2
λ k 02
(3.18)
where we focus on the collinear singularity (p1 + k)2 → 0 and thus (p2 − k)2 is finite. The measure yields,
after scaling a factor λ2 : dd k = dk + dk − dk ⊥ (k ⊥ )d−3 dΩ ∼ λ2 dd k 0 . This results in an overall degree of
P
divergence of ns = j (aj ) − bj ) + c = 0 − 2 + 2 = 0, which results in a divergent integral.
In this chapter we have shown a method for finding singularities in an easy way by using the ColemanNorton method, which together with the Landau equations shows from which momentum modes the
singularities are a result. Keeping track of these singularities is done by introducing a scaling factor λ, and
using powercounting in a small parameter λ. An important observation is that the infrared divergences
which occur are the result of soft and collinear gluons which are radiated off the propagating quark.
These soft and collinear degrees of freedom should therefore be described in an effective field theory,
which is introduced in the following chapter. The powercounting technique discussed in this chapter is
very important for this soft-collinear effective theory (SCET), yet it is used in a slightly different way.
We will now turn to this effective theory.
28
Chapter 3. Landau equations and power counting
Chapter 4
Intro SCET
Many interesting processes which are studied in particle colliders involve hadrons with energies Q much
larger than their mass and thus have Q ΛQCD , the QCD scale. These processes can be described
using SCET [3, 14, 4, 5], an effective theory which treats different momentum modes in a separate way.
It describes the interaction between on collinear quarks on the one hand and soft and/or collinear gluons
on the other. As SCET is an effective theory, is will not be valid for all energy regions. We have to match
the SCET operators onto the full QCD operators at an appropriate scale, where we need both theories
to still be valid.
In this chapter we will lay down the foundations of the soft-collinear effective theory (SCET). To
do this, we will use some of the techniques we already encountered in the previous chapters and use
the appendices B on lightcone coordinates, C on the Heavy Quark Effective Theory (HQET) and D on
Wilson lines. The theory established in this chapter will be used in the next three chapters which discuss
specific processes. We will first show how power counting can be used to classify the various interactions
in QCD. Then the SCET Lagrangian will be derived, and...
4.1
Momentum modes
We distinguish several different momentum modes for particles. Although we already encountered these
different modes in the previous chapters, we will give a brief summary here.
A particle which has a momentum p ∼ Q, with Q the hard scale of the process, is called a hard
particle. If its momentum however is a factor λ 1 smaller, p ∼ Qλ, it is called soft, while for p ∼ Qλ 2
it has ultrasoft momentum. In the beginning of this chapter, we will be mostly concerned with particles
with collinear momenta. Collinear particles have momentum along a specified direction. We choose the
momentum of collinear quarks on the lightcone, thus in the ±z direction, or equivalently the n µ or nµ
direction. These lightcone directions are defined as
nµ = (1, 0, 0, −1),
29
nµ = (1, 0, 0, 1).
(4.1)
30
Chapter 4. Intro SCET
For an arbitrary vector v we have standard + and − components, n·v = v 0 +v 3 ≡ v + , n·v = v 0 −v 3 ≡ v − .
Thus the + direction is equivalent to the n direction while the − direction is equivalent to the n direction. 1
Such a vector v can then be written as
vµ
=
=
nµ
nµ
µ
n · v + v⊥
n·v+
2
2
(v + , v − , v ⊥ ).
(4.2)
(4.3)
This and more on the lightcone coordinates is also found in appendix B, where we find some useful
identities for these coordinates. The momentum of collinear particles is defined as p µc ∼ Q(λ2 , 1, λ), thus
which means p2c ∼ Q2 λ2 . A summary of this is shown in table 4.1.
The values of the factor λ is obviously process dependent. For some processes, like the B → q
Xsγ
Λ
2
2Λ
decay discussed in chapter 5, the collinear momentum goes as p = QΛ = Q Q . This leads to λ = Q
.
2
2
Yet other processes, e.g. the decay of a B meson into a D meson and a pion, which have p = Λ ,
Λ
. These two different values lead to different theories, SCETI and SCETII respectively. We
or λ = Q
will start by introducing SCETI , as SCETII can be derived straightforwardly via this. The differences
between the two theories are summarized here once more:
s
)
Collinear d.o.f.
Λ
SCETI : λ =
Q
Usoft d.o.f.
SCETII
Λ
:λ=
Q
)
Collinear d.o.f.
Soft d.o.f.
p2c ∼ QΛ
p2us ∼ Λ2
(4.4)
p2c ∼ Λ2
(4.5)
p2s ∼ Λ2
The scaling of a n-collinear particle can be understood physically (from here on, when omitting the
direction of the collinear particle, we understand it to be in the n direction.) Let us look at a particle
in rest with momentum pµ ∼ (Λ, Λ, Λ), where we have abbreviated Λ = ΛQCD . When such a particle
with energy Q gets a boost in direction of n, the momentum components transforms as p + → Λ/Q p+ ,
p− → Q/Λ p− , p⊥ → p⊥ and thus the momentum transforms as
pµ ∼ (Λ, Λ, Λ) → pµ ∼ (
Λ2
Λ
, Q, Λ) = Q(λ2 , 1, λ), where λ = .
Q
Q
(4.6)
For the nµ direction the momentum would transform as pµ ∼ Q(1, λ2 , λ). In the above example, the
particle would thus be described by a SCETII field.
4.2
Power Counting in SCET
To classify the various interactions in QCD between a quark and a gluon, it is convenient to analyze the
incoming and outgoing momenta of these interactions in the various regions in momentum space which
SCET covers. There are clearly many different interactions which can occur, we will look at several of
1 An alternative way of writing these directions, which is used in some of the literature (e.g. Beneke [15]), is n
+ and n−
for n and n respectively.
4.2 Power Counting in SCET
31
Mode
Hard
n-Collinear
n-Collinear
Soft
Usoft
pµ
Q(1, 1, 1)
Q(λ2 , 1, λ)
Q(1, λ2 , λ)
Q(λ, λ, λ)
Q(λ2 , λ2 , λ2 )
p2
Q2
Q 2 λ2
Q 2 λ2
Q 2 λ2
Q 2 λ4
Table 4.1: The scaling of the various possible momentum modes.
these in more detail to see how different momentum modes are added after an interaction, and what the
resultant mode is. When a quark emits or absorbs a gluon, the momentum will decrease respectively
increase. This can result in off-shell modes of the outgoing particles.
When a collinear quark with momentum pc interacts with a collinear gluon (kc ) in the same direction,
it clearly stays collinear. This is confirmed by adding the momenta: pc + kc ∼ Q(λ2 , 1, λ) + Q(λ2 , 1, λ) ∼
Q(λ2 , 1, λ), which is again an n-collinear momentum and thus the quark remains to be on-shell. This is
therefore a local interaction. In the same way we can analyze what happens when a usoft quark, like
a heavy quark hv from the Heavy Quark Effective Theory which we will come back to, interacts with
a collinear gluon. The usoft quark will obtain a large momentum from the gluon, throwing it off-shell,
as pus + kc ∼ Q(λ2 , λ2 , λ2 ) + Q(λ2 , 1, λ) ∼ Q(λ2 , 1, λ). The quark can now no longer be represented
by the HQET field hv , and has to be integrated out by using the equation of motion. This will lead to
an auxiliary field ψH = (W − 1)hv [14], where W is a Wilson line, which are discussed in section 4.6.
These and several other interactions between quarks and gluons are given below in a summary, with an
indication of the momentum mode of the outgoing particle.
• A collinear quark with pµ ∼ Q(λ2 , 1, λ) interacts with a collinear gluon, k µ ∼ Q(λ2 , 1, λ). The
momentum (p + k)µ of the outgoing quark will trivially remain to be collinear.
collinear
k
(p + k)µ ∼ Q(λ2 , 1, λ) + Q(λ2 , 1, λ) ∼ Q(λ2 , 1, λ)
collinear
p
(4.7)
p’
• A collinear quark interacting with a usoft gluon, k µ ∼ Q(λ2 , λ2 , λ2 ) yields an outgoing quark which
remains collinear.
usoft
k
(p + k)µ ∼ Q(λ2 , 1, λ) + Q(λ2 , λ2 , λ2 ) ∼ Q(λ2 , 1, λ)
collinear
p
p’
(4.8)
32
Chapter 4. Intro SCET
• A usoft quark with a usoft gluon produces a usoft outgoing quark:
(p + k)µ ∼ Q(λ2 , λ2 , λ2 ) + Q(λ2 , λ2 , λ2 ) ∼ Q(λ2 , λ2 , λ2 ) (4.9)
• A heavy quark hv with usoft momentum interacting with a collinear gluon results in an off-shell
momentum for the outgoing quark.
(p + k)µ ∼ Q(λ2 , λ2 , λ2 ) + Q(λ2 , 1, λ) ∼ Q(λ2 , 1, λ)
(4.10)
• An incoming soft quark interacting with a collinear gluon produces an off-shell quark.
(p + k)µ ∼ Q(λ, λ, λ) + Q(λ2 , 1, λ) ∼ Q(λ, 1, λ)
(4.11)
The last figure shows an incoming soft quark which interacts with a collinear gluon. The soft quark is
described by SCETII and has as mentioned a momentum of ps ∼ Q(λ, λ, λ). When it absorbs a collinear
gluon, the momentum mode is no longer collinear and has to be integrated out. This is not in the scope
of this thesis, but is shown explicitly in [14].
4.3
The collinear SCET Lagrangian
To find the effective Lagrangian for the collinear and usoft degrees of freedom, we will start with the full
QCD Lagrangian. This QCD Lagrangian is composed of a quark and a gluon part,
1
gluon
µν
a
/
LQCD = Lqq
QCD + LQCD = ψiDψ − Tr[Gµν Ga ] + Lgf .
2
(4.12)
4.3 The collinear SCET Lagrangian
33
Here the covariant derivative D µ and the gluon field tensor Gµν are given by
Dµ = ∂ µ − igs Aµ
i
Gµν = [Dµ , Dν ]
gs
(4.13)
(4.14)
with Aµ the gluon field and as usual the gauge fixing Lagrangian Lgf is introduced to have a well defined
gluon propagator.
This QCD Lagrangian is obviously valid for quarks and gluons with every kind of momentum, yet
we can write it in a simpler form for particles with collinear degrees of freedom. Because of the specific
direction these particles have, various terms will be less significant than others by the power counting
regime mentioned. This will become clear in the next subsections, when we derive first the quark Lagrangian in its SCET form and then rewrite the gluon Lagrangian. The (u)soft particles SCET describes
do not have a specific direction. They are therefore correctly described by the usual QCD Lagrangian.
4.3.1
The quark Lagrangian
/n
/
n
nn
For collinear quarks along the n direction, we can use the projection operator /4 + 4/ = 1 derived in
appendix B to split the heavy quark field ψ up into a large and small 4-component field:
ψ=
/
/n
n
n
/
/n
+
4
4
ψ = Pn ψ + Pn ψ ≡ ξn + ξn .
(4.15)
An arbitrary massless QCD spinor is given in Dirac representation [16] by
u
1
u(p) = √
2
!
~
σ ·~
p
p0 u
1
v(p) = √
2
,
~
σ ·~
p
p0 v
v
!
(4.16)
with ~σ · p~ = σ 1 p1 + σ 2 p2 + σ 3 p3 with σ i the Pauli matrices (A.4). Yet collinear spinors have a large
p
3
momentum in n direction, n · p = p0 + p3 = Q/2 + Q/2, thus ~σp·~
0 ∼ σ . The collinear particles then have
u
1
un = √
2
!
(4.17)
σ3 u
while the anti-particles have
σ3 v
1
v(p) = √
2
v
!
.
(4.18)
If we now act on this with projection Pn , of which the matrix form is given in (B.11a), we get
1
Pn u n = √
2 2
1
σ3
σ3
1
!
·
u
σ3 u
!
1
=√
2
u
σ3 u
!
= un ,
(4.19)
34
Chapter 4. Intro SCET
while Pn un = 0. In the same way we see that Pn vn = vn , Pn vn = 0. This indicates that Pn working on
the n-collinear QCD spinors gives n-collinear spinors back, thus Pn working on the QCD quark field ψ
describes the collinear quark in direction n. This is thus the large component, while ξ n will be small. It is
noted that we only discuss an n-collinear particle here. The Lagrangian for n-collinear particles is easily
obtained from this one by substituting n ↔ n in the result. The quark Lagrangian can be rewritten in
terms of these ξn and ξn as
Lqq
=
=
/ (ξn + ξn )
ξ n + ξ n iD
n
/
n
/
/ ⊥ (ξn + ξn ) .
ξn + ξn
in · D + in · D + iD
2
2
(4.20)
(4.21)
This yields twelve terms. By using identities of the projection operator some of these will vanish. The
/ (in · D)ξn = 0, and
nPn ψ = 0 and n
derivative does not contain any γ matrices, thus n
/ (in · D)ξn = (in · D)/
the hermitian conjugates of these vanish as well. However these four terms are not the only vanishing
terms. To see this we write out the hermitian conjugate of ξ using Pn ,
/
n
ξ n i n · D ξn
2
=
=
=
/
n
ψ † Pn† γ 0 i n · Dξn
2
/
n
ψ † Pn γ 0 i n · D ξn
2
/
n
ψ † γ 0 Pn i n · D ξ n
2
(4.22)
(4.23)
(4.24)
/
nn
/ = 0. Thus
/ = ( /4 )n
which equals zero as Pn n
/
n
ξ n i n · D ξn = 0
2
(4.25)
n
/
ξ n i n · D ξn = 0.
2
(4.26)
and similarly
This gets rid of another two terms. Two of the terms containing the perpendicular part of the covariant
derivative vanish as well:
/ ⊥ ξn
ξ n iD
=
/ ⊥ Pn ψ
ψ † Pn† γ 0 iD
(4.27)
/ ⊥ Pn ψ
ψ † γ 0 P n iD
(4.28)
=
/ ⊥ Pn Pn ψ
ψ γ iD
(4.29)
=
0.
(4.30)
=
† 0
/ does
as Pn Pn = 0. In this derivation, the third line comes from the fact that the perpendicular part of D
0
3
0
3
0
3
/ ⊥ and Pn = n
/ = (γ + γ )(γ − γ ) commute.
not contain any γ or γ matrices by definition, thus D
/n
4.3 The collinear SCET Lagrangian
35
This leaves us with
/
n
n
/
/ ⊥ ξn .
/ ⊥ ξn + ξ n i D
Lqq = ξ n in · Dξn + ξ n in · Dξn + ξ n iD
2
2
(4.31)
To integrate the small component ξn out, we need the equation of motion for this field. We have
δ
n
/
/ ⊥ ξn = 0
Lqq = (in · D) ξn + iD
δξn
2
By multiplying this with
/
n
2
/n
n
/
22
(4.32)
and using Pn ξn = Pn Pn ψ = Pn ψ = ξn (equation B.16) we obtain
(in · D) ξn +
/
n
/ ξn
iD
2 ⊥
thus
in · Dξn +
=
(in · D) Pn ξn +
=
(in · D) ξn +
/
n
/ ξn
iD
2 ⊥
/
n
/ ξn
iD
2 ⊥
/
n
/ ξn = 0,
iD
2 ⊥
(4.33)
(4.34)
and the equation of motion becomes
ξn =
/
1
n
/ ⊥ ξn .
iD
in · D
2
(4.35)
The (covariant) derivative in the denominator may look strange, but can be understood in momentum
space as:
Z
e−ip·x
1
= d4 p
.
(4.36)
in · ∂
n·p
By replacing ξn via the equation of motion, the Lagrangian finally becomes
/⊥
Lqq = ξ n in · D + iD
1
/
iD
in · D ⊥
/
n
ξn .
2
(4.37)
We have arrived at a local Lagrangian for the n-collinear quarks. Yet not all of the terms in (4.37) are
equally important. In other words, we should look at how the various terms scale with λ to find the
leading order (LO) quark Lagrangian. To do this, we first need to separate the large from the smaller
momenta and define the label operator P µ .
4.3.2
The label operator
The momentum of an incoming heavy quark can be split up in two parts, a large and a small part,
respectively the label momentum p̃µ and the residual momentum k µ .2 Using this, the quark field can be
P
expanded as ξn = p̃ e−ip̃·x ξn,p̃ . We can now define an operator P µ , the label operator [4], which acts
2 This
[7].
use of label momenta is analogous to the use for non-relativistic quarks in NRQCD [17] and heavy quarks in HQET
36
Chapter 4. Intro SCET
on fields φqi , φpj with momenta qi , pj :
µ
) φ†q1 · · · φ†qm φp1 · · · φpn .
P µ φ†q1 · · · φ†qm φp1 · · · φpn = (p̃µ1 + · · · + p̃µn − q̃1µ − · · · − q̃m
(4.38)
The operator P µ operating on a field φ extracts the large momenta out of this field, but leaves the
(small) residual momentum inside the field. This label operator itself can be written as the sum of two
µ
µ
operators, one in the − and one in the ⊥ direction, P and P⊥ respectively: P µ = n2 P + P⊥
. The
0
operator P scales as λ , while the perpendicular label operator P⊥ produces labels of order λ. From
momentum conservation the label momenta in an interaction must be conserved as well. This will be an
extra restriction on the interaction.
With this operator in hand, a useful identity is found:
∂µ
X
e−ip̃·x ξn,p̃ =
p̃
X
e−ip̃·x (P µ + ∂ µ ) ξn,p̃ .
(4.39)
p̃
The derivative acting here on the fermionic field acts only on the residual momenta and is thus of order
λ2 . This can be used together with the expansion of the collinear field in the label momenta, to arrive
at
Lqq =
X
e
−ip̃·x
ξ n,p̃
p̃
1
/⊥
/
in · D + iD
iD
in · D ⊥
/
n
ξn,p̃
2
(4.40)
or just
/⊥
ξ n,p0 in · D + iD
1
/
iD
in · D ⊥
/
n
ξn,p
2
(4.41)
where we have suppressed the summation and have dropped the exponent. The exponent assures that the
label momentum is conserved, thus as long as we remember to conserve label momentum this suppression
is justified. Also we understand that by the subscript p we mean only dependence of ξ on the label p̃.
4.3.3
The gluon field
Up till now we have ignored with the covariant derivative. Apart from derivatives and label operators
this covariant derivative also contains gluon fields. The gluon field A µ consists of all the momentum
modes, thus (u)soft, collinear and hard modes. As mentioned we will first concentrate on SCET I , hence
at the usoft degrees of freedom. The gluon field can be written as Aµ = Aµus + Aµn,q , where An,q is a gluon
field collinear to the quark moving in the n-direction with label momentum q, as above. Hard gluons are
not included, as these and other hard degrees of freedom are integrated out, into the Wilson coefficients
C(P), which are discussed further on in the thesis.
The covariant derivative was defined as
iDµ = i∂ µ − gs Aµ ∼ i∂ µ − gs Aµus − gs Aµn,q .
(4.42)
This implies that in the quark Lagrangian, equation (4.37), there is an quark-gluon interaction present.
