where we used the notation $s_{ij}=-(k_{i}+k_{j})^{2}$. Using the Fierz
identity (151) and the shorthand notation of (131) and (132) this can be
written as

${\cal M}(1^{+},2^{-},3^{+},4^{-})=2{\rm i}e^{2}\frac{[24]\langle
31\rangle}{\langle 12\rangle[12]}\,.$ (159)

Using momentum conservation this may be further rewritten as

${\cal M}(1^{+},2^{-},3^{+},4^{-})=2{\rm i}e^{2}\frac{[13]^{2}}{[12][34]}\,.$
(160)

It may be readily verified that

${\cal M}(1^{-},2^{+},3^{-},4^{+})=2{\rm i}e^{2}\frac{\langle
13\rangle^{2}}{\langle 12\rangle\langle 34\rangle}\,.$ (161)

The expressions in (160) and (161) are quite compact, and can be transformed
into each other by either a parity transformation or a charge conjugation. For
any other helicity configuration the amplitude actually vanishes.

#### The reaction ${\rm e}^{+}{\rm e}^{-}\rightarrow\mu^{+}\mu^{-}\gamma$

In this second example we study the production of a muon pair together with a
photon in electron positron annihilation

$e^{-}(k_{1})+e^{+}(k_{2})\rightarrow\mu^{-}(k_{3})+\mu^{+}(k_{4})+\gamma(k_{5})$
(162)

The photon can be radiated off any of the four external fermion lines, leading
to the four diagrams shown in Fig. 18.

Figure 18: Feynman diagrams contributing a particular helicity amplitude,
indicated in diagram (a), for ${\rm e}^{+}{\rm
e}^{-}\rightarrow\mu^{+}\mu^{-}\gamma$ at lowest order. All momenta are
outgoing, and time runs from right to left.

Let us list another useful identity, not difficult to prove, for a positive
helicity massless vector boson with polarization $\varepsilon_{+}^{\mu}(k,p)$
emitted from a fermion by

$\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\varepsilon_{+}(k,p)=\frac{{\rm
i}\sqrt{2}}{\langle kp\rangle}\left(|p+\rangle\langle
k\\!+\\!|+|k-\rangle\langle p\\!-\\!|\right)\,.$ (163)

For the negative helicity case one has, in analogy

$\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\varepsilon_{-}(k,p)=\frac{-{\rm
i}\sqrt{2}}{[kp]}\left(|k+\rangle\langle p\\!+\\!|+|p-\rangle\langle
k\\!-\\!|\right)\,.$ (164)

The invariant amplitude reads

${\cal
M}(1^{\lambda_{1}},2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{\lambda_{5}})\,.$
(165)

Given that each external line can have two helicity values, it might seem that
this process allows thirty-two different independent helicity amplitudes.
However, helicity conservation and invariance under charge conjugation and
parity transformation ensure that there is in fact only one independent
amplitude. We thus consider the helicity amplitude

${\cal M}(1^{+},2^{-},3^{+},4^{-},5^{+})\,,$ (166)

and choose as $k_{4}$ as reference momentum for the outgoing photon. Diagram
(a) then reads

${\cal M}_{a}(1^{+},2^{-},3^{+},4^{-},5^{+})=({\rm i}e)^{3}\langle
2\\!-\\!|\gamma^{\mu}\frac{-{\rm i}(\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}1+\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}5)}{-s_{15}}\hbox
to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\varepsilon_{+}(k_{5},k_{4})|1-\rangle\times\frac{-{\rm
i}}{s_{34}}\langle 3\\!+\\!|\gamma_{\mu}|4+\rangle\,.$ (167)

Using the results in eqs. (163) and (164) we find

${\cal M}_{a}(1^{+},2^{-},3^{+},4^{-},5^{+})=-2\sqrt{2}e^{3}\frac{\langle
24\rangle^{2}[23]}{\langle 15\rangle\langle 45\rangle\langle
34\rangle[34]}\,.$ (168)

For diagram (b) we find similarly

${\cal M}_{b}(1^{+},2^{-},3^{+},4^{-},5^{+})=2\sqrt{2}e^{3}\frac{\langle
24\rangle^{2}[13]}{\langle 25\rangle\langle 45\rangle\langle
34\rangle[34]}\,,$ (169)

while for (c) we have

${\cal M}_{c}(1^{+},2^{-},3^{+},4^{-},5^{+})=2\sqrt{2}e^{3}\frac{\langle
24\rangle^{2}}{\langle 12\rangle\langle 35\rangle\langle 45\rangle}\,.$ (170)

Notice that with our choice of reference momentum we have

${\cal M}_{d}(1^{+},2^{-},3^{+},4^{-},5^{+})=0\,.$ (171)

Adding up the contributions we find

${\cal M}(1^{+},2^{-},3^{+},4^{-},5^{+})=2\sqrt{2}e^{3}\frac{\langle
24\rangle^{2}}{\langle 12\rangle}\left(\frac{[34]}{\langle 15\rangle\langle
45\rangle[12]}+\frac{1}{\langle 35\rangle\langle 45\rangle}\right)\,.$ (172)

Again this is a nice, compact result, a complex number fully expressed in
terms of helicity spinors.

Without further proof we can list what is perhaps the most famous result in
tree-level QCD amplitudes calculations [26]: the so-called maximal helicity
violating (MHV) amplitudes (aka. Parke-Taylor amplitudes) for $n$-gluon
scattering. One may first organize the full tree-level invariant amplitude in
the colour quantum number as

$M_{n}(p_{i},\lambda_{i},a_{i})=g^{n-2}\sum_{\sigma\in
S_{n}/Z_{n}}\left(T_{\sigma(a_{1})}\dots
T_{\sigma(a_{n})}\right)A_{n}\left(\sigma(p_{1}^{\lambda_{1}}),\ldots,\sigma(p_{n}^{\lambda_{n}})\right)\,.$
(173)

where the sum is over all permutations $\sigma$ modulo the cyclic ones. The
amplitudes $A_{n}$ are called “colour-ordered”. Such amplitudes [27, 28, 29]
are considerably easier to calculate. First, if all gluons have the same
helicity, say $+$, then the amplitude is zero. The same holds if one of them
has helicity $-$. With two helicities $-$, we have the MHV amplitude. The
stunningly simple expression (a result of millions of Feynman diagrams if $n$
is large enough ) for the colour-ordered amplitude reads

$A_{n}(1^{+},\ldots,i^{-},\ldots,j^{-},\ldots n^{+})={\rm i}\frac{\langle
ij\rangle^{4}}{\langle 12\rangle\langle 23\rangle\ldots\langle
n-1,n\rangle\langle n1\rangle}\,.$ (174)

When flipping all $-$ to $+$ and vice versa, all one has to do is replace the
angled brackets by squared ones.

Helicity spinor methods are now a standard tool in the computation of QCD
scattering amplitudes for the LHC. It is worth mentioning that among the very
interesting developments in QCD in recent years has been the realization of
recursion relations among these amplitudes [30, 31], after new insights where
gained after phrasing them in terms of so-called twistors [32, 33]. Such
recursion relations, besides the still very powerful, and often faster [34,
35], earlier ones by Berends and Gield [28] have been important in extending
analytical and numerical computational power to high-multiplicity amplitudes.

###  4.2 The NLO revolution

I will here briefly touch upon recent ideas that have spurred what is
sometimes referred to as the NLO revolution. An extensive and clear review by
some of the instigators is Ref. [36]. For many years the bottleneck in
computing NLO cross sections for many external lines were the one-loop
diagrams for the virtual part of the cross section. They become increasingly
hard to calculate when the number of external lines grows from 4 to 5, 6 etc.
Because of similar arguments just mentioned for the case of high-multiplicity
tree-level amplitudes one can restrict oneself to a smaller set of diagrams
having a particular colour order. The objects to compute have, besides the
denominator factors due to the propagators in the loop, a numerator containing
dot products among external momenta, polarization vectors, and the loop
momentum. Hence, the integral over the loop momentum has possibly a number of
loop momenta in the numerator, with open Lorentz- indices.

Such tensor integrals can be reduced to scalar integrals in a well-defined
procedure [37]. New stable and efficient reduction techniques for tensor
integrals have been proposed in Refs.[38, 39], and have found much use.

