11institutetext: Department of Physics and MTA-DE Particle Physics Research
Group, University of Debrecen, Debrecen, Hungary

# QCD for collider experiments

Z. Trócsányi<EMAIL_ADDRESS>
###### Abstract

These lectures are intended to provide the theoretical basis of describing
high-energy particle collisions at a level appropriate to graduate students in
experimental high energy physics. They are supposed to be familiar with
quantum electrodynamics, the concept of Feynman rules, Feynman graphs and
computation of the cross section in quantum field theory.

##

_When you measure what you are speaking about and express it in numbers, you
know something about it, but when you cannot express it in numbers, your
knowledge is of a meagre and unsatisfactory kind._

Lord Kelvin

##  1 QCD as quantum field theory of the strong interaction

The Lagrangian of the quark-gluon field based on a non-abelian gauge symmetry
was first proposed in Ref. [1] forty years ago. The paper discussed the
advantages of the colour-octet picture. Since then an immense amount of
research lead to a lot of interesting results and a deep understanding of the
strong interaction based on this quantum field theoretical description of
chromo-dynamics, QCD. Today we are convinced that QCD is the correct
description of the strong interaction, yet we still lack a complete and
satisfactory solution. In such a situation one may set two goals: (i) either
an ambitious one: solve QCD, or (ii) a more pragmatic one: develop tools for
modeling particle interactions in high energy collider experiments. In these
lectures we go for the second one.

Our aim is to understand high-energy particle collisions quantitatively from
first principles. Examples of such events recorded by the CMS experiment at
the LHC are shown in Fig. 1. In these events kinematic characteristics of
particles, such as energy and momentum, are collected. Analyzing many such
events, we can produce distributions of kinematic variables, for instance,
differential distribution of the inclusive jet cross section with respect to
pseudorapidity, ${\rm d}\sigma/{\rm d}\eta$. There is a long way from the QCD
Lagrangian to making predictions for such distributions, full of difficulties.
I clearly do not expect students in high energy experimental physics to be
able to solve those difficulties. Instead I would like to explain the
consequences of solving the difficulties, because an incomplete understanding
of these consequences can easily lead to false interpretation of correct
measurements.

Figure 1: A violent proton-proton collision resulting in two hard jets. Left:
tracks and energy deposits in the calorimeters, right: energy deposits
represented by towers in the pseudorapidity–azimuthal angle plane.

###  1.1 The QCD Lagrangian

The quantum field theory (QFT) of the strong interactions is a part of the
Standard Model (SM) of elementary particle interactions. The SM is based upon
the principle of local gauge invariance. The underlying gauge group is

$SU(3)_{{\rm c}}\,\times\,SU(2)_{{\rm L}}\,\times\,U(1)_{Y}\,,$

where c stands for “colour”, L for “left” (or “weak isospin”) and _Y_ for
“hypercharge”. As we concentrate on QCD, which is based on $SU(3)_{{\rm c}}$
gauge symmetry, we can write the Lagrangian as

${\cal L}_{\rm QCD}={\cal L}_{\rm QCD}^{(0)}+{\cal L}_{\rm sources}\,,$ (1)

where

${\cal L}_{\rm QCD}^{(0)}={\cal L}_{\rm C}+{\cal L}_{\rm GF}+{\cal L}_{\rm
G}\,,$ (2)

while the electroweak sector (based on the $SU(2)_{{\rm L}}\,\times\,U(1)_{Y}$
symmetry) act as sources. In Eq. (2) ${\cal L}_{\rm C}$ is the classical
Lagrangian, while ${\cal L}_{\rm GF}$ is the gauge-fixing term. The last piece
is the ghost Lagrangian, absent if we use physical gauges, which will be our
choice.

To find the classical Lagrangian, one starts with the Lagrangian of free Dirac
fields,

${\cal L}_{q}^{(0)}(q_{f},\,m_{f})=\sum_{k,l=1}^{N_{c}}\bar{q}_{f}^{k}\>({\rm
i}\gamma_{\mu}\partial^{\mu}-m_{f})_{kl}\>q_{f}^{l}\,,$ (3)

where the $\gamma_{\mu}$ matrices satisfy the Clifford algebra,

$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}\,=\,2g^{\mu\nu}\,,\qquad\left\\{\gamma^{\mu},\gamma^{5}\right\\}\,=\,0\,.$
(4)

The matter field content is dictated by the electroweak sector. The fermion
fields are called quark fields: $q_{f}^{k}$ with masses $m_{f}$ and
$f\,=\,1,\dots,n_{f}$, where $n_{f}$ is the number of different flavours. The
quark fields also have an additional degree of freedom: colour, labelled by
$k$, that can take $N_{\rm c}$ values, $k\,=\,1,\dots,N_{{\rm c}}$. The
precise matter content is shown in Table 1.

Table 1:  The six quark flavours as dictated by the electroweak sector. Their baryon number is $B=1/3$. Each quark flavour comes in three colours, not shown. For light flavours (u, d and s) the mass values are not without controversy and still under investigation. For flavours c and b the mass values are running $\overline{\mathrm{MS}}$ quark masses at 2 MeV (see definition below), while for t it is the pole mass. $f$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
$q_{f}$ | u | d | s | c | b | t
$m_{f}$ |  $\approx 3$ MeV |  $\approx 6$ MeV |  $\approx 100$ MeV | 1.2 GeV | 4.2 GeV | 172.6 GeV

If we apply a transformation $q^{k}\,\to\,q^{\prime k}\,=\,U_{kl}q^{l}$, with

$U_{kl}=\exp\left\\{{\rm i}\sum_{a=1}^{N_{{\rm
c}}^{2}-1}\,t^{a}\,\theta^{a}\right\\}_{kl}\equiv\exp\left\\{{\rm
i}{{\mbox{\boldmath$t$}}}\cdot{\mbox{\boldmath$\theta$}}\right\\}_{kl}\,,$

where $\theta^{a}\,\in\,\mathbb{R}$, then the Lagrangian of free Dirac fields
remains invariant,
$\mathcal{L}_{q}^{(0)}\left(q\right)=\mathcal{L}_{q}^{(0)}\left(q^{\prime}\right)$.
The $\left(t^{a}\right)_{kl}$ are $N_{{\rm c}}\times N_{{\rm c}}$ matrices
that constitute the fundamental representation of the generators $T^{a}$
(called colour-charge operators), which satisfy the Lie algebra:

$\left[T^{a},\,T^{b}\right]={\rm i}\,f^{abc}\,T^{c}\,,\quad\textrm{with
normalizaiton}\quad\mathrm{T_{R}}(T^{a}T^{b})=T_{\rm R}\delta^{ab}\,.$ (5)

For $SU(3)$ the matrices $t^{a}$ are the Gell-Mann matrices (see [2]).

Next we ask the question if we can make $\mathcal{L}_{q}^{(0)}\left(q\right)$
invariant under local $SU(N_{{\rm c}})$ transformations. The answer is yes, we
can through the following steps:

  1. 1.

Introduce $A_{\mu}^{a}$ coloured vector field with the following
transformation property under $SU(N_{{\rm c}})$ transformations:

${{\mbox{\boldmath$t$}}}\cdot{{\mbox{\boldmath$A$}}}_{\mu}\longrightarrow\,{{\mbox{\boldmath$t$}}}\cdot{{\mbox{\boldmath$A$}}}^{\prime}_{\mu}=U\left(x\right){{\mbox{\boldmath$t$}}}\cdot{{\mbox{\boldmath$A$}}}_{\mu}U^{-1}\left(x\right)+\frac{{\rm
i}}{g_{s}}(\partial_{\mu}U\left(x\right))U^{-1}\left(x\right),$

where $U\left(x\right)=\exp\left\\{{\rm
i}{{\mbox{\boldmath$t$}}}\cdot{\mbox{\boldmath$\theta$}}(x)\right\\}$.

  2. 2.

Replace $\partial_{\mu}\delta_{kl}$ with
$D_{\mu}\left[A\right]_{kl}=\partial_{\mu}\delta_{kl}+{\rm
i}g_{s}\left({{\mbox{\boldmath$t$}}}\cdot{{\mbox{\boldmath$A$}}}_{\mu}\right)_{kl}$.
This covariant derivative $D_{\mu}\left[A\right]_{kl}q_{l}\left(x\right)$
transforms the same way as the quark field $q_{k}\left(x\right)$.

  3. 3.

Introduce a kinetic term

$\mathcal{L}_{g}(A)\,=\,-\frac{1}{4}F_{\mu\nu}^{a}\left[A\right]F^{a\mu\nu}\left[A\right],$

with the non-abelian field strength $F_{\mu\nu}^{a}$ given by

$F_{\mu\nu}^{a}\left[A\right]=\partial_{[\mu}A^{a}_{\nu]}-g_{s}\underbrace{f^{abc}A_{\mu}^{b}A_{\nu}^{c}}_{``{{\mbox{\boldmath$A$}}}_{\mu}\times{{\mbox{\boldmath$A$}}}_{\nu}"}\,,$

so the Lagrangian contains cubic and quartic terms of the gauge field.

The constants $f^{abc}$ are the structure constants of the Lie algebra. The
structure constants are completely antisymmetric and are related to the
adjoint representation of the generators $F^{a}_{bc}$ by $F^{a}_{bc}=-{\rm
i}f^{abc}$.

Thus we find that the gauge boson field, called gluon field, is a consequence
of the local $SU(N_{\rm c})$ (gauge) invariance. The classical Lagrangian of
QCD is a sum of interacting Dirac Lagrangians for spin 1/2 fermion fields and
a Lagrangian of a gauge field,

${\cal L}_{\rm C}={\cal L}_{f}+{\cal L}_{g}=\sum_{f=1}^{n_{f}}{\cal
L}_{q}(q_{f},\,m_{f})+{\cal L}_{g}(A)\,,$ (6)

where

${\cal L}_{q}(q_{f},\,m_{f})=\sum_{k,l=1}^{N_{c}}\bar{q}_{f}^{k}\>({\rm
i}\gamma_{\mu}D^{\mu}[A]-m_{f})_{kl}\>q_{f}^{l}\,.$ (7)

The gluon field is also coloured and self-interacting. In fact, these self-
interactions are the sources of the main difference between QED and QCD. We
shall see that as a result, QCD is a ‘perfect theory’ in the sense that it is
asymptotically free. Furthermore, among quantum field theories in $d=4$
dimensions only non-Abelian gauge theories are asymptotically free (see
discussion after Eq. (21)). It is also plausible that the self-interactions
are the sources of colour confinement, the colour neutrality of hadrons, but
we do not have a proof based on first principles.

It is clear that there is an unprecedented large number of degrees of freedom
we have to sum over when computing a cross section:

  1. 1.

spin and space-time as in any field theory, not exhibited above,

  2. 2.

flavour, which also appears in electroweak theory, and colour, which is
specific to QCD only.

As a result computations in QCD are rather cumbersome. During the last two
decades a lot of effort was invested and great progress was made to find
“simple” ways of computing QCD cross sections and to automate the
computations.

###### Exercise 1.1

Show that in QED the covariant derivative transforms the same way as the field
itself, i.e., if $f(x)\to U(x)f(x)$ then $D_{\mu}f(x)\to U(x)D_{\mu}f(x)$,
where $D_{\mu}=\partial_{\mu}+{\rm i}\,e\,A_{\mu}$.

###### Exercise 1.2

Show that in QED

$\left[D_{\mu},D_{\nu}\right]={\rm i}\,e\,F_{\mu\nu},$

where $D_{\mu}=\partial_{\mu}+{\rm i}\,e\,A_{\mu}$.

###### Exercise 1.3

Show that the generators of a special unitary group are traceless and
hermitian.

###### Exercise 1.4

The generators in the fundamental representation of SU(2) are the Pauli
matrices divided by two:

$t^{a}_{f}=\frac{\sigma^{a}}{2},\qquad\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$

The adjoint representation of a group is defined as

$(t_{A}^{b})_{ac}={\rm i}f^{abc}\,.$

Compute the generators in the adjoint representation of SU(2).

We define the constant $T_{\rm R}$ for a representation $R$ by the condition

$\mathrm{Tr}[t^{a}_{\rm R}t^{b}_{\rm R}]=\delta^{ab}T_{\rm R}\,.$

Compute this constant from the explicit form of the fundamental ($T_{\rm F}$)
and the adjoint ($T_{\rm A}$) representation.

The quadratic Casimir $C_{2}(R)$ of a representation $R$ is defined by

$C_{2}(R)\mathbbm{1}=\sum\limits_{a}t^{a}_{R}t^{a}_{R}\,.$

Compute the quadratic Casimir in the fundamental ($C_{\rm F}$) and the adjoint
($C_{\rm A}$) representation of $SU(2)$ using the explicit form of the
representation matrices.

###### Exercise 1.5

Show that in SU(N) gauge theories

$\left[D_{\mu},D_{\nu}\right]={\rm i}gF_{\mu\nu}^{a}T^{a}\quad\mbox{with}\quad
F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}-gf^{abc}A_{\mu}^{b}A_{\nu}^{c}\,.$

###### Exercise 1.6

Show that $F_{\mu\nu}^{a}$ transforms according to the adjoint representation
of SU(N):

$F_{\mu\nu}^{a}\to F_{\mu\nu}^{a}-f^{abc}\theta^{b}F_{\mu\nu}^{c}\,.$

###  1.2 Feynman rules

The Feynman rules can be derived from the action,

$S={\rm i}\int\\!{\rm d}^{4}x(\mathcal{L}_{f}\,+\,\mathcal{L}_{g})\equiv
S_{0}\,+\,S_{I}\,,\quad\mathrm{where}\quad S_{0}={\rm i}\int\\!{\rm
d}^{4}x\mathcal{L}_{0}\,,\quad\mathrm{and}\quad S_{I}={\rm i}\int\\!{\rm
d}^{4}x\mathcal{L}_{I}.$

In this decomposition $\mathcal{L}_{0}$ contains the terms bilinear in the
fields and $\mathcal{L}_{I}$ does all other terms, called interactions. The
gluon propagator $\Delta_{g,\mu\nu}(p)$ is the inverse of the bilinear term in
$A_{\mu}$. In momentum space we have the condition (we suppress colour indices
as these terms are diagonal in colour space)

$\Delta_{g,{\mu\nu}}\left(p\right)\,{\rm
i}\left[p^{2}g^{\nu\rho}\,-\,p^{\nu}p^{\rho}\right]=\delta^{\rho}_{\mu}\,.$
(8)

However,

$\left[p^{2}g^{\nu\rho}-p^{\nu}p^{\rho}\right]p_{\rho}=0\,,$ (9)

which means that the inverse does not exist, the matrix
$\left[p^{2}g^{\nu\rho}\,-\,p^{\nu}p^{\rho}\right]$ is not invertible. We can
exploit gauge invariance to rewrite the classical Lagrangian in a physically
equivalent form (action remains the same) such that $\Delta_{g,{\mu\nu}}$
exists, which is called gauge fixing. This amounts to imposing a constraint on
$A_{\mu}$ by adding a term to the Lagrangian with a Lagrange multiplicator
(like in classical mechanics). For example, the covariant gauges are defined
by requiring $\partial_{\mu}A^{\mu}\left(x\right)=0$ for any $x^{\mu}$. Adding

$\mathcal{L}_{\rm
GF}=-\frac{1}{2\lambda}\left(\partial_{\mu}A^{\mu}\right)^{2},\qquad\lambda\,\in\,\mathbb{R},$

to $\mathcal{L}$, the action $S$ remains the same. The bilinear term becomes
in this case

${\rm
i}\left(p^{2}g^{\nu\rho}-\left(1-\frac{1}{\lambda}\right)p^{\nu}p^{\rho}\right),$

with inverse

$\Delta_{g,{\mu\nu}}\left(p\right)=-\frac{{\rm
i}}{p^{2}}\left[g_{\mu\nu}-(1-\lambda)\frac{p_{\mu}p_{\nu}}{p^{2}}\right]\,.$

Of course, physical results must be independent of $\lambda$. It is customary
to choose $\lambda=1$ (called covariant Feynman gauge).

In covariant gauges unphysical degrees of freedom (longitudinal and time-like
polarizations) also propagate. The effect of these unwanted degrees of freedom
is cancelled by the ghost fields (coloured complex scalars with Fermi
statistics). We do not elaborate the details of these fields as the unwanted
degrees of freedom and the ghost fields can be avoided by choosing axial
(physical) gauges, which is our choice. The axial gauge is defined with an
arbitrary, but fixed vector $n^{\mu}$, different from $p^{\mu}$:

$\mathcal{L}_{\rm GF}=-\frac{1}{2\lambda}\left(n^{\mu}A_{\mu}\right)^{2},$

which leads to

$\Delta_{g,{\mu\nu}}\left(p,n\right)=-\frac{{\rm
i}}{p^{2}}\left(g_{\mu\nu}-\frac{p_{\mu}n_{\nu}+n_{\mu}p_{\nu}}{p\,\\!\cdot\\!\,n}+\frac{\left(n^{2}+\lambda\,p^{2}\right)p_{\mu}p_{\nu}}{(p\,\\!\cdot\\!\,n)^{2}}\right).$

Since $p^{2}=0$, we have:

$\Delta_{g,{\mu\nu}}\left(p,n\right)\,p^{\mu}=0\,,\qquad\Delta_{g,{\mu\nu}}\left(p,n\right)\,n^{\mu}=0\,.$

Thus, only 2 degrees of freedom propagate (transverse ones in the
$n^{\mu}+p^{\mu}$ rest-frame). A usual choice is $n^{2}=0,\;\lambda=0$, called
light-cone gauge. The price we pay by choosing this gauge instead of a
covariant one is that the propagator looks more complicated and it diverges
when $p^{\mu}$ becomes parallel to $n^{\mu}$. In this gauge

$\Delta_{g,{\mu\nu}}(p,n)=\frac{{\rm i}}{p^{2}}\,d_{\mu\nu}(p,n)$

with

$d_{\mu\nu}\left(p,n\right)=-g_{\mu\nu}+\frac{p_{\mu}n_{\nu}+n_{\mu}p_{\nu}}{p\,\\!\cdot\\!\,n}=\sum_{\lambda=1,2}\,\epsilon_{\mu}^{(\lambda)}(p)\epsilon_{\nu}^{(\lambda)}(p)^{\ast}\,,$
(10)

where $\epsilon_{\mu}^{(\lambda)}(p)$ is the polarization vector of the gauge
field (photon in QED, gluon in QCD).

###  1.3 Feynman rules for QCD

Propagators (Feynman’s ‘$+{\rm i}\epsilon$’-prescription is assumed, but not
shown):

gluon propagator:
$\Delta_{g,\mu\nu}^{ab}\left(p\right)=\delta^{ab}\,\Delta_{g,\mu\nu}\left(p\right)\longrightarrow$

quark propagator: $\Delta_{q}^{ij}\,\left(p\right)=\delta^{ij}\,{\rm
i}\frac{\slashed{p}+m}{p^{2}-m^{2}}\longrightarrow$

ghost propagator: $\Delta^{ab}\,\left(p\right)=\delta^{ab}\,\frac{{\rm
i}}{p^{2}}\longrightarrow$

(not needed in physical gauges)

Vertices:

quark-gluon: $\Gamma_{gq\bar{q}}^{\mu,\,a}=-{\rm
i}g_{S}\,(t^{a})_{ij}\gamma^{\mu}\longrightarrow$

three-gluon: $\Gamma_{\alpha\beta\gamma}^{abc}\left(p,q,r\right)=-{\rm
i}g_{s}\left(F^{a}\right)_{bc}\,V_{\alpha\beta\gamma}\left(p,q,r\right)\longrightarrow$

$V_{\alpha\beta\gamma}\left(p,q,r\right)=\left(p-q\right)_{\gamma}g_{\alpha\beta}+\left(q-r\right)_{\alpha}g_{\beta\gamma}+\left(r-p\right)_{\beta}g_{\alpha\gamma}\,,\quad\,p^{\alpha}+q^{\alpha}+r^{\alpha}=0$

four-gluon: $\Gamma_{\alpha\beta\gamma\delta}^{abcd}=-{\rm i}g_{\rm
s}^{2}\,\left[\begin{array}[]{l}+f^{xac}\,f^{xbd}\,(g_{\alpha\beta}g_{\gamma\delta}-g_{\alpha\delta}g_{\beta\gamma})\\\
+f^{xad}\,f^{xbc}\,(g_{\alpha\beta}g_{\gamma\delta}-g_{\alpha\gamma}g_{\beta\delta})\\\
+f^{xad}\,f^{xbc}\,(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma})\end{array}\right]\longrightarrow$

ghost-gluon: $\Gamma_{g\eta\bar{\eta}}^{\mu,\,a}=-{\rm
i}g_{S}\,(F^{a})_{ij}\,p^{\mu}\longrightarrow$ (not needed in physical
gauges).