The various terms in this Lagrangian contribute different powers of λ. Using the label operator and thus
4.3 The collinear SCET Lagrangian
37
the phase redefinition i∂ µ → i∂ µ + P µ , we get the following from power counting the terms:
in · D = in · ∂ + gs n · An,q + gs n · Aus ∼ λ2
(4.43)
which are all of order λ2 ,
in · D
/⊥
iD
=
in · ∂ + P + gs n · An,q + gs n · Aus
(4.44)
∼
P + gs n · An,q + O(λ)
(4.45)
=
∼
/ n,q,⊥
/ us,⊥ + gs A
/ ⊥ + gs A
i∂/⊥ + P
(4.46)
/ ⊥ + gs A
/ n,q,⊥ + O(λ2 )
P
(4.47)
(4.48)
The leading order terms in these covariant derivatives are defined as
P + gs n · An,q
/ n,q,⊥
/ ⊥ + gs A
P
≡
≡
in · D c ∼ 1
c
/⊥
iD
(4.49)
∼ λ,
(4.50)
where the order in λ is given. The leading order Lagrangian L(0) must be of order λ4 3 . Using the above
(0)
approximations for the various terms in the Lagrangian, we arrive at the LO Lagrangian L qq :
L(0)
qq
= ξ n,p0
in · D +
c
/⊥
iD
1
c
/⊥
iD
c
in · D
/
n
ξn,p
2
(4.51)
Note once more that this is an effective Lagrangian, only valid for collinear quarks.
4.3.4
Collinear gluon Lagrangian
From the QCD Lagrangian (4.12) it is known that the gluon term is − 12 Tr[Gaµν Gµν
a ], where Gµν =
i
gs [Dµ , Dν ]
is the gluon field tensor, plus the gauge fixing term. Thus
Lcg =
1
1
Tr[Dµ , Dν ]2 + Tr{[iDµ , Aµ ]2 }.
2gs2
α
(4.52)
The SCET form of the gauge-fixing term is derived in the same way as the normal part. This will not
(0)
yield anything interesting, thus we concentrate on the first term. The LO Lagrangian L cg should again
be of order λ4 . Let us first calculate the trace in (4.52) over the gauge group generators T a . These will
not produce factors of λ as seen most easily for the collinear gluon interaction term:
Tr{[gs Aµn,q , gs Aνn,q0 ]2 }
=
∼
0
0
0
gs4 Tr[ta , tb ][T a , T b ]Aµ,a Aν,b Aµ,a Aν,b
0 0
0
0
gs4 f abc tc f a b c T c Aµ,a Aν,b Aµ,a Aν,b .
3 Once more we note that the LO Lagrangian must be of order 1, yet as L =
λ−4 , the LO Lagrangian density L should be of order λ4 .
R
0
(4.53)
(4.54)
d4 x L and the measure yields a factor
38
Chapter 4. Intro SCET
The gluon fields contract pairwise (Aµ Aµ ∼ λ2 ) and thus lead to a factor of λ4 , proving the statement
that the trace does not lead to O(λ) terms.
In equation (4.52) the square of the commutator is taken. This means that while the Lagrangian
is of order λ4 , the commutator itself will be only of order λ2 . However, just like the case of the quark
Lagrangian, not all terms present will contribute to the leading order gluon Lagrangian.
nµ
2 n
Writing out Lcg using D µ =
L=
1
Tr{[
2gs2
+
nµ
2 in
·∂+
nµ
2 gs n
nν
2 in
+
·D+
· Aus +
·∂+
nν
2 gs n
nµ
2 P
nν
2 P
· Aus +
nµ
2 n
+
µ
· D + D⊥
yields a terrible mess:
nµ
2 gs n
nµ
2 gs n
+
nµ
2 gs n
· An,q +
nµ
2 in
·∂+
nµ
2 P
µ
µ
· An,q + i∂⊥
+ P⊥
+ gs Aµus,⊥ + Aµn,q,⊥ ,
nν
2 gs n
nν
2 gs n
· Aus +
· Aus +
nν
2 gs n
· An,q +
nν
2 in
·∂+
nν
2 P
ν
ν
+ gs Aνus,⊥ + Aνn,q,⊥ ]2 }.
+ P⊥
· An,q + i∂⊥
(4.55)
This can be simplified by noting that n2 = n2 = n · V⊥ = n · V⊥ = 0. The only non-vanishing terms are
those which consist of a commutator like [nµ , nν ]2 or [V⊥µ , V⊥ν ]2 (with V a vector):
L
=
+
+
1
nµ
nµ
nµ
nν
nν
nν
nν 2
nµ
P+
Tr{[ in · ∂ +
gs n · Aus +
gs n · An,q ,
in · ∂ +
gs n · Aus +
gs n · An,q +
P]
2gs
2
2
2
2
2
2
2
2
nµ
nµ
nµ
nµ
nν
nν
nν
nν
P+
gs n · Aus +
gs n · An,q ,
in · ∂ +
gs n · Aus +
gs n · An,q ]2
[ in · ∂ +
P+
2
2
2
2
2
2
2
2
µ
µ
ν
ν
(4.56)
+ gs Aνus,⊥ + Aνn,q,⊥ ]2 }.
+ P⊥
[i∂⊥
+ P⊥
+ gs Aµus,⊥ + Aµn,q,⊥ , i∂⊥
As mentioned the commutators should be of order λ2 to contribute to the LO Lagrangian. Thus the two
terms in the commutator should be either both of order λ or one of the of order λ 2 while the other is of
order 1, i.e. commutators like
[i
nµ
nµ
n · D, gs n · An,q ]2 ∼ [λ2 , 1]2 = O(λ4 )
2
2
(4.57)
and
µ
[P⊥
, Aνn,q,⊥ ]2 ∼ [λ, λ]2 = O(λ4 )
(4.58)
(0)
are contained in Lcg , while the commutator
[i
nµ
nµ
n · ∂, i n · ∂]2 ∼ [λ2 , λ2 ]2 = O(λ8 )
2
2
(4.59)
definitely is not. By working straightforward through every combination of terms in (4.56) in this way,
we obtain the final leading order gluon Lagrangian, which can be written as
L(0)
cg
=
iD̂µ
≡
1
1
Tr{[iD̂µ + gs Aµn,q , iD̂ν + gs Aνn,q ]2 } + Tr{[iD̂µ , Aµn,q ]2 } + O(λ5 )
2
2gs
α
µ
nµ
n
µ
i n · Dus + P⊥
+
P,
2
2
(4.60)
(4.61)
4.4 Feynman rules for SCETI
39
where we have included the gauge-fixing term. We note that the next-to-leading terms in λ which where
(0)
(0)
(1)
(1)
left out while constructing Lqq and Lcg will lead to the next-to-leading order Lagrangians Lqq and Lcg .
4.4
Feynman rules for SCETI
Now we have established the SCETI Lagrangian, it is time to derive the Feynman rules following from
it. The first term in the collinear quark Lagrangian,
ξ n,p0 (in · D)
/
/
n
n
ξn,p = ξ n,p0 (in · ∂ + gs n · Aus + gs n · An,q ) ξn,p .
2
2
(4.62)
yields the propagator and the usoft gluon-quark-quark vertex. The collinear quark-collinear gluoncollinear gluon vertex comes from the last term in (4.62) and the second term in the Lagrangian. The
propagator is actually most easily derived from the QCD result. For massless quarks the propagator
∆ξ (p) is given by
∆ξ (p)
ip
/
p2
=
(4.63)
/n · p + p
/n · p + n
/⊥
i n
2 (n · p)(n · p) + p2⊥ + i
i/
n
n·p
.
2 (n · p)(n · p) + p2⊥ + i
=
≈
(4.64)
(4.65)
The usoft gluon vertex is trivially just igs T a nµ n/2 . To find the collinear gluon vertex, we need to expand
1
1
the second term in the quark Lagrangian. Expanding in·D
,
c =
P+g n·A
s
1
P + gs n · An,q
=
∼
=
1
P
1
1+
gs n·An,q
P
n,q
!
n · An,q
1
1 − gs
P
P
n · An,q
1
− gs
,
2
P
P
(4.66)
(4.67)
(4.68)
we find some more interaction terms in the Lagrangian:
c
/⊥
ξ n,p0 iD
/
1
c n
/
iD
ξn,p
in · Dc ⊥ 2
=
=
=
/
1
c n
/ ⊥ ξn,p
iD
(4.69)
2
P + gs n · An,q
/
1
n · An,q
c n
c
/ ⊥ ξn,p
/⊥
iD
ξ n,p0 iD
− gs
(4.70)
2
2
P
P
1
n
/
n · An,q ⊥
⊥
/ n,q
/ ⊥ + gs A
/
/
ξ n,p0 P
A
P
+
g
− gs
ξn,p (. 4.71)
s
⊥
n,q
2
2
P
P
c
/⊥
ξ n,p0 iD
40
Chapter 4. Intro SCET
This results in eight terms:
+
1
1
n · An,q
n · An,q
1
/⊥ A
/⊥
/ ⊥ + gs P
/ ⊥,n,q − gs P
/⊥
/⊥
/ ⊥,n,q P
/ ⊥ − gs P
/ ⊥,n,q + gs A
P
P
A
2
2
P
P
P
P
P /
n · An,q
n · An,q
n
1
/ ⊥,n,q
/ ⊥ − gs3 A
/ ⊥,n,q
/ ⊥,n,q A
/ ⊥,n,q
/ ⊥,n,q − gs2 A
P
A
gs2 A
ξn,p .
(4.72)
2
2
2
P
P
P
ξ n,p0
/⊥
P
In the above are three interaction terms of order g, i.e. with only one collinear gluon field. These terms,
combined with the collinear gluon interaction term in (4.62), yield for the collinear gluon - collinear quark
interaction vertex the following:
igs T
a
µ
µ
0 µ
0
p
γ⊥
p
p
/ γ⊥
/⊥
/⊥ p
/⊥ n
n + ⊥
+
−
n·p
n · p0
(n · p0 )(n · p)
µ
!
/
n
.
2
(4.73)
The terms of order g 2 and higher contribute to the four point vertices and are ignored here. In equation
(4.72) the first term does not have a gluon field. This term should therefore be added to the other
quark-quark term to obtain the full collinear quark-quark Lagrangian Lqq :
Lqq
=
=
/
n
1
/
/
ξ n,p in · ∂ + P ⊥ P ⊥
ξn,p
2
P
/
p2
n
ξ n,p in · ∂ + ⊥
ξn,p .
n·p 2
(4.74)
(4.75)
As we have already derived the propagator through power counting from the QCD Lagrangian, we will
not elaborate on this any further. The gluon propagators originate from the gluon Lagrangian. It is
most convenient to use the full QCD propagators, which were already given in chapter 2. For these gluon
propagators we thus do not discriminate between the usoft and collinear degrees of freedom, yet one can
always use power counting to simplify this propagator.
The above results in the following Feynman rules for the propagators:
=
i/
n
n·p
2 (n · p)(n · p) + p2⊥ + i
(4.76)
(4.77)
=
1
q2
µ ν
g µν − (1 − α) q q2q
δab
(4.78)
4.5 Gauge transformations
41
The Feynman rules for the interactions are:
=
=
igs T a nµ
igs T
a
/
n
2
(4.79)
µ
µ
0
0 µ
γ⊥
p
p
p
/⊥
/⊥ p
/⊥ n
/ γ⊥
+
−
n + ⊥
n·p
n · p0
(n · p0 )(n · p)
µ
!
/
n
2
(4.80)
We will use the Feynman gauge from now on, where α = 1 and the gluon propagator simplifies to
g µν q12 δab . Having these Feynman rules enables us to compute the Wilson lines for the collinear gluons
explicitly, which we will do next.
4.5
Gauge transformations
Let us consider a general local gauge transformation on a collinear field ξn,p ,
(0)
ξn,p → U (x)ξn,p
,
U (x) = eiα
a
(x)T a
.
(4.81)
The first term in the Lagrangian would thus transform as
ξ n,p0 (in · ∂) ξn,p
→
=
(0)
(0)
ξ n,p0 U † (x) (in · ∂) U (x)ξn,p
(0)
ξ n,p0
(0)
(in · ∂) ξn,p
−
(0)
ξ n,p0
(0)
(n · ∂αa (x)T a ) ξn,p
.
(4.82)
(4.83)
The second term can give problems. If ∂ µ α(x) ∼ Qα(x), the momentum would be p2 = .. Thus we need
to consider only gauge transformations which leave us within our EFT, e.g. if the quark has a collinear
momentum, after the gauge transformation it should still be collinear. For the transformation of the
gluon fields the same holds. Thus for SCET I there are two gauge transformations, U us and Uc :
i∂ µ Uus (x)
µ
i∂ Uc (x)
∼ Q(λ2 , λ2 , λ2 )Uus (x)
∼
Q(λ2 , 1, λ)Uc (x)
Ultrasoft
Collinear.
(4.84)
The transformation rules for the fields can be obtained from the usual QCD gauge transformations:
ψ
Aµ
→ U (x)ψ
i
→ U (x)(Aµ (x) + ∂µ )U † (x)
g
(4.85)
(4.86)
42
Chapter 4. Intro SCET
Fields: Collinear transformations Uc
ξn
U c ξn
Aµn
Uc An Uc† + gi Uc [Dus Uc† ]
µ
Aus
Aµus
hv
hv
Ultrasoft transformations Uus
Uus ξn
†
Uus Aµn Uus
†
Uus (Aus + gi ∂ µ )Uus
hv
µ
= ∂ µ − igAµus . The square brackets in
Table 4.2: The gauge transformations. Here Dus
†
Uc [Dus Uc ] imply that the derivative acts only on Uc† .
We factor the large momenta out of U (x), just like we did for the fields ξ:
U (x) =
X
p
e−ip·x Up
(4.87)
Using this we see that
ξn,q
Aµn,q
→
Uq−Q ξn,Q
→
†
UQ Aµn,R UQ+R−q
(4.88)
h
i
1
†
+ UQ P µ UQ−q
.
g
(4.89)
The usoft fields do not transform under collinear transformations. Yet under usoft transformations they
are given by again by the usual QCD transformations. The collinear gluon fields however transform
slightly different under Uus . The derivative ∂ µ acting on Uus give only terms of order λ2 , and can
therefore be ignored. The transformation rules for all the fields are given in table 4.2.
We have not mentioned the heavy quark field hv here as this is not really a part of this theory, yet
for the completeness we mention here that it transforms just like the field q.
From the table it is seen that all fields and operators transform under ultrasoft gauge transformations
Uus . Since ultrasoft gluon momenta can be added to collinear ones without changing the collinear
character of the field, the ultrasoft gluons can be viewed as a background field to the collinear fields.
4.6
Wilson lines in SCET
An important observation of the previous section is that the collinear field is not invariant under collinear
gauge transformations. Yet the operators we are interested which are built from these fields, like currents,
clearly need to be. These operators could contain just one collinear field; the heavy-to-light current which
will be discussed in short in a following section is such an operator: J = ξ n Γhv , with hv a heavy (soft)
quark field and Γ a connection term consisting of gamma matrices. This current is not in a gauge invariant
form! We thus need to define a function W of n · An,q , the order λ0 field, such that the product of it with
the collinear field is an invariant operator.
Now consider the operator
W =
n · An,q
exp −g
P
perms
X
(4.90)
4.6 Wilson lines in SCET
43
with its hermitian conjugate
W† =
n · A∗n,q
exp −g
†
P
perms
X
(4.91)
which satisfy W W † = 1. The label operator P acts to the right on the gluon fields in the expansion. We
will see shortly that there is a physical reason to look at this operator. In appendix D we show it satisfies
the linear equation
in · Dc W (x) = P + gn · An,q (x) W (x) = 0.
(4.92)
This linear equation can be used, together with the transformation rules for A n,q given in (4.89) to find
the transformation properties of W . Let us make the ansatz that W transforms as W → U T W , where
we have decomposed the fields according to (4.87). Then the linear equation (4.92) transforms as
=
=
=
h
i
1
µ
†
µ †
UT W
P + gn · UQ An,R UQ+R−q + UQ P UQ−q
g
i
h
†
†
UT + n · (q − Q)UQ UQ−q
UT W
UT [n · T + P] + gUQ n · An,R U−q
h
i
†
UT [n · T + P] + gUQ n · An,R − n · (Q)UQ U−q
UT W
UT P + gn · An,R W = 0.
(4.93)
(4.94)
(4.95)
(4.96)
Here the unitarity for gauge transformations is used, thus UA+a UA+b = δa,b . UT W is clearly a solution
of the linear equation, and from uniqueness of such a solution, W has to transform as
W → UT W.
(4.97)
This makes the products ξ n W and W † ξn gauge invariant, which was the goal.
The operator W is called a Wilson line [18, 19, 20], and in appendix D we obtain some useful identities
for it. Taking the Fourier transformation of W in (4.90) we obtain the collinear Wilson line in position
space,
Z
Wn (y) ≡ Wn (y, ∞) = P exp ig
y
−∞
dsn · An (sn)
(4.98)
which is a path-ordered exponential. This position space Wilson line is a local operator with respect to x.
This locality is important in the position representation. Wilson lines connect short distance operators
like collinear fields to long distance ones. Consider a (bilinear) operator ξ n q, with q a soft field. In
position space, when we take the collinear field at x = y and the soft field at x = ∞, the gauge invariance
requires a Wilson line between them, so that the operator becomes ξ n (x)W (x, ∞)q(∞).
The combination of W † ξn is sometimes defined as a field χn,P [3] and referred to as the jet field since
it consists of a collinear quark field moving in the n direction and an arbitrary number of collinear gluons
moving in the same direction.
44
4.6.1
Chapter 4. Intro SCET
Finding the form of the Wilson line
We will now show how the operator form of (4.90) is found from integrating the hard propagators out.
Because we at first are interested in the heavy-to-light current, we will compute the collinear Wilson line
by integrating out the off-shell propagators of the heavy quark fields. We will start by showing how this
works for two collinear gluons attached to a heavy quark. When this is established, the result will be
generalized to an infinite number of attached gluons, to find the Wilson line.
In figure 4.1 the attachment of two collinear gluons to a heavy quark is shown. The heavy quark
is represented by the HQET field hv and has a momentum pµ = mv µ + p̂, with p̂ the small residual
momentum, m the momentum of the quark and v the four-velocity, for which v · v = 1. The collinear
gluons have momenta k1 , k2 and gauge labels a and b respectively. The quark propagating between the
first and second gluon therefore has a momentum of
pµ + k1µ
=
mv µ + p̂ +
∼
mv µ +
n
nµ
n · k1 + n · k1 + k1⊥
2
2
nµ
n · k1
2
(4.99)
(4.100)
and the quark propagating after the second gluon attachment has a momentum of
pµ + k1µ + k2µ ∼ mv µ +
nµ
n · (k1 + k2 ).
2
(4.101)
Due to the attachments of the gluons, the heavy quark clearly has gone off-shell. These off-shell propagators have to be integrated out. We will do this by calculating the diagram and using properties of
HQET. The Feynman rules give:
Diagram 4.1(a)
=
=
=
=
i(p
i(p
/ + k/1 + m)
/ + k/1 + k/2 + m)
b µ
a ν
igs T γ
igs T γ hv Aaµ,(n,k1 ) Abν,(n,k2 )
ξnΓ
(p + k1 + k2 )2 − m2
(p + k1 )2 − m2
(p + k/1 + k/2 + m) a
(p
/ + k/1 + m) b
b
a
2 /
/
/
A
A
T T hv
ξ n Γ gs
(p + k1 + k2 )2 − m2 (n,k1 ) (p + k1 )2 − m2 (n,k2 )
"
#
n
n
/
/
n
·
(k
+
k
)
n
·
k
m(/
v
+
1)
+
m(/
v
+
1)
+
n
n
/
/
1
2
1
2
2
ξ n Γ gs2 2
n · Aa(n,k1 ) 2
n · Ab(n,k2 ) T b T a hv
m + mn · vn · (k1 + k2 ) − m2 2
m + mn · vn · k1 − m2 2
!