One may also express external vectors in terms of a basis set of four. In this
procedure also denominators are cancelled, reducing the $n$-point function to
lower-point ones. This leads to an expansion of the amplitude in terms of
scalar functions down from $n$-point ones. Furthermore, up to (here
irrelevant) $\mathcal{O}(\epsilon)$ terms, five- and higher point functions
can be expressed in terms of four-point functions and lower [40, 41, 42]. The
price to pay is that for these lower point functions the external momenta are
not subsets but rather combinations of the original, massless external
momenta. These combinations then are not massless. The upshot is that one has,
schematically

$A_{n}^{\mathrm{one-loop}}=\sum_{j\in B}c_{j}\mathcal{I}_{j}$ (175)

where $B$ is a basis set that consists of a certain set of box-, triangle and
bubble integrals with or without massive external legs [43], and the $c_{j}$
are rational functions of dot products of external momenta and polarization
vectors, see Fig. 19. With a generic representation (175) in hand, the task of
calculating $A_{n}^{\mathrm{one-loop}}$ is then mapped to the task of find the
coefficients $c_{j}$.

For this one may use unitarity methods [44]. In Eq. (175) the elements of the
basis set on the _right hand side_ may have branchcuts in the invariants on
which the logarithms and dilogarithms in the $\mathcal{I}_{j}$ depend. For
instance, a particular integral may have terms of the type
$\ln(-s_{ij}/\mu^{2})$, with $s_{ij}=p_{ij}^{2}=(p_{i}+p_{j})^{2}=2p_{i}\cdot
p_{j}$, which clearly has a branchcut in the $s_{ij}$ variable.

On the other hand, one can also examine a particular discontinuity across a
particular branch cut for a particular invariant, or channel, for _the left-
hand side_ in Eq. (175), which is done by cutting the amplitude and replacing
cut propagators in the loop by delta functions.

Figure 19: Expansion of $n$-leg one-loop amplitude in sum of tadpoles,
bubbles, triangles and boxes.

$\frac{1}{p^{2}+i\epsilon}\rightarrow-i2\pi\,\delta(p^{2})$ (176)

This amounts to taking the imaginary part. From the comparison of both sides
the coefficients $c_{j}$ can then be determined. Essentially, one thus
determines the function $A_{n}^{\mathrm{one-loop}}$ from its poles and cuts.
This is vastly more efficient than computing every term fully by itself, and
is the key insight that spurred the NLO revolution.

However, there are important subtleties. Using four-dimensional momenta in the
cuts leaves an ambiguity in the form of a rational function. Using a
$n=4+\varepsilon$ version of the unitary method [45, 46] avoids this, but this
is somewhat more cumbersome to use. A number of other methods have been
devised to fix this ambiguity, such as using recursion relations [30, 31], or
using $D$-dimensional unitarity [47, 48]. Particularly fruitful is the use of
complex kinematics, which allows non-vanishing, non-trivial three-point
amplitudes. This allows taking multiple cuts of a box integral, such as in
Fig. 20, which goes under the name “generalized unitarity”.

Figure 20: Generalized unitarity

By so doing, one may determine the coefficients $c_{j}$ purely algebraically
[49], since the four delta-functions fix the loop momentum.

An effective way of solving Eq. (175) was proposed in Ref.[50], and is known
as the OPP method. Writing the equivalent of Eq. (175) at the _integrand_
level, the coefficients of the box etc integral can then be extracted by
choosing different values of the loop momentum, and perform the inversion
numerically.

Furthermore, numerical [51, 52] and semi-numerical [53] techniques for loop
integrals have progressed to the level where much work is taken care of for
the user through programs like Blackhat [54], Cuttools [55], or Rocket [56]
and MCFM [57]. The level of automation, including the matching to parton
showers, has now been stepped up tremendously, with the POWHEG Box [58] and
aMC@NLO-MadGraph5 [59] framework. They have brought NLO calculations now to
the general user.

As this snaphot of the NLO revolution suggests, the area of NLO calculations
was a very lively marketplace of ideas and methods. Although it is still a
bustling place, attention is now shifting to exporting the revolution to NNLO.

###  4.3 Aspects of NNLO

Here I will not say much, as this falls out of the scope of the lectures. Many
of the conceptual issues in earlier sections play a role here as well. The
accounting of singularities in a flexible way is much harder at this order. An
equation as (175) does not yet fully exist for this order, though impressive
progress is being made [60]. Nevertheless, results were obtained first already
many years ago for DIS [61], Drell-Yan [62] and some time later for Higgs
production [63, 64, 65]. Recently the latter was even computed to NNNLO using
powerful and clever methods involving threshold expansions [66].

Essential for any NNLO calculation for hadron colliders are the NNLO (3-loop)
Altarelli-Parisi splitting functions. These were calculated some time ago [9,
10] thanks also to the powers of the computer algebra program FORM [67, 68].

For top quark pair production [69] the first full two-to-two QCD process
calculated to NNLO was completed recently (more about this below). Many other
results are now appearing (see e.g. [70] for NNLO results on jet cross
sections), the review of which would take us too far afield, and would anyway
be out of date in very short order.

This concludes our discussions of finite order QCD methods and results. Let us
now turn to aspects of QCD resummation, and all-order results.

##  5 All orders

“Resummation” is shorthand for all-order summation of classes of potentially
large terms in quantum field perturbation theory. To review status and
progress in a field defined so generally is an impossibly wide scope, and I
will restrict myself to certain types in QCD, related of course to observables
at high-energy hadron colliders.

Let us first form an impression of what resummation is and what it does. Let
$d\sigma$ be a (differential) cross section with the schematic perturbative
expansion

$d\sigma=1+\alpha_{s}(L^{2}+L+1)+\alpha_{s}^{2}(L^{4}+L^{3}+L^{2}+L+1)+\ldots$
(177)

where $\alpha_{s}$ is the coupling, also serving as expansion parameter, $L$
is some logarithm that is potentially large. In our discussion we focus on
gauge theories, and on the case with at most two extra powers of $L$ per
order, as Eq. (177) illustrates. An extra order corresponds to an extra
emission of a gauge boson, the two (“Sudakov”) logs resulting from the
situation where the emission is simultaneously soft and collinear to the
parent particle direction.

Denoting $L=\ln A$, we can next ask what $A$ is. In fact, $A$ will in general
depend on the cross section at hand. For example, for a thrust ($T$)
distribution $A=1-T$, while for $d\sigma(p\bar{p}\rightarrow Z+X)/dp_{T}^{Z}$
$A=M_{Z}/p_{T}^{Z}$. It should be pointed out already here that $A$ is not
necessarily constructed out of measured variables but can also be a function
of unobservable partonic momenta that are to be integrated over. E.g. for
inclusive heavy quark production $A$ could be $1-4m^{2}/(x_{1}x_{2}S)$ in
hadron collisions with energy $\sqrt{S}$, where $x_{1},x_{2}$ are partonic
momentum fractions. When $L$ is numerically large so that even for small
$\alpha_{s}$, the convergent behaviour of the series is endangered,
resummation of the problematic terms into an analytic form might provide a
remedy, and thereby extend the theory’s predictive power to the range of large
$L$. In general the resummed form of $d\sigma$ may be written schematically as

$\displaystyle
d\sigma_{res}=C(\alpha_{s})\,\exp\left[Lg_{1}(\alpha_{s}L)+g_{2}(\alpha_{s}L)+\alpha_{s}g_{3}(\alpha_{s}L)+\ldots\right]+R(\alpha_{s})$
(178)

where $g_{1,2,\ldots}$ are computable functions. The series $C(\alpha_{s})$
multiplies the exponential, and $R(\alpha_{s})$ denotes the remainder.

The key aspect of resummation is finding the functions $g_{i}$. With only
$g_{1}$ one has leading logarithmic resummation (LL), with also $g_{2}$ NLL
etc. For NNLL resummation the matching function $C$ must also be known to next
order in $\alpha_{s}$.

###  5.1 Resummation basics, eikonal approximation, webs

Well-developed arguments exist for the exponentation properties of the Drell-
Yan cross section near threshold [71, 72]. Some are based on identifying
further evolutions equations [73, 71] based on refactorizations of the cross
section into different regions only sensitive to either collinear, soft or
hard corrections. This has been made into a formidable systematic programme
based on effective field theory [74, 75, 76, 77].

The connection between refactorization and resummation is already illustrated
by perturbative renormalization, in which the general relation of
unrenormalized and renormalized Green functions of fields $\phi_{i}$ carrying
momenta $p_{i}$ is

$G_{\rm un}(p_{i},M,g_{0})=\prod_{i}Z_{i}^{1/2}(\mu/M,g(\mu))\;G_{\rm
ren}(p_{i},\mu,g(\mu))\,.$ (179)

$M$ is an ultraviolet cutoff, and $g(\mu)$ and $g_{0}$ are the renormalized
and bare couplings respectively. The independence of $G_{\rm un}$ from $\mu$
and $G_{\rm ren}$ from $M$ may be used to derive renormalization group
equations,

$\mu{d\ \ln\;G_{\rm ren}\over d\mu}=-\sum_{i}\gamma_{i}(g(\mu))\,,$ (180)

in which the anomalous dimensions $\gamma_{i}=(1/2)(\mu d/d\mu)\ln Z_{i}$
appear as constants in the separation of variables, free of explicit
dependence on either $\mu$ or $M$. The solution to (180)

$G_{\rm ren}(p_{i},1,g(M))=G_{\rm
ren}(p_{i},M/\mu_{0},g(\mu_{0}))\exp\left[-\sum_{i}\int_{\mu_{0}}^{M}\frac{d\mu}{\mu}\gamma_{i}(g(\mu))\right]\,,$
(181)

is clearly an exponential. While in this example the factorization involves
separation of UV modes from finite energy ones, for the resummation we discuss
in this section one (re)factorizes collinear modes from soft-, anticollinear
and hard modes. In a manner similar to this example differential equations may
be set up whose solution, in terms of appropiate exponentials of (double)
integrals over anomalous dimensions, is the resummed cross section [73].