The four-gluon vertex differs from the rest of the Feynman rules in the sense
that it is not in a factorized form of a colour and a tensor factor. This is
an inconvenient feature because it prevents the separate summation over colour
and Lorentz indices and complicates automation. We can however circumvent this
problem by introducing a fake field with propagator

$=\frac{{\rm
i}}{2}\delta^{ab}(g^{\alpha\beta}g^{\gamma\delta}-g^{\alpha\delta}g^{\beta\gamma})\,,$
that couples only to the gluon with vertex

$={\rm i}\sqrt{2}g_{\rm s}\,f^{xac}g^{\alpha\xi}g^{\gamma\zeta}$.

We can check that a single four-gluon vertex can be written as a sum of three
graphs. This way the summations over colour and Lorentz indices factorize
completely, which helps automation and makes possible for us to concentrate on
the colour algebra independently of the rest of the Feynman rules.

Finally, we have to supply the following factors for incoming and outgoing
particles:

$\begin{array}[]{clccl}\bullet&\textrm{outgoing fermion:
}\bar{u}\left(p\right)&&\bullet&\textrm{outgoing antifermion:
}v\left(p\right)\\\\[2.84526pt] \bullet&\textrm{incoming fermion:
}u\left(p\right)&&\bullet&\textrm{incoming antifermion:
}\bar{v}\left(p\right)\\\ \bullet&\textrm{outgoing photon, or gluon:
}\epsilon_{\mu}^{\left(\lambda\right)}\left(p\right)^{\ast}&&\bullet&\textrm{incoming
photon, or gluon:
}\epsilon_{\mu}^{\left(\lambda\right)}\left(p\right)\,.\end{array}$

###### Exercise 1.7

Show that the four-gluon vertex can be written as a sum of three graphs, with
the help of the fake field such that in each graph the colour and Lorentz
indices are factorized:

###  1.4 Basics of colour algebra

Examining the Feynman rules, we find that there are two essential changes as
compared to QED. One is that there is an additional degree of freedom: colour.
The second is that there are new kind of couplings: the self couplings of the
gauge field. We now explore the effect of the first.

In order to see how to treat the colour degrees of freedom, we set to one all
but the colour part of the Feynman rules and try first to develop an efficient
technique to compute the coefficients involving the colour structure. This is
possible because the colour degrees of freedom factorize from the other
degrees of freedom completely. We use the following graphical representation
for the colour charges in the fundamental representation:

$=\left(t^{a}\right)_{ij}$ .

The normalization of these matrices is given by Tr$\left(t^{a}t^{b}\right)=$

$=T_{\rm R}\,\delta^{ab}$ . The usual choice is $T_{\rm R}=\frac{1}{2}$, but
$T_{\rm R}=1$ is also used often. We shall use both.

In the adjoint representation the colour charge $T^{a}$ is represented by the
matrix $\left(F^{a}\right)_{bc}$ that is related to the structure constants by

$\left(F^{a}\right)_{bc}\,=\,\left(F^{b}\right)_{ca}\,=\,\left(F^{c}\right)_{ab}\,=\,-{\rm
i}\,f_{abc}=$

where $F^{a}$ with $a=1,\dots,(N_{\rm c}^{2}-1)$ are $\left(N_{{\rm
c}}^{2}-1\right)\times\,\left(N_{{\rm c}}^{2}-1\right)$ matrices which again
satisfy the commutation relation (5). The graphical notation in the adjoint
representation is not unique. For the matrix $\left(F^{a}\right)_{bc}$ we
assume an arrow pointing from index $c$ to $b$, opposite to which we read the
indices of $\left(F^{a}\right)_{bc}$ (similarly as for the matrices $t^{a}$).
On the structure constants the indices are not distinguished, therefore arrows
do not appear. However, these are completely antisymmetric in their indices,
therefore, the ordering matters. By convention, in the graphical
representation, the ordering of the indices is counter-clockwise. The
representation matrices are invariant under $SU(N)$ transformations.

The sums $\sum_{a}t^{a}_{ij}t^{a}_{jk}$ and Tr$\left(F^{a}F^{b}\right)$ have
two free indices in the fundamental and adjoint representation, respectively.
These are invariant under $SU(N)$ transformations, therefore, must be
proportional to the unit matrix,

$\sum_{j,a}t^{a}_{ij}t^{a}_{jk}=C_{\rm F}\,\delta_{ik}\,,\qquad{\rm
Tr}\left(F^{a}F^{b}\right)=C_{\rm A}\,\delta^{ab}\,,$

which is depicted graphically as

.

Here $C_{\rm F}$ and $C_{\rm A}$ are the eigenvalues of the quadratic Casimir
operator in the fundamental and adjoint representation, respectively. In the
familiar case of angular momentum operator algebra ($SU(2)$), the quadratic
Casimir operator is the square of the angular momentum with eigenvalues
$j(j+1)$. The fundamental representation is two dimensional, realized by the
(half of the) Pauli matrices acting on two-component spinors, when $j=1/2$ and
$C_{\rm F}=1/2(1/2+1)=3/4$. In the adjoint representation $j=1$ and $C_{\rm
A}=2$. Below we derive the corresponding values for general $SU(N)$.

The commutation relation (5) can be represented graphically by

.

Multiplying this commutator first with another colour charge operator with
summing over the fermion index and then taking the trace over the fermion line
(multiplying with $\delta_{ik}$) we obtain the resolution of the three-gluon
vertex as traces of products of colour charges:

$={\rm Tr}(t^{a}t^{b}t^{c})-{\rm Tr}(t^{c}t^{b}t^{a})={\rm i}\,T_{\rm
R}f^{abc}\,.$

We now show some examples of how one can compute the colour algebra structure
of a QCD amplitude, in particular we will also find an explicit value for
$C_{F}$ and $C_{A}$. Taking the trace of the identity in the fundamental and
in the adjoint representation we obtain

$=N_{{\rm c}}\,,$ $=N_{{\rm c}}^{2}-1$ ,

respectively. Then, using the expressions for the fermion and gluon propagator
corrections, we immediately find

$=C_{\rm F}\,N_{{\rm c}}$ , $=C_{\rm A}\,\left(N_{{\rm c}}^{2}-1\right)$ .

The generators are traceless,

$=\textrm{Tr}\left(t^{a}\right)\,=\,0$ ,
$=\textrm{Tr}\left(F^{a}\right)\,=\,0$ .

We can now find the value of $C_{F}$ as follows. On the one hand we know that

while on the other, the left hand side is also equal to $T_{\rm
R}\left(N_{{\rm c}}^{2}-1\right)$. Thus

$C_{F}=T_{\rm R}\frac{N_{{\rm c}}^{2}-1}{N_{{\rm c}}}\,.$

Analogously one can find

$C_{\rm A}=2\,T_{R}\,N_{{\rm c}}\,.$

As the colour factors $C_{\rm F}$ and $C_{\rm A}$ depend on $N_{{\rm c}}$,
their measurement gives information on the number of colours. The experiments
of the Large Electron Positron collider measured the values of the colour
factors based on fits of theoretical predictions [3] to four-jet angular
distributions that are sensitive to both $C_{\rm F}$ and $C_{\rm A}$. The
result of the simultaneous measurement of the colour factors and the strong
coupling by the OPAL collaboration is shown in Fig. 2 [4]. The values
corresponding to $N_{\rm c}=3$ are marked with the star, just in the middle of
the confidence-ellipses.

Figure 2: Measurement of the colour factors by the LEP collaborations [4]

The expression $\sum_{a}\,t^{a}_{ij}t^{a}_{kl}$ is invariant under $SU(N)$
transformations, therefore has to be expressible as a linear combination of
$\delta_{il}\delta_{kj}$ and $\delta_{ij}\delta_{kl}$ (the third combination
of Kronecker $\delta$’s is not possible, the direction of arrows do not
match). The two coefficients can be obtained by making contractions with
$\delta_{il}\delta_{jk}$ and $\delta_{ij}\delta_{kl}$. Thus we obtain the
Fierz identity,

$\sum_{a}t^{a}_{ij}t^{a}_{kl}=T_{\rm
R}\left(\delta_{il}\delta_{kj}-\frac{1}{N_{{\rm
c}}}\delta_{ij}\delta_{kl}\right)\,,$

or graphically:

.

These graphical rules help in computing colour sums easily. Nevertheless,
nowadays computer algebra codes make computation of colour sums an automated
procedure. For instance, you may try In[1]:=
Import["http://www.feyncalc.org/install.m"] in your Mathematica session to see
one solution.

###### Exercise 1.8

Consider the process $q\bar{q}\rightarrow ggg$. Compute the color structures
that appear in the squared matrix element.

###### Exercise 1.9

Try using the Fierz identity to obtain

.

###### Exercise 1.10

Determine the color factors $A$,$B$,$C$ in the following equations:

,  ,  .

###  1.5 Are we done?

We now have the Feynman rules with colour and the rest factorized, and we
gained some insight how to perform the colour algebra. Thus it seems that we
are in the position to compute the cross section of any process up to the
desired accuracy in perturbation theory (PT), just as we can do in QED. So it
may appear that conceptually we are done. Well, we are going to see big
surprises!

Figure 3: Illustration of the approximation of hadronic final states by
partonic events in electron-positron annihilation: the sprays of hadrons
(called jets) are assumed to originate from primary quarks and gluons, thus
approximated by quarks and gluons as shown by the magnification

The first conceptual challenge is due to a phenomenological observation. In
QED, PT is applicable because the elementary excitations of the quantum
fields, the electrons and photons, can be observed as stable, free particles.
Thus asymptotic states are parts of the physical reality. On the contrary,
free quarks and gluons (usually called simply partons) have never been
observed in nature. This experimental fact can be reformulated saying that the
probability of observing a final state with any fixed number of on-shell
partons is zero. This negative result has been turned into the principle of
‘quark confinement’. Thus it is questionable whether a QFT of quark and gluon
fields can describe the observed world of particles where in addition to
leptons only hadrons have been found. In fact, a main research project at the
LEP was to find an answer to this question in a well controlled quantitative
manner. It turned out that the answer is positive if we make an assumption
that we cannot prove from first principles:

> The result of a low-order perturbative computation in QCD is an
> approximation to sufficiently inclusive hadronic cross section if (i) the
> total centre-of-mass energy $Q$ of partons is much larger than the mass of
> quarks, $Q>>m_{q}$, and (ii) $Q$ is far from hadronic resonances and
> thresholds.

We shall define precisely what ‘sufficiently inclusive’ means later.
Predictions made on the basis of this assumption agree with measurements (e.g.
made at LEP) within the expected accuracy of the prediction, which we are to
define also later.

Based on this assumption, it makes sense to make predictions with quark and
gluon asymptotic states. However, in QCD the complexity of the Feynman rules
will make higher order computations prohibitive. Indeed, the largest effort in
QCD computations during the past 20 years went into devising ever more
efficient methods to decrease the algebraic complexity of the computations.
This research is driven by the observation that the QCD Lagrangian is highly
symmetric, which has to be reflected in the final results. Thus the
complications somehow appear mainly because with our rules we artificially
introduce complications at intermediate steps of the computations, which
cancels to large extent in the final formulae. Learning about the symmetries
of QCD is interesting and useful not only for technical purposes, so let us
make an inventory of those.

###  1.6 Symmetries of the classical Lagrangian

The symmetries can be grouped into two large categories: exact symmetries and
approximate ones. Space-time symmetries are exact. These consist of invariance
against continuous transformations: translations and Lorentz-transformations
(rotations and boosts). In addition ${\cal L}_{\rm cl}$ is invariant under
scale transformation:

$x^{\mu}\to\lambda x^{\mu}\quad A_{\mu}(x)\to\lambda^{-1}A_{\mu}(\lambda
x)\quad q(x)\to\lambda^{-3/2}q(\lambda x)\,,$

and conformal transformations, which we do not detail here. The Lagrangian is
also invariant under charge conjugation (C), parity (P) and time-reversal (T),
in agreement with observed properties of strong interactions (C, P and T
violating strong decays are not observed).

We already discussed exact symmetry in colour space: local gauge invariance.
In addition to the classical Lagrangian of Eq. (6), there exists additional
gauge invariant dimension-four operator, the $\Theta$-term:

${\cal L}_{\Theta}=\frac{\Theta
g_{s}}{32\pi^{2}}\sum_{a}F^{a}_{\mu\nu}\tilde{F}^{a,\mu\nu}\,,\quad\textrm{with}\quad\tilde{F}^{a,\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F^{a}_{\alpha\beta}\,,$

that violates P and T. As experimentally $\Theta<10^{-9}$, we set $\Theta=0$
in perturbative QCD.

Another interesting feature of ${\cal L}_{\rm cl}$ is that it is almost
supersymmetric. For one massless flavour

${\cal L}_{\rm cl}=-\frac{1}{4}\sum_{a}F^{a}_{\mu\nu}F^{a,\mu\nu}+\bar{q}{\rm
i}\slashed{D}q\,,$

which is very similar to the Lagrangian of $N=1$ supersymmetric gauge theory,

${\cal L}^{\rm SUSY}_{\rm
cl}=-\frac{1}{4}\sum_{a}F^{a}_{\mu\nu}F^{a,\mu\nu}+\bar{\lambda}{\rm
i}\slashed{D}\lambda\,.$

The only difference is that the quark $q$ transforms under the fundamental,
while the gluino $\lambda$ under the adjoint representation of the gauge
group.

An important approximate symmetry of the classical Lagrangian is related to
the quark mass-matrix. Let us introduce the quark flavour triplet

$\psi=\left(\begin{array}[]{c}u\\\ d\\\
s\end{array}\right)=\left(\begin{array}[]{c}q_{1}\\\ q_{2}\\\
q_{3}\end{array}\right)\,,$

with each component being a four-component Dirac spinor, and the combinations

$P_{\pm}=\frac{1}{2}\big{(}1\pm\gamma_{5}\big{)}\,.$ (11)

The latter are projections:

$P_{+}\,P_{-}=P_{-}\,P_{+}=0\,,\quad P_{\pm}^{2}=P_{\pm}\,,\quad
P_{+}+P_{-}=1\,.$

It follows from Clifford-algebra that
$\gamma_{\mu}P_{\pm}=P_{\mp}\gamma_{\mu}$. We define $\psi_{\pm}=P_{\pm}\psi$.
Using $\gamma_{5}^{2}=1$, we find that $\psi_{\pm}$ are eigenvectors of
$\gamma_{5}$ with $\pm 1$ eigenvalues:

$\gamma_{5}\psi_{\pm}=\pm\psi_{\pm}\,.$

From the definition of the Dirac adjoint,
$\overline{\psi}=\psi^{\dagger}\gamma_{0}$, we obtain
$\overline{\psi_{\pm}}=\overline{\psi}P_{\mp}$. Thus the quark sector of the
Lagrangian can be rewritten in terms of the chiral fields $\psi_{\pm}$:

$\begin{split}{\cal L}_{\rm cl}&=\overline{\psi}\,{\rm
i}\,\gamma_{\mu}D^{\mu}\,\psi=\overline{\psi}(P_{+}+P_{-})\,{\rm
i}\,\gamma_{\mu}D^{\mu}\,(P_{+}+P_{-})\psi=\overline{\psi}P_{+}\,{\rm
i}\,\gamma_{\mu}D^{\mu}\,P_{-}\psi+\overline{\psi}P_{-}\,{\rm
i}\,\gamma_{\mu}D^{\mu}\,P_{+}\psi=\\\ &=\overline{\psi}_{-}\,{\rm
i}\,\gamma_{\mu}D^{\mu}\,\psi_{-}+\overline{\psi}_{+}\,{\rm
i}\,\gamma_{\mu}D^{\mu}\,\psi_{+}={\cal L}_{-}+{\cal L}_{+}\equiv{\cal L}_{\rm
L}+{\cal L}_{\rm R}\,.\end{split}$

This decomposition would not work if the gluon field in the covariant
derivative were not Lorentz-vector. In this chiral form the left- and right-
handed fields decouple, so the Lagrangian is invariant under separate
$U(N_{\rm f})$ transformations for the left- and right-handed fields, under
$U_{\rm L}(N_{\rm f})\times U_{\rm R}(N_{\rm f})$, hence it is called chiral
symmetry. Indeed, if $(g_{\rm L},g_{\rm R})\in U_{\rm L}(N_{\rm f})\times
U_{\rm R}(N_{\rm f})$, then under the transformation

$\psi_{\rm L}\to g_{\rm L}\psi_{\rm L}\,,\qquad\overline{\psi}_{\rm
L}\to\overline{\psi}_{\rm L}g^{\dagger}_{\rm L}\,,\qquad g_{\rm R}=1$

${\cal L}_{\rm L}$ remains invariant. This symmetry is exact if the quarks are
massless. The group elements can be parametrized using $2N_{\rm f}^{2}$ real
numbers $\\{\alpha,\alpha_{a},\beta,\beta_{b}\\}$ ($a$, $b=1\,,\dots N_{\rm
f}^{2}-1$),

$\split(g_{\rm L},g_{\rm R})&=\Bigg{(}\exp({\rm i}\alpha)\exp({\rm
i}\beta)\exp\left({\rm i}\sum_{a}\alpha_{a}T^{a}\right)\exp\left({\rm
i}\sum_{b}\beta_{b}T^{b}\right),\\\ \qquad\exp({\rm i}\alpha)\exp(-{\rm
i}\beta)\exp\left({\rm i}\sum_{a}\alpha_{a}T^{a}\right)\exp\left(-{\rm
i}\sum_{b}\beta_{b}T^{b}\right)\Bigg{)}\\\ \in U_{\rm V}(1)\otimes SU_{\rm
L}(N_{\rm f})\otimes U_{\rm A}(1)\otimes SU_{\rm R}(N_{\rm f})\,,$

where the matrices $T^{a}$ represent the generators of the group ($N_{\rm
f}\times N_{\rm f}$ matrices). The transformations $(\exp\left({\rm
i}\sum_{a}\alpha_{a}T^{a}\right)$, $\exp\left({\rm
i}\sum_{a}\alpha_{a}T^{a}\right))$, acting as $\psi\to\exp\left({\rm
i}\sum_{a}\alpha_{a}T^{a}\mathrm{I}\right)\psi$, form a vector subgroup
$SU_{\rm V}(N_{\rm f})$. The transformations $(\exp\left({\rm
i}\sum_{b}\beta_{b}T^{b}\right),\exp\left(-{\rm
i}\sum_{b}\beta_{b}T^{b}\right))$, acting as $\psi\to\exp\left({\rm
i}\sum_{b}\beta_{b}T^{b}\gamma_{5}\right)\psi$, however, do not form an axial-
vector subgroup because

$[T^{a}\gamma_{5},T^{b}\gamma_{5}]={\rm
i}\sum_{c}f^{abc}T^{c}\mathrm{I}\qquad(\gamma_{5}^{2}=\mathrm{I}\neq\gamma_{5})\,.$

This chiral symmetry is not observed in the hadron spectrum. Therefore, we
assume that vacuum has a non-zero VEV of the light-quark operator,

$\left\langle 0|\bar{q}q|0\right\rangle=\left\langle 0|\bar{{\rm u}}{\rm
u}+\bar{{\rm d}}{\rm d}|0\right\rangle\simeq(250\,{\rm MeV})^{3}\,,$

a chiral condensate that connect left- and right-handed fields,

$\left\langle 0|\bar{q}q|0\right\rangle=\left\langle 0|\bar{{\rm q}}_{\rm
L}{\rm q}_{\rm R}+\bar{{\rm q}}_{\rm R}{\rm q}_{\rm L}|0\right\rangle\,.$

The condensate breaks chiral symmetry spontaneously to $SU_{\rm V}(N_{\rm
f})\otimes U_{\rm V}(1)$. This remaining symmetry explains the existence of
good quantum numbers of isospin and baryon number, as well as the appearance
of $N_{\rm f}^{2}-1=8$ massless mesons, the Goldstone bosons. As non-zero
quark masses violate the chiral symmetry, which is broken spontaneously, the
Goldstone bosons are not exactly massless. Thus we have natural candidates for
the Goldstone bosons: we can identify those with the pseudoscalar meson octet.
In practice, we assume exact chiral symmetry and treat the quark masses as
perturbation. This procedure leads us to chiral perturbation theory ($\chi$PT)
[5], which is capable to predict the (ratios of) masses of light quarks [6,
7], scattering properties of pions [8] and many more. Although, $\chi$PT is a
non-renormalizable QFT, it can be made predictive order by order in PT if the
measured values of sufficiently many observables are used to fix the couplings
of interaction terms at the given order.