2
n · Aa(n,k1 ) n · Ab(n,k2 )
m(/
v + 1) n
/
2
T bT aΓ
hv
(4.102)
ξ n gs
n · (k1 + k2 )n · k1
m(n · v) 2
n
/ ∼ /2 n · A and in the fourth line that n
where in the third line we have used A
/n
/ = 0. This can be simplified
by seeing that the terms in the second large brackets acting on hv produce 1:
m(/
v + 1) n
/
hv
m(n · v) 2
=
=
=
1
((/
v + 1)/
n) h v
2n · v
1
(/
n + v/n
/ ) hv
2n · v
1
(/
n−n
/ v/ + 2v · n) hv
2n · v
(4.103)
(4.104)
(4.105)
4.6 Wilson lines in SCET
45
(a)
(b)
Figure 4.1: (a) A heavy quark with two collinear gluons with momenta k 1 and k2 attached. The crossed-bubble represents an effective interaction, with an outgoing collinear
quark and other possible outgoing particles. (b) The same diagram with the gluons crossattached.
which is rewritten as
=
=
2v · n
hv
2n · v
hv
(4.106)
(4.107)
as v/n
/ = vµ nν γ µ γ ν = vµ nν (2η µν − γ ν γ µ ) = v · n − n
/ v/ and v/hv = hv . Using this the diagram in figure 4.1
yields:
!
a
b
n
·
A
n
·
A
(n,k
)
(n,k
)
1
2
ξ n gs2
T b T a Γhv .
(4.108)
n · (k1 + k2 )n · k1
Yet there is another two gluon diagram, with the gluons cross-attached, as is shown in the right of figure
4.1. The expression for this diagram is just
=
=
n · Ab(n,k1 ) n · Aa(n,k2 )
2
ξn
gs
ξn
gs2
n · Aa(n,k1 ) n · Ab(n,k2 )
ξn
gs2
n · Aa(n,k1 ) n · Ab(n,k2 )
n · (k1 + k2 )n · k2
n · (k1 + k2 )n · k2
n · (k1 + k2 )n · k2
T bT a
!
Γhv
(4.109)
a
!
Γhv
(4.110)
T T
b
b
a
(T T + f
abc
c
!
T ) Γhv ,
(4.111)
with f abc the structure constant. These are all the diagrams we need to consider for two attached gluons.
In normal QCD we should also consider the case in which two gluons interact with each other, from
which the resulting gluon from the three gluon vertex interacts with the heavy quark. Yet in the SCET
Lagrangian there is no three gluon vertex and we can leave this interaction out of our calculations.
46
Chapter 4. Intro SCET
→
=
Figure 4.2: The collinear gluons attached to a soft quark can be summed into a Wilson
line Wn . This is done by integrating out the off-shell quark propagators between each
attachment. This is shown in a heavy-to-light current, of a b quark to a u quark. The
usoft Wilson line is omitted.
For more gluon attachments, the part in brackets can be generalized as:
1
n
n · Aan,k
· · · n · Aakn
1
Pn
(−gs )
Tna · · · T a1
n
·
k
n
·
(k
+
k
)
·
·
·
n
·
(
k
)
1
1
2
i=1 i
n=0 perms
a
X
n · An,k T
exp −g
n·k
perms
X
n · An,k T a
exp −g
P
perms
∞ X
X
=
=
This gives us as Wilson line
n
n · An (x)T a
W (x) =
,
exp −g
P
perms
X
(4.112)
(4.113)
which is what we had before, only with the gauge group generator T a explicitly in there. This is shown
graphically in figure 4.2.
4.6.2
The ultrasoft Wilson line
Up till now we have only discussed the collinear Wilson line W . To disentangle the usoft and collinear
modes, which is essential for factorized results, we can make an ultrasoft Wilson line in the same manner
as the collinear Wilson line W . This usoft Wilson line Yn is defined by
Z
Yn (x) = P exp ig
x
−∞
ds n · Aaus (n · s)T a .
(4.114)
4.7 The heavy-to-light decay
47
The subscript n shows it originates from an n-collinear quark. For completeness we state here that there
is a soft Wilson line Sn (x) as well, defined by
Z
Sn (x) = P exp ig
x
−∞
ds n · Aas (n · s)T a .
(4.115)
These two are obviously form equivalent, with as only difference the usoft respectively soft gluon field in
the path-ordered exponential. This soft version is not of importance right yet. We will come back to it
when we discuss soft gluon interactions in SCETII , in section 4.8.
We will now turn our attention to gauge transformations. It will becomes apparent that the Wilson
lines in defined in this section are manifest for the gauge invariance of many physical objects, such as
currents.
4.7
The heavy-to-light decay
To understand how all the previous theory works, we will discuss the process of the heavy-to-light decay
here in some detail. The decay will not only act as an example, but is used in some of the following
chapters as well. For example, in the inclusive B → Xs γ decay which is discussed in chapter 5 a heavy b
quark undergoes a weak decay via a penguin diagram into a light s quark. This decay can be written as
an effective heavy-to-light decay.
The physical picture of a heavy-to-light decay is this: a heavy quark emits a collinear gluon, which
leaves the heavy quark off-shell. When further collinear (or usoft) gluons are emitted, the heavy quark
eventually decays into a light quark. There is an infinite number of (tree) diagrams corresponding to this
process. These diagrams have to be summed into the effective current.
This current is given in QCD by
µ
JQCD
= ψΓµ ψ
(4.116)
with Γ the spin structure4 . The effective current is easy to obtain from a physical discussion. The
incoming particle is a heavy quark, given by the HQET field hv . The light outgoing particle can be
chosen, by making the right choice of frame of reference, to be along the n-direction. This quark can then
µ
be represented by the collinear ξn field. The effective current would then be JSCET
= ξ n Γµ hv . Yet from
the discussion on collinear Wilson lines in section 4.6 we see that the collinear gluons can all be summed
into the operator W , by integrating out the off-shell propagators. This is necessary for gauge invariance
of the operator, and just yields
µ
JSCET
= ξ n W Γµ h v .
(4.117)
This effective current is gauge invariant for collinear gauge transformations U c :
ξ n W Γµ hv → ξ n Uc† Uc Γµ hv = ξ n W Γµ hv .
(4.118)
4 A typical expression such a spin structure is Γµ = γ µ (1 − γ ) or with a gauge group operator Γµ = γ µ (1 − γ )T a , which
5
5
we encounter in B decay
48
Chapter 4. Intro SCET
In figure 4.2 such a heavy-to-light current is shown, with the corresponding Wilson line attached to it.
4.7.1
Redefinition of the fields
We still need a grip on the usoft gluons, as these have not yet been summed into an operator. By
introducing the usoft Wilson line Yn as in equation (4.114)
Z
Yn (x) = P exp ig
x
−∞
ds n · Aaus (n · s)T a ,
(4.119)
which obeys the properties
in · Dus = 0Y, Yn† Yn = 1
(4.120)
these usoft gluons are taken care of in the same manner as the collinear gluons. By now redefining the
fields as
†
(0)
ξn,p (x) → Yn (x)ξn,p
(x), An,p (x) → Yn (x)A(0)
n,p (x)Yn (x)
(4.121)
with ξ (0) and A(0) the bare fields without any usoft gluon contributions, we find that the Wilson line
transforms as [4]
W → Yn W (0) Yn†
(4.122)
and the usoft gluons are factored out of the fields. Fields with the superscript (0) do not interact with
the usoft gluons or transform under Uus . This claim can be proved at the level of the Lagrangian by
doing using these redefinitions in the Lqq and Lcg :
L(0)
qq
=
ξ n,p0
c
/⊥
in · D + iD
1
c
/
iD
in · Dc ⊥
/
n
ξn,p
2
(4.123)
/
1
n
/ n,q,⊥ )W W † (P
/ ⊥ + gs A
/ n,q,⊥ )
/ ⊥ + gs A
ξn,p (4.124)
in · Dus + gn · An,q + (P
2
P
(0)
†
ξ n,p0 Y † in · Dus + gY n · A(0)
n,q Y
/ (0)
1
n
(0)
(0)
/ ⊥ + gs Y A
/ n,q,⊥ Y † )Y W (0) Y † Y W (0)† Y † (P
/ ⊥ + gs Y A
/ n,q,⊥ Y † )
(P
Yξ .
(4.125)
2 n,p
P
ξ n,p0
=
=
+
In the second line we have used that f (P̄ + gs n̄ · An,q ) = W f (P̄)W † which is shown in appendix D. Using
now Y † in · Dus Y = in · ∂, this can be written as
L(0)
qq
=
(0)
ξ n,p0
in · ∂ + gn ·
A(0)
n,q
(0)
/ n,q,⊥ )W (0)
/ ⊥ + gs A
+ (P
1 (0)†
(0)
/ n,q,⊥ )
/ ⊥ + gs A
W
(P
P
/ (0)
n
ξ .
2 n,p
(4.126)
In the same way we see that the gluon Lagrangian in (4.60) can be rewritten, where all the usoft Wilson
lines drop out because of the cyclic properties of the trace. Because the usoft covariant derivative changes
to a partial derivative, and the usoft Wilson lines vanish, the usoft gluons no longer couple to the collinear
fields! This is a first step in seeing factorization formulas take shape, which is shown in the next chapters.
4.8 SCETII
49
The effective current is effected by these redefinitions by a factor of Yn :
µ
JSCET
(0)
=
ξ n,p Yn† Yn W (0) Yn† Γµ hv
(4.127)
=
(0)
ξ n,p W (0) Yn† Γµ hv .
(4.128)
We have now arrived at a form of the heavy-to-light current which can be used in other sections where
we consider processes with such a decay.
In this effective current it is manifest that all the contributions of the gluons are factored out of the
fields and are only represented in the Wilson lines, respectively W for the collinear gluons and Y n for
the usoft gluons. This factorization is a property of SCET which allows us to work very easily with the
gluon contributions. It is this property which is one of the main advantages of SCET over other effective
theories for the soft and collinear fields.
4.8
SCETII
Up till now we have only discussed SCETI , which is used for usoft-collinear and hard-collinear factorization. However for soft-collinear factorization this will not work; we have to use SCET II .
Gluons with soft degrees of freedom are to be treated quite differently to usoft ones. When soft gluons
couple to collinear particles, these particles will be taken off their mass shell. In SCET II the momenta
of soft and collinear particles are proportional to
Soft:
ps
Collinear:
pc
∼ Q(η, η, η)
∼ Q(η 2 , 1, η)
(4.129)
Λ
where η = Q
. The soft gluons therefore have the same momenta as usoft gluons in SCET I ! The collinear
particles however have different momentum. It is important to note that the identification of the degrees
of freedom of the various particles is frame dependent. However the relations between these degrees of
freedom are frame independent. Whatever frame we choose, the physics will be the same, e.g. by choosing
another frame of reference the identifications soft and collinear can be interchanged yet the results will
be the same. The invariant mass of collinear and soft gluons in SCETII is the same, namely Q2 η 2 . This
means we use SCETII when the energy of the problem is lower than for SCETI , where the usoft gluons
have a much smaller invariant mass than the collinear particles.
The observation that usoft degrees of freedom in SCETI are principally the same as soft ones in
SCETII leads to an important consequence. When we want to go from SCETI to SCETII , the usoft
degrees of freedom can just be renamed soft, without changing any operators. The usoft Wilson lines are
thus renamed soft by doing the substitution Yn (x) → Sn (x).
As mentioned the coupling of a soft gluon with a collinear particle takes it off-shell. From the momenta
in (4.129) this is quite easily seen. When such a collinear particle interacts with a soft gluon, the resulting
momentum is ps + pc ∼ Q(η, η, η) + Q(η 2 , 1, η) = Q(η, 1, η), which is off-shell.
50
Chapter 4. Intro SCET
Fields:
ξn,p
hv
Wilson lines :
Sn
soft transformations Us
ξn,p
U s hv
U s Sn
Table 4.3: The soft gauge transformations. All trivial gauge transformations have been
omitted.
4.8.1
Wilson lines in SCETII
The soft Wilson lines have already been given in equation (4.115) in section 4.6.2. By integrating out
the off-shell heavy quark propagators, in the same way as in the usoft case, we see that the soft Wilson
line is given by
Sn (x) = Sn (−∞, x) = P exp ig
or
Z
x
−∞
ds n ·
Aas (n
n · As,q (x)T a
Sn (x) = P exp −g
.
n·P
4.8.2
· s)T
a
(4.130)
(4.131)
Gauge transformations in SCETII
In the sections on gauge transformations we have only discussed the transformations for usoft degrees of
freedom.
Again soft gauge invariance determines the appearance of Sn . Let us illustrate with an example. For
the simple current J = ξ n,p Γhv , with Γ the spin structure, we want to know the soft gauge invariant
expression. Under the transformations in table 4.3 and the known collinear gauge transformation we see
that this current transforms as J → J = ξ n,p Uc† ΓUs hv . To make this current gauge invariant, we need
Wilson lines W and Sn to appear: J = ξ n,p W ΓSn hv .
To complete this chapter, we lay out the steps we need to perform when using SCET to describe
energetic processes.
4.9
Using SCET
When calculating matrix elements for a particular process, we should first analyze the process. It is
important to know which particles participate and which interactions occur in the process. There should
be a collinear particle or jet present in the process to make use of the advantages of the effective theory,
therefore one must choose an appropriate frame of reference. Other particles could for instance be hard
or soft heavy particles described by HQET fields. The invariant mass of the particles indicates which
of the two theories, SCETI or SCETII is needed. If the collinear momentum squared is proportional to
QΛ, p2c ∼ QΛ, SCETI will be used. Yet if p2c ∼ Λ2 we only use SCETI as an intermediate step, while
the final theory we use is SCETII . The difference between these two theories is described in section 4.8,
yet as it is important for describing the method, we note again that the difference resides in the collinear
4.9 Using SCET
51
momenta. The usoft and soft momenta squared are both proportional to Λ squared, p 2u ∼ Λ2 , p2s ∼ Λ2 .
From this momentum analysis one also sees what the large momentum scale of the problem is. This has
to be integrated out by matching. When the physics of the process is known, one should find the relevant
operators of the process, e.g. when calculating the cross-section, one needs to know the current. These
QCD operators can usually be looked up in the literature for the specific process. These QCD operators
have to be matched onto the SCET operator, which contain effective fields described in the previous
†
sections, together with the Wilson coefficient C(µ, P, P ), with µ the factorization scale. This coefficient
contains all hard momenta which have been effectively integrated out by this matching procedure. It is
†
not trivial that this coefficient should only depend on P and P , this is shown explicitly in [4]. This
matching coefficient can be calculated perturbatively by comparing the full theory diagrams with SCET
order by order at the matching scale at which both theories should be valid.
(0)
µ(0)
In the SCETI operator, one should make the field redefinitions ξn → Yn ξn , Aµn → Yn An Yn† to
factorize the usoft interactions. This leads to an operator with fields and functions depending explicitly
on particular momentum modes, n-collinear, n-collinear, usoft and/or hard modes. These different field
no longer have an interaction with each other, and by taking the operator between matrix elements the
factorization may happen naturally. We will see this happen in the next chapters.
When the invariant mass is of order Λ2 we only use SCETI as an intermediate step for the final
SCETII theory. Using this intermediate step means that we first integrate out the large degrees of
freedom by matching on SCETI operators. Now the usoft particles can interact with the collinear ones.
Next we integrate out the degrees of freedom of order QΛ by matching onto SCET II operators. This is in
principle done by renaming the usoft operators as soft ones, as these have the same momentum scaling.
The momenta of order QΛ which have again been integrated out result in a new Wilson coefficient.
To summarize this section, we give the method in a short few steps:
1. Analyze the physics of the process. Find out if SCETI or SCETII is needed. Find the relevant
QCD operators, e.g. the appropriate currents.
2. Match QCD onto SCETI at the large scale. The matching coefficient contains the hard momenta
and can be calculated order by order in αs .
(0)
3. Redefine the fields so that the ultrasoft degrees of freedom are in the operators Y : ξ n = Yn ξn and
µ(0)
Aµn → Yn An
Yn† .
4. If needed, match SCETI onto SCETII operators. This is done by renaming the usoft Wilson lines
Yn as soft, Yn → Sn and results in a new matching coefficient containing momenta of order QΛ.
5. Take the matrix elements in the new theory. The different momentum modes are now included in
different operators, which results in a factorization to all orders.
In the next chapters we will apply the theory introduced in this chapter to physical processes. In
chapters 5 and 6 we see how SCET can be used to factorize the decay width of the B → X s γ decay resp.
cross-section of Deep Inelastic Scattering to all orders. In chapter 7 the large logarithms of the Higgs
production via gluon fusion are calculated.
52
Chapter 4. Intro SCET
Chapter 5
The inclusive B → Xsγ decay
The best way to indicate how the details of SCET work and why it is such a useful effective theory, is to
use it on a process. The inclusive B → Xs γ decay1 , where a B meson decays into a meson Xs containing
an s-quark (e.g. Xs = K ∗ ) and a photon, is a good example of a process in which SCET is useful. The
decay is shown in figure 5.1.
This rare B-decay is an important process in current physics. Because it only occurs at loop level,
where the particles propagating in this loop are heavy, the process is very sensitive for ‘new physics’.
Supersymmetric partners of known particles are thought to be very heavy, just like the top and W bosons.
Therefore, if this process would also be mediated through a loop consisting of these supersymmetric
particles, it would contribute in about equal amount to the decay width as the ‘normal’ process. Thus by
finding the decay width of the process, it can be easily checked if any new physics occurred in the loop
as well.
In this chapter we will show that the photon spectrum of the decay rate of this process can be
factorized into hard, collinear and usoft factors:
1 dΓ
= H(mb , µ)
Γ0 dEγ
Z
dk + S(k + )J(k + + mb − 2Eγ ).
(5.1)
The hard amplitude H(mb , µ), usoft structure function S(k + ) and collinear jet function J(k + ) only
interact via k + . This factorization formula simply tells us that short and long distance contributions to
the decay rate can be separated from each other. This factorization is an important property, as it makes
the future calculations less tedious. This will be a slightly technical chapter as we tried to fill in all of
the details in the calculations, yet it should clarify the advantages of SCET.
1 Inclusive processes are processes in which not all end-products are specified. This is in contrast to an exclusive process,
where the end-products of the process are explicitly specified before calculation.
53
54
Chapter 5. The inclusive B → Xs γ decay
5.1
Calculating the decay rate
Following the lines of the previous chapter, we will tackle the calculation of the decay rate of the B → X s γ
decay. The process is shown in figure 5.1(a). We will match the QCD forward scattering amplitude,
(a)
(b)
Figure 5.1: (a) The inclusive decay of a B meson into a meson Xs containing an s
quark, and a photon. The double line for the b and t quark denote that they are heavy
quarks. The second quark is not specified, this can be an anti-up, u, or anti-down quark,
d. (b) The effective decay, where the W -top loop has been approximated by an effective
coupling.
constructed by the heavy-to-light current, onto the SCET one at µ2 ≈ Q2 . The heavy quarks fields will
be matched onto HQET fields, while the light quarks will be matched onto collinear or (u)soft SCET
fields.
The decay of a B meson into a photon and meson Xs is intermediated by a top quark and W bosons,
which results in a penguin diagram, figure 5.1(a). This W -top loop can be approximated by an effective
coupling, shown in figure 5.1(b). The effective Hamiltonian belonging to this process is given by [21]:
Hef f =
with
O7 =
−4GF
√ Vtb Vts∗ C7 O7
2
e
mb sσµν F µν PR b,
16π 2
PR =
(5.2)
1
(1 + γ5 ).
2
(5.3)
Here the masses of the light s quarks have been neglected. F µν is the electromagnetic field strength, the
V ’s are CKM-matrices and PR is the right-handed projection operator defined as above.
The process is a typical example of the heavy-to-light decay discussed in section 4.7. In that section
we showed that the corresponding SCET current is of the form Jµ = ξW Γµ hv . In a moment we will see
what the exact current is for this decay, but first we will introduce the decay rate.