To see the appearance of exponentials in a different way we can observe that
in the refactorization approach the soft or eikonal part of the observable is
isolated in a well-defined way. One may then use the property that moments of
the eikonal DY cross section exponentiate at the level of integrands [78, 79,
80, 72], with exponents consisting of so-called webs. These are selections of
cut diagrams under criteria defined by graphical topology (irreducibility
under cuts of the eikonal lines) and with possibly modified colour weights.
Each web is a cut diagram, and can be integrated over the momentum $k$ that it
contributes to the final state.

To see how webs work, let us first consider the abelian case101010This text is
derived from section 2 in [81].. Webs are phrased in terms of eikonal Feynman
rules. In order to derive these one may consider a single hard massless
external line of final on-shell momentum $p$, originating from some
unspecified hard interaction described by ${\cal M}_{0}(p)$. The hard line may
emit a number $n$ of soft photons with momenta $k_{i}$, as depicted in Fig.
21, where $k_{1}$ is emitted closest to the hard interaction.

Figure 21: Soft photon emission from an energetic line

We shall take the emitting particle to be a scalar (the argument for fermions
is very similar). For this case the hard interaction is dressed according to

${\cal M}^{\mu_{1}\ldots\mu_{n}}(p,k_{i})={\cal
M}_{0}(p)\,\frac{1}{(p+K_{1})^{2}}(2p+K_{1}+K_{2})^{\mu_{1}}\ldots\frac{1}{(p+K_{n})^{2}}(2p+K_{n})^{\mu_{n}}\,,$
(182)

where we have introduced the partial momentum sums
$K_{i}=\sum_{m=i}^{n}k_{m}$.

The eikonal approximation in this case can simply be defined as the leading-
power contribution to the amplitude when the photon momenta
$k_{i}^{\mu_{i}}\to 0$, $\forall i$, in both numerator and denominator. In
this limit, eq. (182) becomes

${\cal M}^{\mu_{1}\ldots\mu_{n}}(p,k_{i})={\cal
M}_{0}(p)\,\frac{p^{\mu_{1}}\ldots p^{\mu_{n}}}{(p\cdot K_{1})\ldots(p\cdot
K_{n})}\,.$ (183)

The eikonal factor is insensitive to the spin of the emitting particles. One
may also notice that the eikonal factor does not depend on the energy of the
emitter, since it is invariant under rescalings of the hard momentum
$p^{\mu}$: at leading power in the soft momenta, one is effectively neglecting
the recoil of the hard particle against soft radiation.

The eikonal factor can be further simplified by employing Bose symmetry.
Indeed for the physical quantity depending on the amplitude ${\cal
M}^{\mu_{1}\ldots\mu_{n}}(p,k_{i})$, one must sum over all diagrams
corresponding to permutations of the emitted photons along the hard line.
Having done this, the eikonal factor multiplying ${\cal M}_{0}(p)$ on the
r.h.s. of (183) will be replaced by the symmetrized expression

$E^{\mu_{1}\ldots\mu_{n}}(p,k_{i})\,=\,\frac{1}{n!}\,p^{\mu_{1}}\ldots
p^{\mu_{n}}\,\sum_{\pi}\frac{1}{p\cdot
k_{\pi_{1}}}\frac{1}{p\cdot(k_{\pi_{1}}+k_{\pi_{2}})}\,\ldots\,\frac{1}{p\cdot(k_{\pi_{1}}+\ldots+k_{\pi_{n}})}\,,$
(184)

where the sum is over all permutations of the photon momenta, and
$k_{\pi_{i}}$ is the $i^{\text{th}}$ momentum in a given permutation. There
are $n!$ permutations, and each gives the same contribution to any physical
observable. This becomes manifest using the eikonal identity

$\sum_{\pi}\frac{1}{p\cdot
k_{\pi_{1}}}\frac{1}{p\cdot(k_{\pi_{1}}+k_{\pi_{2}})}\ldots\frac{1}{p\cdot(k_{\pi_{1}}+\ldots
k_{\pi_{n}})}=\prod_{i}\frac{1}{p\cdot k_{i}}\,.$ (185)

Using (185), the eikonal factor $E^{\mu_{1}\ldots\mu_{n}}(p,k_{i})$ arising
from $n$ soft emissions on an external hard line becomes simply

$E^{\mu_{1}\ldots\mu_{n}}(p,k_{i})\,=\,\prod_{i}\,\frac{p^{\mu_{i}}}{p\cdot
k_{i}}\,,$ (186)

which is manifestly Bose symmetric and invariant under rescalings of the
momenta $\\{p_{i}\\}$. In practice, each eikonal emission can then be
expressed by the effective Feynman rule

${\,=\,\,\,\Large\frac{p^{\mu}}{p\cdot k}}$ (187)

These Feynman rules can be obtained by replacing the hard external line with a
Wilson line along the classical trajectory of the charged particle. In abelian
quantum field theories this is given by the expression

$\Phi_{\beta}(0,\infty)=\exp\left[\,{\rm
i}e\int_{0}^{\infty}d\lambda\,\beta\cdot A(\lambda\beta)\,\right]~{},$ (188)

where $\beta$ is the dimensionless four-velocity corresponding to the momentum
$p$. (For non-abelian gauge theory the gauge field is a matrix
$A_{\mu}=A^{a}_{\mu}T_{a}$ with $T_{a}$ matrices that represent the generators
of the group. Because the exponent is an integral over a matrix-valued
function, the exponential is a path-ordered expression. ) This expresses the
fact that soft emissions affect the hard particle only by dressing it with a
gauge phase.

Figure 22: A process involving two eikonal lines A and B, interacting through
the exchange of soft gluons forming diagram $G$.

Having constructed the effective Feynman rules, one may proceed to demonstrate
the exponentiation of soft photon corrections as follows. As an example, we
consider graphs of the form shown in Fig. 22, at a fixed order in the
perturbative expansion. Fig. 22 consists of two eikonal lines, labelled $A$
and $B$, each of which emits a number of soft photons. One may envisage lines
$A$ and $B$ as emerging from a hard interaction, and one may consider the
graph $G$ either as a contribution to an amplitude, or to a squared amplitude
(in which case some of the propagators in $G$ will be cut).

Diagram $G$ can be taken as consisting only of soft photons and fermion loops.
Photons originating from one of the two eikonal lines must land on the other
one, or on a fermion loop inside $G$. Indeed, a photon cannot land on the same
eikonal line, as in that case the diagram is proportional to
$p^{\mu}p_{\mu}=0$.

Using eikonal Feynman rules, one finds that graphs of the form of Fig. 22
contribute to the corresponding (squared) amplitude a factor

${\cal F}_{AB}\,=\,\sum_{G}\,\left[\prod_{i}\frac{p_{A}^{\mu_{i}}}{p_{A}\cdot
k_{i}}\right]\left[\prod_{j}\frac{p_{B}^{\nu_{j}}}{p_{B}\cdot
l_{j}}\right]G_{\mu_{1}\ldots\mu_{n};\nu_{1}\ldots\nu_{m}}(k_{i},l_{j})\,,$
(189)

where $k_{i}$, $l_{j}$ are the momenta of the photons emitted from lines $A$
and $B$ respectively, with $i=1,\ldots,n$ and $j=1,\ldots,m$.

