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The Pennsylvania State University
The Graduate School
Eberly College of Science
HELICITY AMPLITUDES ON THE LIGHT-FRONT
A Dissertation in
Physics
by
Christian A. Cruz Santiago
© 2015 Christian A. Cruz Santiago
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2015
The dissertation of Christian A. Cruz Santiago was reviewed and approved∗ by the
following:
Anna M. Stasto
Associate Professor of Physics
Dissertation Advisor, Chair of Committee
Eugenio Bianchi
Assistant Professor of Physics
John C. Collins
Distinguished Professor of Physics
Victor Nistor
Professor of Mathematics
Richard W. Robinett
Professor of Physics
Associate Head for Undergraduate and Graduate Students
∗Signatures are on file in the Graduate School.
Abstract
Significant progress has been made recently in the field of helicity amplitudes. Currentlythere are on-shell recursion relations with shifted complex momenta, geometric interpre-tations of amplitudes and gauge invariant off-shell amplitudes. All this points to helicityamplitudes being a rich field with much more to say. In this work we take initial stepsin understanding amplitudes through the light-front formalism for the first time.
We begin by looking at crossing symmetry. In the light-front it is not obvious thatcrossing symmetry should be present as there are non-local energy denominators thatmix energies of different states. Nevertheless, we develop a systematic approach to relate,for example, 1 → N gluon processes to 2 → N − 1 processes. Using this method, wegive a perturbative proof of crossing symmetry on the light-front. One important caveatis that the proof requires the amplitudes to be on-shell. We also saw that the analyticcontinuation from outgoing to incoming particle produces a phase that’s dependent onthe choice of polarizations.
Next, we reproduce the Parke-Taylor amplitudes. For this purpose we found a re-cursion relation for an off-shell object called the fragmentation function. This recursionrelies on the factorization property of the fragmentation functions, and it becomes appar-ent that this recursion is the light-front analog of the Berends-Giele recursion relation.We also found this object’s connection to off-shell and on-shell amplitudes. The solutionfor the off-shell amplitude, which does reproduce the Parke-Taylor amplitudes in theon-shell limit, turns out to be very interesting. It can be written as a linear sum ofoff-shell objects with the same structure as MHV amplitudes.
Finally, we look at the Wilson line approach to generate gauge invariant off-shellamplitudes. It turns out that the exact same recursion relation appears on both frame-works, thereby providing the interpretation that our recursion relation has it’s origins ingauge invariance. This also proved that the interesting, off-shell, MHV-like object thatappeared algebraically in our solution is gauge invariant. We also show that for a Wardidentity calculation the light-front rules must be modified. The Ward identity involvesan extra instantaneous term that has the effect of conserving full four-momentum in thenumerator of the amplitude.
iii
Table of Contents
List of Figures vii
List of Tables ix
Chapter 1Introduction 1
Chapter 2Basics of tree level amplitudes 62.1 Helicity amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Polarization vectors . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Color ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Tree level examples: Parke-Taylor amplitudes . . . . . . . . . . . . . . . 152.4 Berends-Giele recursion relations for off-shell currents . . . . . . . . . . 17
Chapter 3Introduction to light-front 213.1 Light-front coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Poincare group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Light-front rules for the gauge theory . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.2 Gluon propagator in the light-front theory . . . . . . . . . . . . . 263.3.3 Analysis of a gluon scattering diagram . . . . . . . . . . . . . . . 293.3.4 Formal light-front rules . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Modified light-front rules . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Factorization of fragmentation tree amplitudes . . . . . . . . . . . . . . 38
iv
Chapter 4Light-front wavefunctions and fragmentation functions 414.1 Conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Fragmentation function . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.1 Recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.2 Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.4 T (−→−+...+) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Connection to amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.1 M(+→+...+)N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Chapter 5Crossing symmetry on the light-front 575.1 Crossing symmetry example . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Crossing symmetry proof . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Guiding example . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.2 General proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.3 Equal contributions from all Gi,j for fixed i . . . . . . . . . . . . 70
Chapter 6Off-shell (+→ −+ . . .+) amplitudes on the light-front 74
6.1 M(+→−++)3, wf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 General tree-level off-shell amplitudes:
M(+→−+...+)N, ff and MHV . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.1 Recursion relations for (+→ −+ . . .+) . . . . . . . . . . . . . . 806.2.2 Pattern and solution . . . . . . . . . . . . . . . . . . . . . . . . . 816.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2.4 Proof of identity (6.41) . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter 7Gauge Invariance of Off-Shell Amplitudes 957.1 The Ward identity for light-front amplitudes . . . . . . . . . . . . . . . 96