The differential decay rate can be written in terms of the effective Hamiltonian,
dΓ =
X
d3 ~q
(2π)4 δ(pB − q − pX )|hXs (pX )γ(q)|Hef f |B v (pB )i|2 .
(2π)3 2Eγ
(5.4)
Xs
The delta function ensures momentum conservation. This decay rate can be written in terms of the
5.1 Calculating the decay rate
55
forward scattering amplitude T (Eγ ), in which all the hadronic degrees of freedom are contained, by using
the optical theorem
XZ
2ImM(a → b) =
dΠf M∗ (b → f )M(a → f ),
(5.5)
f
or
2ImM(p → p) =
XZ
f
dΠf |M(p → f )|2 .
(5.6)
Here we have used the notation for the forward scattering amplitude M from Peskin & Schroeder [16].
In words this just says that the sum over all final states and flavours of the absolute squared of the
matrix element from initial state |pi to final state hf |, hf |Hef f |pi, equals two times the imaginary part of
the matrix element of initial state hp| to itself, which is the forward scattering amplitude. After Fourier
transforming to coordinate space, equations (5.4) and (5.6) together become
dΓ =
d3 ~q
|Hef f |2 Im T (Eγ ),
4(2π)3 Eγ
(5.7)
with T (Eγ ) the forward scattering amplitude. We will come back to this amplitude, with the precise
expression for this decay, in due course. For now we only use that we can use it to write the photon
spectrum of the decay rate as
1 dΓ
4Eγ
= 3
Γ0 dEγ
mb
−1
π
Im T (Eγ ),
(5.8)
|Hef f |2
G2F m5b
|Vtb Vts∗ |2 |C7 (mb )|2 .
32π 4
(5.9)
with Γ0 defined by
Γ0
=
=
(5.10)
It can be interpreted as the partonic decay rate for b → sγ, and contains all the, for our discussion,
irrelevant factors.
5.1.1
The physics of the decay
We consider the endpoint region of this decay, where nearly all energy is in the photon; the remaining
energy is in the collinear jet. In the rest frame of the B meson its momentum is given by p B = m2B (nµ +nµ )
The photon momentum can be described by q µ = Eγ nµ , which is obviously in the opposite direction of
the collinear jet. This jet of collinear Xs mesons has momentum
pµX
=
pµB − q µ = mB (
=
mB
nµ
nµ
+
) − E γ nµ
2
2
nµ
nµ
+
(mB − 2Eγ ).
2
2
(5.11)
For the energy of the photon in the endpoint region, where almost all energy resides in the final state
meson, we can take m2B − Eγ . ΛQCD . Thus the part in brackets in (5.11) is approximately Λ, and we
56
Chapter 5. The inclusive B → Xs γ decay
have
p2X
nµ
nµ
+
(mB − 2Eγ ))2
2
2
= mB (mB − 2Eγ )
= (mB
= mB Λ
(5.12)
Collinear particles have, in both SCETI and SCETII , an invariant q
mass of p2 ∼ Q2 λ2 . For mB = Q,
Λ
Λ
the invariant mass of the Xs meson is p2X ∼ Q2 Q
, and we have λ = Q
. This means we have to use
SCETI for the decay and the B meson has usoft degrees of freedom, i.e. the gluons attached will have
momenta scaling as pµ ∼ Qλ2 . Now we have established which theory to use, we can start the calculation
of the photonic spectrum of the B → Xs γ decay rate.
5.2
Matching QCD onto SCETI
The inclusive B decay is an example of a heavy-to-light decay. Therefore we can use the results for the
heavy-to-light current found in the previous chapter, where the heavy bottom quark is described by a
HQET field. To indicate that the field for the b quark is dictated by HQET, we represent this field by
bv . After the decay, the 4s4 quark will have a very large energy compared to its mass. It can therefore
(s)
be described by a collinear field, which is represented as ξn , for an s quark in collinear n-direction. The
QCD current must be matched onto the SCETI current, which was given in section 4.7.
5.2.1
The operators
The QCD current for the decay is given by
µ
JQCD
(x) = s(x)(iσ µν qν PR )b(x).
(5.13)
The operator PR is the usual right projection operator, which is defined, together with the left projection
operator PL , as
PR =
1
(1 + γ5 ),
2
PL =
1
(1 − γ5 ).
2
(5.14)
As noted before, all the hadronic degrees of freedom resides in this current. In figure 5.2(a) the forward
scattering amplitude is shown. This amplitude T (Eγ ) is given by
T (Eγ ) =
i
mB
Z
d4 xe−iq·x hB|T{Jµ† (x)J µ (0)}|Bi,
(5.15)
where T is the usual time-ordered product and where we have introduced the B-meson state |Bi. Meson
states |M (p, s)i with momentum p and spin s are usually normalized in the full QCD theory by the
relativistic normalization
hM (p0 , s0 )|M (p, s)i = 2(2π)3 Eδs,s0 δ(~
p − p~0 ).
(5.16)
5.2 Matching QCD onto SCETI
57
(a)
(b)
Figure 5.2: (a) The forward scattering amplitude without gluon attachments for the
B → Xs γ decay. As always, the double line represents the heavy b quark and the dashed
line denotes the collinear s quark. The crossed circle denotes the insertion of the current.
(b) Example of a graph contributing to the forward scattering amplitude. After redefinition
of the fields, we have factorization of the usoft and collinear fields. The usoft gluons
(spring lines) are only attached to the heavy fields, while the collinear gluons (springs
with a line) are attached to the collinear quark.
†
From the matching of the QCD current onto SCET at a scale µ we obtain a Wilson coefficient C(P , µ).
The coefficient contains all the factors which the matching generates. It therefore depends on the matching
scale µ, as well as on the label operator. Using this Wilson coefficient, the SCET current can be written
as
†
n
(s)
Jµ (x) = −Eγ ei(P 2 +P⊥ −mb v)·x C(P , µ)ξ n,p (x)W (x)γµ⊥ PL bv (x).
(5.17)
n
The exponent ei(P 2 +P⊥ −mb v)·x ensures momentum conservation; the heavy quark has momentum mb v,
while the momentum of the collinear quark is given by the label operators P n2 + P. The large and small
momenta should be conserved separately as well, therefore the label operators in this exponent obey label
and residual momentum conservation. This yields P = mb while P⊥ = 0. Another way of putting it, is
that the total momentum of the collinear s quark jet, pX , is given by P = n·pX = mb . The momentum in
the perpendicular direction is P ⊥ = 0. We see that the current, and therefore the time-ordered product
in (5.15), only depends on residual momenta. The exponent in (5.17) becomes, using these values of the
momenta:
i(P
n
+ P⊥ − mb v) · x
2
=
imb
n
−v ·x
2 0
x + x3
0
−x
= imb
2
−x0 + x3
= imb
2
n
= −imb · x.
2
(5.18)
(5.19)
(5.20)
(5.21)
58
Chapter 5. The inclusive B → Xs γ decay
The next step is to redefine the fields, so to decouple the usoft gluons from the collinear fields. This is
done by making the substitutions
(s,0)
(s)
(x),
(x) → Y (x)ξn,p
ξn,p
W (x) → Y (x)W (x)(0) Y † (x),
(5.22)
which was discussed in section 4.7. The current transforms as
(s)
ξ n,p (x)W (x)γµ⊥ PL bv (x)
→
(s,0)
ξ n,p Y † (x)
(s,0)
Y (x)W (0) (x)Y † (x) γµ⊥ PL bv (x)
(5.23)
=
ξ n,p (x)W (0) (x)γµ⊥ PL Y † (x)bv (x)
(5.24)
≡
Jµef f (x).
(5.25)
In the above we have defined the effective current,
n
Jµ = −Eγ e−imb 2 ·x Jµef f C(mb , µ).
(5.26)
The usoft gluons have been decoupled from the collinear degrees of freedom using the procedure which
SCET dictates.
5.2.2
Hard kernel and jet function
The effective currents can be introduced in the forward scattering amplitude,
T (Eγ )
=
=
i
mB
Z
d4 xe−iq·x hB|T{Jµ† (x)J µ (0)}|Bi
Z
n
2
2 i
d4 xei(mb 2 −q)·x hB|T{Jµef f,† (x)J µ,ef f (0)}|Bi
Eγ |C(mb , µ)|
mB
(5.27)
(5.28)
All the hard interactions are now in the terms in front of the integral, and we can define the hard
amplitude H(mb , µ) as
4Eγ3
H(mb , µ) ≡ 3 |C(mb , µ)|2 .
(5.29)
mb
We can now write the forward scattering amplitude as a product of this hard amplitude with an effective
forward scattering amplitude,
4Eγ mB
T (Eγ ) = H(mb , µ)T ef f (Eγ , µ).
m3b
(5.30)
This effective forward scattering amplitude is defined by
T ef f ≡
Z
n
µ
d4 xei(mb 2 −q)·x hB v |T{Jµef f † (x)Jef
f (0)}|B v i
(5.31)
5.2 Matching QCD onto SCETI
59
so
T
ef f
Z
n
µ
(0)
d4 xei(mb 2 −q)·x hB v |T{[bv Y PR γµ⊥ W (0)† ξn,p
](x)[ξ n,p0 W (0) γ⊥
PL Y † bv ](0)}|B v i (5.32)
=
i
=
i
×
µ
(0)
PL (Y † bv )(0)}|B v i. (5.33)
](x)[ξ n,p0 W (0) ](0)}|0iγ⊥
hB v |T{(bv Y )(x)PR γµ⊥ h0|T{[W (0)† ξn,p
Z
n
d4 xei(mb 2 −q)·x
The last step is allowed because the distinct regions do not interact, and the only contribution to the
collinear part comes from the vacuum states. Differently stated, the B meson states do not contain
collinear particles, hence the collinear operators do not couple to them. Introducing a collinear jetfunction JP (k),
Z
d4 k −ik·x
n
/
(0)
e
JP (k) ≡ h0|T[W (0)† ξn,p
](x)[ξ n,p0 W (0) ](0)}|0i
i
(5.34)
(2π)4
2
we can write (5.33) as
T ef f
=
=
=
=
n
d4 k 4 i(mb n −q−k·x)
/ µ
2
PL (Y † bv )(0)}|B v iJP (k) (5.35)
d xe
hB v |T{(bv Y )(x)PR γµ⊥ γ⊥
(2π)4
2
Z
d4 k 4 i(mb n −q−k)·x
/
µ n
2
−
d xe
hB v |T{(bv Y )(x)PR (−γµ⊥ γ⊥
) PL (Y † bv )(0)}|B v iJP (k)(5.36)
(2π)4
2
Z
4
d k 4 i(mb n −q−k)·x
2
(5.37)
d xe
hB v |T{(bv Y )(x)PR n
/ PL (Y † bv )(0)}|B v iJP (k)
(2π)4
Z
d4 k 4 i(mb n −q−k)·x
2
hB v |T{(bv Y )(x)/
d xe
nPL (Y † bv )(0)}|B v iJP (k).
(5.38)
(2π)4
−
Z
(0)
The label P on the jet function is the sum of the label momenta of the collinear fields in W (0)† ξn,p and
(0)
ξ n,p0 W (0) . In the last line we used that PR n
/ PL = PR (γ0 + γ3 )PL = n
/ (PL )2 = n
/ PL . Thus
T
ef f
=
=
d4 k 4 i(mb n −q−k)·x
2
nPL (Y † bv )(0)}|B v iJP (k)
hB v |T{(bv Y )(x)/
d xe
(2π)4
Z
d4 k 4 i(mb n −q−k)·x
1
2
d xe
hB v |T{(bv Y )(x)(Y † bv )(0)}|B v iJP (k)
2
(2π)4
Z
(5.39)
(5.40)
The last step is not trivial. To show this we use that the B meson is a pseudoscalar, by introducing
so-called interpolating fields:
√
B v = m B bv γ 5 lv ,
(5.41)
so that the meson state |B v i becomes
|B v i =
√
mB bv γ5 lv |0i.
(5.42)
The field lv is a spinor field which destroys the light degrees of freedom, which is needed as we are dealing
with heavy quarks. The square root of the mass comes from the normalization (5.16). Putting this into
60
Chapter 5. The inclusive B → Xs γ decay
the expression for the effective amplitude T ef f , we obtain
T
ef f
= mB
Z
1
d4 k 4 i(mb n −q−k)·x
2
n (1 − γ5 )(Y † bv )(0)}|bv γ5 lv |0iJP (k). (5.43)
d xe
h0|lv γ5 bv T {(bv Y )(x)/
(2π)4
2
We will omit the time-ordered product and usoft Wilson lines Y for now, as these are not relevant for
this discussion. It is important to note that the fields in (5.42), which are denoted without the space
dependence, are evaluated in the origin. The matrix element in (5.43) can now be written as
1
n (1 − γ5 )bv }bv γ5 lv |0i
h0|lv γ5 bv bv (x)/
2
1
h0|lv γ5 bv bv (x)(γ0 + γ3 )(1 − γ5 )bv }bv γ5 lv |0i
2
1
h0|lv γ5 bv bv (x)(/
v + γ3 )(1 − γ5 )bv }bv γ5 lv |0i.
2
=
=
(5.44)
(5.45)
This matrix element can be seen as a real expectation value. The heavy quark propagator can be found
from the HQET Lagrangian L∞ = h̄v iv · Dhv , which is the M → ∞ limit of (C.29);
h0|T{bv (x)bv (x)}|0i =
Z
d4 p 1
(2π)4 v · p
1 + v/
2
eip·x .
(5.46)
If we take the limit of x → 0 of this equation, we obtain the equal time “propagator” h0|b v bv |0i:
h0|bv bv |0i =
1 + v/
2
.
(5.47)
This is, up to a constant, the most general form of the propagator which is consistent with the constraint
v/bv = bv . We can take the trace ‘for free’, since the matrix element is an expectation value and thus a
real number. This element is
v + γ3 )PL bv bv γ5 lv |0i
h0|lv γ5 bv bv (x)(/
=
Tr{h0|lv γ5 bv bv (x)(/
v + γ3 )PL bv bv γ5 lv |0i}
Z
4
d p 1 ip·x
1 + v/
1 + v/
e Tr{h0|lv γ5
(/
v + γ3 )PL
γ5 lv |0i}
=
(2π)4 v · p
2
2
Z
1 + v/
d4 p 1 ip·x
1 + v/
e Tr{γ5
=
(/
v + γ3 )PL
γ5 M } (5.48)
4
(2π) v · p
2
2
In the last line the invariance under cyclic permutations of the trace is used, and the matrix M is defined
as
M = h0|lv lv |0i.
(5.49)
It is shown in [22] that the most general form of this matrix is M = A(v)1, where A(v) is a function of
v. As this is not a very relevant discussion for this thesis, we will omit the calculation and just use the
result that is can be written in this way.
To show the equivalence of (5.39) and (5.40), we will first show that the term with γ 5 from the projec1+/
v
1+/
v
5
tion operator vanishes. Using that 2 v/ = 2 , the part of the matrix element with γ5 , hB v |bv (x)/
n( −γ
2 )bv bv |B v i,
5.2 Matching QCD onto SCETI
61
becomes
n(
hB v |bv (x)/
−γ5
)bv bv |B v i
2
Z
1
2
Z
1
= −
2
Z
1
= −
2
Z
1
= −
2
= 0,
=
−
1 + v/
1 + v/
d4 p eip·x
(/
v
+
γ
)γ
γ5 M } (5.50)
Tr{γ
3
5
5
(2π)4 v · p
2
2
1 + v/
1 − v/
d4 p eip·x
Tr{γ
(1
+
γ
)
γ52 M }
5
3
(2π)4 v · p
2
2
1 − v/
1 − v/
1 + v/
d4 p eip·x
+
γ
M}
Tr{γ
3
5
(2π)4 v · p
2
2
2
d4 p eip·x
1 − v/
M}
Tr{γ
γ
5 3
(2π)4 v · p
2
(5.51)
because of the identity Tr(γ5 γµ ) = Tr(γ5 γµ γν ) = Tr(γ5 γµ γν γρ ) = 0. Yet the other term, the one without
the γ5 matrix from the projection operator, has an even number of γ5 matrices, so that we cannot follow
this reasoning. It can be rewritten as
1
hB v |bv (x)/
n bv bv |B v i
2
Z
1
2
Z
1
2
Z
1
2
Z
1
2
=
=
=
=
d4 p eip·x
1 + v/
1 + v/
Tr{γ5
(/
v + γ3 )
γ5 M }
(2π)4 v · p
2
2
1 + v/
1 + v/
d4 p eip·x
Tr{γ5
(1 + γ3 )
γ5 M }
(2π)4 v · p
2
2
1 − v/
1 − v/
d4 p eip·x
Tr{
(1 − γ3 )
γ5 M }
(2π)4 v · p
2
2
d4 p eip·x
1 − v/
γ5 M },
Tr{
4
(2π) v · p
2
(5.52)
(5.53)
(5.54)
(5.55)
which is non-zero. This non-zero part are the terms without γ3 matrices in (5.53), and we can write it
back into the form we want,
Non-zero part of (5.53)
=
=
=
=
Z
1 + v/
1 + v/
1
d4 p eip·x
Tr{γ5
γ5 M }
2
(2π)4 v · p
2
2
Z
1 + v/
1 + v/
1
d4 p eip·x
Tr{h0|lv γ5
γ5 lv |0i}
2
(2π)4 v · p
2
2
1
Tr{h0|lv γ5 bv bv (x)bv bv γ5 lv |0i}
2
1
hB v |bv (x)bv |B v i.
2
(5.56)
(5.57)
(5.58)
(5.59)
When we reinsert the usoft Wilson lines and the time-ordered product which we left out of the calculations,
we see that this is indeed exactly the matrix element in equation (5.40).
Let us now continue with the evaluation of (5.40). There are two 4 dimensional integrals, which can
be simplified significantly by realizing that JP (k) does depend in fact only on the residual momentum
k + = n · k. In the collinear quark and gluon Lagrangians, (4.126) and (4.52) respectively, the derivative
only appears in terms of n · ∂. This implies that the collinear fields only have residual momenta and thus
JP (k) only depends on k + , JP (k) = JP (k + ). We can now do the remaining integrals over k − and k ⊥ in
62
Chapter 5. The inclusive B → Xs γ decay
equation (5.40):
T
ef f
Z
Z
d4 k
1
−q)·x
†
4
i(mb n
2
exp [−ik · x] JP (k)
d xe
hB v |T{(bv Y )(x)(Y bv )(0)}|B v i
2
(2π)4
Z
n
1
d4 xei(mb 2 −q)·x hB v |T{(bv Y )(x)(Y † bv )(0)}|B v i
2
+
Z
dk + dk − d2 k ⊥
k − k− +
⊥ ⊥
JP (k + )
exp
−i
x
+
x
+
k
x
(2π)4
2
2
Z
Z
1
n
k+
dk +
d4 x exp i mb − q · x hB v |T{(bv Y )(x)(Y † bv )(0)}|B v iδ(x+ )δ(~x⊥ )
exp −i x− JP (k + )
2
2
2π
2
Z
Z
+
+
+
q
−
n
1
dk
k
dx− ei(mb − 2 )x hB v |T{(bv Y )( x− )(Y † bv )(0)}|B v i
exp −i x− JP (k + )
(5.60)
2
2
2π
2
=
=
×
=
=
which becomes, with q + = n · q = n · nEγ = 2Eγ and the multiplication identity for Wilson lines
Y (x)Y † (y) = Y (y, x), see eq. (D.20),
T
ef f
=
Z
dk +
JP (k + )
4π
Z
i
dx− e− 2 (2Eγ −mb +k
+
)x−
n
n
hB v |T{bv ( x− )Y (0, x− )bv (0)}|B v i.