Given that we have already summed over permutations in order to obtain the
eikonal Feynman rules, each diagram $G$ can be uniquely specified by the set
of connected subdiagrams it contains, as indicated schematically in Fig. 23,
where each possible connected subdiagram $G_{c}^{(i)}$ occurs $N_{i}$ times.
According to the standard rules of perturbation theory, diagram $G$ has a
symmetry factor corresponding to the number of permutations of internal lines
which leave the diagram invariant. This symmetry factor is given by

$S_{G}=\prod_{i}\,S_{i}^{N_{i}}\,(N_{i})!\,,$ (190)

where $S_{i}$ is the symmetry factor associated with each connected subdiagram
$G_{c}^{(i)}$, and the factorials account for permutations of identical
connected subdiagrams, which must be divided out. Contracting Lorentz indices
as in (189), the eikonal factor ${\cal F}_{AB}$ may be written as

${\cal
F}_{AB}\,=\,\sum_{\\{N_{i}\\}}\,\prod_{i}\,\frac{1}{N_{i}!}\,\left[{\cal
F}_{c}^{(i)}\right]^{N_{i}}\,,$ (191)

where

${\cal
F}_{c}^{(i)}\,=\,\frac{1}{S_{i}}\,\left(\prod_{q}\frac{p_{A}^{\mu_{q}}}{p_{A}\cdot
k_{q}}\right)\left(\prod_{r}\frac{p_{B}^{\nu_{r}}}{p_{B}\cdot
l_{r}}\right)G^{(i)}_{\mu_{1}\ldots\mu_{n_{q}};\nu_{1}\ldots\nu_{m_{r}}}(k_{q},l_{r})\,,$
(192)

is the expression for each connected subdiagram, including the appropriate
symmetry factor. Recognising (191) as an exponential series, it follows that

${\cal F}_{AB}\,=\,\exp\left[\sum_{i}{\cal F}_{c}^{(i)}\right]\,.$ (193)

We conclude that soft photon corrections exponentiate in the eikonal
approximation, and the exponent is given by the sum of all connected
subdiagrams.

Figure 23: Decomposition of a soft photon graph into connected subdiagrams
$G_{c}^{(i)}$, each of which occurs $N_{i}$ times.

Having seen the abelian case, the non-abelian case for the eikonal cross
section is not all that much more difficult, though we shall not treat it
here.

Following arguments similar to that for the abelian case [79][80, 78, 82, 83,
84] one may in fact arrive at the result that the eikonal cross section is a
sum over eikonal diagrams $D$

$\sigma^{\rm(eik)}=\exp\left[\sum_{W}\sum_{D,D^{\prime}}{\cal
F}(D)R^{(W)}_{DD^{\prime}}C(D^{\prime})\right],$ (194)

where a mixing matrix $R$ connects the momentum space parts of the diagrams
${\cal F}(D)$ and their colour factors $C(D)$ in an interesting way. A very
recent, pedagogical review of this and other aspects of webs can be found in
[85].

The functions $g_{i}$ that constitute the resummation are usually not only
defined by the eikonal cross section. Hard collinear modes in the higher-order
corrections can be resummed in different ways, through so-called jet
functions. These are in fact also universal, so that by now constructing a
resummed cross section is often a matter of putting the right set of all-order
functions together. Some automation of this has already been undertaken [86,
87] and is at present being worked on further by various groups.

###  5.2 Some applications in threshold resummation: heavy quark production,
Higgs production

The very general arguments in the previous section can be applied to
transverse momentum resummation and threshold resummation. Here we focus on
the latter. An illuminating study of the effects of threshold resummation was
given in Ref.[88]. One can representing the partonic resummed cross section in
moment space as

$\displaystyle\sigma_{DY}(N,Q^{2})$ $\displaystyle=$ $\displaystyle
H(Q)\,\exp\left[G_{DY}(N,Q)\right]$ (195) $\displaystyle G_{DY}$
$\displaystyle=$ $\displaystyle 2\ln
Ng_{1}(2\lambda)+g_{2}(2\lambda)+\alpha_{s}g_{3}(2\lambda)+\ldots$ (196)
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle\beta_{0}\alpha_{s}\ln
N\,,$ (197)

which was already more schematically given in (178), and where

$g_{1}(\lambda)=\frac{C_{F}}{\beta_{0}\lambda}\left[\lambda+(1-\lambda)\ln(1-\lambda)\right]\,.$
(198)

In Fig. 24 [88] convergence properties for both the exponent and the resummed
cross section are shown (for toy PDF’s) when increasing the logarithmic
accuracy of the exponent, and of the hadronic $K$ factor.

Figure 24: Convergence behavior of Drell-Yan partonic and hadronic cross
section [88]

One observes good convergence as the logarithmic accuracy of the resummation
is increased. For the inverse Mellin transform, required to compute the
hadronic cross section in momentum space, one may use the so-called minimal
prescription [89].

Very similar to Drell-Yan is Higgs production, in the large top mass limit
where there is essentially a pointlike gluon-gluon-Higgs coupling. A recent
N3LL threshold resummed result [90] is shown in Fig. 25.

Figure 25: A recent result [90] for the N3LL cross section for Higgs
production. One observes the notably less dependence of the result on the
renormalization scale $\mu_{R}$.

We conclude this section with the already mentioned NNLO top quark cross
section results, matched to a NNLL threshold resummed calculation for this
observable [91]. For the resummed part an added complication is the accounting
for colour, as all four external particles are coloured. This issue was solved
in Refs. [92, 73, 93], we shall not go into the technical aspects of this. The
result of the very impressive, and important calculation [69] is shown in Fig.
26

Figure 26: Theoretical prediction for the LHC as a function of the collider
c.m. energy, compared to available measurement from ATLAS and/or CMS at 7 and
8 TeV.

##  6 Conclusions

In these lectures I have discussed many aspects of QCD, from the conceptual to
the practical, that are relevant for understanding and appreciating its role
in the physics of particle colliders. These aspects are often quite technical
in nature, and no doubt I have done poor justice to them in the limited space
available. Nevertheless, some attention must be given to these, in order to
assess the strengths and weaknesses of theoretical results. This is of crucial
importance when confronting these results with data. With the focus of theory
and measurement turning towards precision, having paid this attention should
be all the more valuable.

Nevertheless I hope that readers are not blinded by the technicalities, but
are able to sharpen their intuition regarding how QCD behaves a bit further.
Both technical understanding and intuition will be fruitful in the theory-
experiment collaborations, joint workshops etc in which they may find
themselves at times, and upon which much of the success of the LHC research
programme depends.

## Acknowledgements

I would like to thank the organizers of the CERN Summer School for the
excellent environment they created at the school, and for their forebearance
towards these notes. I thank the students of the school for their interest and
lively participation in the lectures and question sessions.

##  Appendix A Conventions and useful formulae

### Units

We use here $\hbar=1$ and $c=1$. Energy can be converted to inverse distance
and vice versa by the relation $1=197.3\mathrm{MeV}\,\mathrm{fm}$, with 1 fm
(“fermi” or “femtometer”) equal to $10^{-15}$m, and $c=2.998\cdot
10^{8}\;\mathrm{m/s}$.

Cross sections are expressed in nanobarns (nb), picobarns (pb) etc, where
$1\,\mathrm{b}=10^{-24}cm^{2}$.

The metric tensor we use in this course is

$\eta^{\mu\nu}=\mathrm{diag}(-1,1,1,1)$ (199)

### Cross sections and decay rates

The cross section for the scattering of two incoming massless particles with
momenta $k_{1}$ and $k_{2}$ to $n$ particles with momenta $\\{p_{i}\\}$ is
given by

$\sigma=\frac{1}{2s}\overline{\sum_{spins}}\int|{\cal M}|^{2}\,dPS(n)$ (200)

where the bar indicates initial spin averaging and

$dPS(n)=\prod_{j=1}^{n}\frac{d^{3}p_{j}}{(2\pi)^{3}\,2E_{j}}\times(2\pi)^{4}\delta\left(k_{1}+k_{2}-\sum_{i}^{n}p_{i}\right)$
(201)

If there are $j$ identical particles among the $n$, there is an extra factor
$1/j!$. For the case $n=2$, in the center of momentum frame, and with both
outgoing particles having equal mass $m$

$dPS(2)=\frac{1}{16\pi}\sqrt{1-\frac{4m^{2}}{s}}\,d\cos\theta\,,$ (202)

where $\theta$ is the polar angle of one of the outgoing particles with
respect to the collision axis.

The width for the decay of a particle with mass $Q$ and 4-momentum $k$ to $n$
particles with 4-momenta $\\{p_{i}\\}$ reads

$\Gamma=\frac{1}{2Q}\overline{\sum}_{\mathrm{spins},(\mathrm{colours}..)}\int|{\cal
M}|^{2}\,dPS(n)\,.$ (203)

If there are $j$ identical particles among the $n$, there is an extra factor
$1/j!$. For the case $n=2$, in the center of momentum frame

$dPS(2)=\frac{1}{16\pi
Q^{2}}\sqrt{\lambda(Q^{2},m_{1}^{2},m_{2}^{2})}d\cos\theta\,,$ (204)

where $\theta$ is the polar angle of one of the outgoing particles with
respect to some arbitrary axis, and
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$.