The QCD Lagrangian was written forty years ago. Since then many attempts were
tried to solve it and mature fields emerged that aim at solving the theory in
a limited range of physical phenomena. For instance, $\chi$PT is a PT that
uses low-energy information (in the MeV range) to explain the world of hadrons
and masses of light quarks. In the same energy range non-perturbative
approaches, notably lattice QCD and sum rules have been developed for the same
purpose. By now it is possible to explain the light hadron spectrum from first
principles using lattice results [9]. The main goal at colliders, our focus in
these lectures, is different. We shall prove that PT can give reliable
predictions for scattering processes at high energies, which is the topic of
jet physics.

We have seen that the classical QCD Lagrangian shows many interesting symmetry
properties that can be utilized for (i) easing computations, (ii) checking
results, (iii) hinting on solving QCD. We shall see that some of these
symmetries are violated by quantum corrections, which leads to important
physical consequences. In QCD an important example is scaling violations.
Another example is the axial anomaly which provides strong constraints on
possible QFT’s, but it is discussed within the electroweak theory usually.

###  1.7 What is scaling?

Let us consider a dimensionless physical observable $R$ that depends on a
large energy scale $R=R(Q^{2})$. Large means that $Q$ is much bigger than any
other dimensionful parameter, for instance, masses of quarks. Thus we assume
that these other dimensionful parameters can be set zero.111We shall study the
validity of this assumption in the next subsection. Classically, dim $R\,=\,0$
and, since $Q$ is dimensionful, it follows that $\frac{dR}{dQ}=0$. So
$\lim_{Q^{2}\rightarrow\infty}R=$constant, which is called scaling.

In these lectures we do not have room for a complete description of
ultraviolet (UV) renormalization of QCD. We simply state that in a
renormalized QFT $R$ depends also on another scale, the renormalization scale
$\mu_{\rm R}$. So

$R=\lim_{Q^{2}\to\infty}R\left(\frac{Q^{2}}{\mu^{2}_{{\rm R}}},\,\alpha_{\rm
s}\left(\mu_{\rm R}^{2}\right)\right)\neq\textrm{constant},$

$R$ need not be a constant. This is called scaling violation. The first term
in parenthesis is the only dimensionless combination of $Q$ and $\mu_{\rm R}$.
However, $\mu_{\rm R}$ is arbitrary. If $R$ depended on $\mu_{\rm R}$, then
its value could not be predicted. For simplicity from now on we drop the
subscript “R” from $\mu_{\rm R}$. As $\mu$ is an arbitrary, un-physical
parameter (the classical Lagrangian did not contain $\mu$), we expect that
measurable (physical) quantities cannot depend on it, which is expressed by
the renormalization group equation (RGE):

$0=\mu^{2}\frac{\rm d}{{\rm
d}\mu^{2}}\,R\left(\frac{Q^{2}}{\mu^{2}},\alpha_{\rm
s}\big{(}\mu^{2}\big{)}\right)=\left(\mu^{2}\frac{\partial}{\partial\mu^{2}}+\mu^{2}\frac{\partial\alpha_{\rm
s}}{\partial\mu^{2}}\frac{\partial}{\partial\alpha_{\rm s}}\right)R\,.$

We can simplify this equation a bit by introducing the new variable $t$ and
the function $\beta(\alpha_{\rm s})$,

$t=\ln\frac{Q^{2}}{\mu^{2}}\,,\qquad\beta(\alpha_{\rm
s})=\mu^{2}\frac{\partial\alpha_{\rm s}}{\partial\mu^{2}}\bigg{|}_{\alpha_{\rm
s}\>\mathrm{fixed}}\,.$ (12)

Then the RGE becomes

$\left(-\frac{\partial}{\partial\,t}+\beta(\alpha_{\rm
s})\frac{\partial}{\partial\alpha_{\rm s}}\right)R\big{(}e^{t},\alpha_{\rm
s}\big{)}=0\,.$ (13)

To present the solution of this partial differential equation, we introduce
the running coupling $\alpha_{\rm s}(Q^{2})$, defined implicitly by

$t=\int_{\alpha_{\rm s}}^{\alpha_{\rm s}\big{(}Q^{2}\big{)}}\\!\frac{{\rm
d}x}{\beta(x)}\,,\quad\mathrm{with}\quad\alpha_{\rm s}\equiv\alpha_{\rm
s}\big{(}\mu^{2}\big{)}\,,$ (14)

where $\alpha_{\rm s}\equiv\alpha_{\rm s}\left(\mu^{2}\right)$ is an
arbitrarily fixed number. The derivative of Eq. (14) with respect to the
variable $t$ gives

$1=\frac{1}{\beta\left(\alpha_{\rm
s}\left(Q^{2}\right)\right)}\,\frac{\partial\alpha_{\rm
s}\left(Q^{2}\right)}{\partial\,t}\,,\quad\mbox{which
implies}\quad\beta\left(\alpha_{\rm
s}\left(Q^{2}\right)\right)=\frac{\partial\alpha_{\rm
s}\left(Q^{2}\right)}{\partial\,t}\,.$

The derivative of Eq. (14) with respect to $\alpha_{\rm s}$ gives

$0=\frac{1}{\beta\left(\alpha_{\rm
s}\left(Q^{2}\right)\right)}\,\frac{\partial\alpha_{\rm
s}\left(Q^{2}\right)}{\partial\alpha_{\rm s}}-\frac{1}{\beta(\alpha_{\rm
s})}\,\frac{\partial\alpha_{\rm s}}{\partial\alpha_{\rm s}}\,,$

from which it follows that

$\frac{\partial\alpha_{\rm s}\left(Q^{2}\right)}{\partial\alpha_{\rm
s}}=\frac{\beta\left(\alpha_{\rm
s}\left(Q^{2}\right)\right)}{\beta(\alpha_{\rm s})}\,.$

It is now easy to prove that the value of $R$ for $\mu^{2}=Q^{2}$,
$R\left(1,\alpha_{\rm s}\left(Q^{2}\right)\right)$ solves Eq. (13):

$-\frac{\partial}{\partial\,t}R\left(1,\alpha_{\rm
s}\left(Q^{2}\right)\right)=-\frac{\partial\,R}{\partial\alpha_{\rm
s}\left(Q^{2}\right)}\,\frac{\partial\alpha_{\rm
s}\left(Q^{2}\right)}{\partial\,t}=-\beta\left(\alpha_{\rm
s}\left(Q^{2}\right)\right)\,\frac{\partial\,R}{\partial\alpha_{\rm
s}\left(Q^{2}\right)}$

and

$\beta(\alpha_{\rm s})\,\frac{\partial}{\partial\alpha_{\rm
s}}\,R\left(1,\alpha_{\rm s}\left(Q^{2}\right)\right)=\beta(\alpha_{\rm
s})\,\frac{\partial\alpha_{\rm s}\left(Q^{2}\right)}{\partial\alpha_{\rm
s}}\,\frac{\partial\,R}{\partial\alpha_{\rm
s}\left(Q^{2}\right)}=\beta\left(\alpha_{\rm
s}\left(Q^{2}\right)\right)\,\frac{\partial\,R}{\partial\,\alpha_{\rm
s}\left(Q^{2}\right)}.$

It then follows that the scale-dependence in $R$ enters only through
$\alpha_{\rm s}\big{(}Q^{2}\big{)}$, and we can predict the scale-dependence
of $R$ by solving Eq. (14), or equivalently,

$\frac{\partial\alpha_{\rm
s}\big{(}Q^{2}\big{)}}{\partial\,t}=\beta\left(\alpha_{\rm
s}\big{(}Q^{2}\big{)}\right)\,.$ (15)

So far our analysis was non-perturbative. Assuming that PT is applicable,
which we shall discuss at the end of this subsection, we may try to solve Eq.
(15) in PT where the $\beta$-function has the formal expansion:

$\beta(\alpha_{\rm s})=-\alpha_{\rm
s}\,\sum_{n=0}^{\infty}\beta_{n}\left(\frac{\alpha_{\rm
s}}{4\pi}\right)^{n+1}\,.$ (16)

The first four coefficients are known from cumbersome computations [10]

$\split\beta_{0}&=\frac{11}{3}\,C_{A}-\frac{4}{3}\,T_{R}\,n_{f}\,>\,0\,,\qquad\beta_{1}=\frac{34}{3}C_{A}^{2}-\frac{20}{3}C_{A}T_{R}\,n_{\rm
f}-4C_{F}T_{R}\,n_{\rm f}\,,\\\ \beta_{2}=\frac{2857}{2}-\frac{5033}{18}n_{\rm
f}+\frac{325}{54}n_{\rm f}^{2}\,,\qquad\beta_{3}=29243-6946.3n_{\rm
f}+405.9n_{\rm f}^{2}+1.5n_{\rm f}^{3}\,.$ (17)

The first two coefficients in the expansion of the $\beta$ function are
independent of the renormalization scheme. The second two coefficients in Eq.
(17) are valid in the $\overline{\mathrm{MS}}$ renormalization scheme.222As we
have not gone through the renormalization procedure, we cannot define
precisely what we mean by ‘renormalization scheme’. Various schemes differ by
finite renormalization of the parameters and fields in the Lagrangian.

Another often used convention is

$\beta(\alpha_{\rm s})=-b_{0}\alpha_{\rm
s}^{2}\left[1+\sum_{n=1}^{\infty}b_{n}\,\alpha_{\rm s}^{n}\right],$ (18)

where $b_{0}=\frac{\displaystyle\beta_{0}}{\displaystyle 4\pi}$ and
$b_{0}b_{1}=\frac{\displaystyle\beta_{1}}{\displaystyle\left(4\pi\right)^{2}}$,
thus $b_{1}=\frac{\displaystyle\beta_{1}}{\displaystyle 4\pi\beta_{0}}$.

If $\alpha_{\rm s}\left(Q^{2}\right)$ is small we can truncate the series. The
solution at leading-order (LO) accuracy is

$\displaystyle Q^{2}\,\frac{\partial\alpha_{\rm
s}}{\partial\,Q^{2}}=\frac{\partial\alpha_{\rm
s}}{\partial\,t}=-b_{0}\alpha_{\rm
s}^{2}\quad\Rightarrow\quad-\left[\frac{1}{\alpha_{\rm
s}\left(Q^{2}\right)}-\frac{1}{\alpha_{\rm
s}\left(\mu^{2}\right)}\right]=-b_{0}t\quad\Rightarrow\quad\alpha_{\rm
s}\left(Q^{2}\right)=\frac{\alpha_{\rm
s}\left(\mu^{2}\right)}{1+b_{0}t\,\alpha_{\rm s}\left(\mu^{2}\right)}\,,$ (19)

which gives $\alpha_{\rm s}\left(Q^{2}\right)$ as a function of $\alpha_{\rm
s}\left(\mu^{2}\right)$ if both are small; $\alpha_{\rm
s}\left(\mu^{2}\right)$ is a number to be measured. We observe that:

$\alpha_{\rm s}\left(Q^{2}\right)\stackrel{{\scriptstyle
Q^{2}\rightarrow\,\infty}}{{\longrightarrow}}\,\frac{1}{b_{0}t}\,\stackrel{{\scriptstyle
Q^{2}\rightarrow\,\infty}}{{\longrightarrow}}\,0\,.$ (20)

This behaviour is called asymptotic freedom. The sign of $b_{0}$ (positive for
QCD) plays a crucial role in establishing whether or not a theory is
asymptotically free. If it is, then the use of PT is justified: the higher
$Q^{2}$, the smaller the coupling. The coefficient $b_{0}$ is easiest to
compute in background field gauge [11] where only three graphs contribute, the
quark and gluon loops:

$\includegraphics[width=173.44534pt]{figures/beta0graphs.pdf}\,,$ (21)

and a similar ghost loop. The contribution of the quark loop is negative
$-\frac{4}{3}T_{\rm R}n_{\rm f}$, while that of the gluon+ghost loop is
positive $\frac{11}{3}C_{\rm A}$. (We knew the colour factors immediately,
only the coefficients have to be computed!) The net result is positive up to
$n_{\rm f}<17$ in QCD. In 2004 D.J. Gross, H.D. Politzer and F. Wilczek were
awarded the Nobel prize for their discovery of asymptotic freedom in QCD [12,
13].

Clearly, it is the gluon self-interaction that makes QCD perfect in PT. In
QED, in the absence of photon self-interaction, $b_{0}<0$, hence the coupling
increases with energy, but remains perturbative up to the Planck scale
($\simeq 10^{19}$ GeV) where we expect that any known physics breaks down.

Asymptotic freedom gives rationale to perturbative QCD, but we shall see that
LO accuracy is not enough. The analysis is also simple at next-to-leading
order (NLO):

$\left[\alpha_{\rm s}^{2}(1+b_{1}\alpha_{\rm
s})\right]^{-1}\,\frac{\partial\alpha_{\rm s}}{\partial\,t}=-b_{0}.$

$\alpha_{\rm s}\left(Q^{2}\right)$ is then given implicitly by the equation

$\frac{1}{\alpha_{\rm s}\left(Q^{2}\right)}-\frac{1}{\alpha_{\rm
s}\left(\mu^{2}\right)}+b_{1}\ln\frac{\alpha_{\rm
s}\left(Q^{2}\right)}{\alpha_{\rm
s}\left(\mu^{2}\right)}-b_{1}\ln\frac{1+b_{1}\alpha_{\rm
s}\left(Q^{2}\right)}{1+b_{1}\alpha_{\rm s}\left(\mu^{2}\right)}=bt\,,$

which can be solved numerically.

Using the formula for the sum of the geometric series,
$(1+x)^{-1}=\sum_{j=0}^{\infty}(-x)^{j}$ and recalling Eq. (19), we find that
the running coupling sums logarithms,

$R\big{(}1,\alpha_{\rm s}\big{(}Q^{2}\big{)}\big{)}=R_{0}+R_{1}\alpha_{\rm
s}\big{(}\mu^{2}\big{)}\sum_{j=0}^{\infty}\big{[}-\alpha_{\rm
s}\big{(}\mu^{2}\big{)}b_{0}t\big{]}^{j}\,.$

The NLO term $R_{2}\alpha_{\rm s}^{2}$ gives logarithms with one power less in
each term.

Figure 4: Results of measurements of the strong coupling at different scales.
The theoretical prediction with four-loop running, fixed at $\mu=M_{Z}$ is
marked as ‘QCD’.

###  1.8 Measuring $\alpha_{\rm s}(\mu^{2})$

We know $\alpha_{\rm s}\left(Q^{2}\right)$ if $\alpha_{\rm
s}\left(\mu^{2}\right)$ is known. We therefore, have to measure $\alpha_{\rm
s}$ at some scale $\mu$. The perturbative solution of the renormalization
group equation (RGE, Eq. (13)) is never unique. The difference between two
solutions at $\mathcal{O}\left(\alpha_{\rm s}^{n}\right)$ is suppressed by
$\alpha_{\rm s}$, i.e. at $\mathcal{O}\left(\alpha_{\rm s}^{n+1}\right)$.
Nevertheless, this difference can lead to significant difference in
$\alpha_{\rm s}\left(Q^{2}\right)$ if $\mu^{2}$ and $Q^{2}$ are far from each
other, which is important in present day precision measurements. Therefore,
the scale $\mu$ is chosen to be $\mu=M_{Z}$ because $M_{Z}=91,2$ GeV is not
far from the scales where $\alpha_{\rm s}(Q^{2})$ is used in current
experimental analyses. In Figs. 4 and 5 we show the present status of
$\alpha_{\rm s}$ measurements from Ref. [14].

Figure 5:  Results of different measurements of the strong coupling run to
$\mu=M_{Z}$

Another approach to solving the RGE is to introduce a refence scale $\Lambda$
by

$\ln\frac{Q^{2}}{\Lambda^{2}}=\int_{\alpha_{\rm
s}\big{(}Q^{2}\big{)}}^{\infty}\\!\frac{{\rm d}x}{\beta(x)}\,.$

The scale $\Lambda$ indicates where the coupling becomes strong. The following
exercise is to explore the characteristics of this choice.

###### Exercise 1.11

The running of the strong coupling constant is given by Eq. (12). The
perturbative expansion of the QCD beta function is given by Eq. (18) with
$b_{0},b_{1}\geq 0$. Determine (i) the expression for the coupling constant in
leading order ($b_{0}\not=0$, $b_{1}=0$) and the corresponding scale
$\Lambda_{0}$ (see below) (ii) the expression for the coupling constant in
next-to-leading order ($b_{0}\not=0$, $b_{1}\not=0$) and the corresponding
scale $\Lambda_{1}$ (see below).

Hints:

  1. 1.

Solve the differential equation for $\alpha(\mu)$; you’ll get an integration
constant.

  2. 2.

Express your result in the form

$\alpha(\mu)=\frac{1}{K\ln(\frac{\mu^{2}}{\Lambda_{0}^{2}})}$

where $K$ is a constant.

  3. 3.

Solve the differential equation using $b_{1}\not=0$

$\int\mathrm{d}\alpha\frac{1}{-b_{0}\alpha^{2}-b_{1}\alpha^{3}}=\frac{b_{0}+b_{1}\alpha\log(\alpha)-b_{1}\alpha\log(b_{0}+b_{1}\alpha)}{b_{0}^{2}\alpha}+K$

  4. 4.

This time the solution cannot be solved for $\alpha$ analytically. One can
nevertheless find an approximate solution by expanding $\alpha$ in
$\log\frac{\mu^{2}}{\Lambda_{1}^{2}}$. The constant $K$ is not equal to the
one in the first part of this exercise.

  5. 5.

Cast your equation for $\alpha$ into the form

$\alpha=\frac{1}{K\ln\frac{\mu^{2}}{\Lambda_{1}^{2}}}\quad\frac{1}{1+c_{1}\frac{\ln(c_{2}+b_{0}\alpha)}{\ln\frac{\mu^{2}}{\Lambda_{1}^{2}}}}$

with a suitable choice of $\Lambda_{1}$.

  6. 6.