2
2
(5.61)
We will see that the matrix element, which only consists of ultrasoft degrees of freedom, can be rewritten
as the universal structure function.
5.2.3
The structure function
The structure function S(l + ), or shape function as it is sometimes referred to, can be defined by [23]:
S(l+ ) ≡
1
hB v |bv δ(in · D − l+ )bv |B v i.
2
(5.62)
This is the same as the integral over x− in (5.61),
1
S(l ) =
2
+
Z
dx− − i l+ x−
n
n
hB v |T{bv ( x− )Y (0, x− )bv (0)}|B v i.
e 2
4π
2
2
(5.63)
To see that these expressions are equal, we use that for an operator O(x) a translation x → x + a can be
µ
generated by O(x + a) = ea ∂µ O(x), which is essentially a Taylor expansion. The field bv at space-time
x−
point n2 x− is then given by e 2 n·∂ bv (0). To emphasize that the field is still dependent on the space-time
coordinate, yet taken at point zero, we will introduce a coordinate y, which will be set to zero after the
calculations. Now
(5.63)
=
=
=
dx− − i l+ x−
n −
n −
2
e
hB v |T{bv (y + x )Y (y, y + x )bv (y)}|B v i
4π
2
2
y=0
Z
−
+
−
1
n
dx − i l x
e 2
hB v |T{ bv Y (y + x− ) Y † bv (y) }|B v i
2
4π
2
y=0
Z
−
x−
1
dx − i l+ x−
†
n·∂y
2
2
hB v |T{ e
(bv Y (y) Y bv (y) }|B v i
e
2
4π
y=0
1
2
Z
(5.64)
(5.65)
(5.66)
5.2 Matching QCD onto SCETI
63
R
R
If we now use partial integration, so that dy(e∂ f (y))g(y) = dy((1 + ∂ + 12 ∂ 2 + · · · )f (y))g(y) =
R
R
dyf (y)(1 − ∂ + 21 ∂ 2 + · · · )g(y) = dyf (y)e−∂ g(y), we see the above is written as
1
2
Z
−
dx− − i l+ x−
− x2 n·∂y
†
2
Y bv (y) }|B v i .
hB v |T{ bv Y (y)e
e
4π
y=0
(5.67)
Now we will use a trick to find the final form. In the appendix about Wilson lines it is shown that, for
collinear degrees of freedom, in · ∂ = W † (x)in · DW (x). An equivalent identity is however also true for
our ultrasoft degrees of freedom:
in · ∂ = Y † (x)in · Dus Y (x).
(5.68)
By just doing a Taylor series expansion it is seen that the exponential version is also true:
ea(in·∂) = Y † (x)ea(in·Dus ) Y (x).
(5.69)
Using this for the exponent in (5.67), we have
(5.67)
=
1
2
=
1
2
=
=
Z
−
dx− − i l+ x−
i x2 in·∂y
†
2
e
hB v |T{ bv Y (y)e
Y bv (y) }|B v i
4π
y=0
x−
dx− − i l+ x−
e 2
hB v |T{ bv Y (y)Y † (y)ei 2 in·Dus Y (y) Y † bv (y) }|B v i
4π
y=0
Z
+
x−
1
dx−
hB v |T{bv (y)ei 2 (in·Dus −l ) bv (y)}|B v i
2
4π
y=0
1
hB v |T{bv δ(in · Dus − l+ )bv }|B v i
2
Z
(5.70)
(5.71)
(5.72)
(5.73)
The fields in the fourth line are understood to be evaluated in the origin. Thus we have shown that
the structure function defined in (5.62) is the same as the form in (5.63). The physical interpretation of
this definition of the structure function is the probability of finding a b quark with residual momentum
l+ inside the B meson. The support of S(l + ) is over the range −∞ ≤ l+ ≤ ΛQCD , because the higher
momenta are no longer usoft, while it peaks at l + ' 0.
5.2.4
The final factorization
Comparing (5.61) with (5.63) we see that we should have l + = 2Eγ + k + − mb , hence
T ef f =
Z
dk + S(k + )JP (k + + mb − 2Eγ ),
(5.74)
and
1 dΓ
Γ0 dEγ
=
=
4Eγ −1
ImT (Eγ )
m3b
π
−1
Im[H(mb , µ)T ef f (Eγ , µ)].
π
(5.75)
(5.76)
64
Chapter 5. The inclusive B → Xs γ decay
We have related the decay rate of the B meson to the imaginary part of the hard amplitude and effective
forward scattering amplitude, which itself is a function of the structure function and the jet function
J(k + ).
The hard amplitude H(mb , µ) as well as the structure function S(k + ) are real. The former is obvious
from the definition of H(mb , µ) in equation (5.29), yet from the definition in (5.62) it is not clear that the
latter is real. To prove that it is, we use the translation operator e−iP x bv (0)eiP x = bv (x) and completeness
P
of states, m |mihm| = 1. The momentum operator P working on the state |mi has the eigenvalue p m ,
the momentum of the state. Working on the meson state |B v i it therefore gives the momentum pB .
Starting with S(k + ) given in equation (5.63), we can rewrite it as
S(k + )
=
=
=
=
=
=
Z
1
n
n
dx− − i l+ x−
hB v |T{bv ( x− )Y (0, x− )bv (0)}|B v i
e 2
2
4π
2
2
Z
n −
n −
dx− − i l+ x−
1
e 2
hB v |T{e−iP 2 x (bv Y )(0)eiP 2 x (Y † bv )(0)}|B v i
2
4π
Z
X
n −
n −
dx− − i l+ x−
1
e 2
hB v |T{e−iP 2 x (bv Y )(0)eiP 2 x
|mihm|(Y † bv )(0)}|B v i
2
4π
m
Z
1 X dx− − i (l+ +pB n−pm n)x−
hB v |T{(bv Y )(0)|mihm|(Y † bv )(0)}|B v i
e 2
2 m
4π
Z
1X
dx− − i (l+ +pB n−pm n)x−
e 2
|hB v |(bv Y )(0)|mi|2
2 m
4π
1X
|hB v |(bv Y )(0)|mi|2 δ(l+ + pB n − pm n),
2 m
(5.77)
(5.78)
(5.79)
(5.80)
(5.81)
(5.82)
which is real. We have used here again that Y (0, n2 x− ) = Y (−∞, n2 x− )Y † (−∞, 0). In the result we only
have usoft Wilson lines connecting −∞ with 0. This leaves
−1
π
Im[H(mb , µ)T ef f (Eγ , µ)] =
If we define the jet function J(k + ) as
−1
π
−1
π
H(mb , µ)
Z
dk + S(k + )ImJP (k + + mb − 2Eγ ).
(5.83)
times the imaginary part of the function JP ,
+
J(k ) ≡
−1
π
ImJP (k + ),
(5.84)
we find as the final result,
1 dΓ
= H(mb , µ)
Γ0 dEγ
Z
ΛQCD
−∞
dk + S(k + )J(k + + mb − 2Eγ ),
(5.85)
where the bounded integral comes from the structure function which has support over −∞ ≤ k + ≤ ΛQCD .
In this form all hard, usoft and collinear momenta have been factorized into different functions! This could
be done without too much effort, by combining known HQET with SCET. We have used SCET to decouple
the collinear, usoft and hard degrees of freedom. The calculation of the decay rate is made significantly
5.2 Matching QCD onto SCETI
65
less tedious by this factorization. The structure function is a universal non-perturbative function which
can be interpreted as the probability of finding a b quark inside the B meson with momentum k + . This
universality means that once it is determined, it can be used in all processes involving B mesons. The Jet
function J is not process dependent, as seen from its definition in (5.34), hence it is a universal function
as well. The hard kernel can be calculated perturbatively, as it is essentially the squared of the Wilson
coefficient. This Wilson coefficient is calculated by comparing the QCD diagrams with SCET order by
order in the coupling constant. This is illustrated for the production of a Higgs boson in chapter 7.
In this chapter we have proved the factorization formula for the photonic spectrum of the decay rate
of the B → Xs γ decay. This is not a new result, it has already been proved by using the full QCD theory,
yet this was done order by order in the coupling constant αs , while we have proved it using SCET for all
orders in αs , and for leading order in λ. While it is not proven for all λ, it can be seen as an improvement
as it is a quick and easy proof.
66
Chapter 5. The inclusive B → Xs γ decay
Chapter 6
Deep Inelastic Scattering
In this chapter we will discuss how to use SCET for deeply inelastic scattering (DIS) processes. Deep
inelastic scattering is used to probe the insides of hadrons using electrons, muons and neutrinos. It has
been used to prove the existence of quarks, which had been a purely mathematical concept up till then.
The aim of this chapter is to show that we can factorize the differential cross-section of DIS, and
specifically the hadronic tensor, into an integral over hard coefficients times parton distribution functions
up to leading in λ and up to all orders of αs . In the full QCD theory this factorization has to be shown
order by order in αs , yet using SCET we do this by matching on local operators, which will turn out
to be much less work. As DIS is a ‘standard’ subject, we will use known results from DIS, without
elaborating on them. Most of these results can be found in Peskin and Schroeder [16] or any other book
about Quantum Field Theory.
6.1
The kinematics of the process
Deep Inelastic Scattering ’consists’ of a lepton emitting a photon, which interacts with a proton to produce
a hadron jet. This is shown in figure 6.1. The photon has momentum q, which defines the hard scale,
q 2 = −Q2 . We use the Bjorken scaling variable x, which is defined as
x≡
Q2
,
2p · q
(6.1)
with p being the momentum of the proton. Because of momentum conservation we know the momentum
of the jet is pµX = pµ + q µ , with pµX the momentum of the hadronic jet and pµ the momentum of the
incoming proton.
6.1.1
The Breit frame
We will view DIS in a frame in which the exchanged photon is completely space-like q µ = (0, ~q). This
frame is called the Breit frame. The Breit frame is obtained by boosting the target rest frame along the
67
68
Chapter 6. Deep Inelastic Scattering
Figure 6.1: The graphical representation of Deep Inelastic Scattering. A lepton, in
this case an electron, with momentum k µ emits a photon with momentum q = k − k 0
which interacts with an incoming proton, resulting in a jet of collinear particles X with
momentum pX . From momentum conservation it is clear that pµX = pµ + q µ .
µ
µ
z-axis. In the target rest frame we have pµ = Q(1, 0, 0, 0) = Q
2 (n + n ) and q⊥ = 0. Boosting this along
µ
µ
the z axis, we get q µ = Q(0, 0, 0, 1) = Q
2 (n − n ). All perpendicular components are still zero, thus
p2 = p+ p− = m2p ∼ Λ2QCD . In this Breit frame the proton momentum p is given by
pµ
=
=
=
=
as x =
Q2
2p·q
=
Q2
2( 21 n·pn·q)
=
Q
n·p .
nµ
n·p
2
nµ m2p
2 2n · p
µ m2
µ
n Q n
p
+
2 x
2 n·p
2
nµ Q n µ Λ
+
Qx
2 x
2 Q
nµ
n·p+
2
nµ
n·p+
2
(6.2)
(6.3)
(6.4)
(6.5)
From this we find for the jet momentum:
pµX = pµ + q µ
=
=
nµ Q Q µ
+ (n − nµ )
2 x
2 Q µ Q µ 1−x
n + n
,
2
2
x
(6.6)
(6.7)
and p2X = Q2 ( 1−x
x ). For different values of the variable x we have different regions:
• When
1−x
x
∼ 1, we have p2X ∼ Q2 .
• When
1−x
x
∼
Λ
Q,
• When
1−x
x
∼
Λ2
Q2 ,
p2X ∼ QΛ. This is the endpoint region of DIS.
p2X ∼ Λ2 , which is called the resonance region.
We will look at the first region, where (1−x)
∼ 1. It will become clear that this means we will use SCETII
x
for the analysis, where as always we use SCETI as an intermediate step. However, not all of the power
of SCETII will not be needed, as the soft Wilson lines S vanish trivially.
6.2 The DIS cross-section
6.1.2
69
Powercounting
Λ
in
To find out what the momentum modes of the various particles are, we will look at the factors of Q
the momentum components. In the Breit frame the photon is the hard scale, while the proton has an
x 2
invariant mass of p2 = Λ2 , and has as small and large components n·p = Q
Λ , respectively n·p = Q
x ∼ Q.
Λ
2
2 2
With λ = Q , the invariant mass of the proton is p = Q λ ,hence we will use SCETII . In this process
the proton is an n-collinear particle.
On the other hand, the final state hadronic jet has, as mentioned above, an invariant mass of p 2X =
1−x
2
2x
Q2 1−x
x ∼ Q , and has components n · pX = Q + Λ Q ∼ Q and n · pX = Q x ∼ Q. This means that is
a hard particle, just like the photon.
6.2
The DIS cross-section
We consider the differential cross section for DIS, which can be written as [16]
dσ =
πe4 µν
d3 k~0
L (k, k 0 )Wµν (p, q).
2|~k 0 |(2π)3 sQ4
(6.8)
The tensors Lµν and Wµν are respectively the leptonic and hadronic tensors. As the names suggest, these
have only leptonic resp. hadronic degrees of freedom. The leptonic tensor L µν is dependent only on k
and k 0 , the momenta of the leptons, while Wµν depends on p and q, which automatically give pX as well.
We concentrate on the hadronic tensor, as this can be simplified using SCET. The leptonic part is not
discussed here.
The hadronic tensor can be written as the imaginary part of the forward scattering amplitude,
Wµν
=
=
1 XX
(2π)4 δ(p + q − pX )hp|Jµ (0)|XihX|Jν (0)|pi
2
spin X
Z
1X
d4 xeiq·x hp|T{Jµ (x)Jν (0)}|pi,
2
(6.9)
(6.10)
spin
or
Wµν =
1
ImTµν ,
π
(6.11)
where the forward scattering amplitude Tµν is given by
Tµν
1X
=i
2
spin
Z
d4 xeiq·x hp|T{Jµ (x)J µ (0)}|pi.
(6.12)
The sum over the spins is only over the initial proton spins. The spins of the final state hadrons where
included in the sum over X, which vanished. The current Jµ is the quark electromagnetic current,
Jµ = ψγµ ψ. As always with electromagnetic currents we have current conservation, ∂ µ J µ = 0. This
can be used to find an identity for Tµν . The forward scattering amplitude consists of the time-ordered
product of two currents. When we act with the partial derivative on this, it must vanish. In momentum
70
Chapter 6. Deep Inelastic Scattering
→
+
Figure 6.2: The matching of the QCD forward scattering amplitude onto SCET. The
gray blob in the right graph denotes the matching operator O (i) .
space this means that we find the Ward identity q µ Tµν = q ν Tµν = 0. There are only two possible tensors
built from the momenta p and q which satisfy these constraints, thus T µν must be a linear combination
of these:
qµ qν
p·q
p·q
T1 (Q2 , x) + pµ − qµ 2
pν − qν 2 T2 (Q2 , x)
(6.13)
Tµν =
−gµν + 2
q
q
q
qµ qν
qν qµ =
−gµν + 2
pν +
T2 (Q2 , x).
(6.14)
T1 (Q2 , x) + pµ +
q
2x
2x
Looking at the expression for the cross-section in equation (6.8), we see that the hadronic tensor which is
essentially built from Tµν , is contracted with the leptonic tensor Lµν . It is known that Lµν qµ = Lµν qν = 0.
Hence the terms which contain a qµ or qν vanish in the expression for Tµν , and we are left with
Tµν = −gµν T1 (Q2 , x) + pµ pν T2 (Q2 , x).
6.2.1
(6.15)
SCET operators
Following the procedure we discussed in section 4.9 we have to match these QCD results onto SCET
operators at leading order in λ. The virtuality of the intermediate state in the forward scattering, which
is in this case the hadronic jet, is of order Q2 , as p2X ∼ Q2 . Therefore we can perform an operator product
expansion (OPE), in which we expand the current in local operators in powers of 1/Q. The SCET current
on which we match is J µ = ψ X γ µ W ξn,p . The field ψX describes the hadronic jet with momentum pX ,
while the collinear field ξn,p describes the collinear proton. This is shown pictorially in figure 6.2.
In stead of calculating the hadronic tensor directly we calculate the forward scattering and finally
take the imaginary part. While the forward scattering is a straightforward amplitude without final state
particles, Wµν uses on-shell particles. The forward scattering can be matched onto SCET operators at
a scale where we want the QCD and SCET results to both be valid. The Wilson coefficients C(Q, µ)
are calculated at this point, and from here on the running of the matching scale µ determines C(Q, µ)
at other SCET scales. The most general form of leading order operators, with collinear quark and gluon
fields attached, is
µν
TSCET
=
1X
µν
hp|T̂SCET
|pi
2
spin
(6.16)
6.2 The DIS cross-section
71
with
µν
T̂SCET
g µν
=
Q
X
(i)
O1
i
Og
+ 1
Q
!
X (i) Og
1
qµ ν qν
+ (pµ +
O2 + 2
)(p +
)
Q
2x
2x
Q
i
!
,
(6.17)
(i)
where the form of the operators Oj and OJg are easily seen to be
(i)
Oj
Ojg
where
/ (i)
n
(i)
(i)
ξ n,p0 W Cj (P + , P − )W † ξn,p
2
= nµ nν Tr W † Gµλ W Cjg (P + , P − )W † Gνλ W
(6.18)
=
(6.19)
igGµν ≡ [iD̂µ + gAµn,q , iD̂λ + gAλn,q0 ],
(6.20)
†
1
with P ± = P ± P. We have put the factor of Q
in equation (6.17) there to make the Wilson coefficients
dimensionless: the fields ξn are fermion fields and are hence of dimension 32 , while the Wilson lines
(i)
(i)
obviously are dimensionless. This makes the operator Oj of dimension 3 − [Cj ]. The dimension of Tµν
(i)
(i)
is 2[J] − 4 = 2, so we need a factor of 1/Q in front of Oj to have [Cj ] = 0. From (6.19) it is then
obvious that to have dimensionless gluon Wilson coefficients Cjg we should have an extra 1/Q term in
front of the gluon operators Ojg . In equation (6.17) we have both quark and gluon operators. Yet at tree
level, we match the forward scattering (6.12) which has only quark components and we can forget about
the gluon operators Ojg . This is seen in figure 6.2 as well. The SCET forward scattering then becomes
µν
T̂SCET
=
µν X
g⊥
(nµ + nµ )(nν + nν ) X (i)
(i)
O1 +
O2
Q i
Q
i
(6.21)
µ
µ
µ
µ
Q n −n
=
Here we have rewritten the term in front of the second set of operators as pµ + q2x = n2 Q
x + 2
2x
Q
µ
µ
4x (n + n ). The perpendicular part of the metric tensor comes from just such an analysis: we know
that gµν =
nµ nν
2
+
n µ nν
2
− gµν +
⊥
+ gµν
, thus
qµ qν
q2
=
=
=
nµ n ν
Q2 (nµ − nµ )(nν − nν )
nµ nν
⊥
−
− gµν
−
2
2
4Q2
nµ n ν
nµ nν
n
n
n
nµ nν
n µ nν
µ ν
µ nν
⊥
−
−
− gµν
−
+
+
−
2
2
4
4
4
4
nµ nν
nµ nν
nµ nν
nµ n ν
⊥
−
− gµν −
−
.
−
4
4
4
4
−
Working on the above with q µ , it should vanish, thus q µ (−gµν +
using n · q = −n · q, we get
− gµν q µ +
qµ qν µ
q
q2
=
=
=
qµ qν
q2 )
(6.23)
(6.24)
= 0. Performing this action, and
n · qnν
n· qnν
n · qnν
n · qnν
⊥
−
− q µ · gµν
−
−
4
4
4
4
n · qnν
n · qnν
n· qnν
n · qnν
⊥
−
− q µ · gµν
−
+
4
4
4
4
⊥
−q µ · gµν
.