7.1.1 Example: the Ward identity check for the lowest order amplitude 967.1.2 Ward identity and the recursion relation for the lowest order am-
plitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.2 Proof of gauge invariance of the amplitude M from Wilson lines . . . . 100
7.2.1 Matrix elements with Wilson lines and off-shell amplitudes . . . 1007.2.2 Light-front recursion relation from Wilson lines . . . . . . . . . . 106
v
7.2.2.1 Off-shell (+→ + . . .+) amplitude from Wilson lines . . 108
7.2.2.2 Relationship between M(−+···+)
ε+1...N(k1...N ) and
M(−+···+)
ε+1...m(k1...m) . . . . . . . . . . . . . . . . . . . . . 112
7.2.2.3 Arriving at (7.37) . . . . . . . . . . . . . . . . . . . . . 113
Chapter 8Conclusions 117
Appendix AVertex factors 119A.1 Polarization vector relations . . . . . . . . . . . . . . . . . . . . . . . . . 119A.2 3-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.3 4-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.4 Instantaneous interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Bibliography 124
vi
List of Figures
2.1 Non-planar and planar graphs . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Color-ordered gluonic off-shell current . . . . . . . . . . . . . . . . . . . 182.3 Berends-Giele recursion relation . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Simple gluon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Light-front time ordered diagrams . . . . . . . . . . . . . . . . . . . . . 313.3 Instant vertex interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 Wavefunction decomposition of an amplitude . . . . . . . . . . . . . . . 444.2 Wavefunction recursion relation . . . . . . . . . . . . . . . . . . . . . . . 454.3 3 gluon wavefunction Ψ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Fragmentation function . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Factorization of fragmentation functions . . . . . . . . . . . . . . . . . . 50
5.1 2→ 2 gluon scattering amplitude . . . . . . . . . . . . . . . . . . . . . . 595.2 Leg crossing: from transition to scattering amplitude . . . . . . . . . . . 635.3 2→ 3 scattering: topological group G1 . . . . . . . . . . . . . . . . . . . 645.4 2→ 3 scattering: topological group G2 . . . . . . . . . . . . . . . . . . . 645.5 2→ 3 scattering: topological group G3 . . . . . . . . . . . . . . . . . . . 655.6 2→ 3 scattering: topological groups G4 and G5 . . . . . . . . . . . . . . 655.7 Crossing of denominator . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.8 Relative ordering between trees . . . . . . . . . . . . . . . . . . . . . . . 705.9 Final state splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.10 Initial state splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.1 Vertices for wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . 766.2 Wavefunction recursion relation revisited . . . . . . . . . . . . . . . . . . 766.3 Fragmentation function recursion relation . . . . . . . . . . . . . . . . . 806.4 Different representation of instantaneous and 4-gluon graphs . . . . . . 836.5 Diagrammatic representation of (6.44) . . . . . . . . . . . . . . . . . . . 87
7.1 Ward identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2 M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.3 Recursion of M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
vii
A.1 3-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.2 4-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.3 Instantaneous interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 122
viii
List of Tables
5.1 Physical conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.1 Vertex factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
ix
Chapter 1Introduction
In recent years there has been much progress in the computation of helicity amplitudes
for the scattering of on-shell gluons. It all started when, in 1986, Parke and Taylor con-
jectured that the multi-gluon amplitudes with special helicity configurations have a very
simple form when expressed in terms of the spinor products [1]. One immediate question
is, why were they interested in these quantities when gluons are known to be confined in-
side hadron and cannot appear as incoming and outgoing particles in a scattering event?
It turns out that if one uses suitable factorization theorems [2, 3] and parametrizations
of the non-perturbative parton densities [4, 5] and fragmentation functions [6], on-shell
gluon amplitudes can be used to evaluate cross sections for various observables at high
transverse momenta in the processes that occur in high energy collisions. In fact, the
process gg → gg is one of the major contributors to these cross sections [7, 8].
Later, Berends and Giele constructed a recursion relation for an off-shell current from
which amplitudes could be obtained [9, 10]. This allowed the computation of amplitudes
for arbitrary number of external legs and, with this method, they were able to prove
Parke-Taylor’s conjecture. Recently, another method which uses recursion relations has
surfaced. In their work [11, 12], Britto, Cachazo, Feng, and Witten (BCFW) established
a recursion relation which uses gauge invariant on-shell amplitudes with shifted complex
momenta to calculate amplitudes for higher number of legs [11, 12, 13, 14, 15]. This
progress led to a better implementation of current automated tools that calculate multi-
parton amplitudes for different species of particles, see for example [16, 17, 18, 19, 20,
21]. Furthermore, with these recursive procedures came a better understanding of the
2
underlying structures of the scattering amplitudes [15, 22, 23, 24, 25, 26] and even a
geometrical point of view has arisen [27, 28, 29].