### Dirac algebra

Dirac equation in x-space:

$(\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial+m)\psi(x)=0$ (205)

Dirac equation in momentum space for $u$ and $v$ spinors:

$(i\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p+m)\,u(p,s)=0,\qquad(i\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p-m)\,v(p,s)=0$ (206)
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$ (207)

where on the right hand side the 4 by 4 unit matrix in spinor space is
implied. An often-used identity based on this is

$\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}p=p^{2}\,.$ (208)

Other useful relations:

$\gamma^{0}=-i\,\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),\qquad\vec{\gamma}=-i\,\left(\begin{array}[]{cc}0&\vec{\sigma}\\\
-\vec{\sigma}&0\end{array}\right)\,,$ (209)
$(\gamma^{\mu})^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}\,,$ (210)
$\gamma_{5}=i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}\,,$ (211)
$\gamma_{5}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\,,$ (212)
$\left\\{\gamma_{5},\gamma^{\mu}\right\\}=0\,,$ (213) $\gamma^{\mu}\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}a\gamma_{\mu}=-2\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}a,\quad\gamma^{\mu}\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}a\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}b\gamma_{\mu}=4a\cdot
b,\quad\gamma^{\mu}\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}a\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}b\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}c\gamma_{\mu}=-2\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}c\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}b\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}a\,,$ (214)
$\displaystyle\mathrm{Tr}(\gamma^{\mu})=\mathrm{Tr}(\gamma_{5})=0\,,\quad\mathrm{Tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho})=0\,,$
(215)
$\displaystyle\mathrm{Tr}(\gamma_{5}\gamma_{\mu})=\mathrm{Tr}(\gamma_{5}\gamma_{\mu}\gamma_{\nu})=\mathrm{Tr}(\gamma_{5}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho})=0\,,$
$\mathrm{Tr}(\gamma^{\mu}\gamma^{\nu})=4\,\eta^{\mu\nu},\quad\mathrm{Tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=4\,(\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho})\,,$
(216)
$\mathrm{Tr}(\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{5})=-4i\epsilon_{\mu\nu\rho\sigma}\,,\quad\epsilon_{0123}=+1\,.$
(217)

Conjugate spinor:

$\bar{\psi}=i\psi^{\dagger}\gamma^{0}\,.$ (218)

With this definition the term

$\bar{\psi}\psi$ (219)

is hermitean, since $\gamma^{0}$ is anti-hermitean. The chiral (left- and
righthanded) projections of a fermion are defined by

$\displaystyle\psi_{L}$ $\displaystyle=$
$\displaystyle\left(\frac{1+\gamma_{5}}{2}\right)\psi\,,$
$\displaystyle\psi_{R}$ $\displaystyle=$
$\displaystyle\left(\frac{1-\gamma_{5}}{2}\right)\psi\,.$ (220)

A parity transform on a spinor is defined by

$P:\psi(x)\rightarrow\gamma_{0}\psi(\tilde{x})\,,$ (221)

with $\tilde{x}=(x^{0},-\vec{x})$. Complex-conjugation of general spinor-
trace:

$(\bar{u}(p)\gamma_{\mu_{1}}\cdots\gamma_{\mu_{n}}u(p^{\prime}))^{*}=(-)^{n}\;(\bar{u}(p^{\prime})\gamma_{\mu_{n}}\cdots\gamma_{\mu_{1}}u(p))\,,$
(222)
$(\bar{u}(p)\gamma_{\mu_{1}}\cdots\gamma_{\mu_{n}}\gamma_{5}u(p^{\prime}))^{*}=-(-)^{n}\;(\bar{u}(p^{\prime})\gamma_{5}\gamma_{\mu_{n}}\cdots\gamma_{\mu_{1}}u(p))\,.$
(223)

### Unitary groups and their Lie algebras

The U(1) Lie algebra has only 1 generator $t$, which we choose hermitean.
Acting on a d-dimensional vector $t$ may be represented as the d-dimensional
unit matrix. The group elements are then

$\exp\left(i\alpha\,t\right)\,.$ (224)

The SU(2) Lie algebra has 3 generators $t_{i},\,i=1,2,3$. If we choose the
$t_{i}$ hermitean, then in the fundamental representation
$t^{(F)}_{i}=\sigma_{i}/2$, with $\sigma_{i}$ the Pauli matrices

$\displaystyle\begin{array}[]{ccc}\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)&\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\par&\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\,.\end{array}$ (232)

The group elements are

$U=\exp\left(i\xi^{i}\,t_{i}\right)\,.$ (233)

Note that for the fundamental representation

$\sigma_{2}U\sigma_{2}=U^{*}=(U^{\dagger})^{T}\,.$ (234)

Note that the SU(2) Lie-algebra is isomorphic to the SO(3) Lie-algebra, whose
generators $t_{i}=-iS_{i}$ read, in the fundamental representation:

$\displaystyle\begin{array}[]{ccc}S_{1}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&-1&0\end{array}\right)&S_{2}=\left(\begin{array}[]{ccc}0&0&-1\\\
0&0&0\\\ 1&0&0\end{array}\right)\par&S_{3}=\left(\begin{array}[]{ccc}0&1&0\\\
-1&0&0\\\ 0&0&0\end{array}\right)\end{array}\,.$ (245)

The SU(3) Lie algebra has 8 generators $t_{i},\,i=1,\ldots 8$, which are not
needed explicitly. Lie algebra generators in general obey the commutation
relations

$\left[t_{i},t_{j}\right]=i\,f_{ijk}\,t_{k}\,,$ (246)

with the $f_{ijk}$ the structure constants for the given group. The $kj$
matrix element of the generator $t^{(A)}_{i}$ in the adjoint representation is
defined as

$\left[t^{(A)}_{i}\right]_{kj}=i\,f_{ijk}\,.$ (247)

Note that we can always choose anti-hermitean generators $t^{\prime}$ by
multiplying the hermitean versions $t$ by $i$. In that case the group elements
for SU(2) e.g. are

$\exp\left(\xi^{i}\,t^{\prime}_{i}\right)\,.$ (248)

Representations of SU(2), etc. groups are often indicated by
$\underline{2},\ldots$ indicating the size of the matrices of that
representation. Trivial or singlet representation are then indicated with
$\underline{1}$.

Some other group theory factors (the so-called Casimir factors):

$\displaystyle\mathrm{SU(2)}:\quad\quad C_{A}$ $\displaystyle=$ $\displaystyle
2,\quad C_{F}=\frac{3}{4}\,,$ $\displaystyle\mathrm{SU(3)}:\quad\quad C_{A}$
$\displaystyle=$ $\displaystyle 3,\quad C_{F}=\frac{4}{3}\,.$

For the fundamental representation of $SU(N)$ we have

$\mathrm{Tr}\left[t^{(F)}_{i}t^{(F)}_{j}\right]=\frac{1}{2}\delta_{ij}\,.$
(249)

### Standard Model quantities

The amount of electric charge (in units of $e$) $Q$, the hypercharge $Y$ and
the third component of weak isospin $t_{3}$ are related by

$Y=2(Q-t_{3})\,.$ (250)

In the Standard Model the generator of the U(1) of hypercharge is
conventionally written as

$\frac{1}{2}Y\,,$ (251)

and is then represented on d-dimensional vectors as $1/2$ times the
hypercharge eigenvalue times the unit matrix.

The gauge couplings associated with the $\mathrm{SU}_{I_{W}}(2)$ and
$\mathrm{U}_{Y}(1)$ gauge groups are traditionally denoted $g$ and
$g^{\prime}$ respectively. In terms of these couplings the unit of electric
charge is

$e=\frac{g\,g^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}=g\,\sin\theta_{W}=g^{\prime}\,\cos\theta_{W}\,,$ (252)

with $\sin^{2}\theta_{W}\simeq 0.226$.

The gauge fields are mixed as follows:

$\displaystyle B_{\mu}$ $\displaystyle=$
$\displaystyle\cos\theta_{W}A_{\mu}-\sin\theta_{W}Z_{\mu},\quad
W^{3}_{\mu}=\cos\theta_{W}Z_{\mu}+\sin\theta_{W}A_{\mu},$ (253) $\displaystyle
W^{1}_{\mu}-iW^{2}_{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{2}W_{\mu}^{+},\quad
W^{1}_{\mu}+iW^{2}_{\mu}=\sqrt{2}W_{\mu}^{-}\,.$ (254)

The Fermi constant is defined by

$G_{F}=\frac{g^{2}}{4\sqrt{2}m_{W}^{2}}=\frac{1}{\sqrt{2}v^{2}}\simeq
1.2\times 10^{-5}\,\mathrm{GeV}^{2}\,.$ (255)

Vector boson masses in GeV:

$m_{Z}=\frac{gv}{2\cos\theta_{W}}=91.1876\pm 0.0021,\quad
m_{W}=\frac{gv}{2}=80.385\pm 0.015\,.$ (256)

The photon and gluon are massless.