−
(6.22)
(6.25)
(6.26)
(6.27)
72
Chapter 6. Deep Inelastic Scattering
Only the perpendicular component of the metric tensor remains, the other terms vanish trivially, so we
⊥
in front of the operator.
only need gµν
When we redefine the fields to decouple from the usoft gluons in the usual way, ξ → Y ξ (0) , these usoft
Wilson lines vanish trivially,
(i)
Oj
→
=
/ (i)
n
(i)
(i)
(ξ n,p0 Y † )(Y W Y † ) Cj (P + , P − )(Y W † Y † )(Y ξn,p
)
2
(i),(0)
Oj
.
(6.28)
(6.29)
The Y lines commute with the operators P ± , as the latter only act on label momenta while the former
consist of only usoft, residual, momenta. This is where the real power of SCET comes in again. By
redefining the fields in the above way, we have decoupled the usoft gluons from the fields. However, not
only have they been decoupled, they have vanished completely! The usoft gluons hence do not couple
in any way to the fields in the forward scattering amplitude. From now on we will therefore omit the
superscript (0) for the decoupled fields.
From the section on powercounting it is known that we have to use SCET I , with the usoft degrees of
freedom, only as an intermediate step. When we scale up to SCETII , the usoft gluons are replaced by
soft gluons. By the same kind of analysis it is seen that soft gluons also decouple from the fields. The
fields are redefined by W † ξ → SW † ξ, which causes the soft Wilson lines to cancel just like the usoft ones.
(i)
By taking a convolution of the operators Oj
hard coefficients from the long distance operators:
(i)
Oj =
Z
†
with δ(ω1 − P ) and δ(ω2 − P) we can separate the
/
† n
(i)
dω1 dω2 Cj (ω+ , ω− )[(ξ n W )ω1 δ(ω1 − P ) (W † ξn )ω2 δ(ω2 − P)]
2
(6.30)
(i)
with ω± = ω1 ± ω2 . The hard degrees of freedom are all in the Wilson coefficients Cj (ω+ , ω− ), while
the collinear operators which act on long distances, are between the brackets. This is the first important
step in getting the required factorized result.
6.2.2
Charge conjugation
Under charge conjugation, the electromagnetic current transforms as Jµ → −Jµ . This means that the
forward scattering, which is a time-ordered product of two currents, is invariant under these transfor(i)
mations. By (6.17) the operators Oj have to be invariant as well. This leads to an identity for the
Wilson coefficients, C(−ω+ , ω− ) = −C(ω+ , ω− ), which we will prove here. Under charge conjugation C,
a fermion field transforms as CψC −1 = −iγ 2 ψ ∗ , or CψC −1 = −(ψC)t , and CψC −1 = −(Cψ)t , where
C ≡ iγ 0 γ 2 . To find the transformation rules for the collinear fields ξn,p we go back to the definition of this
/
nn
field. The collinear quark field was defined by ξn,p = /4 ψp , where ψp is defined from the usual fermion
6.2 The DIS cross-section
field as ψ =
P
p̃
73
e−ip̃·x ψp . Thus under charge conjugation Cξn C −1 = C(
Cξn C −1
=
=
=
=
/
n
/n
ψC −1
4
/
n
/n
− iγ 2 ψ ∗
4
/
n
/n
− iγ 2 (ψ † )t
4
X
/
n
/n
ψp† eip̃·x )t
− iγ 2 (
4
P
†
ip̃·x
)C −1 ,
p̃ ξn,p e
and we have
C
(6.31)
(6.32)
(6.33)
(6.34)
p̃
=
=
=
−
/ 2 0 X
n
/n
ψ p eip̃·x )t
iγ γ (
4
(6.35)
p̃
/ 2 0 X
n
/n
ψ −p e−ip̃·x )t
iγ γ (
4
p̃
X
ξ n,−p e−ip̃·x C)t
−(
−
(6.36)
(6.37)
p̃
which means that Cξn,p (x)C −1 = −[ξ n,−p (x)C]t and Cξ n,p (x)C −1 = −[ξn,−p (x)C]t = [ξn,−p (x)C −1 ]t .
The gluon fields transform in the same way as in usual QCD, thus CAµn,p (x)C −1 = −[Aµn,p (x)]t . Using
this and the expression for W in (4.113), it that the transformation properties of the collinear Wilson line
are CW C −1 = [W † ]t . With these conjugation properties for the fields, we can find out how the operators
(i)
Oj transform.
(i)
COj C −1
=
=
/ (i)
n
(i)
(i) −1
Cξ n,p0 W Cj (P + , P − )W † ξn,p
C
2
/ (i)
n
(i)
(i) −1
Cξ n,p0 C −1 CW C −1 C Cj (P + , P − )C −1 CW † C −1 Cξn,p
C .
2
(6.38)
(6.39)
It is convenient to write this out with indices, and rearrange the terms,
/
n
(i)
† −1
(i)
C (P + , P − )C −1 CWkl
C C(ξn,p
)l C −1
2 jk j
/
n
(i)
†
−1
C (P + , P − )Wlk Clβ (ξ n,−p )β (x)
= −(ξn,−p )α (x)Cαi Wji
2 jk j
/
−n
(i)
†
= (ξ n,−p )β (x)Wlk
(ξn,−p )α (x)
C (P + , P − )Wji
2 kj j
(6.41)
=
(6.43)
(i)
C(ξ n,p0 )i C −1 CWij C −1 C
/ (i)
n
−ξ n,−p (x)W Cj (P + , P − )W † ξn,−p (x),
2
where we have used that C −1 γ µ C −1 = −(γ µ )t .
(6.40)
(6.42)
74
Chapter 6. Deep Inelastic Scattering
(i)
Doing the convolution, this form of COj C −1 yields
/
† n
(i)
dω1 dω2 Cj (ω+ , ω− )[(ξ n W )ω1 δ(ω1 − P ) (W † ξn )ω2 δ(ω2 − P)]
2
Z
/
† n
(i)
− dω1 dω2 Cj (ω+ , ω− )[(ξ n W )−ω2 δ(−ω2 − P ) (W † ξn )−ω1 δ(−ω1 − P)].
2
Z
C
→
(6.44)
(6.45)
Substituting ω1 → −ω2 , ω2 → ω1 , under which ω+ → −ω+ and ω− → ω− , we get
−
Z
/
† n
(i)
dω1 dω2 Cj (−ω+ , ω− )[(ξ n W )ω1 δ(ω1 − P ) (W † ξn )ω2 δ(ω1 − P)].
2
(6.46)
Invariance of the forward scattering under charge conjugation hence requires the Wilson coefficients to
obey the identity C(ω+ , ω− ) = −C(−ω+ , ω− ), which relates the Wilson coefficient for quarks to that for
anti quarks.
6.2.3
The parton distribution function
We want to use the parton distribution function fi/p (z), which gives the probability of finding a quark i
in proton p with momentum fraction zpµ , where 0 < z < 1. This parton distribution is given by [24, 25]
fi/p (z) =
Z
/ ξn (y)|pi
dye−2iz(n·p)y hp|ξ n (y)W (y, −y)n
(6.47)
and for anti-partons f i/p (z) = −fi/p (−z). We can use the Fourier transform of these PDFs to rewrite the
forward scattering amplitude in terms of these PDFs. To do that, we note first that the Fourier transform
of ξ n (y)W (y) with respect to R is ξ n,p W δP † ,R , thus the Fourier transform of hp|ξ n (y)W (y, −y)ξn (−y)|pi
is:
Z
Z
dy −iωy
dy −iωy
(6.48)
e
hξ n (y)W (y, −y)ξn (−y)i =
e
h(ξ n W )(y)(W † ξn )(−y)i
2π
2π
Z
dy X i(R+T −ω)y
†
)(δP,R W † ξn,p )i (6.49)
=
e
h(ξ n,p0 W δP,R
2π
R,T
X
†
=
)(δP,R W † ξn,p )i
δ(ω − R − T )h(ξ n,p0 W δP,R
(6.50)
R,T
=
X
T
=
†
)(δP,R W † ξn,p )i
h(ξ n,p0 W δP,ω−T
hξ n,p0 W δ(ω − P + )W † ξn,p i
(6.51)
(6.52)
The parton distribution function can now be rewritten in momentum space as
1X
/ ξn (y)|pi
hp|ξ n (y)W (y, −y)n
2
=
spin
−
4n · p
Z
1
0
Z
1
0
dzδ(ω− )δ(ω+ − 2n · pz)fi/p (z)
dzδ(ω− )δ(ω+ + 2n · pz)f i/p (z) .
(6.53)
6.2 The DIS cross-section
75
The delta function over ω− sets ω1 = ω2 . The other delta function, δ(ω+ − 2n · pz) ensures that positive
ω+ = ω1 + ω2 gives particles with parton distribution function fi/p (z), while, due to second delta function
over ω+ , negative values of ω1 = ω2 give rise to anti-particles with parton distribution function f i/p (z).
(i)
Using this and (6.52), the operator Oj between proton states and summed over spins can be written as
1X
(i)
hp|Oj |pi
2
=
Z
/
† n
(i)
dω1 dω2 Cj (ω+ , ω− )[(ξ n W )ω1 δ(ω1 − P ) (W † ξn )ω2 δ(ω2 − P)]
2
=
Z
dω1 dω2 Cj (ω+ , ω− )4n · p
spin
Z
×
Z
=
(6.54)
(i)
1
dzδ(ω− )δ(ω+ + 2n · pz)fi/p (z) −
0
(i)
dω1 Cj (2ω1 , 0)4n
Z
=
4n · p
0
dzδ(2ω1 − 2n · pz)fi/p (z) −
Z
1
0
1
0
dzδ(ω− )δ(ω+ + 2n · pz)f i/p (z) (6.55)
·p
1
×
Z
Z
1
0
dzδ(2ω1 + 2n · pz)f i/p (z)
i
h
(i)
(i)
dz Cj (2n · pz, 0)fi/p (z) − C1 (−2n · pz, 0)f i/p (z) .
(6.56)
(6.57)
Yet we have also, from charge invariance, C(−ω+ , ω− ) = −C(ω+ , ω− ), thus the final form of the forward
scattering is
1X
(i)
hp|Oj |pi = 4n · p
2
spin
Z
1
0
h
i
(i)
dzCj (2n · pz, 0) fi/p (z) + f i/p (z) .
(6.58)
As the forward scattering is a function of these matrix elements, equation (6.21), this forward scattering
can also be written as a function of the PDFs.
6.2.4
The matching
What is left to do is the matching itself. On the one hand we have the QCD forward scattering
Tµν
=
=
=
+
≈
=
=
−gµν T1 (Q2 , x) + pµ pν T2 (Q2 , x)
nµ nν
nµ nν
⊥
−gµν
T1 −
T 1 + p µ pν T 2
+
2
2
1
⊥
−gµν
T1 − ((nµ + nµ )(nν + nν ) − (nµ − nµ )(nν − nν )) T1
4
(n · p)2
nµ nν
+ O(λ2 ) T2
4
1
Q2
⊥
T1 − (nµ + nµ )(nν + nν )T1 + nµ nν 2 T2
−gµν
4
4x
1
Q2
⊥
−gµν
T1 − ((nµ − nµ ) + 2nµ ) ((nν − nν ) + 2nν ) T1 + nµ nν 2 T2
4
4x
2
Q
⊥
−gµν
T1 (Q2 , x) + nµ nν
T2 (Q2 , x) − T1 (Q2 , x) ,
4x2
(6.59)
(6.60)
(6.61)
(6.62)
(6.63)
(6.64)
76
Chapter 6. Deep Inelastic Scattering
2q
where we have used on several occasions that (nµ − nµ ) = Qµ and that all terms proportional to qµ
vanish. Also the decomposition of the metric tensor, given in appendix B, is used. On the other hand
the SCET forward scattering is given by equation (6.17), or
SCET
Tµν
=
⊥
gµν
1 X
(nµ + nµ )(nν + nν ) 1 X
(i)
(i)
hp|O1 |pi +
hp|O2 |pi
Q 2
Q
2
i,spin
=
(6.65)
i,spin
⊥
gµν
1 X
4nµ nν 1 X
(i)
(i)
hp|O1 |pi +
hp|O2 |pi.
Q 2
Q 2
i,spin
(6.66)
i, spin
From equations (6.64) and (6.66) it is clear that at the matching scale we have to identify
−1 X
(i)
hp|O1 |pi
2Q
=
T1 (Q2 , x)
=
(6.67)
i,spin
2 X
(i)
hp|O2 |pi
Q
i,spin
Q2
2
2
T
(Q
,
x)
−
T
(Q
,
x)
,
2
1
4x2
(6.68)
or
T1 (Q2 , x)
=
=
=
−1 X
(i)
hp|O1 |pi
2Q
i,spin
Z
h
i
−4n · p X 1
(i)
dzC1 (2n · pz, 0) fi/p (z) + f i/p (z)
Q
0
i
Z 1
h
i
X
−4
(i)
dzC1 (2n · pz, 0) fi/p (z) + f i/p (z)
x i 0
(6.69)
(6.70)
(6.71)
and
T2 (Q2 , x)
8x2 X
2x2 X
(i)
(i)
hp|O
|pi
−
hp|O1 |pi
(6.72)
2
Q3
Q3
i,spin
i,spin
h
i
X Z 1 4x2 (i)
4x2 (i)
= 4n · p
dz
C
(2n
·
pz,
0)
−
C
(2n
·
pz,
0)
fi/p (z) + f i/p (z) (6.73)
2
1
3
3
Q
Q
0
i
Z
h
i
4x X 1 (i)
(i)
(6.74)
=
dz 4C2 (2n · pz, 0) − 4C1 (2n · pz, 0) fi/p (z) + f i/p (z) .
2
Q i 0
=
With these expressions for T1 and T2 , we can finally put everything together.
6.3
Results
The hadronic tensor was defined as Wµν = π1 Im Tµν . If we define the hard functions Hj as Hj (z) ≡
1
π Im Cj (2Qz, 0), where the dependence on the hard scale Q and the matching scale µ is implied, we can
6.3 Results
77
write the components W1 and W2 , where Wi ≡
W1 (Q2 , x)
=
W2 (Q2 , x)
=
4X
x i
1
π ImTi ,
as
1
h
i
(i) z
dzH1 ( ) fi/p (z) + f i/p (z)
x
0
Z
1
h
i
4x X
(i) z
(i) z
)
−
H
(
)
f
(z)
+
f
(z)
.
dz
4H
(
i/p
i/p
1
2
Q2 i 0
x
x
−
Z
(6.75)
(6.76)
The hadronic tensor can be constructed from W1 and W2 , and the thus the expression for the differential
cross-section can. We will not go into this deeper, as the main result has already been found: we have
again a factorized result. The hard functions are computable, by computing the matching coefficients
precisely, and contain all hard degrees of freedom. Meanwhile the parton distribution functions contain
the collinear degrees of freedom. These distributions are universal non-perturbative functions, and appear
in many different processes. Hence they cannot be calculated, but once the precise distribution is found
by experiment, it can be used for all other processes.
The Wilson coefficients were constructed to be dimensionless, therefore the hard functions H j are as
µ
well. This means they can only depend on Q by αs (Q) ln( Q
). Another property of the used method is
that as we look upwards in energy, the matching coefficient µ is the same as the renormalization scale
(i)
µR . As an aside, we note that from equation (6.68) we see that when the second set of operators O 2
vanish, we have H2 = 0 and T1 =
get
Q2
4x2 T2 .
For the components of the hadronic tensor, Wi =
W1 (Q2 , x)
Q2
=
W2 (Q2 , x)
4x2
1
π Im
Ti , we
(6.77)
which is the well known Callan-Gross relation [26]. Thus in our description of the process using SCET
the Callan-Gross relation holds true.
We will now look at the final example of a process which is simplified by use of SCET, the production
of a Higgs boson by gluon fusion. We will not only look at the factorization, but also at the resummation
of the large logarithms. What this means will become clear in due course.
78
Chapter 6. Deep Inelastic Scattering
Chapter 7
Higgs production through gluon
fusion
The last process we discuss is the production of a Higgs boson by gluon fusion via a top quark loop. At
high energies the Higgs cross-section is dominated by this process, pp → gg → H, hence it is the most
significant Higgs production process at the LHC. The quark loop which produces the Higgs particle could
in principle contain all sorts of quarks, yet because the top quark mass is some 35 times heavier than
the bottom quark, while the other quarks are even lighter still, the top Higgs boson couples 35 times
stronger to the top quark. The contribution of the bottom quark is thus 35 2 less than that of the top
quark. Hence we will only consider production via a top quark loop. This loop can be seen as an effective
coupling, just like the W -top loop in the B → Xs γ decay in chapter 5. The effective Lagrangian density
for this process is
1
(7.1)
LHgg = − GH φTr[Gµν Gµν ]
4
with φ is the scalar Higgs field and GH the effective coupling coming from integrating out the top loop.
We consider a scalar Higgs boson φ, as opposed to a pseudo-scalar Higgs. The effective theory could be
used to deal with the pseudo-scalar Higgs as well, yet the Lagrangian and corresponding currents would
be slightly different.
The cross-section of the process is again an integral over the forward scattering amplitude,
σab
=
π
s
Tab
=
G2H
Z
dn q + 2
δ (q − m2 )Tab
(2π)n
Z
d4 yeiqy hpp|J † (y)J(0)|ppi.
(7.2)
(7.3)
The amplitude is the product of two gluon currents J(x) = Tr[Gµν Gµν ] between the proton states. We
will have to match this onto SCET operators, after the kinematics of the process are known.
79
80
Chapter 7. Higgs production through gluon fusion
(a)
(b)
Figure 7.1: The production of a Higgs boson via gluon fusion. The incoming protons
which produce the quarks have been omitted. Diagram (a) shows the gluons attached to
the top-quark loop. In (b) the top loop is seen as an effective coupling. The gluon with
momentum k1 is n-collinear, while k2 is n-collinear.
7.1
Kinematics of Higgs production
This channel of Higgs production is very similar to the Drell-Yan process described in chapter 2. In
the center of mass frame the incoming gluons propagate in opposite (lightlike) directions. We take one
gluon moving in the nµ direction, while the other will be then moves in the nµ direction. The resulting
2
Higgs boson has an invariant mass of p2H = MH
which is the hard scale in this process, hence we take
2
2
Q = MH . The incoming protons have momentum P1 , P2 and center of mass energy of s = (P1 + P2 )2 ,
while the gluons have a fraction of their momentum, qi = ξi Pi with ξi smaller than 1. The momentum
of the final state hadrons is the difference of that of the incoming protons and the outgoing Higgs boson,
pX = P1 + P2 − pH , and
p2X
=
=
=
(P1 + P2 )2 + p2H − 2pH · (P1 + P2 )
s + Q2 − 2pH · P1 − 2pH · P2
1
1
1
Q2 1 + −
−
z
x1
x2
with
z=
Q2
,
s
xi =
Q2
.
2pH · Pi
(7.4)
(7.5)
(7.6)
(7.7)
The variables x1 and x2 are like the Bjorken scaling variables encountered in DIS.
We consider the threshold region where the produced Higgs boson has nearly all incoming energy.
As the invariant mass of the Higgs boson approaches the center of mass energy of the incoming proton
pair, there is less and less energy available for producing gluon radiation. This implies that only gluon
emissions will occur with low momenta. The threshold limit is given by z → 1, or 1 − z ∼ λ 2 .