On-shell scattering amplitudes, however, have some limitations, since, as we men-
tioned before gluons are never on-shell particles, and thus are never observed as free
states in the experiments. The off-shell matrix elements are more general objects which
can be used for the construction of the on-shell scattering amplitudes, like for example
in the above-mentioned Berends-Giele recursion [9, 10]. Furthermore, the use of the
off-shell matrix elements in the phenomenology together with the unintegrated parton
densities and appropriate kT factorization approaches is the alternative method for the
computation of the cross sections, see for example [30, 31, 32]. This approach, albeit
more theoretically challenging, has the benefit of taking into account kinematics more
accurately. This can be essential for example, when computing more exclusive processes
which do require information about the details of the kinematics. One complication
though in using off-shell matrix elements is the condition of the gauge invariance. Re-
cently, progress has been made [33, 34, 35] in the construction of off-shell amplitudes
which do satisfy Ward identities and hence obey gauge invariance. The general method
[34] utilizes infinite Wilson line operators corresponding to the off-shell gluons, whose
directions are defined by the polarization vectors perpendicular to the momenta of the
off-shell gluons. It has been shown that such a definition of the off-shell matrix elements
satisfies the corresponding Ward identities with respect to the remaining on-shell states
and, as such, is gauge invariant.
With these new physical insights coming from the different reformulations of the gluon
amplitudes and the recent progress on off-shell amplitudes, we then wonder if there is
any new physical understanding we could gain by studying off-shell gluon amplitudes in
a different framework. Specifically, we are referring to the light-front formalism.
The quantization procedure on the the light-front (or the null-plane) was first pro-
posed a long time ago by Dirac [36] as an alternative approach to the more standard
instant-time quantization. One of the interesting features of the light-front quantiza-
tion is the presence of only three dynamical Poincare generators which describe the
evolution of a system in light-front time, see for example [37]. Thus, one may hope
that the light-front formalism may lead to a simpler solution of problems in relativis-
tic quantum mechanics than other quantization schemes which typically possess larger
number of dynamical operators. It can be also shown that, there exists a subgroup on
the light-front which exhibits algebraic structure isomorphic to the Galilean symmetry
group of non-relativistic quantum mechanics in two dimensions [38, 39]. It has been
3
also argued, see for example [40], that the vacuum on the light-front is essentially struc-
tureless (the arguments about simplicity of the vacuum have been provided earlier by
analysis of graphs in the infinite momentum frame [41]). This stems from the fact that
the lines in the diagrams for amplitudes only have positive longitudinal momenta which
dramatically reduces the number of diagrams which are needed to be considered and
eliminates vacuum graphs. To be precise, the vacuum on the light-front is structureless
up to zero modes, for which special treatment may be necessary, like discrete quanti-
zation which isolates these modes, see for example [42]. It was also shown that zero
modes contribute to the Higgs vacuum expectation value in the standard model [43].
Due to the apparent simplicity of vacuum, light-front methods have been also used to
study the chiral symmetry breakdown, for a recent nice review see [44]. In any case,
this property of the light-front vacuum allows one to define unambiguously the partonic
content of hadrons and of hadronic wave functions and has been used to argue about the
presence of in-hadron quark condensates [45]. The light-front framework has been used
to investigate the hadron dynamics from AdS/CFT correspondence [46] and, in the high
energy approximation, to compute the soft gluon component of the heavy onium wave
function and to obtain a correspondence with the hard Pomeron in QCD [47].
Thus, we see that the light-front formalism, thanks to the simpler vacuum structure
and the iso-morphism with the Galilean subgroup, is an ideal choice to provide us with a
unique insight into some convoluted aspects of QCD. As an added benefit, Motyka and
Stasto have laid down the groundwork for us. In their paper [48], they used the light-
front formalism to construct recursive off-shell objects for certain helicities. In addition,
they obtained the Parke-Taylor amplitudes [1] for the process of an evolved projectile
gluon separated in rapidity from a target gluon.
The objective of this dissertation is then to find off-shell amplitudes–with exact
kinematics–on the light-front that reduce to the Parke-Taylor amplitudes [1] in the on-
shell limit, and to learn all we can along the way. This work is spread across three papers
[49, 50, 51] and we present their content here.
The structure of this dissertation is then as follows. Chapters 2, 3 and 4 provide us
with the tools and background necessary to carry out our calculations in Chapters 5, 6,
and 7. Specifically, in Chapter 2 we introduce helicity and color-ordered amplitudes.