Heavy quark masses in GeV:

$m_{c}=1.5,\quad m_{b}=5,\quad m_{t}=175\,.$ (257)

$u,d,s$ are massless.

Lepton masses in GeV:

$m_{\tau}=1.7,\quad m_{\mu}=0.105,\quad m_{e}=0.0005\,.$ (258)

QCD scale in GeV:

$\Lambda_{QCD}\simeq 0.2\,.$ (259)

### Assorted Feynman rules

All momenta in vertex Feynman rules are incoming here.

  * •

Fermion propagator $\longrightarrow$p$\alpha$$\beta$

$\frac{1}{i(2\pi)^{4}}{(-i\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}p+m)_{\beta\alpha}\over p^{2}+m^{2}-i\epsilon}$ (260)

  * •

Electron-electron photon vertex

$\begin{array}[]{cc}&\begin{picture}(110.0,105.0) \put(0.0,0.0){}
\put(0.0,0.0){} \put(0.0,0.0){} \raise 70.0pt\hbox to0.0pt{\kern
60.0pt\makebox(0.0,0.0)[l]{$p_{3}$}\hss} \ignorespaces \raise 25.0pt\hbox
to0.0pt{\kern 20.0pt\makebox(0.0,0.0)[lb]{$p_{1}$}\hss} \ignorespaces \raise
25.0pt\hbox to0.0pt{\kern 80.0pt\makebox(0.0,0.0)[rb]{$p_{2}$}\hss}
\ignorespaces \raise 5.0pt\hbox
to0.0pt{\kern-5.0pt\makebox(0.0,0.0)[b]{$\alpha$}\hss} \ignorespaces \raise
5.0pt\hbox to0.0pt{\kern 105.0pt\makebox(0.0,0.0)[b]{$\beta$}\hss}
\ignorespaces \raise 105.0pt\hbox to0.0pt{\kern
50.0pt\makebox(0.0,0.0)[b]{$\mu$}\hss} \ignorespaces \end{picture}\end{array}$
$i(2\pi)^{4}\delta(p_{1}+p_{2}+p_{3})(-ie)\,\gamma^{\mu}_{\beta\alpha}$ (261)

where the three momentum vectors (not drawn) are pointing to the vertex.

  * •

Outgoing fermion:  $p$ $\overline{u}(p,s)$ Row spinor

  * •

Outgoing anti-fermion:  $p$ $v(p,s)$ Column spinor

  * •

Incoming fermion:  $u(p,s)\;\;$ $p$ Column spinor

  * •

Incoming antifermion:  $\overline{v}(p,s)\;\;$ $p$ Row spinor

  * •

Outgoing vector boson:  $k$ $\epsilon^{*}_{\mu}(k,\lambda)$

  * •

Incoming vector boson:  $\epsilon_{\mu}(k,\lambda)$ $k$

In Feynman diagrams, always start where the charge vector top of the fermion
lines ends (i.e. start with a row spinor), and work your way back against the
charge flow.

Integrate over internal momenta: $\int d^{4}k_{i}$

Completeness relations for spin sums over polarization spinors and
polarization vectors, associated with external particles

$\sum_{s}u_{\alpha}(p,s)\overline{u}_{\beta}(p,s)=(-i\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p+m)_{\alpha\beta}\,,$ (262)
$\sum_{s}v_{\alpha}(p,s)\overline{v}_{\beta}(p,s)=(-i\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p-m)_{\alpha\beta}\,,$ (263)

with $m$ the fermion mass, $\alpha,\beta$ are spinor indices.

For photons the sum over the two physical polarizations gives

$\sum_{\lambda}\epsilon^{\mu}(k,\lambda)\epsilon^{*\,\nu}(k,\lambda)=\eta^{\mu\nu}-\frac{k^{\mu}\bar{k}^{\nu}+k^{\nu}\bar{k}^{\mu}}{k\cdot\bar{k}}\,,$
(264)

where the sum is over the physical spin only, and we define

$k^{\mu}=(k^{0},\vec{k}),\qquad\bar{k}^{\mu}=(-k^{0},\vec{k})\,.$ (265)

The photon propagator is ($\kappa$ is the gauge parameter here)

$\Delta_{\mu\nu}(k)=\frac{1}{i(2\pi)^{4}}\frac{1}{k^{2}-i\epsilon}\left(\eta_{\mu\nu}-\left(1-\frac{1}{\kappa^{2}}\right)\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\,.$
(266)

The scalar field propagator is simply

$\Delta(p)=\frac{1}{i(2\pi)^{4}}\frac{1}{p^{2}+m^{2}-i\epsilon}\,.$ (267)

For massive vector bosons the sum over the three physical polarizations gives

$\sum_{\lambda}\epsilon^{\mu}(k,\lambda)\epsilon^{*\,\nu}(k,\lambda)=\eta^{\mu\nu}+\frac{k^{\mu}k^{\nu}}{M^{2}}\,.$
(268)

### Loop integrals

$\frac{1}{A_{1}A_{2}}=\int_{0}^{1}dx_{1}\int_{0}^{1}dx_{2}\frac{\delta(1-x_{1}-x_{2})}{\left[x_{1}A_{1}+x_{2}A_{2}\right]^{2}}\,.$
(269)

The result for the integral

$I(n,\alpha)=\int d^{n}q\frac{1}{(q^{2}+m^{2}-i\epsilon)^{\alpha}}$ (270)

is

$I(n,\alpha)=i\pi^{n/2}\frac{\Gamma(\alpha-(n/2))}{\Gamma(\alpha)}\left(m^{2}\right)^{(n/2)-\alpha}\,.$
(271)

The Euler gamma function $\Gamma(z)$ has the following relevant properties:

$\displaystyle\Gamma(z+1)$ $\displaystyle=$ $\displaystyle
z\Gamma(z),\quad\Gamma(1)=1\,,$ (272) $\displaystyle\ln\Gamma(1+z)$
$\displaystyle\simeq$ $\displaystyle-z\gamma_{E}+{\cal
O}(z^{2}),\quad\gamma_{E}=0.577\ldots$ (273)

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# Revisiting photon-statistics effects on multiphoton ionization. II.

G. Mouloudakis1<EMAIL_ADDRESS>P. Lambropoulos1,2 1Department of
Physics, University of Crete, P.O. Box 2208, GR-71003 Heraklion, Crete, Greece
2Institute of Electronic Structure and Laser, FORTH, P.O.Box 1527, GR-71110
Heraklion, Greece

###### Abstract

In this paper, we extend the results of an earlier paper in which we had
demonstrated the limitations of the notion of nonresonant multiphoton
ionization, in the exploration of photon statistics effects in non-linear
processes. Through the quantitative analysis of specific realistic processes,
we provide the connection to conditions of intensity and pulse duration
necessary in relevant experiments, including a recent seminal experiment
demonstrating the effect of superbunching found in squeezed radiation.

PACS numbers

32.80.Rm, 42.50.Ar

††preprint: APS/123-QED

##  I Introduction

In a recent paper ref1 , referred to hereafter as I, we examined the effect of
field fluctuations on near-resonant 2- and 3-photon ionization. Through a
quantitative analysis of the role of intensity, we found that the enhancement
of the process due to bunching, more often than not, will be reduced owing to
the onset of Rabi oscillations, even at seemingly large detuning from
resonance. This makes the notion of non-resonant ionization rather tenuous and
possibly misleading in the planning of an experiment. For processes of order
higher than 3, it is practically inevitable that near resonances will modify
the enhancement, even at moderate intensities; simply because the level
spacing decreases with increasing level excitation.

That paper came at an auspicious time as it practically coincided with an
experimental breakthrough by Spasibko et al. ref2 , in which enhancement in
processes of order up to 4 due to superbunched light was reported. This
represents a break in the rather long period of experimental drought in this
field. Because, despite the rather extensive relevant theoretical literature
over the last 50 years ref3 ; ref4 ; ref5 ; ref6 ; ref7 ; ref8 ; ref9 ; ref10
, experimental results have been quite sparse ref11 ; ref12 ; ref13 ; ref14 ;
ref15 ; ref16 ; ref17 . But most significantly because photon statistics
enhancement due to squeezed (superbunched) light has been observed for the
first time; a development that opens the route towards the experimental
investigation of a number of challenging open problems.

Owing to computational demands, the numerical results in I, illustrating the
involvement of near-resonances and the ensuing modification of the bunching
effect with increasing intensity, were obtained with scaled intensity and
atomic parameters; such as Rabi frequencies and ionization cross sections. The
need for scaling arose from the necessity of formulating the problem in terms
of a quantized field, which then led to summations over the photon number
distributions of the various states of the field we examined. The concomitant
drawback of that approach is the difficulty in translating photon numbers to
intensity of a pulsed source, which is what enters an experiment ref2 . Our
purpose in this article is to provide a bridge between the two, through a
sample of realistic calculations in atomic systems.