Expand the right hand side of your equation in
$t=\frac{1}{\ln\frac{\mu^{2}}{\Lambda_{1}^{2}}}$ and keep only the first order
term. Use the expansion

$\frac{1}{1+C_{1}\,t\ln(C_{3}\frac{1}{t}+C_{2})}=1+t\,C_{1}\ln(\frac{1}{t})+O(t)\,.$

###  1.9 Quark masses and massless QCD

Quark masses $m_{q}$ are parameters of $\mathcal{L}_{QCD}$ like the gauge
coupling, which need to be renormalized. In QED the electron mass is measured
in the laboratories at $\mu_{\rm R}^{2}=0$ (classical limit). We cannot
similarly isolate a quark at $\mu_{\rm R}^{2}=0$ (at low scale quarks are
confined). Instead, we can perform a similar RGE analysis as with $\alpha_{\rm
s}$. For simplicity we assume one quark flavour with mass $m$, which is yet
another dimensionful parameter, so the RGE becomes:

$\left[\mu^{2}\frac{\partial}{\partial\mu^{2}}+\beta(\alpha_{\rm
s})\frac{\partial}{\partial\alpha_{\rm s}}-\gamma_{m}(\alpha_{\rm
s})m\frac{\partial}{\partial\,m}\right]\,R\left(\frac{Q^{2}}{\mu^{2}},\alpha_{\rm
s},\frac{m}{Q}\right)=0\,,$ (22)

where $\gamma_{m}$ is called the mass anomalous dimension and the minus sign
before $\gamma_{m}$ is a convention. In PT we can write the mass anomalous
dimension as

$\gamma_{m}(\alpha_{\rm s})\,=\,c_{0}\alpha_{\rm s}\left(1+c_{1}\alpha_{\rm
s}+\mathcal{O}\left(\alpha_{\rm s}^{2}\right)\right),$

with known coefficient up to $c_{3}$. At NLO accuracy we need only
$c_{0}=\frac{1}{\pi}$ and $c_{1}=\frac{303-10\,n_{f}}{72\pi}$. As $R$ is
dimensionless, the dependence on the dimensionful parameters has to cancel

$\left(Q^{2}\frac{\partial}{\partial\,Q^{2}}+\mu^{2}\frac{\partial}{\partial\mu^{2}}+m^{2}\frac{\partial}{\partial\,m^{2}}\right)\,R\left(\frac{Q^{2}}{\mu^{2}},\alpha_{\rm
s},\frac{m}{Q}\right)=0\,.$ (23)

The difference of Eqs. (22) and (23) gives the dependence of $R$ on $Q$:

$\left[Q^{2}\frac{\partial}{\partial\,Q^{2}}-\beta\left(\alpha_{\rm
s}\right)\frac{\partial}{\partial\alpha_{\rm
s}}+\left(\frac{1}{2}+\gamma\left(\alpha_{\rm
s}\right)\right)m\frac{\partial}{\partial\,m}\right]\,R\left(\frac{Q^{2}}{\mu^{2}},\alpha_{\rm
s},\frac{m}{Q}\right)=0\,.$ (24)

This equation is solved by introducing the running mass (in addition to the
running coupling) $m\left(Q^{2}\right)$ obeying

$Q^{2}\frac{\partial\,m}{\partial\,Q^{2}}=-\gamma_{m}\left(\alpha_{\rm
s}\right)m\left(Q^{2}\right)\,,\quad\Rightarrow\quad\ln\frac{m\left(Q^{2}\right)}{m\left(\mu^{2}\right)}\,=\,-\int_{\mu^{2}}^{Q^{2}}\frac{{\rm
d}q^{2}}{q^{2}}\,\gamma_{m}\left(\alpha_{\rm s}\left(q^{2}\right)\right)\,.$
(25)

Exponentiating, changing integration variable from $q^{2}$ to $\alpha_{\rm s}$
and using the definition of the $\beta$ function, we obtain

$m\left(Q^{2}\right)\,=\,m\left(\mu^{2}\right)\exp\left[-\int_{\alpha_{\rm
s}\left(\mu^{2}\right)}^{\alpha_{\rm s}\left(Q^{2}\right)}\\!{\rm
d}\alpha_{\rm s}\frac{\gamma_{m}\left(\alpha_{\rm
s}\right)}{\beta\left(\alpha_{\rm s}\right)}\right]\stackrel{{\scriptstyle
Q^{2}\rightarrow\infty}}{{\longrightarrow}}\,0\,,$ (26)

which means that asymptotically free QCD is a massless theory at
asymptotically large energies. At LO in PT theory the solution of (26) is
given by

$-\frac{\gamma_{m}\left(\alpha_{\rm s}\right)}{\beta\left(\alpha_{\rm
s}\right)}=\frac{c_{0}}{b_{0}\alpha_{\rm
s}}\quad\Rightarrow\quad\,m\left(Q^{2}\right)=\overline{m}\left[\alpha_{\rm
s}\left(Q^{2}\right)\right]^{\frac{c_{0}}{b_{0}}}\,,$

where we introduced the abbreviation
$\overline{m}={m\left(\mu^{2}\right)\left[\alpha_{\rm
s}\left(\mu^{2}\right)\right]^{-\frac{c_{0}}{b_{0}}}}$. At NLO the solution
becomes

$m\left(Q^{2}\right)=\overline{m}\left[\alpha_{\rm
s}\left(Q^{2}\right)\right]^{\frac{c_{0}}{b_{0}}}\left(1+\frac{c_{0}}{b_{0}}\left(c_{1}-b_{1}\right)\left(\alpha_{\rm
s}\left(Q^{2}\right)-\alpha_{\rm
s}\left(\mu^{2}\right)\right)+\mathcal{O}\left(\alpha_{\rm
s}^{2}\right)\right)\,.$

In terms of the running coupling and mass, $R\left(1,\alpha_{\rm
s}(Q^{2}),\frac{m(Q^{2})}{Q}\right)$ is a solution of Eq. (24), proven
similarly as $R\left(1,\alpha_{\rm s}(Q^{2})\right)$ being the solution of Eq.
(13). Expanding around $m(Q^{2})=0$, we obtain

$R\left(1,\alpha_{\rm
s}(Q^{2}),\frac{m(Q^{2})}{Q}\right)=R\left(\frac{Q^{2}}{\mu^{2}},\alpha_{\rm
s},0\right)+\sum_{n=1}^{\infty}\frac{1}{n!}\left(\frac{m(Q^{2})}{Q}\right)^{n}R^{(n)}\left(\frac{Q^{2}}{\mu^{2}},\alpha_{\rm
s},0\right)\,.$ (27)

We see from Eq. (27) that derivative terms are suppressed by factors of
$1/Q^{n}$ at large $Q^{2}$. From the dependence of $R$ on
$\frac{m\left(Q^{2}\right)}{Q}$ we can conclude that the effect of mass is
suppressed at high $Q^{2}$ by its physical dimension and also by its anomalous
dimension, which justifies the assumption about negligible quark masses. The
expansion in Eq. (27) has a deeper consequence. The dimensionless observable
$R$ may depend on $\ln\frac{m\left(Q^{2}\right)}{Q}$ that can become large
when $Q^{2}$ is large. If we want to avoid such large logarithms, we should
consider physical observables (that is physically measurable quantities) that
have a finite zero-mass limit.

##  2 Predictions in perturbative QCD

Figure 6: An event with four hard muons in the CMS detector

In a typical collider experiment we collect collision events with something
interesting in the final state. For instance, in searching for the Higgs
boson, events with four hard muons such as in Fig. 6 are interesting. Counting
the event rate of such events we obtain measured cross sections, which compare
to theoretical predictions. Following our assumption about the use of low-
order perturbative predictions in QCD, for such comparisons we need
predictions for cross sections with partons. We start with the simplest
possible case when partons appear only in the final state: electron-positron
annihilation into hadrons (and possibly other particles).

Let us consider a measurable quantity $O$, that has non-vanishing value for at
least $m$ partons in the final state. At LO accuracy the basic formula for the
differential cross section in $O$ is

$\split\frac{{\rm d}\sigma}{{\rm d}O}=\mathcal{N}\,\int\\!&{\rm
d}\phi_{m}(p_{1},\ldots,p_{m};Q)\,\frac{1}{S_{\left\\{m\right\\}}}\left|\mathcal{M}_{m}(p_{1},\ldots,p_{m})\right|^{2}\delta\big{(}O-O_{m}^{(m)}(p_{1},\ldots,p_{m})\big{)}\,,$
(28)

where $\mathcal{N}$ contains non-QCD factors (the flux factor), ${\rm
d}\phi_{m}$ is the phase space of $m$ particles, $S_{m}$ is a symmetry factor,
$\left|\mathcal{M}_{m}(p_{1},\ldots,p_{m})\right|^{2}$ is the squared matrix
element (SME), and $O_{m}^{(m)}$ is the value of $O$ computed from the $m$
final state momenta. The integration is usually done by Monte Carlo
integration and the hard part of the computation is to obtain the SME. In
these lectures we can compute hardly any SME explicitly. Fortunately, there
are freely available computer programs [15, 16, 17, 18, 19] that can be used
to check the formulae. Even more, these programs can often be used to obtain
the cross sections at LO accuracy, too.

We now use Eq. (28) to make predictions for the cross section of electron-
positron annihilation into hadrons.

###  2.1 $R$ ratio at lowest order

The leading-order (LO) perturbative contribution to the cross section
$\sigma\left(e^{+}e^{-}\rightarrow\textrm{hadrons}\right)$ is
$e^{+}e^{-}\rightarrow\,q\bar{q}$. The calculation is like in the case of
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$, supplemented with colour and fractional
electric charge of $q_{j}$. The colour diagram is a loop in the fundamental
representation which corresponds to a factor $N_{{\rm c}}$ as we have seen in
the previous chapter. While the annihilation into $\mu^{+}\mu^{-}$ contains
only one flavour in the final state, quarks can have three, four or five
flavours depending on the centre-of-mass energy.333The sixth flavour, the top
is so heavy that it cannot contribute at CM energies attained in $e^{+}e^{-}$
experiments so far. We have, therefore, to sum over all possible flavours
which can appear. The ratio of the two cross sections is thus given by

$R\,\equiv\,\frac{\sigma\left(e^{+}e^{-}\rightarrow\,q\bar{q}\right)}{\sigma\left(e^{+}e^{-}\rightarrow\,\mu^{+}\mu^{-}\right)}\,=\,\left(\sum_{q}\,e_{q}^{2}\right)\,N_{{\rm
c}},$ (29)

where $e_{u}=e_{c}=\frac{2}{3}$ and $e_{d}=e_{s}=e_{b}=-\frac{1}{3}$. If we
consider only the up, down, strange and charm quarks
$\sum_{q}e_{q}^{2}=2\frac{4}{9}+2\frac{1}{9}=\frac{10}{9}$. Considering also
the bottom quark $\sum_{q}e_{q}^{2}=\frac{11}{9}$. This step-wise increasing
behavior of the $R$-ratio was observed (see Fig. 7), providing an experimental
confirmation of the existence of 3 families of quarks and of the $SU(N_{\rm
c})$ gauge-symmetry of QCD with $N_{\rm c}=3$.

Figure 7: Experimental measurements of the $R$-ratio as a function of the
total centre-of mass energy (taken from Ref. [14]).

According to our basic assumption, pQCD cannot give predictions for the
resonances in Fig. 7. However, there is one exception, the impressive $Z$
peak. The LO prediction uses the cross section for the $e^{+}e^{-}\to
q\bar{q}$ process. A $2\to 2$ process has only a single free kinematic
variable, the scattering angle $\vartheta$. In the full SM the differential
cross section for electron-positron annihilation into a massless and
colourless fermion pair $f\bar{f}$ is obtained from the square of a single
Feynman graph, $\bigg{|}$ $\bigg{|}^{2}$ , as

$\frac{{\rm d}\sigma}{{\rm
d}\cos\vartheta}=\frac{\pi\alpha^{2}}{2s}\bigg{\\{}\big{(}1+\cos^{2}\vartheta\big{)}\bigg{[}e_{f}^{2}+\big{(}A_{e}^{2}+V_{e}^{2}\big{)}\big{(}A_{f}^{2}+V_{f}^{2}\big{)}\frac{\kappa^{2}s^{2}}{(s-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}}+\dots\bigg{]}\bigg{\\}}\,,$
(30)

where we neglected terms that vanish at centre-of-mass energy
$\sqrt{s}=M_{Z}$, or after integration. $e_{f}$, $A_{f}$ and $V_{f}$ denote
the fractional charge, axial-vector and vector electroweak couplings of the
fermions and $\kappa=\sqrt{2}G_{\rm F}M_{Z}^{2}/(16\pi\alpha_{\rm em})\simeq
0.374$ is a number. Well below the $Z$ peak the $Z$ propagator becomes
negligible and the total cross section is obtained by integrating over the
scattering angle and we find the LO prediction $\sigma_{\rm
LO}(s)=\sigma_{0}(s)e_{f}^{2}$, where
$\sigma_{0}(s)=\frac{4\pi\alpha^{2}}{3s}$. On the $Z$ peak the same
integration results in $\sigma_{\rm
LO}(M_{Z}^{2})=\sigma_{0}(M_{Z}^{2})\bigg{[}e_{f}^{2}+\big{(}A_{e}^{2}+V_{e}^{2}\big{)}\big{(}A_{f}^{2}+V_{f}^{2}\big{)}\kappa^{2}\frac{M_{Z}^{2}}{\Gamma_{Z}^{2}}\bigg{]}$.
Then we can make prediction for the hadronic $R$ ratio at LO accuracy by
simply counting the contributing final states and relating their total charge
factors to that of the muon and find $R_{\rm LO}=3\sum_{q}e_{q}^{2}$ away from
the $Z$ peak and $R_{Z,{\rm
LO}}=3\sum_{q}(A_{q}^{2}+V_{q}^{2})/(A_{\mu}^{2}+V_{\mu}^{2})$ on the $Z$
peak. The factor three is due to the three colours of quarks. Considering five
quark flavours, $m_{\rm b}<<s<<m_{\rm t}$, we find $R_{\rm LO}=11/3$ and
$R_{Z,{\rm LO}}=20.09$. We have seen on Fig. 7 that 11/3 is fairly close to
the measured value away from the $Z$ peak. The measured value of $R_{Z}$ at
LEP is $R_{Z}=20.79\pm 0.04$ [20]. The LO prediction works amazingly well. The
3.5% difference is mainly due to QCD radiation effects that we call NLO
corrections. Our next goal is to understand the origin of those corrections.

###### Exercise 2.1

Derive the result in Eq. (30) (at least below the $Z$ peak, where you consider
only photon intermediate state) and integrate it over $\vartheta$.

###### Exercise 2.2

Use Mathematica and the Package Tracer.m (or FORM) to compute the following
traces:

$\displaystyle\mathrm{Tr}\left(\slashed{p}_{2}\gamma^{\nu}(\slashed{p}_{1}-\slashed{k}_{1})\gamma^{\mu}\slashed{p}_{1}\gamma_{\mu}(\slashed{p}_{1}-\slashed{k}_{1})\gamma_{\nu}\right)$
$\displaystyle\mathrm{Tr}\left(\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}\gamma^{\mu_{4}}\gamma^{\mu_{5}}\gamma^{\mu_{6}}\gamma^{\mu_{7}}\gamma^{\mu_{8}}\gamma^{\mu_{9}}\gamma^{\mu_{10}}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}\gamma_{\mu_{4}}\gamma_{\mu_{5}}\gamma_{\mu_{6}}\gamma_{\mu_{7}}\gamma_{\mu_{8}}\gamma_{\mu_{9}}\gamma_{\mu_{10}}\right)$

###  2.2 Ultraviolet renormalization of QCD

The strong coupling is rather large as compared to the other couplings in the
SM, and as a result, the QCD radiative corrections are also large. Therefore,
it is always important to compute at least the NLO accuracy, but if possible,
even higher order corrections.444There is even a more severe reason that we
shall discuss later.

The computation of QCD radiative corrections is technically quite involved and
a good organization of the calculations is very important. Thus, first we
introduce some notation. The tensor product of the ket vectors
$\left|c_{1},\dots,c_{m}\right\rangle\otimes\left|s_{1},\dots,s_{m}\right\rangle$
denotes a basis vector in colour and helicity space,
$\left|\mathcal{A}_{m}\left(p_{1},\dots,p_{m}\right)\right\rangle$ is a state
vector of $n=m-2$ final-state particles in colour and helicity space. The
amplitude for producing $n$ final-state particles of colour
$\left(c_{1},\dots,c_{n}\right)$, spin $\left(s_{1},\dots,s_{n}\right)$,
momentum $\left(p_{1},\dots,p_{n}\right)$ is

$\mathcal{A}_{m}^{c_{1}\dots\,c_{m},s_{1}\dots\,s_{m}}\left(p_{1}\dots\,p_{m}\right)\equiv\left\langle
c_{1}\dots\,c_{m}\right|\otimes\left\langle
s_{1}\dots\,s_{m}|\mathcal{A}_{m}\left(p_{1},\dots,p_{m}\right)\right\rangle$
(31)

($m=n+2$), so

$\sum_{\textrm{colour}}\sum_{\textrm{helicity}}\left|\mathcal{A}_{m}^{{c_{1}\dots\,c_{m},s_{1}\dots\,s_{m}}}\left(\\{p_{i}\\}\right)\right|^{2}=\left\langle\mathcal{A}_{m}\left(\\{p_{i}\\}\right)|\mathcal{A}_{m}\left(\\{p_{i}\\}\right)\right\rangle.$
(32)

The loop expansion in terms of the bare coupling, the coupling that appears in
the classical Lagrangian, $g_{\rm
s}^{\left(0\right)}\,\equiv\,\sqrt{4\pi\alpha_{\rm s}^{\left(0\right)}}$ is:

$\left|\mathcal{A}_{m}\right\rangle=\left(\frac{\alpha_{\rm
s}^{\left(0\right)}\,\mu^{2\epsilon}}{4\pi}\right)^{\frac{q}{2}}\left[\left|\mathcal{A}_{m}^{\left(0\right)}\right\rangle+\left(\frac{\alpha_{\rm
s}^{\left(0\right)}\,\mu^{2\epsilon}}{4\pi}\right)\left|\mathcal{A}_{m}^{\left(1\right)}\right\rangle+\mathcal{O}\left((\alpha_{\rm
s}^{\left(0\right)})\right)^{2}\right]\,,$ (33)

where $q\,\in\,\mathbb{N}$, $\mu$ is the dimensional regularization scale,
introduced to keep $\alpha_{\rm s}^{(0)}$ dimensionless in $d\,=\,4-2\epsilon$
dimensions. The exponent $\frac{q}{2}$ in the prefactor takes account of the
power of $\alpha_{\rm s}$ at LO, the loop-expansion is an expansion in the
strong coupling $\alpha_{\rm s}$. For instance, $q=0$ for $e^{+}e^{-}\to
q\bar{q}$, while $q=1$ for $e^{+}e^{-}\to q\bar{q}g$. The tree amplitude
$\left|\mathcal{A}_{m}^{(0)}\right\rangle$ is finite, while the one-loop
correction $\left|\mathcal{A}_{m}^{\left(1\right)}\right\rangle$ is divergent
in $d=4$ dimensions, which is manifest in terms of $1/\epsilon^{2}$ and
$1/\epsilon$ poles if dimensional reglarization is used. These poles have both
ultraviolet (UV) and infrared (IR) origin.