We will start by matching on SCETI at Q2 , to absorb the virtuality of order Q2 in the Wilson
coefficients. We note that the usoft scale has p2us ∼ Q2 λ4 ∼ Q2 (1 − z)2 , which is of the same order as the
final hadronic state. The collinear gluons have qi2 ∼ Q2 λ2 ∼ Q2 (1 − z). At Q2 (1 − z)2 this is seen as a
large scale and has to be integrated out. Therefore the process will be described by this effective theory
7.2 Matching at Q2
81
from Q2 up to momentum scale Q2 (1 − z)2 , where we match onto SCETII operators.
7.2
Matching at Q2
We make the operator expansion
JQCD = CHgg
Q
, αs (µ) Jef f (µ)
µ
(7.8)
where the QCD current is the usual
JQCD = Tr[Gµν Gµν ]
(7.9)
and CHgg (Q2 /µ2 ) is the matching condition. The effective current and matching coefficient depend on the
renormalization or factorization scale µ. The Wilson coefficient, or matching condition, C Hgg contains the
contribution between Q and µ, while Jef f contains the soft and collinear contributions below momentum
scale µ. The effective current we match onto can be found by looking at the theory we want to describe.
We want to make contact with the gluon distribution functions, which are given by [25]
φg/p
Q2
ξ, 2
µ
1
=
2πξp−
Z
dx+ e−iξp
−
x+
nµ nν hp|Gµλa (x+ )Y ab (x+ , 0)Gbλν (0)|pi
spin avg.
.
(7.10)
For collinear gluons, we know from the powercounting identities derived in chapter 4 that the field strength
Gµν can be approximated by
Gµν
n
=
iD̂µ
≡
[iD̂µ + gs Aµn,q , iD̂ν + gs Aνn,q ]
nµ
n̄µ
µ
+
P̄.
i n · Dus + P⊥
2
2
(7.11)
(7.12)
The effective current we match onto should thus have a trace of the product two operators of the form
nλ Gn,µλ . However, just as in the case with collinear quark fields we need Wilson lines to obey gauge
invariance. The effective current we match on is then
Jef f (x) =
with
1
nν nλ Tr [Gnµν Gn,µλ ]
Q4
i
h
i
Gnµν = − Wn† iD̂µ + gs Aµn,q , iD̂ν + gs Aνn,q Wn , Gn = n ↔ n.
g
(7.13)
(7.14)
The fourth power of Q which is in the denominator in front, is put in to make the Wilson coefficient
dimensionless, just like we did in the previous chapter. This SCET current is the continuation of the
QCD current, written in the most convenient form to be identified with SCET distribution functions,
which we define by
Z
Q2
1
+ −iξp− x+
ν
µλ +
ξ,
φSCET
=
dx
e
n
n
hp|Tr[G
(x
)G
(0)|pi
.
µ
n,λν
n
g/p
µ2
2πξp−
spin avg.
(7.15)
82
Chapter 7. Higgs production through gluon fusion
This is the distribution of finding a gluon in parton p, which is in our case either an n-collinear gluon or
an n-collinear one.
µ (0)
Since we have found the SCET current, we have to make the usual field redefinitions A µn → Yn An,q Yn†
(0)
and Wn → Yn Wn Yn† , and their n equivalent. Under these redefinitions the operator Gn transforms as
Gnµν
i
h
i
→ − Yn Wn(0) † Yn† iD̂µ + gs Yn Aµn,q(0) Yn† , iD̂ν + gs Yn Aνn,q(0) Yn† Yn Wn(0) Yn†
g
i
h
i
= − Yn W (0) † iD̂µ + gs Aµn,q(0) , iD̂ν + gs Aνn,q(0) Yn Wn(0) Yn†
g
=
Yn Gnµν
(0)
Yn† .
(7.16)
(7.17)
(7.18)
These redefinitions result in a form of the effective coupling in which the usoft degrees of freedom are all
summed into the Wilson lines Y ,
Jef f (x)
=
=
=
1
(0)
nν nλ Tr[Yn Gnµν(0) (x)Yn† Yn Gn, µλ (x)Yn† ]
Q4
1
(0)
nν nλ Tr[Yn† Yn Gnµν(0) (x)Yn† Yn Gn, µλ (x)]
Q4
1
(0)
nν nλ Tr[Gnµν(0) Gn, µλ ](x).
Q4
(7.19)
(7.20)
(7.21)
In the second line we have used the cyclic property of the trace to bring the Y n† in front. We see that all
the usoft Wilson lines vanish, in the same way as in the DIS process! Again we omit the superscript (0)
from hereon.
The cross-section can now be written as
Z
dσ
(0)
βα(0) (0)
2
=
σ
C
Gn, βγ ]|pn pn i.
d4 xeiqx hpn pn |nν nλ Tr[Gnµν(0) Gn, µλ ]nα nγ Tr[Gn
0 Hgg
dQ2
(7.22)
This form is clearly correct, as we need two operators in n-direction, and two moving in n-direction, like
we have in (7.22). The proof of this form is given in [27]. The n-collinear operators only interact with
the n-collinear states, and likewise for the n collinear operators. Thus we can write the matrix element
as
(0)
βα(0) (0)
γ
Gn, βγ ]|pn pn i
µλ ]nα n Tr[Gn
βα(0) (0)
λ
µν(0) (0)
γ
Gn, µλ ]|pn ihpn |nα n Tr[Gn
hpn |nν n Tr[Gn
Gn, βγ ]|pn i.
hpn pn |nν nλ Tr[Gnµν(0) Gn,
=
(7.23)
(7.24)
We have again written the cross-section in factorized form, namely with the hard contributions contained
in CHgg , and the n and n-collinear parts separated.
7.2.1
The matching coefficient CHgg
The matching coefficient C(µ) is found by comparing the effective theory diagrams with the full QCD
result at the matching scale µ2 = Q2 . It is valid around this scale; to determine the physics between
7.2 Matching at Q2
83
Q2 and Q2 (1 − z)2 we need to know the anomalous dimension. We consider the calculation of the
coefficient in pure dimensional regularization (DR). Then the coefficients are obtained by computing the
on-shell diagrams in full QCD and taking only the finite terms. These will be the coefficient function.
This procedure works as the ultraviolet 1/ divergences which occur will be cancelled by the appropriate
counterterms in dimensional regularization, while the remaining infrared 1/ terms cancel between the
two theories, as these must be equal in both theories. That is essentially the whole point of effective
theories, that the divergent behaviour around the degrees of freedom which are described by the theory
is equal to that of the full theory! This leaves the finite terms in both theories to compare.
Here we come with a simplification. In the effective theory, the incoming on-shell particles as well as
the gluons which interact between them are all massless. The integrals which we have to do are therefore
R
scaleless; they are integrals of the form dd k k12 and vanish in DR, as mentioned earlier in the chapter 2
on the Drell-Yan process. Therefore these on-shell diagrams vanish in DR and the matching coefficient
consists of only the finite terms of the full QCD result.
The diagrams we need to consider to find the coefficient function at the matching scale Q 2 up to
order αs are the one-loop diagrams, as these are the ones which contribute a factor g s2 = αs /4π. These
diagrams are not calculated here, but we will give the result up to first order in the coupling constant.
(1)
The result found in the literature for CHgg is [28]
(1)
CHgg
Q2
αs (µ), 2
µ
2
2 Q
αs (µ2 )
Q
2
2
CA − ln
=
+ 7ζ(2) + 2iπ ln
,
2
4π
µ
µ2
(7.25)
(0)
while CHgg is given by the tree diagram, which is obviously equal in both theories, and thus this matching
(0)
coefficient equals CHgg = 1. The coefficient CA = Nc is the number of colours. At Q2 the logarithms in
the coefficient vanish and up to first order CHgg = 1 +
αs (Q2 )
4π CA 7ζ(2).
The anomalous dimension γ(µ) of the effective current is a function of Q 2 /µ2 and αs (µ2 ) [29] and can
be found from this matching coefficient. It is defined as
γ1 (µ) = −µ
d ln Je f f (µ)
dµ
(7.26)
where the subscript 1 indicates that it is the anomalous dimension which belongs to the first matching
step. The full QCD current clearly does not depend on the factorization scale, thus from (7.8) we can
write
d ln C
γ1 (µ) = µ
(7.27)
dµ
or
γ1 (µ)C = µ
dC
,
dµ
(7.28)
where the dependance of C on µ and Q is dropped for notational convenience. From here it is straight-
84
Chapter 7. Higgs production through gluon fusion
forward to compute γ1 up to order αs :
γ1 (µ)C(µ)
2
2 αs (µ2 )
Q
Q
d
2
2
1+
+ 7ζ(2) + 2iπ ln
CA − ln
= µ
dµ
4π
µ2
µ2
2
2
2
4iπ
αs (µ )
−4
Q
= µ
−
CA
ln
4π
µ
µ2
µ
2
2
−αs (µ )
Q
=
+ iπ 2 .
CA ln
π
µ2
(7.29)
(7.30)
(7.31)
As C(µ) = 1 + O(αs ), up to first order the anomalous dimension is given by equation (7.31),
2
Q
−αs (µ2 )
2
CA ln
γ1 (µ) =
+ iπ .
π
µ2
(7.32)
This anomalous dimension determines the physics between the two important scales, Q 2 and Q(1 − z)2
RQ
by use of the evolution exp[− Q(1−z) γ1 dµ/µ]. The coefficient CHgg (Q2 /µ2 ) is most easily defined by the
running
#
" Z
2
Q
dµ
Q
γ1 (µ)
.
(7.33)
= CHgg (αs (Q)) exp −
CHgg
µ2
µ
Q(1−z)
This exponent contains all logarithms originating from the diagrams involved in the matching. The
coefficient CHgg (αs (Q)) is the matching coefficient at µ2 = Q2 , which as we mentioned earlier is, up to
first order in αs ,
CHgg (αs (Q)) = 1 +
αs (µ2 )
CA 7ζ(2).
4π
(7.34)
Below Q2 the Higgs production process is thus completely described by SCETI with the above coefficient
function.
7.3
Matching at Q2 (1 − z)2
The next step is to match at the second scale, Q2 (1 − z)2 . We integrate the QΛQCD modes out by
matching onto SCETII operators.
The matching coefficient has been calculated by others and is given by [28]
(1)
MHgg (z)
=
( 2
ln(1 − z)
Q
1
αs
CA θ(0 ≤ z ≤ 1) 8
− 4 ln
2π
1−z
µ2
1−z +
+
2
Q
+ δ(1 − z) ln2
− 3ζ(2)
µ2
(7.35)
The theta function ensures that the coefficient only has support for 0 ≤ z ≤ 1. We want to look at this
in moment space, where we take the Mellin transform of functions defined on z ∈ [0, 1],
MN [f (z)] =
Z
1
dzz N −1 f (z)
0
(7.36)
7.3 Matching at Q2 (1 − z)2
85
In moments, the variable N is significant variable. It will dominate the logarithms which will come up.
In appendix E the moment space is explained, and large N limits of various functions are given. The
Mellin transform of (7.35) in the large N limit is
(1)
MN
µ2
αs (µ), 2
Q
=
=
"
"
(
#
2
#
αs
µ
1
ln(1 − z)
− 4 ln
MN
CA θ(0 ≤ z ≤ 1) 8MN
2π
1−z
Q2
1−z +
+
2
µ
− 3ζ(2)
(7.37)
+ MN [δ(1 − z)] ln2
Q2
2
αs
µ
CA θ(0 ≤ z ≤ 1) 4 ln2 N + 4ζ(2) + 4 ln
ln N
2π
Q2
2
µ
+ ln2
− 3ζ(2)
(7.38)
Q2
This can then be simplified by completing the squares of ln N and ln
2
4 ln N + 4 ln
µ2
Q2
ln N + ln
2
µ2
Q2
=
=
=
µ2
Q2
:
2 2
µ
1
4 ln N + ln
2
Q2
2 2
1
1
µ
2
4
ln N + ln
2
2
Q2
!
2
µ2 N
.
ln2
Q2
(7.39)
(7.40)
(7.41)
The moment of (7.35) finally becomes
(1)
MN
µ2
αs (µ), 2
Q
"
αs
=
CA ln2
2π
µ2 N
Q2
2
!
#
+ ζ(2) .
(7.42)
where N = N eγE , with γE the Euler constant. To identify the variable N with the scale Q(1 − z) we
look at the logarithms. For µ = µI , the intermediate matching scale Q(1 − z), the large logarithms
should vanish, resulting in only small finite terms. In moment space the large logarithms are minimized
by setting µ = Q/N . Therefore the identification between the factor 1 − z and N is 1 − z ∼ 1/N . The
threshold limit z → 1 hence corresponds to large N and the large N limit we did is valid. At µ I = Q/N
the matching coefficient is just, up to order αs ,
MN (αs (µI )) = 1 +
αs
CA ζ(2)
2π
(7.43)
Just as after the first matching, we need to know the running of the effective operators which we used to
perform the matching at Q2 (1 − z)2 . Yet as mentioned these are just the conventional parton distribution
functions at and below µ2I = Q2 (1 − z)2 . Hence we can use the well-known DGLAP (Dokshitzer-Gribov-
86
Chapter 7. Higgs production through gluon fusion
Lipatov-Altarelli-Parisi) evolution [30]
Z
exp 2
µI
µF
dµ g
γ
µ 2,N
(7.44)
with the anomalous dimension
γ2,N
=
Large N
−→


N
X
αs
2
1
− CA 4
−
+ 1
2π
j
N
(N
+
1)
j=2
−αs
CA 4 ln N − 3 .
2π
(7.45)
(7.46)
Here µF is the factorization scale for parton distributions.
The differential cross-section can now be written in its final form in moment space. We start at Q 2 ,
2
where the first matching has taken place, resulting in the squared of the matching coefficient C Hgg
at
R
Q
2
2
2
2
2
2
µ = Q . Between Q and µ = Q (1 − z) we run the scale by the evolution exp[− µI γ1 (µ)dµ/µ], where
2
we multiply with the moments of the matching coefficient at µ2 = Q2 /N . From here we the running of
the PDFs is governed by the DGLAP evolution with anomalous dimension γ2 to the factorization scale
µF where the moments of the local PDFs are defined as g(µF , N ). In summary,
dσ
dQ2
N
2
(Q2 )e−I1 (Q,Q/N ) MN αs (Q/N ) e−I2 (Q/N ,µF ) g 2 (µF , N )
= σ0 CHgg
where
(7.47)
Z Q/N
dµ
dµ
γ1 (µ), I2 (Q/N , µF ) = 2
γ2 (µ).
(7.48)
µ
µ
Q/N
µF
2
(Q2 ) and MN αs (Q/N ) have been given in (7.34) and (7.43). We have thus done
The coefficients CHgg
a resummation of the large logarithms. They have been removed from the coefficient functions, which
I1 (Q, Q/N ) = 2
Z
Q
thus no longer contain any large logarithms, into the exponents; I1 contains the logarithms due to the
evolution between Q and Q(1 − z) while I2 contains the logarithms coming from the evolution between
Q(1 − z) and µF .
We have found a factorized form for the cross-section, where the coefficient functions have been given
up to order αs . These coefficient functions do not contain any large logarithms, as we have resummed
these. The effective theory results are equivalent to the conventional approach, yet it is technically much
less complicated. We did not need to introduce new non-perturbative operators, which is usually the
case, while all the needed quantities are easily obtained. This is therefore a very useful technique for
doing these resummations.
Chapter 8
Conclusions
In this thesis we have seen how soft and collinear degrees of freedom can lead to infrared divergences in
QCD processes. To control these, we have introduced soft-collinear effective theory (SCET). SCET is
an effective field theory, which separates the soft and collinear degrees of freedom, and uses an effective
Lagrangian for collinear quarks and gluons. Using this new Lagrangian for the collinear particles, and
the usual QCD Lagrangian for soft particles, one can find the Feynman rules for the effective theory.
We have shown in this thesis that SCET can be used to reproduce the factorization formulas to all
orders in αs in a quick and simple way. This is illustrated for the inclusive decay of a B meson into
a photon and a meson containing an s-quark, for the process of Deep Inelastic Scattering and for the
production of a Higgs boson via gluon fusion. In all these three processes the factorization was proven
via operator product expansions, thus to all orders in the coupling constant, and to first order in the
p
small parameter λ = ΛQCD /Q1 . This is an improvement over conventional methods for obtaining
factorization, where it has to be proven order by order in αs . For the case of Higgs production, we have
also done a resummation of the large logarithms.
The development of the theory is far from over and is thus still evolving. Some problems still haven’t
been solved, or are being worked on. One such subject we have completely omitted from the discussion
in this thesis are Glauber gluons, which are soft gluons with transverse momenta much larger than
the momenta in collinear direction. These Glauber gluons have been a serious problem in proving the
factorization of the Drell-Yan processes in QCD, hence we may expect problems with them. Recent work
on this by Liu and Ma [31] shows however that these gluons can be added to the effective theory, and
that their effects cancel in the full factorization.
As mentioned SCET is a theory which is still in development. There is still a lot to learn about the
theory, and more research is needed to discover the full usefulness of SCET and all its aspects.
1 This
value of λ is for SCETI . For the somewhat different theory of SCETII one has λ = ΛQCD /Q.
87
88
Chapter 8. Conclusions
Appendix A
Conventions
In this thesis we have used ~ = 1 and c = 1. The metric signature which is used is (1, −1, −1, −1). Thus
the Minkowski metric is


1 0
0
0


0 −1 0
0
µν


η =
(A.1)

0
0
−1
0


0
A.1
0
0
−1
Dirac Algebra
The Dirac matrices obey the Clifford Algebra or Dirac Algebra which has the following anti-commutation
relation:
{γ µ , γ ν } = 2η µν 1
(A.2)
where 1 is the identity matrix. In this thesis the Dirac matrices are written in the Dirac basis and are
given by
!
!
1 0
0
σi
0
i
(A.3)
γ =
, γ =
−σ i 0
0 −1
where the σ i s are Pauli matrices,
1
σ =
0
1
!
1
,
0
2
σ =
0
i
!
−i
,
0
3
σ =
1
0
!
0
.
−1
(A.4)
We define the hermitian matrix γ5 as
γ5 = iγ0 γ1 γ2 γ3 ,
89
(A.5)
90
Chapter A. Conventions
or in it’s explicit form derived from the above,
!
(A.6)
{γ5 , γ µ } = 0.
(A.7)
γ5 =
0
1
1
0
It anti-commutes with all other γ µ :
Appendix B
Lightcone Coordinates
We will introduce the light-cone coordinates and in particular the way they are used throughout this
paper.
B.1
Vectors
The basisvectors of this coordinate system must obey n2 = n̄2 = 0, n · n̄ = 2. Let us define the vectors
n, n̄ as follows:
nµ
=
(B.1)
n̄µ
(1, 0, 0, −1)
=
(1, 0, 0, 1).
(B.2)
Obviously other choices of these basisvectors which obey n2 = n̄2 = 0, n · n̄ = 2 will work just as well,
yet this particular one will be useful for our purposes. The choice above coincides with vectors pointing
in the −~z and ~z direction respectively.
So in the lightcone coordinate system there are three specific directions: along the n direction, along
the n̄ direction and perpendicular to these directions. Vectors V in this system will be written in the
following way:
Vµ
=
=
nµ
n̄µ
n̄ · V +
n · V + V⊥µ
2
2
(V + , V − , V ⊥ ),
(B.3)
(B.4)
where V⊥ · n, V⊥ · n̄ = 0. The plus and minus directions of the vector are defined as
V+
V
−
= n·V =V0+V3
(B.5)
= n̄ · V = V − V .
(B.6)
0
3
This definition is the standard definition of the lightcone coordinates, apart from a normalization factor
91
92
of
Chapter B. Lightcone Coordinates
√1 .