For that purpose, we have chosen the cases of 2-photon ionization in Cesium in
the vicinity of the intermediate 7p state and 3-photon ionization in Sodium,
in the vicinity of 2-photon intermediate resonance with the 4d(3/2) and
4d(5/2) states. It is common knowledge that, for a non-resonant N-photon
process, the ionization yield as a function of laser intensity, in a log-log
plot, will be a straight line of slope N ref3 . When, with increasing
intensity, an intermediate state near resonance begins playing a role, the
slope begins diminishing. The intensity at which this change of slope becomes
noticeable is the intensity at which the bunching enhancement will also begin
decreasing.

The most convenient formal tool for the calculation of the dependence of ion
yields on intensity is the time-dependent density matrix of the atomic system,
driven by a classical electromagnetic (EM) field. This formulation, in
addition to allowing for realistic calculations which include the source
temporal pulse shape, it does also allow for the incorporation of the laser
bandwidth; an experimentally important aspect not accessible in the formalism
of I. Evaluating the role of the bandwidth addresses the possibility that,
even with the carrier frequency tuned far from resonance, the wings of the
spectral shape of the pulse may involve real excitation of a near resonant
state, thereby altering the slope and therefore the enhancement due to
bunching.

A short clarification of the issue of bandwidth is needed at this point. The
bandwidth of a pulsed source involves two separate contributions. One is due
to the finite duration of the pulse and is referred to as the Fourier
bandwidth. The second is due to stochastic phase and/or amplitude fluctuations
of the field; the latter being the cause of bunching. Depending on the
duration of the pulse, the stochastic bandwidth may become important and
therefore would have to be included in the formulation. For the sake of
completeness we have included the effects of the stochastic bandwidth in our
calculations.

For pulses of extremely short duration of the order of a few femtoseconds, it
is the Fourier bandwidth that dominates. For longer pulses of hundreds of
femtoseconds or longer, it is the stochastic bandwidth that dominates, which
is in fact the case with sources in which bunching is significant ref2 . In
the examples discussed in this paper, the pulses we have chosen are of
relatively long duration, so that it is the stochastic bandwidth that
dominates. Nevertheless, the Fourier bandwidth is inherently included in the
calculation, through the solution of the time-dependent density matrix
differential equations.

In order to avoid repetition of formal aspects and derivations readily
available in the literature ref18 , we have chosen to simply provide the basic
differential equations governing the time evolution of the density matrix for
the 3-photon process, as a special case of which the 2-photon process can be
obtained. After a brief discussion and explanation of the basic equations in
section II, a collection of illustrative examples with discussion and
conclusions are presented in section III.

##  II Theory

Consider the atom to be initially in its ground state, denoted by
$\left|1\right\rangle$, in the presence of an external electric field of the
form $E(t)=\mathcal{E}(t){e^{i\omega t}}+c.c.$, where $\omega$ is the
frequency of the field and $\mathcal{E}(t)$ its complex amplitude. The field
amplitude can be expressed as
$\mathcal{E}(t)=\left|{\mathcal{E}(t)}\right|\exp[i\varphi(t)]$, where both
$\left|{\mathcal{E}(t)}\right|$ and $\varphi(t)$ can in general be stochastic
quantities, owing to fluctuations of the field. The absorption of two photons
excites the atom to the vicinity of an intermediate state
$\left|2\right\rangle$, whose energy is denoted by
${\omega_{2}}\left({\hbar=1}\right)$. In addition to the coherent coupling
between the ground and excited state induced by the external field, the state
$\left|2\right\rangle$ can either decay spontaneously or be ionized by a
single photon absorption, with a rate denoted by $\Gamma(t)$. When state
$\left|2\right\rangle$ is connected to the initial state by a two-photon
transition, spontaneous decay back to $\left|1\right\rangle$ is allowed only
through a cascade. For the sake of completeness we include in our equations
that rate, in the sense of an effective rate denoted by $\gamma$. In any case,
for the type of experiments pertaining to our problem, that rate is too small
to be of relevance. The excited state $\left|2\right\rangle$, in addition to
its coupling to the initial state and to the continuum, it does also undergo a
Stark shift through virtual transitions to all dipole allowed bound and
continuum states. It is denoted here by $S(t)$, where the time dependence is
due to the fact that the shift is proportional to the instantaneous intensity.
Since in this case, the two-photon Rabi frequency, as well as the ionization
rate, are also proportional to the intensity, the shift needs to be included
in the formalism.

If ${\rho_{ij}}$, $i,j=1,2$ are the matrix elements of the density matrix of
our effective two-level model of the three-photon process, in the rotating
wave approximation (RWA), then the differential equations governing the time
evolution of the respective slowly-varying matrix elements, defined by
${\rho_{ii}}(t)={\sigma_{ii}}(t)$, $i=1,2$ and
${\rho_{12}}(t)={\sigma_{12}}(t){e^{i\omega t}}$ are ref1 :

${\partial\over{\partial
t}}{\sigma_{11}}\left(t\right)=\gamma{\sigma_{22}}\left(t\right)+2{\mathop{\rm
Im}\nolimits}\left[{\Omega_{12}^{*}\left(t\right){\sigma_{12}}\left(t\right)}\right]$
(1) ${\partial\over{\partial
t}}{\sigma_{22}}\left(t\right)=-\left[{\gamma+\Gamma\left(t\right)}\right]{\sigma_{22}}\left(t\right)-2{\mathop{\rm
Im}\nolimits}\left[{\Omega_{12}^{*}\left(t\right){\sigma_{12}}\left(t\right)}\right]$
(2) ${\partial\over{\partial
t}}{\sigma_{12}}\left(t\right)=\left\\{{i\left[{\Delta-S(t)}\right]-{\gamma_{12}}\left(t\right)}\right\\}{\sigma_{12}}\left(t\right)\\\
+i{\Omega_{12}}\left(t\right)\left[{{\sigma_{22}}\left(t\right)-{\sigma_{11}}\left(t\right)}\right]$
(3)

where $\Delta=2\omega-({\omega_{2}}-{\omega_{1}})$ is the detuning from the
intermediate resonance, ${\gamma_{12}}\left(t\right)={1\over
2}\left[{\gamma+\Gamma\left(t\right)}\right]$ is the off-diagonal relaxation
and ${\Omega_{12}}\left(t\right)={\hbar^{-2}}{\mu_{12}}\mathcal{E}^{2}(t)$ is
the effective two-photon Rabi frequency of the
$\left|1\right\rangle\leftrightarrow\left|2\right\rangle$ transition, given by
the product of the effective two-photon matrix element ${{\mu_{12}}}$ of the
dipole operator and the square of the electric field amplitude.

All parameters depending on the applied EM field, namely the ionization rate,
the Stark shift and the Rabi frequency, are in general stochastic quantities,
as they are subjected to the fluctuations of the field. As a result, the
equations of motion of the density operator’s matrix elements are also
stochastic. The next step is to average the differential equations (1) to (3)
over the field fluctuations, denoting such averaged quantities by angular
brackets. In the process of averaging the above set of differential equations,
we encounter atom-field products of the form
$\left\langle{\Gamma(t){\sigma_{ij}}\left(t\right)}\right\rangle$,
$\left\langle{{\Omega_{12}}(t){\sigma_{ij}}\left(t\right)}\right\rangle$,
etc., $i,j=1,2$. The rigorous evaluation of such products require the exact
model of the stochastic properties of the field, entailing considerable
mathematical complexity, which is beyond the scope of this paper. However, a
detailed treatment of how this can be accomplished can be found in reference
ref5 . Since our purpose in this paper is to simply assess the effect of the
laser bandwidth, in an approximate fashion, we adopt the decorrelation
approximation, which amounts to replacing the stochastic averages of products
such as the above, by the products of their averages. Assuming that the
deterministic real field amplitude is constant and equal to $\mathcal{E}_{0}$
(square pulse shape), the resulting differential equations governing the time-
evolution of the averaged density matrix elements are:

${\partial\over{\partial
t}}\left\langle{{\sigma_{11}}(t)}\right\rangle=\gamma\left\langle{{\sigma_{22}}(t)}\right\rangle+2{\mathop{\rm
Im}\nolimits}\left\\{{i\bar{\Omega}_{12}^{2}\int\limits_{0}^{t}{dt^{\prime}{e^{i\left[{\Delta-\left\langle
S\right\rangle-\left\langle{{{\tilde{\gamma}}_{12}}}\right\rangle}\right]\left({t-t^{\prime}}\right)}}\left[{\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle-\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle}\right]}}\right\\}$
(4) ${\partial\over{\partial
t}}\left\langle{{\sigma_{22}}(t)}\right\rangle=-\left[{\gamma+\left\langle\Gamma\right\rangle}\right]\left\langle{{\sigma_{22}}(t)}\right\rangle-2{\mathop{\rm
Im}\nolimits}\left\\{{i\bar{\Omega}_{12}^{2}\int\limits_{0}^{t}{dt^{\prime}{e^{i\left[{\Delta-\left\langle
S\right\rangle-\left\langle{{{\tilde{\gamma}}_{12}}}\right\rangle}\right]\left({t-t^{\prime}}\right)}}\left[{\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle-\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle}\right]}}\right\\}$
(5)

where
$\left\langle{{{\tilde{\gamma}}_{12}}}\right\rangle\equiv\left\langle{{\gamma_{12}}}\right\rangle+{\gamma_{L}}={1\over
2}\left({\gamma+\left\langle\Gamma\right\rangle+2{\gamma_{L}}}\right)$ and
${{\bar{\Omega}}_{12}}\equiv{\hbar^{-1}}{\mu_{12}}{\mathcal{E}_{0}^{2}}$ the
average Rabi frequency of the
$\left|1\right\rangle\leftrightarrow\left|2\right\rangle$ transition.

Note that $\left\langle{\Gamma(t)}\right\rangle$ and
$\left\langle{S(t)}\right\rangle$ have been replaced by
$\left\langle{\Gamma}\right\rangle$ and $\left\langle{S}\right\rangle$,
respectively, since we assumed that the deterministic field amplitude is
constant. The above averaged equations contain the additional term
${{\gamma_{L}}}$ in the off-diagonal relaxation which represents the bandwidth
of the radiation source. This result is the consequence of relations ref19 ;
ref20 ; ref21  that connect higher-order field correlation functions, to the
second-order field correlation function:

$\left\langle{\mathcal{E}\left({{t_{1}}}\right){\mathcal{E}^{*}}\left({{t_{2}}}\right)}\right\rangle=\left\langle{{\mathcal{E}^{2}}}\right\rangle\exp\left[{-{1\over
2}{\gamma_{L}}\left|{{t_{1}}-{t_{2}}}\right|}\right]$ (6)

where $\left\langle{{\mathcal{E}^{2}}}\right\rangle$ is the variance of the
electric field. Such field correlation functions appear in the process of
averaging over field’s stochastic fluctuations and introduce the laser
bandwidth in our model.

The solution of the equations of motion of the averaged density matrix
elements, provide the probability of ionization through the expression

$P(t)=1-\left\langle{{\sigma_{11}}(t)}\right\rangle-\left\langle{{\sigma_{22}}(t)}\right\rangle$
(7)

which can be obtained either numerically or analytically with the use of
Laplace transform, if the field amplitude is not pulsed but can be assumed
constant.

We are particularly interested in the behavior of the ionization probability
as a function of the intensity, for various detunings from the intermediate
resonance. In order to provide results pertaining to a realistic model, we
apply our theory to the $3s\to\to 4d\to Continuum$ process in Na, as described
in the next section.

The above theoretical model, with minor modifications, reduces to the case of
two-photon near-resonant ionization. The form of the density matrix equations
is the same. Since the coupling between the initial and excited state is
mediated by a single-photon transition, the Rabi frequency now is proportional
to the field amplitude and not the intensity. For the same reason the
relaxation constant of the off-diagonal matrix element is now given by
${\tilde{\gamma}_{12}}={1\over
2}\left[{\gamma+\Gamma\left(t\right)+{\gamma_{L}}}\right]$, where the laser
bandwidth ${{\gamma_{L}}}$ is not multiplied by 2, as it was in the previous
3-photon case.

As discussed in the next section, the model is applied to the realistic two-
photon process $6s\to 7p\to Continuum$ process in Cs.

##  III Results and Discussion

In this section, we present quantitative results from the application of our
theoretical model to the two realistic processes described above, namely the
$3s\to\to 4d\to Continuum$ transition in Na and the $6s\to 7p\to Continuum$ in
Cs. The atomic parameters ref18 ; ref22  obtained through quantum defect
theory are: for the Na transition, ${\omega_{21}}=4.2845eV$,
$\Gamma\left[{Hz}\right]=9.4I\left[{W/c{m^{2}}}\right]$,
$\Omega\left[{Hz}\right]=1.46\times{10^{3}}I\left[{W/c{m^{2}}}\right]$,
$S\left[{Hz}\right]=179I\left[{W/c{m^{2}}}\right]$ while for the Cs
transition, ${\omega_{21}}=2.665eV$,
$\Gamma\left[{Hz}\right]=11I\left[{W/c{m^{2}}}\right]$,
$\Omega\left[{Hz}\right]=0.75\times{10^{7}}\sqrt{I\left[{W/c{m^{2}}}\right]}$,
where $I$ is the field intensity. Note that the Stark shift for an ”1+1”
photon process such as the $6s\to 7p\to Continuum$, as well as the spontaneous
decay rates for both transitions in the range of intensities considered are
negligible and can be neglected. The laser bandwidth used in our calculations
is ${\gamma_{L}}=2\times{10^{13}}Hz$.

Figure 1: Ionization probability of Cs as a function of the intensity for
various detunings from the intermediate resonance and $T=100fs$. Solid line:
$\Delta/{\omega_{2}}=0.01$, dashed line: $\Delta/{\omega_{2}}=0.05$, dotted
line: $\Delta/{\omega_{2}}=0.1$.

The illustrative results are summarized in figures 1 through 4. It should
first be noted that the pulse duration is an additional parameter affecting
the slope of the ionization signal as a function of intensity. Obviously, for
any peak intensity, if the pulse duration is sufficiently long, complete
ionization will occur. In that limit, the slope will become zero, which will
eventually be reached through a gradual decrease of the slope with increasing
intensity, as shown in figures 1 and 2. A similar behaviour is shown in
figures 3 and 4 with the difference that the slope becomes zero for
intensities below the complete ionization regime. This indicates that higher-
order processes are more prone to depart from the non-resonant condition as we
increase the intensity, compared to lower order ones.

Figure 2: Ionization probability of Cs as a function of the intensity for
various detunings from the intermediate resonance and $T=1ps$. Solid line:
$\Delta/{\omega_{2}}=0.01$, dashed line: $\Delta/{\omega_{2}}=0.05$, dotted
line: $\Delta/{\omega_{2}}=0.1$.

Aside from this difference, the feature shared by all cases is that for larger
detuning (dotted as compared to solid lines) from the intermediate resonance,
the unperturbed slope reflecting the overall order of the process, persists
for higher intensities. This behavior reflects the onset of the effect of the
increasing Rabi frequency, signaling the onset of departure from the notion of
a non-resonant process. It is precisely the effect which in reference 1 was
shown to herald the distortion of the photon statistics enhancement; although
the process nominally is non-resonant. Therefore the message emerging from
these results is that up to ${10^{10}}$ ${W/c{m^{2}}}$ or so, it can be
assumed that the non-resonant behavior will persist. And that is the type of
assessment that served as the motivation for this paper.

It could be argued that the results may be of limited usefulness as they
pertain to specific atomic transitions. However, they may not be as limited as
they might seem at first sight. For one thing, parameters such as matrix
elements entering in non-linear transitions do not differ by orders of
magnitude. Cognizant of the specificity of our results, we do not claim exact
limiting values of the intensities, but only a range of intensities. The
validity of that range is further underscored by comparison with the results
of the recent data by Spasibko et al. ref2  who observed non-resonant slopes
in harmonic generation up to order 4, for intensities in the above range. And
that was in a completely different material.

In view of the above estimates, in combination with the results in ref2 , and
the dramatic progress in the potential for observation of non-linear processes
induced by squeezed light, it appears that novel effects can be observed under
intensities in the range of ${10^{9}}$ to ${10^{10}}$ ${W/c{m^{2}}}$; as it is
intensities in that range that are needed for the observation of such
processes and are luckily becoming available even for squeezed radiation.

Figure 3: Ionization probability of Na as a function of the intensity for
various detunings from the intermediate resonance and $T=100fs$. Solid line:
$\Delta/{\omega_{2}}=0.01$, dashed line: $\Delta/{\omega_{2}}=0.05$, dotted
line: $\Delta/{\omega_{2}}=0.1$. Figure 4: Ionization probability of Na as a
function of the intensity for various detunings from the intermediate
resonance and $T=1ps$. Solid line: $\Delta/{\omega_{2}}=0.01$, dashed line:
$\Delta/{\omega_{2}}=0.05$, dotted line: $\Delta/{\omega_{2}}=0.1$.

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