The UV poles can be removed by multiplicative redefinition of the fields and
parameters in the Lagrangian, systematically order by order in PT. This is a
hard task even at one loop, but presently known up to four loops [21] – a
truly remarkable computation! It turns out that when computing scattering
amplitudes in massless QCD at one-loop accuracy, the renormalization amounts
to the simple substitution

$\alpha_{\rm s}^{\left(0\right)}\mu^{2\epsilon}\,\longrightarrow\,\alpha_{\rm
s}\left(\mu_{{\rm R}}^{2}\right)\mu_{{\rm
R}}^{2\epsilon}S_{\epsilon}^{-1}\left[1-\frac{\alpha_{\rm s}\left(\mu_{{\rm
R}}^{2}\right)}{4\pi}\,\frac{\beta_{0}}{\epsilon}+\mathcal{O}\left(\alpha_{\rm
s}^{2}\right)\right]\,,$ (34)

with
$S_{\epsilon}=\frac{\left(4\pi\right)^{\epsilon}}{\Gamma\left(1-\epsilon\right)}$.
Note that on the left of this substitution $\mu$ is the dimensional
regularization scale to keep $\alpha_{\rm s}^{\left(0\right)}$ dimensionless,
while on the right $\mu_{{\rm R}}$ is the renormalization scale. We discussed
in Sect. 1.8 when we extract $\alpha_{\rm s}$ from measurements, we have to
define $\mu_{{\rm R}}$. The dimensional regularization scale turns into the
renormalization scale through the substitution (34).

Why does the substitution (34) work? Each Feynman graph consists of vertices
with propagators connecting those and external lines. Moreover,

  * $\bullet$

each vertex receives a factor $Z_{g}$ (or $Z_{g}^{2}$ for quartic vertex) and
factors of $\sqrt{Z_{i}}$, ($i=q,\,A$) for each field connected to the vertex,

  * $\bullet$

each propagator of field $i$ receives a factor of $Z_{i}^{-1}$,

  * $\bullet$

each external leg of field $i$ receives a factor of $Z_{i}^{-\frac{1}{2}}$.

Thus the renormalization field factors cancel from each graph and only the
charge renormalization ($Z_{g}$) is needed in practice! This can be seen as a
consequence of the fact that in massless QCD the only free parameter besides
the gauge-fixing parameter $\lambda$ is $\alpha_{\rm s}$. The scattering
amplitudes are physical, and any physical quantity has to be independent of
$\lambda$, so the only remaining parameter, which the amplitudes may depend
on, is the coupling. The renormalization factor $Z_{g}$ is most easily
computed in background field gauge, defined by

$\mathcal{L}_{\rm
GF}=-\frac{1}{2\lambda}\sum_{a}\left(\partial^{\mu}A_{\mu}^{a}+g\,f^{abc}\,\mathcal{A}_{\mu}^{b}A^{\mu\,c}\right)^{2},$
(35)

where $\mathcal{A}_{\mu}^{b}$ is a background field and $A_{\mu}^{c}$
describes the quantum fluctuations on this background. It can be shown [11]
that in this gauge the field and coupling renormalization factors are related
by the Ward identity $Z_{A}^{-\frac{1}{2}}=Z_{g}$, and $Z_{A}$ can be computed
from loop insertions into the propagator shown in (21).

The simple substitution rule (34) for the coupling leads to a simple shift in
the amplitude. As

$\left[1-\frac{\alpha_{\rm s}\big{(}\mu_{\rm
R}^{2}\big{)}}{4\pi}\,\frac{\beta_{0}}{\epsilon}\right]^{\frac{q}{2}}=1-{\frac{q}{2}}\frac{\alpha_{\rm
s}\big{(}\mu_{\rm
R}^{2}\big{)}}{4\pi}\,\frac{\beta_{0}}{\epsilon}+O(\alpha_{\rm s}^{2})\,$

we obtain for the renormalized amplitude $\big{|}\mathcal{M}_{m}\big{\rangle}$

$\split\big{|}\mathcal{M}_{m}\rangle&=\left(\frac{\alpha_{\rm s}\left(\mu_{\rm
R}^{2}\right)\mu_{\rm
R}^{2\epsilon}}{4\pi}\,S_{\epsilon}^{-1}\right)^{\frac{q}{2}}\\!\left(\big{|}\mathcal{M}_{m}^{\left(0\right)}\rangle+\frac{\alpha_{\rm
s}\left(\mu_{\rm
R}^{2}\right)}{4\pi}\,S_{\epsilon}^{-1}\big{|}\mathcal{M}_{m}^{\left(1\right)}\rangle\right)\quad
q\in\mathbb{N},\\\
\big{|}\mathcal{M}_{m}^{(0)}\big{\rangle}=\big{|}\mathcal{A}_{m}^{(0)}\big{\rangle},\quad\big{|}\mathcal{M}_{m}^{(1)}\big{\rangle}=\mu_{\rm
R}^{2\epsilon}\big{|}\mathcal{A}_{m}^{(1)}\big{\rangle}-\frac{q}{2}\,\frac{\beta_{0}}{\epsilon}\,S_{\epsilon}\big{|}\mathcal{A}_{m}^{(0)}\big{\rangle}\,.$
(36)

The renormalized theory is UV finite, yet
$\left|\mathcal{M}_{m}^{\left(1\right)}\right\rangle$ is still infinite in
$d=4$ dimensions, as it is divergent also in the infrared. After UV
renormalization is achieved we can use dimensional regularization to regulate
the amplitudes in the IR by continuing into $d>4$ ($\epsilon<0$). The
integrals that are scaleless in $d=4$ have $\left(q^{2}\right)^{-\epsilon}$
mass dimension in $d=4-2\epsilon$ dimensions. Therefore, in the massless limit
all integrals can depend only on momentum invariants raised to a positive
fractional power $\left(\epsilon\,<\,0\right)$. We conclude that when all
external invariants vanish, the continued integral must also vanish
(“scaleless integrals vanish in dimensional regularization”).

For IR-safe observables these IR poles vanish and we can set $d=4$ at the end
of the computations, and we obtain the UV finite, IR regularized SME that can
be used to compute cross sections.

###### Exercise 2.3

Compute the contribution to the beta function from the fermion loop:

  1. 1.

Write down carefully the amplitude and compute the trace.

  2. 2.

The following types of integrals occur:

$I_{2}^{\mu}=\int\\!\frac{{\rm
d}^{d}\ell}{(2\pi)^{d}}\frac{\ell^{\mu}}{\ell^{2}\left(\ell-p\right)^{2}}\,,\qquad
I_{2}^{\mu\nu}=\int\\!\frac{{\rm
d}^{d}\ell}{(2\pi)^{d}}\frac{\ell^{\mu}\ell^{\nu}}{\ell^{2}\left(\ell-p\right)^{2}}$
(37)

Express these as linear combination of

$I_{2}(p)=\int\\!\frac{{\rm
d}^{d}\ell}{(2\pi)^{d}}\frac{1}{\ell^{2}\left(\ell-p\right)^{2}}\,.$ (38)

  3. 3.

Obtain $I_{2}$ from

$I_{2}(p,m)=\int\\!\frac{{\rm
d}^{d}l}{\left(2\pi\right)^{d}}\,\frac{1}{\left[\left(l-p\right)^{2}-m^{2}\right]l^{2}}=\frac{{\rm
i}}{\left(4\pi\right)^{2-\epsilon}}\Gamma\left(\epsilon\right)\left(p^{2}\right)^{-\epsilon}\int_{0}^{1}\\!{\rm
d}x\left(x\frac{m^{2}}{p^{2}}-x\left(1-x\right)-{\rm
i}\varepsilon\right)^{-\epsilon}$ (39)

and find the divergent pieces.

The contribution to $\beta_{0}$ is the coefficient of the $1/\epsilon$ pole
without the coupling factor.

###  2.3 $R$ ratio at NLO accuracy

This is by far the simplest example of computing QCD radiative corrections. As
we saw in Sect. 2.1 it requires the total hadronic cross section that depends
only on a single kinematic invariant, the total centre-of-mass energy
$\sqrt{s}$. As a result, the emerging integrals in this computation can be
evaluated exactly. Nevertheless, the complete computation is still too
lengthy, and we shall be able to present the main step and filling the details
is left to the student.

There are two kinds of corrections that contribute at NLO accuracy. One is the
real correction, with an additional gluon in the final state, so the SME is
computed from Feynman graphs as

$\big{\langle}\mathcal{M}_{3}^{(0)}\big{|}\mathcal{M}_{3}^{(0)}\big{\rangle}=\bigg{|}$
$\bigg{|}^{2}$ , which gives an $\mathcal{O}(\alpha_{\rm s})$ correction. The
other kind of contribution

is the virtual correction, with an additional gluon providing a loop in the
final state,

$\big{\langle}\mathcal{M}_{2}^{(1)}\big{|}\mathcal{M}_{2}^{(0)}\big{\rangle}+\big{\langle}\mathcal{M}_{2}^{(0)}\big{|}\mathcal{M}_{2}^{(1)}\big{\rangle}=2{\rm R}{\rm e}\bigg{\langle}$ | $\bigg{\rangle}$ .

The real correction has three particles in the final state. The three-particle
phase space has five independent variables: two energies and three angles. As
we are looking for the total cross section, we integrate over the angles and
use $y_{ij}=(p_{i}+p_{j})^{2}/s=2p_{i}\cdot p_{j}/s$ scaled two-particle
invariants to write both the phase space and the SME. Momentum conservation
implies $1=(p_{1}+p_{2}+p_{3})^{2}/s=y_{12}+y_{13}+y_{23}$. The complete real
contribution to the total cross section is

$\sigma^{\rm R}=\sigma_{0}R_{0}\int_{0}^{1}\\!{\rm d}y_{13}\int_{0}^{1}\\!{\rm
d}y_{23}\,C_{\rm F}\,\frac{\alpha_{\rm
s}}{2\pi}\left(\frac{y_{23}}{y_{13}}+\frac{y_{13}}{y_{23}}+\frac{2y_{12}}{y_{13}y_{23}}\right)\Theta(1-y_{13}-y_{23})\,.$
(40)

Figure 8: Region of integration for real correction

This integral is divergent along the boundaries at $y_{13}=0$, $y_{23}=0$ as
well as in the point $y_{13}y_{23}=0$, so the singularities are in the IR
parts of the phase space. As $y_{i3}s=2E_{i}E_{3}(1-\cos\vartheta_{i3})$, the
divergence occurs either when $E_{3}\to 0$, which is called soft-gluon
singularity, or when $\vartheta_{i3}\to 0$, which is called collinear
singularity (the gluon is collinear to either of the quarks). The region of
integration with the singular places is shown in Fig. 8.

To make sense of the integral, we use dimensional regularization, which
amounts to the computation of the phase space and the SME in $d=4-2\epsilon$
dimensions. The result is

$\displaystyle\sigma^{\rm R}(\epsilon)\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=\sigma_{0}R_{0}H(\epsilon)\int_{0}^{1}\\!\frac{{\rm
d}y_{13}}{y_{13}^{\epsilon}}\int_{0}^{1}\\!\frac{{\rm
d}y_{23}}{y_{23}^{\epsilon}}\Theta(1-y_{13}-y_{23})$
$\displaystyle\qquad\qquad\times\,C_{\rm F}\frac{\alpha_{\rm
s}}{2\pi}\left[(1-\epsilon)\left(\frac{y_{23}}{y_{13}}+\frac{y_{13}}{y_{23}}\right)+\frac{2y_{12}}{y_{13}y_{23}}-2\epsilon\right]\,,$

where $H(\epsilon)=1+\mathcal{O}(\epsilon)$ (the exact form of this function
will turn out to be irrelevant). The integrals can be evaluated exactly, but
actually the Laurent-expansion around $\epsilon=0$ is sufficient,

$\sigma^{\rm R}(\epsilon)=\sigma_{0}R_{0}H(\epsilon)\,C_{\rm
F}\,\frac{\alpha_{\rm
s}}{2\pi}\left[\frac{2}{\epsilon^{2}}+\frac{3}{\epsilon}+\frac{19}{2}-\pi^{2}+\mathcal{O}(\epsilon)\right]\,.$
(42)

The computation of the virtual correction is even more cumbersome due to the
loop integral. We present only the result:

$\sigma^{\rm V}(\epsilon)=\sigma_{0}R_{0}H(\epsilon)\,C_{\rm
F}\,\frac{\alpha_{\rm
s}}{2\pi}\left[-\frac{2}{\epsilon^{2}}-\frac{3}{\epsilon}-8+\pi^{2}+\mathcal{O}(\epsilon)\right]\,,$
(43)

We now see that the sum of the real and virtual contribution is finite in
$d=4$, so for the sum we can set $\epsilon\to 0$ and find the famous
$\alpha_{\rm s}/\pi\simeq 0.037$ correction:
$R=R_{0}\big{(}1+\frac{\alpha_{\rm s}}{\pi}+\mathcal{O}(\alpha_{\rm
s}^{2})\big{)}$. The correction is the same for $R_{Z}$.

Actually there is a much easier way of computing the radiative corrections to
the total cross section from the imaginary part of the hadronic vacuum
polarization, using the optical theorem ($\sigma\propto{\rm
Im}f(\gamma\to\gamma))$). The state of the art is $R$ at
$\mathcal{O}(\alpha_{\rm s}^{4})$ [22]. The result of the computation at next-
to-next-to-leading order (NNLO) accuracy,

$\split\frac{R}{R_{0}}=1+c_{1}\alpha_{\rm
s}(\mu)&+\left[c_{2}+c_{1}b_{0}\ln\frac{\mu}{Q^{2}}\right]\alpha_{\rm
s}(\mu)^{2}\\\
+\left[c_{3}+\left(2c_{2}b_{0}+c_{1}b_{1}+c_{1}b_{0}^{2}\ln\frac{\mu^{2}}{Q^{2}}\right)\ln\frac{\mu^{2}}{Q^{2}}\right]\alpha_{\rm
s}(\mu)^{3}+\mathcal{O}(\alpha_{\rm s}^{4})$ (44)

satisfies the RGE to order $\alpha_{\rm s}(\mu)^{4}$. The coefficients
$c_{1}=1/\pi$, $c_{2}=1.409/\pi^{2}$, $c_{3}=-12.85/\pi^{3}$ suggest that the
perturbation series is convergent. Our more complicated way of computing $R$
is instructive for our studies in the next section.

Figure 9: Dependence on the renormalization scale of the hadronic ratio on the
$Z$ pole

The predictions at the first three fixed orders in PT for $R_{Z}$ are shown in
Fig. 9. The $R$ ratio at LO accuracy does not depend on the strong coupling,
hence it is independent of the scale. The figure is meant to show the general
pattern of QCD predictions which, with the exception of the $R$ ratio, depend
on the scale already at LO. The NLO curve shows the typical feature of LO
predictions: it depends on the renormalization scale in a monotonically
decreasing way. As this scale is unphysical, in principle, its value can be
arbitrary. Thus the prediction at LO is in general only an order of magnitude
indication of the cross section, but not a precision result. (In the case of
the hadronic ratio the QCD corrections are actually quite small as compared to
many other QCD cross sections and the precision is actually better than
usual.) As a result, if we want to make reliable predictions in pQCD, the NLO
accuracy (NNLO for $R$) is absolutely necessary unless we have some way to fix
the scale.

However, there is no theorem that tells us the proper scale choice. The usual
practice is to set the scale at a characteristic physical scale of the
process. A reasonable assumption that the strength of the QCD interaction for
a process involving a momentum transfer $Q$ is given by $\alpha_{\rm s}(Q)$,
so $\mu=Q$ is the proper scale choice, to minimize logarithmic contributions
$\ln(\mu/Q)$ in higher-order terms. For instance, in case of electron-positron
annihilation the total centre-of-mass energy is the usual choice, while for a
jet cross section in proton-proton collisions the transverse momentum of the
jet555We discuss jets in the next section. is used. The application of this
recipe appears clear as long as there is only one hard scale in the process.
In the state of the art computations there are complex processes with several
scales and it is not obvious which one to choose. For instance, in vector
boson hadroproduction in association with $m$ jets ($m\leq 5$) [23], in
addition to the transverse momentum of the vector boson
$E_{V,\perp}=\sqrt{p_{V,x}^{2}+p_{V,y}^{2}}$ there are the transverse momenta
of the jets. In this example, the choice $\mu=E_{\perp}^{V}$ was found to
result in a badly behaving perturbation series with corrections driving the
$E_{\perp}$ distribution of the second hardest jet at NLO accuracy even
unphysically negative for $m=3$ and $E_{\perp}>475$ GeV at the LHC. Choosing a
dynamical scale, set event by event, appears a better choice. For instance,
half the total transverse energy of the final-state particles (both QCD
partons and leptons from the decay of the vector boson), $\mu=\hat{H}_{\rm
T}/2$, leads to much milder scale dependence and a similar shape of the
distributions at LO and NLO accuracies.

There are suggestions on making educated guesses for the best scale. Among
those are the principle of fastest apparent convergence (FAC), that of minimal
sensitivity (PMS), or the BLM scale choice [24, 25, 26], beyond the scope of
these lectures. The experience is that in hadron collisions there is no choice
that works well for any process and it is best to choose a dynamical scale
chosen by examining the process.

As there is no unique scale, the standard procedure is to choose a default
scale $\mu_{0}$, related to the typical momentum transfer in the process, and
to assign a theoretical uncertainty by varying the scale within a certain
range around the default choice $\mu_{0}$. The usual range is between half and
twice the default choice. However, this is again an indication only of the
scale uncertainties and there is no mathematical theorem that states this
procedure yields the true theoretical uncertainty due to neglected higher
order terms. In order to have a measure on the effect of neglected higher
orders, to understand the reliability of the assigned theoretical uncertainty
one has to compute the NNLO corrections. The latter are very demanding
computations both technically and numerically and predictions at NNLO accuracy
for some fairly simple processes, with one or two final-state particles in the
prediction at LO, constitute the state of the art of pQCD.

###### Exercise 2.4

Show that the $d$-dimensional three-particle phase space for $q\to
p_{1}+p_{2}+p_{3}$ can be expressed in terms of the Lorentz-invariants
$s_{ij}=(p_{i}+p_{j})^{2}$

${\rm d}\phi_{3}=(2\pi)^{3-2d}\,2^{-1-d}\,(q^{2})^{\frac{2-d}{2}}\,{\rm
d}^{d-2}\Omega{\rm d}^{d-3}\Omega\,\\\
\left(s_{12}s_{13}s_{23}\right)^{\frac{d-4}{2}}\,{\rm d}s_{12}\,{\rm
d}s_{13}\,{\rm d}s_{23}\,\delta\left(q^{2}-s_{12}-s_{13}-s_{23}\right)\;.$

where ${\rm d}^{d}\Omega$ is the measure of the hypersurface element in $d$
dimensions, $\int\\!{\rm d}^{d-3}\Omega=\Omega_{d}=2\pi^{d/2}/\Gamma(d/2)$.
Hints:

  1. 1.

The $d$-dimensional volume measure in spherical coordinates is recurisvely
given by

${\rm d}^{d+1}p=E\,{\rm d}E\,d^{\rm d}\,p_{E}=E\,{\rm d}\,E\,E^{d-1}{\rm
d}^{d}\Omega\\\ \,,\qquad{\rm d}^{d}\Omega=(\sin\theta_{1})^{d-1}{\rm
d}\theta_{1}{\rm d}^{d-1}\Omega\,.$

  2. 2.

Show that

$\sin^{2}\theta_{1}=\frac{1}{4}\frac{s_{12}\,s_{13}\,s_{23}}{q^{2}\,E_{1}^{2}\,E_{2}^{2}}\,,$

where $\theta_{1}$ is the angle between $p_{1}$ and $p_{2}$.

###### Exercise 2.5

Let $y_{ij}=\frac{s_{ij}}{q^{2}}$. Using the previous exercise, compute the
real correction to the process $e^{+}\,e^{-}\,\rightarrow\,q\bar{q}$ given in
Eq. (42). Hint: Transform the triangular integration region into the unit
square and evaluate the $B$ (Euler $\beta$) functions.