2
In a similar way we can define the metric g µν as:
g µν =
B.2
nµ n̄ν
n̄µ nν
µν
+
+ g⊥
2
2
(B.7)
Identities
It is convenient to write out the contraction of nµ and n̄µ explicitly here:
n
/ = γ0 + γ3 =
0
1
σ3
−σ 3
−1
−σ 3
1
3
/̄ = γ − γ =
n
σ3
−1
!
!
.
(B.8a)
(B.8b)
There are a few useful identities which follow from the above:
†
n
/ =n
/,
†
/̄ = n,
/̄
n
†
/̄ = n
/̄
(/
nn)
/n
(B.9)
Using these we can define a projection operator:
1 = Pn + Pn̄ =
/̄ n
/̄
n/
n
/n
+
4
4
as
/̄
n
1
/n
=
4
2
/̄ n
1
n/
=
4
2
1
σ3
σ3
1
1
−σ 3
!
−σ 3
1
!
(B.10)
(B.11a)
.
(B.11b)
From the definition of these projection operators it is clear that
/̄
Pn n
=
/̄ n̄ = 0
n
/ = nP
/ Pn = Pn̄ n
(B.12)
Pn†
=
Pn
(B.13)
Pn̄†
=
Pn̄
(B.14)
Pn Pn
=
Pn
(B.15)
Pn̄ Pn̄
=
Pn̄
(B.16)
Some other useful identities for these projection operators come from the anticommutator of the γ
B.2 Identities
93
matrices, {γ µ , γ ν } = η µν :
Pn γ 0
=
=
=
=
1 0
(γ + γ 3 )(γ 0 − γ 3 )γ 0
4
1
(2 − γ 0 γ 3 + γ 3 γ 0 )γ 0
4
1 0
γ (2 + γ 0 γ 3 − γ 3 γ 0 )
4
γ 0 Pn̄
(B.17a)
and by a similar calculation
Pn̄ γ 0 = γ 0 Pn .
(B.17b)
94
Chapter B. Lightcone Coordinates
Appendix C
HQET
In this chapter we will introduce the heavy quark effective theory, HQET [6, 7]
The HQET theory is used for heavy quarks with low energies at which the effects of these heavy
quarks are negligible. It is then useful to leave these heavy quarks out of the theory, as it will then be
easier to work with. To achieve a theory without these heavy quarks three steps should be followed. First
the heavy quarks should be integrated out of the theory. This results in a non-local theory, which can
be resolved by rewriting the effective action as a Operator Product Expansion (OPE). Finally the shortand long-distance effects should be added in a perburtative way via renormalization-group equations. In
the sections which follow the construction of HQET is (roughly) shown, with the resulting Feynman rules
which are used throughout this thesis when dealing with heavy fields.
C.1
Constructing HQET
As a heavy quark with mass M bound in a hadron has approximately the same momentum as the hadron
itself, its momentum can be written as
pµ = M v µ + k µ
(C.1)
where k µ is the residual momentum, which is much smaller than M and v is a four-velocity, thus having
v 2 = 1.
The quark field ψ(x) can be rewritten in by introducing large- and small- component fields h v and
Hv as
hv (x) = eiM v·x P+ ψ(x),
Hv (x) = eiM v·x P− ψ(x)
(C.2)
where P+ , P− are projection operators like those defined in the appendix B (see Eq. B.10), yet in this
case defined by
1 ± v/
2
P+ + P− = 1.
P± ≡
95
(C.3)
(C.4)
96
Chapter C. HQET
It is easily checked that these projection operators obey the identities
v/P+
=
P+
(C.5)
v/P−
=
−P−
(C.6)
P− P+
=
P + P− = 0
(C.7)
The quark field is therefore decomposed as
ψ(x) = e−iM v·x (hv (x) + Hv (x)) .
(C.8)
Furthermore the new fields hv and Hv satisfy
v/hv (x) =
e−iM v·x v/P+ ψ(x) = e−iM v·x P+ ψ(x)
v/Hv (x) =
e−iM v·x v/P− ψ(x) = −e−iM v·x P− ψ(x)
= hv (x)
= −Hv (x)
(C.9)
(C.10)
and
P− hv ∼
P − P+ ψ =
0
(C.11)
P + Hv ∼
P + P− ψ =
0.
(C.12)
/ − M ψ rewritten in the new heavy quark fields is
The QCD Lagrangian LQCD = ψ̄ iD
LQCD
=
=
=
=
=
/ −M ψ
ψ̄ iD
(C.13)
/ − M )[hv + Hv ]
[h̄v + H̄v ](M v/ + iD
(C.15)
/ − M )e−iM v·x [hv + Hv ]
[h̄v + H̄v ]eiM v·x (iD
/ − M )[hv + Hv ] + [h̄v + H̄v ](M hv − M Hv )
[h̄v + H̄v ](iD
/ v + Hv ] − H̄v 2M Hv .
[h̄v + H̄v ](iD)[h
(C.14)
(C.16)
(C.17)
/ =D
/ ⊥ + v/v · D and a somewhat lengthy calculation this can be simplified even further to
Using D
/ v + H̄v (iD)h
/ v + h̄v iv · Dhv − H̄v (iv · D + 2M )Hv
LQCD = h̄v (iD)H
(C.18)
From this Lagrangian we can find the equations of motion for Hv in the usual way,
δ
LQCD
δ H̄v
=
→
/ ⊥ hv = 0
−(iv · D + 2M )Hv + iD
Hv =
1
/ hv .
iD
iv · D + 2M ⊥
(C.19)
(C.20)
C.1 Constructing HQET
97
The effective Lagrangian which is obtained from this is
/⊥
Lef f = h̄v iv · Dhv + h̄v iD
1
/ hv
iD
iv · D + 2M ⊥
(C.21)
or
Lef f
=
=
1
1
/ ⊥ hv
/⊥
iD
h̄v iD
2M
1 + iv·D
2M
n
∞
1 X
−iv · D
/
/ ⊥ hv ,
h̄v iD⊥
h̄v iv · Dhv +
iD
2M n=0
2M
h̄v iv · Dhv +
(C.22)
(C.23)
an expansion in powers of D/M . This is allowed as in momentum space the derivative acting on h v
produces only factors of the residual momentum k, which is much smaller than the mass M .
/ ⊥ iD
/ ⊥ hv . The
When approximating this effective Lagrangian up to order 1/M , we get a term ∼ h̄v iD
covariant derivatives acting on the operators P+ in the quark fields hv can be written such that the gluon
interaction becomes clear:
/ ⊥ iD
/ ⊥ P+
P + iD
=
=
=
=
µ
ν
P+ iD⊥
γµ iD⊥
γ ν P+
1
1
µ
ν
]
P+ ([iD⊥
, iD⊥
[γµ , γν ] + {γmu , γν } +
2
2
µ
ν
({γµ , γnu } − γν γµ ) )P+
iD⊥
D⊥
ν
ν
P+ (igs Gµν (−iσµν + ηµν ) + iD⊥
iD⊥
(2ηµν − γν γµ )) P+
2
µν
µ
/ ⊥ iD
/ ⊥ P+ .
P+ gs G σµν + igs Gµ + 2i(D⊥ ) − iD
(C.24)
(C.25)
(C.26)
(C.27)
Here we have used that the gluon field strength is defined by igs Gµν = [iDµ , iDν ], σµν = 2i [γµ , γν ] and
the anti-commutation relation {γµ , γν } = 2ηµν . Using that the trace of the field strength is zero we
obtain the identity
g
/ ⊥ iD
/ ⊥ P+ = P+ s Gµν σµν + (iD⊥ )2 P+
P + iD
(C.28)
2
and thus
Lef f = h̄v iv · Dhv +
C.1.1
g
1
1
s µν
h̄v
G σµν + (iD⊥ )2 hv + O( 2 ).
2M
2
M
(C.29)
Feynman rules for HQET
For M → ∞ the only surviving term is the first term, L∞ = h̄v iv · Dhv , where iv · Dhv = iv · (M v +
k)hv + igs v · Ata . The Feynman rules for the infinite mass limit are
98
Chapter C. HQET
=
=
i(p
i(M v/ + M )ji
/ + M )ji
∼
p2 + i
2M v · k + i
i
1 + v/
δji
v · k + i 2
igs v α (ta )ji
(C.30)
(C.31)
The 1/M terms will not be elaborated in the remainder of this thesis, we make the approximation
M → ∞, as in comparison with the quarks in SCET they are very heavy. This comes down to using
the Feynman rules (C.30) and (C.31) when computing diagrams with heavy quarks. HQET obviously
contains much more than the above, yet this is all we will need from the theory for our purposes. For
more information about HQET and its techniques, we refer to the (abundant) literature on the subject.
Appendix D
Properties of Wilson lines
The Wilson lines introduced in chapter 4 are known in the full QCD theory as the functions which lead to
parallel transport of gauge invariance. These Wilson lines obey various properties which are very useful
in our discussions. Therefore we will give several properties which we will use in chapters 5. 6, 7.
D.1
Definition
The Wilson line is defined by (see [15]),
Z
W (x) = P exp ig
0
−∞
or equivalently
W (x) = P exp ig
Z
ds n̄ · A(x + n̄ · s)
x
−∞
ds n̄ · A(n̄ · s)
(D.1)
(D.2)
where P denotes the path-ordering defined by P[A(x)A(x + n̄ · s) = A(x)A(x + n̄ · s) if s < 0. The Wilson
line obeys W W † = W † W = 1 and is thus a unitary operator. From the properties
in̄ · DW
(in̄ · D + i)
−1
= W n̄ · ∂
= W (in̄ · ∂ + i)
(D.3)
−1
W
†
(D.4)
we can find in operator description that
in̄ · Dc W (x, ∞)ψ(x)
=
(P̄ + gs n̄ · An,q ) W (x, ∞)ψ(x)
= (P̄ + gs n̄ · An,q ) W (x, ∞) ψ(x) + W (x, ∞)P̄ψ(x)
=
0 + W (x, ∞)P̄ψ(x)
(D.5)
(D.6)
(D.7)
or
in̄ · Dc W = W P̄,
99
(D.8)
100
Chapter D. Properties of Wilson lines
where Dc is the collinear part of the covariant derivative, in̄ · Dc = P̄ + gs n̄ · An,q . This last equation can
be rewritten to find the identity:
in̄ · Dc W
=
in̄ · Dc
=
(in̄ · Dc )k
=
W P̄
W P̄W
(D.9)
†
(D.10)
(W P̄W † )k = W (P̄)k W †
(D.11)
Any function f (P̄ + gs n̄ · An,q ) can be expanded as f (P̄ + gs n̄ · An,q ) =
P
k
k ak (in̄ · Dc ) , where in the last equality we are in momentum space. Now
f (P̄ + gs n̄ · An,q )
=
X
k
=
X
k
= W
P
k
ak (P̄ + gs n̄ · An,q )k =
ak (in̄ · Dc )k
(D.12)
ak W (P̄)k W †
(D.13)
X
k
ak (P̄)k W †
(D.14)
= W f (P̄)W †
(D.15)
This implies that gauge invariant combinations of n · An,q only appear in the Wilson line W. This is an
important observation which is used to construct gauge invariant operators.
D.2
Properties of Wilson lines
Wilson lines obey unitarity, causality and are multiplicative of nature. The unitarity W † W = W W † = 1
is easily seen from the definition of the Wilson line. Yet the other two properties we will show here
explicitly.
D.2.1
causality
Causality yields
W (a, b)W (c, a) = W (c, b)
(D.16)
as we have
W (a, b)W (c, a)
=
=
=
which proves the causality.
P exp ig
P exp ig
W (c, b)
Z
Z
ds n̄ · A(n̄ · s)
!
ds n̄ · A(n̄ · s)
!
b
a
b
c
Z
+ P exp ig
a
c
ds n̄ · A(n̄ · s)
(D.17)
(D.18)
(D.19)
D.2 Properties of Wilson lines
D.2.2
101
Multiplication of Wilson lines
For Wilson lines we have the following identity:
W (x)W † (y) = W (y, x).
(D.20)
This follows from
†
W (x)W (y)
x
Z
ds n̄ · A(n̄ · s) P exp −ig
= P exp ig
−∞
Z x
= P exp ig
ds n̄ · A(n̄ · s)
Z
y
−∞
ds n̄ · A(n̄ · s)
(D.21)
(D.22)
y
≡ W (y, x),
(D.23)
or
W (x)W † (y)
=
W (−∞, x)W † (−∞, y)
(D.24)
=
W (−∞, x)W (y, −∞)
(D.25)
=
W (y, x).
(D.26)
102
Chapter D. Properties of Wilson lines
Appendix E
Moments
The large logarithms which are under consideration in this chapter are discussed in moment space.
Let f be a function of z, defined on [0, 1]. Then the moments of f (z) are defined by the Mellin
transform 1
Z 1
MN [f (z)] =
dzz N −1 f (z).
(E.1)
0
They satisfy the relation
MN [f ? g] = MN [f ]MN [g]
(E.2)
where the ? product denotes the convolution of functions f and g,
h(z)
=
=
Z
Z
1
dx
0
1
z
Z
1
0
dyδ(z − xy)f (x)g(y)
z
dx
f xg( ).
x
x
(E.3)
(E.4)
From this definition it is straightforward to calculate moments of functions. In chapter 7 we will need
the moments given below. Not all are calculated here, as they can be looked up in e.g. [32].
1 Moments
purposes.
∞ =
are also defined for functions on (−∞, ∞) by MN
103
R∞
−∞
dzz N −1 f (z), yet we do not need this for our
104
Chapter E. Moments
MN [1]
=
=
=
MN [δ(1 − z)]
=
1
dzz N −1
0
1 N 1
z N
0
1
N
Z 1
dzz N −1 δ(1 − z)
(E.5)
0
=
MN [1 − z]
Z
=
=
=
=
1
Z
Z
(E.6)
1
dzz N −1 (1 − z)
0
1
0
(z N −1 − z N )
MN [1]MN +1 [1]
1
N (N + 1)
(E.7)
and likewise
"
#
ln(1 − z)
=
MN
1−z
+
"
#
zr
MN
=
1−z +
N
−1
X
j=1
j
1X1
j i=1 i
r
X
1
j=1
j
−
NX
−1+r
j=1
(E.8)
1
j
(E.9)
(E.10)
We will use the large N limits for these moments. The non-trivial are seen to be
"
#
ln(1 − z)
MN
1−z
+
"
#
r
z
MN
1−z +
→
→
1 2
1
ln N + ζ(2)
2
2
r
X
1
j=1
j
− ln N .
(E.11)
(E.12)
Bibliography
[1] L. D. Landau. On analytic properties of vertex parts in quantum field theory. Nucl. Phys., 13:181–192
(1959).
[2] S. Coleman and R. E. Norton. Singularities in the physical region. Nuovo Cim., 38:438–442 (1965).
[3] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart. An effective field theory for collinear and soft
gluons: Heavy to light decays. Phys. Rev., D63:114020 (2001). hep-ph/0011336.
[4] C. W. Bauer and I. W. Stewart. Invariant operators in collinear effective theory. Phys. Lett.,
B516:134–142 (2001). hep-ph/0107001.
[5] C. W. Bauer, D. Pirjol and I. W. Stewart. Power counting in the soft-collinear effective theory. Phys.
Rev., D66:054005 (2002). hep-ph/0205289.
[6] E. Eichten and B. R. Hill. An effective field theory for the calculation of matrix elements involving
heavy quarks. Phys. Lett., B234:511 (1990).
[7] M. Neubert. Heavy-quark effective theory (1996). hep-ph/9610266.
[8] B. Pötter. Calculational techniques in perturbative QCD: The drell-yan process (1997).
[9] J. A. M. Vermaseren. New features of FORM (2000). math-ph/0010025.
[10] E. Laenen. The standard model.
[11] G. Altarelli and G. Parisi. Asymptotic freedom in parton language. Nucl. Phys., B126:298 (1977).
[12] T. Eynck. Soft gluons and hard scales in QCD - Heavy quarks at finite and all orders. Ph.D. thesis,
Utrecht University (2003).
[13] G. Sterman. An introduction to quantum field theory. Cambridge, UK: Univ. Pr. (1993) 572 p.
[14] C. W. Bauer, D. Pirjol and I. W. Stewart. Soft-collinear factorization in effective field theory. Phys.
Rev., D65:054022 (2002). hep-ph/0109045.
[15] M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann. Soft-collinear effective theory and heavyto-light currents beyond leading power. Nucl. Phys., B643:431–476 (2002). hep-ph/0206152.
105
106
Bibliography
[16] M. E. Peskin and D. V. Schroeder. An introduction to quantum field theory. Reading, USA:
Addison-Wesley (1995) 842 p.
[17] M. E. Luke, A. V. Manohar and I. Z. Rothstein. Renormalization group scaling in nonrelativistic
QCD. Phys. Rev., D61:074025 (2000). hep-ph/9910209.
[18] R. Frederix. Wilson lines in QCD (2005).
[19] K. G. Wilson. Confinement of quarks. Phys. Rev. D, 10(8):2445–2459 (1974). doi:10.1103/PhysRevD.
10.2445.
[20] J. Chay, C. Kim, Y. G. Kim and J.-P. Lee. Soft wilson lines in soft-collinear effective theory. Phys.
Rev., D71:056001 (2005). hep-ph/0412110.
[21] B. Grinstein, R. P. Springer and M. B. Wise. Effective hamiltonian for weak radiative b meson
decay. Phys. Lett., B202:138 (1988).
[22] N. Isgur and M. B. Wise. Heavy quark symmetry. Adv. Ser. Direct. High Energy Phys., 10:549–572
(1992).
[23] M. Neubert. Analysis of the photon spectrum in inclusive B → X(s) gamma decays. Phys. Rev.,
D49:4623–4633 (1994). hep-ph/9312311.
[24] D. E. Soper. Parton distribution functions. Nucl. Phys. Proc. Suppl., 53:69–80 (1997). hep-lat/
9609018.
[25] J. C. Collins and D. E. Soper. Parton Distribution and Decay Functions. Nucl. Phys., B194:445
(1982).
[26] J. Callan, Curtis G. and D. J. Gross. High-energy electroproduction and the constitution of the
electric current. Phys. Rev. Lett., 22:156–159 (1969).
[27] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein et al. Hard scattering factorization from effective
field theory. Phys. Rev., D66:014017 (2002). hep-ph/0202088.
[28] A. Idilbi, X.-d. Ji and F. Yuan. Resummation of threshold logarithms in effective field theory for
DIS, Drell-Yan and Higgs production. Nucl. Phys., B753:42–68 (2006). hep-ph/0605068.
[29] A. V. Manohar, T. Mehen, D. Pirjol and I. W. Stewart. Reparameterization invariance for collinear
operators. Phys. Lett., B539:59–66 (2002). hep-ph/0204229.
[30] A. Idilbi and X.-d. Ji. Threshold resummation for Drell-Yan process in soft- collinear effective theory.
Phys. Rev., D72:054016 (2005). hep-ph/0501006.
[31] F. Liu and J. P. Ma. Glauber Gluons in Soft Collinear Effective Theory and Factorization of DrellYan Processes (2008). arXiv:0802.2973[hep-ph].
BIBLIOGRAPHY
107
[32] J. Blumlein and S. Kurth. Harmonic sums and Mellin transforms up to two-loop order. Phys. Rev.,
D60:014018 (1999). hep-ph/9810241.
[33] T. Becher, M. Neubert and G. Xu. Dynamical Threshold Enhancement and Resummation in DrellYan Production (2007). 0710.0680.
[34] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables. Dover, New York, ninth dover printing, tenth gpo printing edition (1964).
ISBN 0-486-61272-4.
[35] M. Kaku. Quantum field theory: A Modern introduction. New York, USA: Oxford Univ. Pr. (1993)
785 p.