##  3 Jet cross sections

In the first two sections we established our theoretical playground to make
predictions for hadronic cross sections. Based on RGE analysis we showed that
PT can only be fully consistent in an asymptotically free QFT, like QCD. We
found that predictions can be made only for those quantities that remain
finite in the limit of vanishing masses of light quarks. We computed the
radiative corrections for such a quantity, the total hadronic cross section in
electron-positron annihilation. We found that at intermediate steps of the
computations there are singular contributions of two types: of UV and IR
origin. The UV singularities can be removed by renormalization, and the
remaining IR ones can be regularized in dimensional regularization where IR
singularities appear as $1/\epsilon$ poles. When adding all contributions,
these poles cancel and we obtain the finite correction after setting
$\epsilon=0$. Our question in this section is whether there are more exclusive
observables than the totally inclusive one for which this procedure can be
applied.

Figure 10: Two events observed in the OPAL detector

It is clear from experiments that typical final states have structures. For
instance, Fig. 10 shows two events, one with two and the other with three
sprays of hadrons, called hadron jets. If we count the relative number of
events with two, three, four jets, an interesting pattern emerges:

# of events with 2 jets : # of events with 3 jets : # of events with 4 jets
$\simeq\mathcal{O}(\alpha_{\rm s}^{0})$ : $\mathcal{O}(\alpha_{\rm s}^{1})$ :
$\mathcal{O}(\alpha_{\rm s}^{2})$.

Recalling our basic assumption and Fig. 3 we find that jets reflect the
partonic structure of the events. We now use our pQCD formalism to describe
these structures theoretically. For this purpose, we need a function of the
final state momenta $J_{m}(\\{p_{i}\\})$ that quantifies the structure of the
final state in some ways (we give examples below). This function is called jet
function.

Let us consider again the process $e^{+}e^{-}\to$ hadrons. If we are not
interested in the orientation of the final state events, we can average over
the orientation and find that the SME $|\mathcal{M}_{2}|^{2}$ has no
dependence on the parton momenta. Then the two-particle phase space is ${\rm
d}\phi_{2}={\rm d}y_{12}\delta(1-y_{12})$, and contribution of the process
$e^{+}e^{-}\to q\bar{q}$ to the cross section is

$\sigma_{\rm LO}=|\mathcal{M}_{2}|^{2}\int_{0}^{1}\\!{\rm
d}y_{12}\delta(1-y_{12})J_{2}(p_{1},p_{2})\,,$ (45)

which sets our normalization of $|\mathcal{M}_{2}|^{2}$. The two kinds of NLO
corrections are

$\split{\rm d}\sigma^{\rm R}&=|\mathcal{M}_{2}|^{2}S_{\epsilon}\frac{{\rm
d}y_{13}}{y_{13}^{\epsilon}}\frac{{\rm d}y_{23}}{y_{23}^{\epsilon}}\,C_{\rm
F}\frac{\alpha_{\rm
s}}{2\pi}\left[(1-\epsilon)\left(\frac{y_{23}}{y_{13}}+\frac{y_{13}}{y_{23}}\right)+\frac{2y_{12}}{y_{13}y_{23}}-2\epsilon\right]J_{3}(p_{1},p_{2},p_{3})\,,\\\
{\rm d}\sigma^{\rm V}=|\mathcal{M}_{2}|^{2}S_{\epsilon}\,C_{\rm
F}\frac{\alpha_{\rm
s}}{2\pi}\left(\frac{\mu^{2}}{s}\right)^{\epsilon}\left[-\frac{2}{\epsilon^{2}}-\frac{3}{\epsilon}-8+\pi^{2}+\mathcal{O}(\epsilon)\right]{\rm
d}y_{12}\delta(1-y_{12})J_{2}(p_{1},p_{2})\,.$ (46)

Contrary to the case of the total cross section, where $J_{m}=1$, we cannot
simply perform the integration analytically and combine the results, neither
we can combine the integrands. The general method to deal with this problem is
to regularize both with a properly chosen subtraction,

${\rm d}\sigma_{3}^{\rm NLO}={\rm d}\sigma^{\rm R}J_{3}-{\rm d}\sigma^{\rm
A}J_{2}\,,\quad\mbox{and}\quad{\rm d}\sigma_{2}^{\rm NLO}=\big{(}{\rm
d}\sigma^{\rm V}+{\rm d}\sigma^{\rm A}\big{)}J_{2}\,,$

such that both terms are separately integrable in $d=4$ dimensions. This
requires a special property of the jet function $J_{n}$, called IR safety,
expressed analytically as

$\lim_{y_{13},y_{23}\to 0}J_{3}=J_{2}\,.$ (47)

Qualitatively IR safety means that the jet function is insensitive to an
additional soft particle, or to a collinear splitting in the final state.

How can we construct such an approximate cross section? For this simple
process we can follow the steps:

$\split\frac{y_{23}}{y_{13}}+\frac{y_{13}}{y_{23}}+\frac{2y_{12}}{y_{13}y_{23}}=\frac{y_{23}}{y_{13}}+\frac{1}{y_{13}}&\frac{2y_{12}}{y_{13}+y_{23}}+(1\leftrightarrow
2)\\\
=\frac{1}{y_{13}}\bigg{[}y_{23}+\bigg{(}2\frac{\overbrace{y_{12}+y_{13}+y_{23}}^{=1}}{y_{13}+y_{23}}-2\bigg{)}\bigg{]}+(1\leftrightarrow
2)\,.$ (48)

Then introduce the new variable $z_{1}\equiv
z_{1,2}=\frac{y_{12}}{y_{12}+y_{23}}$, so that
$y_{13}+y_{23}=1-y_{12}=1-z_{1}(1-y_{13})$ and

$\frac{y_{23}}{y_{13}}-\frac{1}{y_{13}}=\frac{y_{23}(1-y_{13})-y_{12}-y_{23}}{y_{13}(y_{12}+y_{23})}=\frac{-y_{23}y_{13}-y_{12}}{y_{13}(y_{12}+y_{23})}=-\frac{y_{23}}{y_{12}+y_{23}}-\frac{z_{1}}{y_{13}}\,,$

and substitute these into Eq. (48):

$\frac{y_{23}}{y_{13}}+\frac{y_{13}}{y_{23}}+\frac{2y_{12}}{y_{13}y_{23}}=\bigg{[}\frac{1}{y_{13}}\left(\frac{2}{1-z_{1}(1-y_{13})}-1-z_{1}\right)-\frac{y_{23}}{y_{12}+y_{23}}\bigg{]}+(1\leftrightarrow
2)\,.$ (49)

The term $y_{23}/{y_{12}+y_{23}}$ never becomes infinite, thus the approximate
cross section

${\rm d}\sigma^{\rm A}=|\mathcal{M}_{2}|^{2}S_{\epsilon}\frac{{\rm
d}y_{13}}{y_{13}^{\epsilon}}\frac{{\rm d}y_{23}}{y_{23}^{\epsilon}}\,C_{\rm
F}(V_{13,2}+V_{23,1})\,,\quad\mbox{with}$ (50) $\quad
V_{ij,k}=\frac{\alpha_{\rm
s}}{2\pi}\bigg{[}\frac{1}{y_{ij}}\left(\frac{2}{1-z_{i,k}(1-y_{ij})}-1-z_{i,k}\right)-\epsilon(1-z_{i,k})\bigg{]}$
(51)

is a proper subtraction term that regularizes the real contribution in all of
its singular limits in $d$ dimensions. Consequently, the difference ${\rm
d}\sigma_{3}^{\rm NLO}={\rm d}\sigma^{\rm R}J_{3}-{\rm d}\sigma^{\rm A}J_{2}$
can be integrated in any dimensions, in particular, we can set $\epsilon=0$
and integrate in $d=4$ numerically.

To obtain ${\rm d}\sigma_{2}^{\rm NLO}$ we integrate the two terms separately.
For $V_{13,2}$ we change variables in the phase space to $y_{13}$ and $z_{1}$,
and find

${\rm d}\sigma^{\rm
A}=|\mathcal{M}_{2}|^{2}S_{\epsilon}\left(\frac{\mu^{2}}{s}\right)^{\epsilon}\int_{0}^{1}\\!{\rm
d}y_{13}\int_{0}^{1}\\!{\rm
d}z_{1}y_{13}^{-\epsilon}(1-y_{13})^{1-2\epsilon}\,C_{\rm
F}V_{13,2}+(1\leftrightarrow 2)\,.$ (52)

We shall see that this factorization of the singular terms is universal. We
can now perform the integration over the factorized one-particle phase space,
independently of the jet function, and obtain the integrated subtraction term
in the form ${\rm d}\sigma^{\rm A}=|\mathcal{M}_{2}|^{2}\bm{I}(\epsilon)$ with
insertion operator

$\bm{I}(\epsilon)=C_{\rm F}\frac{\alpha_{\rm
s}}{2\pi}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{s}\right)^{\epsilon}\left[\frac{2}{\epsilon^{2}}+\frac{3}{\epsilon}+10-\frac{\pi^{2}}{3}+\mathcal{O}(\epsilon)\right]\,.$
(53)

Comparing this integrated subtraction to Eq. (46), we see that the sum ${\rm
d}\sigma_{2}^{\rm NLO}=\big{(}{\rm d}\sigma^{\rm V}+{\rm d}\sigma^{\rm
A}\big{)}J_{2}$ is finite if $\epsilon=0$,

$\sigma_{2}^{\rm NLO}=|\mathcal{M}_{2}|^{2}C_{\rm F}\frac{\alpha_{\rm
s}}{\pi}\left(1+\frac{\pi^{2}}{3}\right)\int_{0}^{1}\\!{\rm
d}y_{12}\delta(1-y_{12})J_{2}(p_{1},p_{2})+\mathcal{O}(\epsilon)\,,$ (54)

and so can be integrated in $d=4$ dimensions.

###  3.1 Infrared safety

A natural question is if we can construct the approximate cross section
universally, independently of the process and observable. Our presentation
above suggests the affirmative answer. To understand how, we have to study the
origin of the singular behaviour in the SME. This singularity arises from
propagator factors that diverge
$\propto\frac{1}{(p_{i}+p_{s})^{2}}=\frac{1}{2\,p_{i}\cdot
p_{s}}=\frac{1}{2E_{i}E_{s}(1-\cos\theta)}\simeq\frac{1}{E_{i}E_{s}\theta^{2}}$

In the collinear limit, $\theta\,\rightarrow\,0$ and
$\mathcal{M}_{m+1}\simeq\mathcal{M}_{m}/{\theta}+$ less singular terms (a
factor of $\theta$ appears in the numerator factors). In the soft limit,
$E_{s}\,\rightarrow\,0$ and $\mathcal{M}_{m+1}\simeq\mathcal{M}_{m}/{E_{s}}+$
less singular terms. The gluon phase space is

$\frac{{\rm d}^{3}p_{s}}{2E_{s}}=\frac{1}{2}E_{s}\,{\rm d}E_{s}\,{\rm
d}\\!\cos\theta\,{\rm d}\phi\simeq\frac{1}{4}E_{s}\,{\rm d}E_{s}\,{\rm
d}\theta^{2}\,{\rm d}\phi\,,$

so in the cross section we find logarithmic singularities in both the soft and
the collinear limits: $\frac{{\rm d}E_{s}}{E_{s}}$ or $\frac{{\rm
d}\theta^{2}}{\theta^{2}}$. These are the IR singular limits. In dimensional
regularization the logarithmic singularities appear as poles:

$\int\\!{\rm d}y_{is}\,y_{is}^{-1-\epsilon}\,=\,-\frac{1}{\epsilon}\,.$

Thus, the singular behaviour arises at kinematically degenerate phase space
configurations, which at the NLO accuracy means that one cannot distinguish
the following configurations: (i) a single hard parton, (ii) the single parton
splitting into two nearly collinear partons, (iii) the single parton emitting
a soft gluon (on-shell gluon with very small energy). Then an answer to the
question posed at the beginning of Sect. 3 is given by the Kinoshita-Lee-
Nauenberg (KLN) theorem [27, 28]:

> _In massless, renormalized field theory in four dimensions, transition rates
> are IR safe if summation over kinematically degenerate initial and final
> states is carried out._

For the $e^{+}e^{-}\rightarrow$ hadrons process, the initial state is free of
IR singularities. Typical IR-safe quantities are (i) event shape variables and
(ii) jet cross sections.

###  3.2 Event shapes

Figure 11: Distribution of thrust as measured at LEP compared to pQCD
predictions obtained with vector and scalar gluon

Thrust, thrust major/minor, C- and D-parameters, oblateness, sphericity,
aplanarity, jet masses, jet-broadening, energy-energy correlation,
differential jet rates are examples of event shape variables. The value of an
event shape does not change if a final-state particle further splits into two
collinear particles, or emits a soft gluon, hence it is (qualitatively) IR
safe. As as example we consider the thrust $T$, which is defined by

$T=\max_{\vec{n}}\,\frac{\sum_{i=1}^{m}\,\left|\vec{p}_{i}\,\\!\cdot\\!\,\vec{n}\right|}{\sum_{i=1}^{m}\left|\vec{p}_{i}\right|},$
(55)

where $\vec{n}$ is a three-vector (the direction of the thrust axis) such that
$T$ is maximal. The particle three-momenta $\vec{p}_{i}$ are defined in the
$e^{+}e^{-}$ centre-of-mass frame. $T$ is an example of the jet function
$J_{m}$. It is infrared safe because neither $p_{j}\to 0$, nor replacing
$p_{i}$ with $zp_{i}+(1-z)p_{i}$ change $T$. At LO accuracy it is possible to
perform the phase space integrations and

$\frac{1}{\sigma}\frac{{\rm d}\sigma}{{\rm d}T}=C_{\rm F}\frac{\alpha_{{\rm
s}}}{2\pi}\left[\frac{2\big{(}3T^{2}-3T+2\big{)}}{T(1-T)}\ln\left(\frac{2T-1}{1-T}\right)-3(3T-2)\frac{2-T}{1-T}\right]\,.$
(56)

We see that the perturbative prediction for the thrust distribution is
singular at $T=1$. In addition to the linear divergence in $1-T$ there is
logarithmic divergence, too. The latter is characteristic to events shape
distributions. In PT at $n$th order logarithms of $1-T$ in the form
$\alpha_{\rm s}^{n}\ln^{m}(1/(1-T))\,,m\leq 2n$ appear. These spoil the
convergence of the perturbation series and call for resummation if we want to
make reliable prediction near the edge of the phase space, for large values of
$T$ where the best experimental statistics are available. Resummations of
leading ($m=2n$) and next-to-leading ($m=2n-1$) logarithms are available for
many event shape variables, but the discussion of this technique is beyond the
scope of these lectures.

###### Exercise 3.1

Verify that $T$ as defined in Eq. (55) is infrared und collinear safe. What is
the range of values that $T$ can take if (i) there are only two particles in
the final state, or (ii) $m\rightarrow\infty$ and all $\vec{p}_{i}$ are
distributed spherically?

###  3.3 Jet algorithms

Jets are sprays of energetic, on-shell, nearly collinear hadrons. The number
of jets does not change if a final-state particle further splits into two
collinear particles, or emits a soft gluon, hence it is again qualitatively IR
safe. To quantify the jet-like structure of the final states jet algorithms
have been invented. These have a long history with rather slow convergence.
The reason is that the experimental and theoretical requirements posed to a
jet algorithm are rather different. Experimentally we need cones that include
almost all hadron tracks at cheap computational price. Theoretically the
important requirements are IR safety, so that PT can be employed to make
predictions and resummability, so that we can make predictions in those region
of the phase space where most of the data appear.

For many years experimenters preferred cone jet algorithms (according to the
‘Snowmass accord’) [29]. These start from a cone seed (centre of the cone) in
pseudorapidity ($\eta$) and azimuthal-angle ($\phi$) plane:
($\eta_{c},\phi_{c}$). We define a distance of a hadron track $i$ from the
seed by $d_{ic}=\sqrt{(\eta_{i}-\eta^{c})^{2}+(\phi_{i}-\phi^{c})^{2}}$. A
track belongs to the cone if $d_{ic}<R$, with a predefined value for $R$
(usually 0.7). It turned out however, that (i) this is an IR unsafe definition
and (ii) there is a problem how to treat overlapping cones, so the cone jet
definition has been abandoned.

Theoreticians prefer iterative jet algorithms, consisting of the following
steps. (i) First we define a distance between two momenta (of partons or
hadron tracks) and a rule to combine two momenta, $p_{i}$ and $p_{j}$ into
$p_{(ij)}$. (ii) Then we select a value for resolution $d_{\rm cut}$ and
consider all pairs of momenta. If the minimum of $\\{d_{ij}\\}$ is smaller
than $d_{\rm cut}$, then we combine the momenta $p_{i}$ and $p_{j}$ and start
the algorithm again. If the minimum is larger than $d_{\rm cut}$, then the
remaining momenta (after the combinations) are considered the momenta of the
jets, and the algorithm stops. The drawback of this algorithm is that it
becomes very expensive computationally for many particles in the final state.
This is not an issue in pQCD computations because according to our basic
assumption there are only few partons, but a major problem for the final
states in the detectors where hundreds of hadrons may appear in a single
event.

At LEP theory won and the Durham (or $k_{\perp}$) algorithm was used. It was
invented so that resummation of large logarithms could be achieved [30]. The
distance measure is $d_{ij}=2\frac{\min(E_{i}^{2},E_{j}^{2})}{s}R_{ir}$, where
$R_{ij}=1-\cos\theta_{ij}$ and the recombination scheme is simple addition of
the four momenta, $p_{(ij)}^{\mu}=p_{i}^{\mu}+p_{j}^{\mu}$. The resolution
parameter $y_{\rm cut}=d_{\rm cut}/s$ can take values in $[0,1]$. The pQCD
prediction contains logarithmically enhanced terms of the form $\alpha_{\rm
s}^{n}\ln^{m}(1/y_{\rm cut})$, at any order, which has to be resummed if we
want to use small value of $y_{\rm cut}$, where we find the bulk of the data
(see Figure 12). Predictions are available with leading- ($m=2n$) and next-to-
leading ($2n-1$) logarithms (LL and NLL) summed up to all orders [30].

Figure 12 shows the fixed order LO and NLO predictions, as well as predictions
where NLO and NLL are matched. The curve at NLO accuracy gives a good
description of the measure data by the ALEPH collaboration [31], but only for
$y_{\rm cut}>0.01$. As $\alpha_{\rm s}\ln^{2}(100)=2.5$, for smaller values of
$y_{\rm cut}$ resummation is indispensable. The resummed prediction however,
is not expected to give a good description at large $y_{\rm cut}$ because in
the resummation formula only the collinear approximation of the matrix element
is used. Matching the two predictions gives a remarkably good description of
the data over the whole phase space.

Figure 12: Comparison of pQCD predictions to data for three- and four-jet
leptoproduction [32]. The data points include corrections from hadrons to
partons based on Monte Carlo simulations. The ’+K’ term indicates the
inclusion of a well-defined subleading (NNLL) term in the resummation formula.

At hadron colliders the $k_{\perp}$ algorithm needs modifications. First,
instead of energy, the boost-invariant measure of hardness, transverse
momentum is used to define the distance between tracks,
$d_{ij}=\min(p_{\perp,i}^{2},p_{\perp,j}^{2})\frac{R_{ij}^{2}}{R^{2}}$ where
$R_{ij}^{2}=(y_{i}-y_{j})^{2}+(\phi_{i}-\phi_{j})^{2}$ (distance in
rapidity–azimuthal-angle plane), $R$ is a small positive real number, and we
need a distance from the beam $d_{iB}=p_{\perp,i}^{2n}$, too. Also, the
algorithm needs modification because either $d_{ij}$ or $d_{iB}$ can be the
smallest distance. If a $d_{ij}$ is the smallest value, then $i$ and $j$ are
merged, while if the smallest is a $d_{iB}$, then momentum $p_{i}$ becomes a
jet momentum and is removed from the tracks to be clustered. We then call jet
candidates with transverse momentum $p_{\perp,i}>E_{R}$ resolved jets. The
merging rule may change as well. In the usual merging we add four-momenta, but
another option is to add transverse momenta,
$p_{\perp,(ij)}=p_{\perp,i}+p_{\perp,j}$, and add rapidities $y$ and azimuthal
angles $\phi$ weighted, $y_{(ij)}=(w_{i}y_{i}+w_{j}y_{j})/(w_{i}+w_{j})$ and
$\phi_{(ij)}=(w_{i}\phi_{i}+w_{j}\phi_{j})/(w_{i}+w_{j})$, where the weight
can be $w_{i}=p_{\perp,i}$, $p_{\perp,i}^{2}$, $E_{\perp,i}$, or
$E_{\perp,i}^{2}$. Such a merging is boost invariant along the beam. The
parameter $R$ plays a similar role as $d_{\rm cut}$ in electron-positron
annihilation or the cone radius $R$ in the cone algorithms: the smaller $R$,
the narrower the jet.

The iterative $k_{\perp}$-algorithm is infrared safe and resummation of large
logarithmic contributions of the form $\alpha_{\rm s}^{n}\ln^{2n}$ and
$\alpha_{\rm s}^{n}\ln^{2n-1}$ is possible, which is a clear advantage from
the theoretical point of view. The logarithms are those of $1/R$ and/or
$Q/E_{R}$, $Q$ being the hard process scale. By taking $E_{R}$ sufficiently
large in hadron-hadron collisions, we avoid such leading contributions from
initial-state showering and the underlying event, so these terms are
determined by the time-like showering of final-state partons (when the
virtuality of the parent parton is always positive). Particles within angular
separation $R$ tend to combine and particles separated by larger distance than
$R$ from all other particles become jets. The algorithm assigns a clustering
sequence to particles within jets, so we can look at jet substructure.

Nevertheless, at the TeVatron experiments the $k_{\perp}$-algorithm did not
become a standard for several reasons. The jets have irregular, often weird
shapes as seen on Fig. 13(a) because soft particles tend to cluster first
(even arbitrary soft particles can form jets). As a result there is a non-
linear dependence on soft particles, energy calibration and estimating
acceptance corrections are more difficult. The underlying event correction
depends on the area of the jet (in $\eta-\phi$ plane). It was also very
expensive computationally, so experimenters had a clear preference of cone
algorithms.

Figure 13: Jets in a proton-proton scattering event obtained with the (a)
$k_{\perp}$, (b) anti-$k_{\perp}$ clustering algorithms

The breakthrough occurred with Refs. [33, 34] where variants of the
$k_{\perp}$ algorithm and an improved, fast implementation was introduced. The
distance formula was modified to
$d_{ij}=\min(p_{\perp,i}^{2n},p_{\perp,j}^{2n})\frac{R_{ij}^{2}}{R^{2}}$
($n=-1,0,1$). IR-safety is independent of $n$, as well as NLL resummation of
large logarithms. It was found that with $n=-1$ (called
anti-$k_{\perp}$-alogrihtm) particles close in angle cluster first, which
results in regular cone-like shapes as seen on Fig. 13(b) without using stable
cones. As a result it became the standard jet algorithm at the LHC
experiments. Yet, one should keep in mind that there is no ‘perfect’ jet
algorithm. For instance, the anti-$k_{\perp}$ one does not provide useful
information on jet substructure. It is important to remember that in pQCD
theoretical prediction can be made only with IR-safe jet functions, but among
those the goal of the study may help decide which algorithms to use.

##  4 Towards a general method for computing QCD radiative corrections

We have seen that (i) in pQCD the computation of radiative corrections at NLO
accuracy is indispensable, (ii) the NLO corrections are of two kinds: real and
virtual, that are separately divergent and contain different number of
particles in the final state, (iii) these singularities cancel for IR-safe
cross sections. To find the finite NLO corrections we have to develop a method
for combining the real and virtual corrections. In order to be able to
automate the NLO computations such a method has to be general, independent of
the measurable quantity and the process. To devise such a general method, we
need to study the origin of the singularities in a more precise way than we
did in the previous section. We shall find factorization formulae of the SME’s
that find many important applications in QCD, and so belong to the most
important features of QCD.

###  4.1 Factorization of $\left|\mathcal{M}_{n}\right|^{2}$ in the soft limit

The soft limit is defined by $p_{s}^{\mu}=\lambda\,q^{\mu}$, with
$\lambda\,\in\,\mathbb{R}^{+}$ and $\lambda\,\rightarrow\,0$ for $q^{\mu}$
fixed. In this limit the emission of the soft gluon from (internal)
propagators is IR finite. If we consider the emission of a soft gluon off an
external quark we find

$\propto\mathcal{M}_{m}T_{i}^{s}g_{\rm
s}\,\bar{u}(p_{i},s_{i})\gamma^{\mu}\,\frac{\slashed{p}_{i}+\slashed{p}_{s}}{s_{is}}\stackrel{{\scriptstyle
p_{s}\to 0}}{{\simeq}}\mathcal{M}_{m}T_{i}^{s}g_{\rm
s}\,\frac{p_{i}^{\mu}}{p_{i}\cdot p_{s}}\,\bar{u}(p_{i},s_{i})$.
In taking the limit, we used the anti-commutation relation (4) to write
$\gamma^{\mu}\slashed{p}_{i}=-\slashed{p}_{i}\gamma^{\mu}+2p_{i}^{\mu}$ and
the Dirac equation of the massless bi-spinor,
$\bar{u}(p_{i})\slashed{p}_{i}=0$. The factor $\frac{p_{i}^{\mu}}{p_{i}\cdot
p_{s}}$ is the “square root” of the eikonal factor
$S_{ik}\left(s\right)=\frac{2s_{ik}}{s_{is}s_{ks}}$. In the same limit, we can
derive after a bit more algebra the factorization formula for soft-gluon
emission off a gluon line. The emission of a soft gluon off an external gluon
(in light-cone gauge) is given by

$\propto\,\mathcal{M}_{m}\varepsilon^{\mu}\left(p_{s},n\right)\frac{1}{s_{is}}\,d^{\lambda\lambda^{\prime}}\left(p_{i}+p_{s},n\right)\Gamma_{\nu\mu\lambda^{\prime}}^{asb}\left(-p_{i},-p_{s},p_{i}+p_{s}\right)\varepsilon^{\nu}\left(p_{i},n\right)$,
where in the three-gluon vertex

$\displaystyle
V_{\nu\mu\lambda}(-p_{i},-p_{s},p_{i}+p_{s})\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=-\left(p_{i}+2p_{s}\right)_{\nu}g_{\mu\lambda}+\left(2p_{i}+p_{s}\right)_{\mu}g_{\nu\lambda}-\left(p_{i}-p_{s}\right)_{\lambda}g_{\mu\nu}$
$\displaystyle=2p_{i\mu}\,g_{\lambda\nu}+\left[-\left(p_{i}+p_{s}\right)_{\lambda}\,g_{\mu\nu}-p_{i\nu}\,g_{\mu\lambda}\right]+\left[p_{s\mu}\,g_{\nu\lambda}+2p_{s\,\lambda}\,g_{\mu\nu}-2p_{s\nu}\,g_{\mu\lambda}\right]$
$\displaystyle\stackrel{{\scriptstyle
p_{s}\rightarrow\,0}}{{\simeq}}2p_{i\mu}\,g_{\nu\lambda}-\left[\left(p_{i}+p_{s}\right)_{\lambda}\,g_{\mu\nu}+p_{i\nu}\,g_{\mu\lambda}\right].$

We use
$d^{\lambda\lambda^{\prime}}\left(p_{i}+p_{s},n\right)\left(p_{i}+p_{s}\right)_{\lambda}=0$
and $\varepsilon^{\nu}\left(p_{i},n\right)p_{i\,\nu}=0$, thus

$\frac{1}{s_{is}}d^{\lambda\lambda^{\prime}}\left(p_{i}+p_{s},n\right)\Gamma_{\nu\mu\lambda^{\prime}}^{asb}\left(-p_{i},-p_{s},p_{i}+p_{s}\right)\varepsilon^{\nu}\left(p_{i},n\right)\stackrel{{\scriptstyle
p_{s}\rightarrow\,0}}{{\>\simeq\>}}-T_{b}^{s}g_{\rm
s}\frac{p_{i\mu}}{p_{i}\\!\cdot\\!p_{s}}\,\underbrace{\left[d^{\lambda\lambda^{\prime}}\left(p_{i},n\right)g_{\lambda^{\prime}\nu}\varepsilon^{\nu}\left(p_{i},n\right)\right]}_{-\varepsilon^{\lambda}\left(p_{i},n\right)}.$

These two results can be unified and formalized by

$\hat{S}_{s}\left\langle\,c_{s}\right|\left.\mathcal{M}_{m+1}\left(p_{s},\ldots\right)\right\rangle=g_{\rm
s}\epsilon^{\mu}\left(p_{s}\right)\,{{\mbox{\boldmath$J$}}}_{\mu}\left(s\right)\left|\mathcal{M}_{m}\left(\ldots\right)\right\rangle,$

where $c_{s}$ is the colour index of the soft gluon $s$, $\hat{S}_{s}$ is an
operator which takes the soft limit and keeps the leading $\frac{1}{\lambda}$
singular term, and the soft gluon current
${{\mbox{\boldmath$J$}}}_{\mu}\left(s\right)$ is given by

${{\mbox{\boldmath$J$}}}_{\mu}\left(s\right)=\sum_{k=1}^{m}\,{{\mbox{\boldmath$T$}}}_{k}^{s}\,\frac{p_{k\,\mu}}{p_{k}\,\\!\cdot\\!\,p_{s}}\,.$

The soft gluon can be emitted from any of the external legs, therefore the sum
in the previous formula runs over all external partons. A soft quark leads to
an integrable singularity because the fermion propagator is less singular than
that of the gluon. Colour conservation implies that the current
${{\mbox{\boldmath$J$}}}_{\mu}\left(s\right)$ is conserved,

$p_{s}^{\mu}\,{{\mbox{\boldmath$J$}}}_{\mu}\left(s\right)\left|\mathcal{M}_{m}\right\rangle=\sum_{k=1}^{m}{{\mbox{\boldmath$T$}}}_{k}^{s}\left|\mathcal{M}_{m}\right\rangle=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=0\,.$

Then the soft limit of the SME
$\left\langle\mathcal{M}_{m}^{\left(0\right)}\right|\left.\mathcal{M}_{m}^{\left(0\right)}\right\rangle$
is as follows:

$\displaystyle\hat{S}_{s}\left|\mathcal{M}_{m+1}\left(p_{s},\dots\right)\right|^{2}\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=4\pi\alpha_{\rm
s}\mu^{2\epsilon}\sum_{i=1}^{m}\sum_{k=1}^{m}\,\underbrace{\epsilon_{\mu}\left(s\right)\epsilon_{\nu}^{\ast}\left(s\right)}_{d_{\mu\nu}\left(p_{s},n\right)}\frac{p_{i}^{\mu}p_{k}^{\nu}}{p_{i}\cdot\,p_{s}\,p_{k}\cdot\,p_{s}}\left\langle\mathcal{M}_{m}\right|{{\mbox{\boldmath$T$}}}_{i}\cdot{{\mbox{\boldmath$T$}}}_{k}\left|\mathcal{M}_{m}\right\rangle$
$\displaystyle=-8\pi\alpha_{\rm
s}\mu^{2\epsilon}\sum_{i,k=1}^{m}\frac{1}{2}S_{ik}\left(s\right)\left|\mathcal{M}_{m\,\left(i,k\right)}^{\left(0\right)}\right|^{2}\,+\,\textrm{gauge
terms}=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dots\,.$

The gauge terms give zero contribution on on-shell matrix elements due to
gauge invariance.

###  4.2 Factorization of $\left|\mathcal{M}_{n}\right|^{2}$ in the collinear
limit

The collinear limit of momenta $p_{i}$ and $p_{r}$ is defined by Sudakov
parametrization:

$p_{i}^{\mu}=z_{i}p^{\mu}+k_{i\perp}^{\mu}-\frac{k_{i\perp}^{2}}{z_{i}}\,\frac{n^{\mu}}{2\,p\,\\!\cdot\\!\,n}\,,\qquad
p_{r}^{\mu}=z_{r}p^{\mu}+k_{r\perp}^{\mu}-\frac{k_{r\perp}^{2}}{z_{r}}\,\frac{n^{\mu}}{2\,p\,\\!\cdot\\!\,n}$

where $k_{i\perp}^{\mu}+k_{r\perp}^{\mu}\,=\,0$ and $z_{i}+z_{r}\,=\,1$. The
momentum $p^{\mu}$ is the collinear direction and

$p^{2}=p_{i}^{2}=p_{r}^{2}=n^{2}=0\,,\qquad k_{i\perp}\cdot p=k_{r\perp}\cdot
n=0\,,$

In the collinear limit $k_{i\perp}^{\mu},\,k_{r\perp}^{\mu}\,\rightarrow\,0$
and $s_{ir}=-\frac{k_{r\perp}^{2}}{z_{i}z_{r}}$. We now state the following
theorem

> _In a physical gauge, the leading collinear singularities are due to the
> collinear splitting of an external parton._

This means that we need to compute in the collinear limit. There are three
cases:

$f_{ir}$ | $\rightarrow$ | $f_{i}\,+\,f_{r}$
---|---|---
$q$ | $\rightarrow$ | $q\,+\,g$
$g$ | $\rightarrow$ | $q\,+\,\bar{q}$
$g$ | $\rightarrow$ | $g\,+\,g$

We compute explicitly the first case and leave the second and the third as
exercise.

For the case of a quark splitting into a quark and a gluon we have

$\split&=C_{F}g_{\rm
s}^{2}\mu^{2\epsilon}\,\frac{\slashed{p}_{i}+\slashed{p}_{r}}{s_{ir}}\,\gamma^{\mu}\slashed{p}_{i}\gamma^{\nu}\,d_{\mu\nu}\left(p_{r},n\right)\frac{\slashed{p}_{i}+\slashed{p}_{r}}{s_{ir}}\\\
=C_{F}\,4\pi\alpha_{\rm
s}\mu^{2\epsilon}\,\frac{\slashed{p}_{i}+\slashed{p}_{r}}{s_{ir}}\left(-\gamma^{\mu}\slashed{p}_{i}\gamma_{\mu}\,+\,\frac{\slashed{p}_{r}\slashed{p}_{i}\slashed{n}+\slashed{n}\slashed{p}_{i}\slashed{p}_{r}}{p_{r}\,\\!\cdot\\!\,n}\right)\,\frac{\slashed{p}_{i}+\slashed{p}_{r}}{s_{ir}}.$
(58)

Using

$-\gamma^{\mu}\slashed{p}_{i}\gamma_{\mu}=\left(d-2\right)\slashed{p}_{i}\,,\qquad\slashed{p}_{i}\slashed{p}_{i}=p_{i}^{2}\,\mathbbm{1}\,,\quad\mathrm{and}\quad\slashed{p}_{i}\slashed{p}_{r}\slashed{p}_{i}=s_{ir}\slashed{p}_{i}-p_{i}^{2}\slashed{p}_{r}=s_{ir}\slashed{p}_{i}\,,$

we find

$\displaystyle\left(\slashed{p}_{i}+\slashed{p}_{r}\right)\left(-\gamma^{\mu}\slashed{p}_{i}\gamma_{\mu}\right)\left(\slashed{p}_{i}+\slashed{p}_{r}\right)=\left(d-2\right)s_{ir}\slashed{p}_{r}\,,$
$\displaystyle\slashed{p}_{r}\slashed{p}_{i}\slashed{n}=-\slashed{p}_{i}\slashed{p}_{r}\slashed{n}+s_{ir}\slashed{n}\,=\,\slashed{p}_{i}\slashed{n}\slashed{p}_{r}-2\slashed{p}_{i}p_{r}\\!\cdot\\!n+s_{ir}\slashed{n}==-\slashed{n}\slashed{p}_{i}\slashed{p}_{r}+2p_{i}\\!\cdot\\!n\slashed{p}_{r}-2\slashed{p}_{i}p_{r}\\!\cdot\\!n+s_{ir}\slashed{n}\,.$

Then

$\displaystyle\left(\slashed{p}_{i}+\slashed{p}_{r}\right)\left(\slashed{p}_{r}\slashed{p}_{i}\slashed{n}+\slashed{n}\slashed{p}_{i}\slashed{p}_{r}\right)\left(\slashed{p}_{i}+\slashed{p}_{r}\right)\,=$
$\displaystyle\qquad\qquad=\,2\left(\slashed{p}_{i}+\slashed{p}_{r}\right)\left(p_{i}\\!\cdot\\!n\,\slashed{p}_{r}-p_{r}\\!\cdot\\!n\,\slashed{p}_{i}+p_{i}\\!\cdot\\!p_{r}\,\slashed{n}\right)\left(\slashed{p}_{i}+\slashed{p}_{r}\right)$
$\displaystyle\qquad\qquad=2\left[p_{i}\\!\cdot\\!n\,s_{ir}\slashed{p}_{i}-p_{r}\\!\cdot\\!n\,s_{ir}\slashed{p}_{r}+p_{i}\cdot
p_{r}\left(2\left(p_{i}+p_{r}\right)\\!\cdot\\!n\left(\slashed{p}_{i}+\slashed{p}_{r}\right)-\left(p_{i}+p_{r}\right)^{2}\slashed{n}\right)\right]$
$\displaystyle\qquad\qquad=s_{ir}\left[4p_{i}\\!\cdot\\!n\,\slashed{p}_{i}+2p_{i}\\!\cdot\\!n\,\slashed{p}_{r}+2p_{r}\\!\cdot\\!n\,\slashed{p}_{i}-s_{ir}\slashed{n}\right].$

Substituting these results and then the Sudakov parametrization of the momenta
into Eq. (58) we obtain

$\split&\stackrel{{\scriptstyle
p_{i}\|p_{r}}}{{\simeq}}\,\frac{1}{s_{ir}}\,C_{F}\,4\pi\alpha_{\rm
s}\mu^{2\epsilon}\,\left[2\left(1-\epsilon\right)z_{r}+4\frac{z_{i}^{2}}{z_{r}}+4z_{i}+\mathcal{O}\left(k_{\perp}\right)\right]\slashed{p}\\\
~{}~{}=\frac{1}{s_{ir}}\,C_{F}\,8\pi\alpha_{\rm
s}\mu^{2\epsilon}\,\left[2\frac{z_{i}}{z_{r}}+\left(1-\epsilon\right)z_{r}\right]\slashed{p}=\frac{1}{s_{ir}}\,C_{F}\,8\pi\alpha_{\rm
s}\mu^{2\epsilon}\left[\frac{1+z_{i}^{2}}{1-z_{i}}-\epsilon\left(1-z_{i}\right)\right]\slashed{p}$

Similarly to the soft case we can define an operator $\hat{C}_{ir}$ which
takes the collinear limit and keeps the leading singular
($\mathcal{O}(1/k_{\perp}^{2})$) terms:

$\hat{C}_{ir}\left|\mathcal{M}_{m+1}^{\left(0\right)}\right|^{2}=8\pi\alpha_{\rm
s}\mu^{2\epsilon}\frac{1}{s_{ir}}\left\langle\mathcal{M}_{m}^{\left(0\right)}\left(p,\ldots\right)\right|\hat{P}_{qg}^{\left(0\right)}\left(z_{i},z_{r},k_{\perp};\epsilon\right)\left|\mathcal{M}_{m}^{\left(0\right)}\left(p,\ldots\right)\right\rangle\,.$
(59)

The kernel $\hat{P}_{qg}$, called Altarelli-Parisi splitting function for the
process $q\rightarrow\,q\,+\,g$, is diagonal in the spin-state of the parent
(splitting) parton:

$\left\langle\,s\right|\hat{P}_{qg}\left|s^{\prime}\right\rangle=C_{F}\left[2\frac{z_{i}}{z_{r}}+\left(1-\epsilon\right)z_{r}\right]\delta_{ss^{\prime}}.$

Similar calculations give the splitting kernels for the gluon splitting
processes, which however, contain azimuthal correlations of the parent parton

$\displaystyle\left\langle\mu\right|\hat{P}_{q\bar{q}}^{(0)}\left(z_{i},z_{r},k_{\perp};\epsilon\right)\left|\nu\right\rangle=T_{R}\left[-g^{\mu\nu}+4z_{i}z_{r}\,\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{\perp}^{2}}\right]$
(60)
$\displaystyle\left\langle\mu\right|\hat{P}_{gg}^{(0)}\left(z_{i},z_{r},k_{\perp};\epsilon\right)\left|\nu\right\rangle=2C_{A}\left[-g^{\mu\nu}\left(\frac{z_{i}}{z_{r}}+\frac{z_{r}}{z_{i}}\right)-2\left(1-\epsilon\right)z_{i}z_{r}\,\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{\perp}^{2}}\right].\quad~{}$
(61)

The soft and collinear limits overlap when the soft gluon is also collinear to
its parent parton:

$\hat{C}_{jr}\,\hat{S}_{r}\,\left|\mathcal{M}_{m+1}^{\left(0\right)}\left(p_{r},\ldots\right)\right|^{2}=-8\pi\alpha_{\rm
s}\mu^{2\epsilon}\sum_{k\neq
j}\,\frac{2z_{j}}{s_{jr}\,z_{r}}\left|\mathcal{M}_{m\left(j,k\right)}^{\left(0\right)}\left(\ldots\right)\right|^{2}\
=8\pi\alpha_{\rm
s}\mu^{2\epsilon}\,{{\mbox{\boldmath$T$}}}_{j}^{2}\,\frac{2}{s_{jr}}\,\frac{z_{j}}{z_{r}}\left|\mathcal{M}_{m}^{\left(0\right)}\right|^{2}\,.$

The notation for the splitting kernels in these lectures is different from the
usual notation in the literature. Usually,
$\hat{P}_{ij}^{(0)}\left(z,k_{\perp};\epsilon\right)$ denotes the splitting
kernel for the process $f_{i}(p)\to f_{j}(zp)+f_{k}((1-z)p)$, which does not
lead to confusion for $1\to 2$ splittings because the momentum fraction of
parton $j$ determines that of parton $k$ as their sum has to be one. For
splittings involving more partons, it is more appropriate to introduce as many
momentum fractions $z_{i}$ as the number of offspring partons, with the
constraint $\sum_{i}z_{i}=1$, and use the flavour indices to denote the
offspring partons in the order of the momentum fractions in the argument. For
$1\to 2$ splittings this means the use of
$\hat{P}_{ir}^{(0)}\left(z_{i},z_{r},k_{\perp};\epsilon\right)$ for the
splitting process $f_{k}(p)\to f_{i}(z_{i}p)+f_{r}(z_{r}p)$. The flavour of
the parent parton $f_{k}$ is determined uniquely by the flavour summation
rules, $q+g=q$, $q+\bar{q}=g+g=g$. These flavour summation rules are unique
also for $1\to 3$ splittings.

###### Exercise 4.1

Compute the Altarelli-Parisi-splitting function $\hat{P}_{qg}(z)$ for the
process $q\rightarrow qg$ from the collinear limit of the matrix element for
the process $e^{+}e^{-}\rightarrow q\bar{q}g$:

$\left|{\cal M}\left(e^{+}e^{-}\rightarrow
q\bar{q}g\right)\right|^{2}\propto\left(\left(1-\epsilon\right)\left(\frac{y_{23}}{y_{13}}+\frac{y_{13}}{y_{23}}\right)+2\left(\frac{y_{12}}{y_{13}y_{23}}-\epsilon\right)\right)\,.$

###### Exercise 4.2

The Altarelli-Parisi splitting function $\hat{P}_{q\bar{q}}\left(z\right)$ for
the process $g\rightarrow q\bar{q}$ is defined by the following collinear
limit:

$\displaystyle\langle\mathcal{M}_{n+1}^{\left(0\right)}\left(p_{i},p_{r},\ldots\right)\bigl{.}\bigr{|}\mathcal{M}_{n+1}^{\left(0\right)}\left(p_{i},p_{r},\ldots\right)\rangle\,\stackrel{{\scriptstyle
p_{i}\|p_{r}}}{{\simeq}}\frac{1}{s_{ir}}8\pi\alpha_{\rm
s}\mu^{2\epsilon}\,\langle\mathcal{M}_{n}^{\left(0\right)}\left(p,\ldots\right)\bigl{|}\hat{P}^{(0)}_{q\bar{q}}\left(z,k_{\perp}\right)\bigr{|}\mathcal{M}_{n}^{\left(0\right)}\left(p,\ldots\right)\rangle\,\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\quad=\frac{1}{s_{ir}}8\pi\alpha_{\rm
s}\mu^{2\epsilon}\,\langle\mathcal{M}_{n}^{\left(0\right)}\left(p,\ldots\right)\bigl{|}\bigr{.}\mu\rangle\langle\mu\left|\hat{P}^{(0)}_{q\bar{q}}\left(z,k_{\perp}\right)\right|\nu\rangle\langle\nu\bigr{|}\mathcal{M}_{n}^{\left(0\right)}\left(p,\ldots\right)\rangle$
$\displaystyle\quad=\frac{1}{s_{ir}}8\pi\alpha_{\rm
s}\mu^{2\epsilon}\,\langle\mathcal{M}_{n}^{\left(0\right)}\left(p,\ldots\right)\bigl{|}\bigr{.}\mu\rangle\,\frac{d_{\mu\rho}}{s_{ir}}\>\Pi^{\rho\sigma}\>\frac{d_{\sigma\nu}}{s_{ir}}\langle\nu\bigr{|}\mathcal{M}_{n}^{\left(0\right)}\left(p,\ldots\right)\rangle\,.$

Compute
$\langle\mu\left|\hat{P}^{(0)}_{q\bar{q}}\left(z,k_{\perp}\right)\right|\nu\rangle$
in leading order in $k_{\perp}$. Hint: In which sense does
$\Pi_{\mu\nu}=d_{\mu\rho}\Pi^{\rho\sigma}d_{\sigma\nu}$ hold?

###### Exercise 4.3

Derive the flavour summation rules for $1\to 3$ splittings.

###### Exercise 4.4

Compute the soft limit of Eq. (59) and the collinear limit of Eq. (4.1).

###  4.3 Regularization of real corrections by subtraction

The cross section at NLO accuracy is a sum of two terms, the LO prediction and
the corrections at one order higher in the strong coupling,

$\sigma_{{\scriptscriptstyle\rm NLO}}=\sigma^{{\scriptscriptstyle\rm
LO}}+\sigma^{{\scriptscriptstyle\rm NLO}}\,,$

where $\sigma^{{\scriptscriptstyle\rm LO}}$ is the integral of the fully
differential Born cross section over the available phase space defined by the
jet function, while $\sigma^{{\scriptscriptstyle\rm NLO}}$ is the sum of the
real and virtual corrections:

$\sigma^{{\scriptscriptstyle\rm LO}}=\int_{m}\\!{\rm
d}\sigma^{B}\,J_{m}{\left(\\{p\\}_{m}\right)}\,,\qquad\sigma^{{\scriptscriptstyle\rm
NLO}}=\int_{m+1}\\!{\rm
d}\sigma^{R}\,J_{m}{\left(\\{p\\}_{m+1}\right)}+\int_{m}\\!{\rm
d}\sigma^{V}\,J_{m}{\left(\\{p\\}_{m}\right)}\,.$

Both contributions to $\sigma^{{\scriptscriptstyle\rm NLO}}$ are divergent in
four dimensions, but their sum is finite for IR-safe jet functions.

The factorization of the squared matrix elements in the soft and collinear
limits allows for a process and observable independent method to regularize
the real corrections in their singular limits. The essence of the method is to
devise an approximate cross section ${\rm d}\sigma^{A}$ that matches the
singular behaviour of the real cross section ${\rm d}\sigma^{R}$ in all
kinematically degenerate regions of the phase space when one parton becomes
soft or two partons become collinear. Then we subtract this approximate cross
section from the real one and the difference can be integrated in four
dimensions. Next, we integrate ${\rm d}\sigma^{A}$ over the phase space of the
unresolved parton and we add it to ${\rm d}\sigma^{V}$. The integrated
subtraction term cancels the explicit poles in the virtual correction and the
sum can also be integrated in four dimentions. The key for this procedure is a
proper mapping of the $(m+1)$-parton phase space to the $m$-parton one which
respects the limits, thus the approximate cross section is defined with the
$m$-parton jet function. This way we can rewrite the NLO correction as a sum
of two finite terms,

$\sigma^{{\scriptscriptstyle\rm NLO}}=\int_{m+1}\left[{\rm
d}\sigma^{R}\,J_{m}{\left(\\{p\\}_{m+1}\right)}-{\rm
d}\sigma^{A}\,J_{m}{\left(\\{\tilde{p}\\}_{m}\right)}\right]_{\epsilon=0}\,+\,\int_{m}\left[{\rm
d}\sigma^{V}+\int_{1}\\!{\rm
d}\sigma^{A}\right]_{\epsilon=0}\,J_{m}{\left(\\{p\\}_{m}\right)}\,.$ (62)

The definition of the approximate cross section is not unique and the best
choice may depend on further requirements that we do not discuss here. We also
skip the precise definition of the momenta $\tilde{p}^{\mu}$ which are
obtained by mapping the $(m+1)$-particle phase space onto an $m$-particle
phase space times a one-particle phase space. A widely used general
subtraction scheme that can be used also for processes including massive
partons with smooth massless limits is presented in Ref. [35], where these
definitions are given explicitly. This method uses the factorization of the
SME in the soft and collinear limits. The challange posed by the overlapping
singularity in the soft-collinear limit is solved by a smooth interpolation
between these singular regions.

The factorization properties of Eqs. (4.1) and (59) play other very important
roles in pQCD. The numerical implementation of the SME is in general a process
prone to errors. Testing the factorization in the kinematically degenerate
phase space regions serves a good check of the implementation. The computation
is even more difficult for the virtual corrections. Similar factorization
holds for those. The factorized form of the SME can be used in resumming
logarithmically enhanced terms at all orders, or in devising a parton shower
algorithm for modelling events (see Sect. 6.3). The splitting kernels that
appear in the collinear factorization have a role in the evolution equations
of the parton distribution functions (see Sect. 5.6).

The state of the art in making precision predictions assaults on the one hand
the full automation of computations at NLO, and on the other the realm of
next-to-next-to-leading order (NNLO) corrections. The automation of computing
jet cross sections at NLO accuracy has been accomplished and several programs
are available with the aim to facilitate automated solutions for computing jet
cross sections at NLO accuracy:

  * •

aMC@NLO (http://amcatnlo.web.cern.ch)

  * •

BlackHat/Sherpa (https://blackhat.hepforge.org)

  * •

FeynArts/FormCalc/LoopTools (http://www.feynarts.de)

  * •

GoSam (https://gosam.hepforge.org)

  * •

HELAC-NLO (http://helac-phegas.web.cern.ch)

  * •

MadGolem (http://www.thphys.uni-heidelberg.de/ lopez/madgolem-corner.html).

In the NNLO case the IR singularity structure is much more involved than in
the case of NLO computations due to complicated overlapping singly- and
doubly-unresolved configurations. Several subtraction methods have been
proposed for the regularization of the IR divergences and there is intense
research to find a general one that can be automated. To provide an impression
about the importance of NNLO corrections, we present QCD predictions at
various accuracies for the three-jet rate computed with $\alpha_{\rm s}=0.118$
and at a centre-of-mass energy of $\sqrt{s}=35$ GeV in Fig. 14. Figure 14(a)
shows comparison of prediction at NLO with that at matched NLO and resummed
next-to-leading logarithmic (denoted by NLLA in the figure) accuracy, while
Fig. 14(b) presents comparison of prediction at NNLO with that at matched NNLO
and resummed NLL accuracy. The inserts in both cases show the ratio between
the matched and the unmatched predictions. For all calculations the
uncertainty band reflects the uncertainty due to the variation of the
renormalization scale around the default scale $\mu=\sqrt{s}$ by factors of 2
in both directions.

Figure 14: QCD predictions for the three-jet rate in electron-positron
annihilation [36]

##  5 Deeply inelastic lepton-proton scattering

Perturbative QCD stems from the parton model that was developed to understand
deeply inelastic lepton-hadron scattering (DIS). The purpose of those
experiments was to study the structure of the proton by measuring the
kinematics of the scattered lepton. In Fig. 15(a) we show a real event in the
H1 experiment at the HERA collider. The value of $Q^{2}$, which is the modulus
squared of the momentum transfer between the lepton and the proton is 21475
GeV${}^{2}>>1$ GeV2, signifying that the scattering is well in the deeply
inelastic region. The parton model interpretation of the event is shown in
Fig. 15(b): the lepton is scattered by an angle $\theta$ due to the exchange
of a virtual photon with one of the constituents of the proton (a parton). The
measurement is inclusive from the point of view of hadrons ($X$ means any
number of hadrons that are not observed separately), thus the process can be
described in pQCD.

Figure 15: Deeply inelastic lepton-proton scattering (a) in the H1 detector
and (b) parton model interpretation of such an event

The DIS kinematics is described by the following varibales

$\split\mbox{centre-of-mass energy}^{2}&=s=(P+k)^{2}\,,\\\
\mbox{{\color[rgb]{1.000000,0.000000,0.000000} momentum
transfer}}={\color[rgb]{1.000000,0.000000,0.000000}q^{\mu}}=k^{\mu}-k^{\prime\mu}\,,\\\
{\color[rgb]{1.000000,0.000000,0.000000}|\mbox{ momentum
transfer}|^{2}}={\color[rgb]{1.000000,0.000000,0.000000}Q^{2}}=-q^{2}=2MExy\,,\\\
\mbox{{\color[rgb]{1.000000,0.000000,0.000000} scaling
variable}}={\color[rgb]{1.000000,0.000000,0.000000}x}=Q^{2}/(2P\cdot q)\,,\\\
\mbox{energy loss}=\nu=(P\cdot q)/M=E-E^{\prime}\,,\\\
\mbox{{\color[rgb]{1.000000,0.000000,0.000000} relative energy
loss}}={\color[rgb]{1.000000,0.000000,0.000000}y}=(P\cdot q)/(P\cdot
k)=1-E^{\prime}/E\,,\\\ \mbox{recoil
mass}^{2}=W^{2}=(P+q)^{2}=M^{2}+\frac{1-x}{x}Q^{2}\,,$

where we set the more important ones for these lectures in red.

###  5.1 Parametrization of the target structure

The cross section for $e(k)+p(P)\to e(k^{\prime})+X$ reads

${\rm d}\sigma=\sum_{X}\frac{1}{4ME}\int\\!{\rm d}\phi\,\frac{1}{4}\sum_{{\rm
spin}}|{\cal M}|^{2}\,.$ (63)

We factorize the phase space and the SME into two parts, one for the lepton
and one for the hadrons:

${\rm d}\phi=\frac{{\rm d}^{3}k^{\prime}}{(2\pi)^{3}2E^{\prime}}\,{\rm
d}\phi_{X}\,,\qquad\frac{1}{4}\sum_{{\rm spin}}|{\cal
M}|^{2}=\frac{e^{4}}{Q^{4}}L^{\mu\nu}H_{\mu\nu}\,.$

Then the hadron part of the cross section is the dimensionless Lorentz tensor
$W_{\mu\nu}=\frac{1}{8\pi}\sum_{X}\int\\!{\rm d}\phi_{X}H_{\mu\nu}$ (the
factor of $\frac{1}{8\pi}$ is included here by convention). As it depends on
two momenta $P^{\mu}$ and $q^{\mu}$, the most general gauge invariant
combination of the Lorentz tensor can be written as

$W_{\mu\nu}(P,q)=\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)W_{1}(x,Q^{2})+\left(P_{\mu}-q_{\mu}\frac{P\cdot
q}{q^{2}}\right)\left(P_{\nu}-q_{\nu}\frac{P\cdot
q}{q^{2}}\right)\frac{W_{2}(x,Q^{2})}{P\cdot q}\,,$

where the structure functions $W_{i}(x,Q^{2})$ are dimensionless functions of
the scaling variable and the momentum transfer.

For the lepton part we express the kinematical relations $E^{\prime}=(1-y)E$,
$\cos\vartheta=1-\frac{xyM}{(1-y)E}$ to change variables to scaling variable
and relative energy loss:

$\frac{{\rm d}^{3}k^{\prime}}{(2\pi)^{3}2E^{\prime}}=\frac{{\rm
d}\varphi}{2\pi}\frac{E^{\prime}}{8\pi^{2}}\,{\rm d}E^{\prime}\,{\rm
d}\cos\vartheta=\frac{{\rm d}\varphi}{2\pi}\,\frac{yME}{8\pi^{2}}\,{\rm
d}y\,{\rm d}x\,,$

and compute the trace
$L^{\mu\nu}=\frac{1}{2}Tr[\slashed{k}\gamma^{\mu}\slashed{k}^{\prime}\gamma^{n}]=k^{\mu}k^{{}^{\prime}\nu}+k^{\nu}k^{{}^{\prime}\mu}-g^{\mu\nu}k\cdot
k^{\prime}\,.$ Then the differential cross section in $x$ and $y$ is obtained
from Eq. (63) as

$\frac{{\rm d}^{2}\sigma}{{\rm d}x\,{\rm
d}y}=\frac{4\pi\alpha^{2}}{y\,Q^{2}}\left[y^{2}W_{1}(x,Q^{2})+\left(\frac{1-y}{x}-xy\frac{M^{2}}{Q^{2}}\right)W_{2}(x,Q^{2})\right]\,,$

which we rewrite in the scaling limit, defined by $Q^{2}\to\infty$ with $x$
fixed, as

$\frac{{\rm d}^{2}\sigma}{{\rm d}x\,{\rm
d}y}=\frac{4\pi\alpha^{2}}{y\,Q^{2}}\left[\big{(}1+(1-y)^{2}\big{)}F_{1}+\frac{1-y}{x}\big{(}F_{2}-2xF_{1}\big{)}\right]\,.$
(64)

Figure 16: Measured value of the $F_{2}$ structure function at several
different values of $Q^{2}$

The dimensionless functions $F_{1}$ and $F_{2}$ were first measured by the
SLAC-MIT experiment [37]. The result of that measurement supplemented by some
later ones is shown in Fig. 16. The interesting feature is that in the scaling
limit $F_{2}$ becomes independent of $Q^{2}$, $F_{2}(x,Q^{2})\to F_{2}(x)$ (in
fact, the independence starts at quite low values of $Q^{2}$).

###  5.2 DIS in the parton model

Let us now describe the same scattering process by assuming the proton is a
bunch of free flying quarks and the lepton exchanges a hard virtual photon
with one of those quarks as shown in Fig. 15(b). The struck quark carries a
momentum $p^{\mu}$, which is a fraction of the proton momentum, $p^{\mu}=\xi
P^{\mu}$, so we consider the process $e(k)+q(p)\to
e(k^{\prime})+q(p^{\prime})$. The corresponding cross section is

${\rm d}\hat{\sigma}=\frac{1}{2\hat{s}}\int\\!{\rm
d}\phi_{2}\,\frac{1}{4}\sum_{{\rm spin}}|{\cal M}|^{2}\,,$

with $\hat{s}=(p+k)^{2}$. The SME is proportional to the product of the lepton
tensor $L^{\mu\nu}$ and a similar quark tensor