These are two techniques that greatly simplify the calculations of amplitudes. After-
wards, we give a quick example that showcases their power by calculating Parke-Taylor
amplitudes [1] for a low number of legs with relative ease. We end the chapter by pre-
senting the Berends-Giele recursion relation–the first recursion relation we will see out of
4
many–from which the Parke-Taylor amplitudes can be derived for an arbitrary number
of legs.
In Chapter 3 we introduce the light-front: its coordinates, algebra and perturbation
rules. The light-front rules are sketched out by looking at a gluon scattering example.
We begin with Feynman rules and then perform the coordinate change and integrals
required to arrive at a light-front time ordered series. From there the light-front rules will
emerge. The biggest changes from the Feynman rules being that all particles must have
a positive longitudinal momenta and the propagator gets replaced by a non-local energy
denominator that mixes energies from different states. In addition, as a consequence of
the rules, it turns out that a certain off-shell object, which will later be identified as a
fragmentation function, can be factorized. This result, first obtained in [48], is rederived
in this chapter. Finally, one of our new results from [51] appears in this chapter as well.
In our interest to calculate Ward identities, we found that we had to modify the light-
front rules. Whereas the light-front rules specify that each particle is on-shell, including
the intermediate state particles, and therefore energy is not conserved from state to state,
for a Ward identity calculation one must effectively conserve full four momentum in the
numerator of the expression.
In Chapter 4 the off-shell wavefunctions and fragmentation functions from [48] are
introduced, along with their recursion relations and solutions for two specific helicity
configurations. We also calculate a new, but similar, fragmentation function that will be
utilized in the later chapters. However, its form seems to indicate a possible generaliza-
tion. Next, we show that these off-shell objects are inherently connected to the off-shell
amplitudes we are interested in and will therefore act as building blocks to them. We
also see that reaching the on-shell limit involves setting one of the energy denominators
to zero. At the end we show that the two simplest Parke-Taylor amplitudes [1]were
encoded in Motyka and Stasto’s solutions. Furthermore, the vanishing of one of these
appears a consequence of energy conservation.
Chapter 5 discusses crossing symmetry. As a result of the light-front rules, it turns
out that it is not obvious to see if there is crossing symmetry on the light-front. In
this chapter we show that for on-shell amplitudes crossing symmetry does hold. This
allows us to make connections between 1 → N processes (transition amplitudes) and
2 → N − 1 processes (scattering amplitudes). This is important since our work focuses
on transition amplitudes, as they are easier to compute. We want to stress again that
crossing symmetry does not hold for off-shell amplitudes.
The less trivial (+ → − + · · ·+) helicity configuration is studied in Chapter 6. We
5
obtain an off-shell amplitude for arbitrary number of legs that, in the on-shell limit,
reduces to the last of the Parke-Taylor amplitudes [1], the so-called Maximally Helicity
Violating (MHV) amplitude [52, 53]. We also note that even though the correct limit can
be obtained from either the wavefunctions or the fragmentation functions, the latter have
a structure that makes them the better choice. Two things are of particular interest in
this chapter. First, the off-shell amplitudes were obtained through a recursion relation
that is the light-front analog of the Berends-Giele recursion discussed in Chapter 2.
Secondly, the off-shell amplitude can be written as a linear sum of MHV-like amplitudes.
But, even though these interesting objects appear like MHV amplitudes, they are not,
as they don’t have the correct kinematics, i.e. they are still off-shell.
In Chapter 7 we discover that the MHV-like objects that appeared in Chapter 6 are
actually gauge invariant. To show this we begin with a simple example, recalling to use
the modified light-front rules discussed in Chapter 3. Afterwards, we show the gauge-
invariance more generally by making a connection with the Wilson line method [34] of
obtaining gauge-invariant off-shell amplitudes. The connection in this case being that
both off-shell objects satisfy exactly the same recursion relation and must, therefore, be
equal themselves. For the purpose of understanding this, we also give a brief introduction
into the topic of generating gauge-invariant off-shell amplitudes via matrix elements of
a certain Wilson line [34] .
Finally, we give conclusions in Chapter 8.
Chapter 2Basics of tree level amplitudes
Throughout this entire thesis the main objects of interest are amplitudes. Specifically, the
focus is on tree level gluon amplitudes. To calculate these one may technically proceed
with drawing all Feynman diagrams for the process and calculate their contributions
from Feynman rules. If one tries this, even for a relatively low number of external legs,
one would be immediately convinced that this is impractical to do. Gluons can self-
interact in three or four gluon interactions. Furthermore, they present a color structure
that we have to take into account. Both of these cause a great proliferation of diagrams
and of terms in the diagrams, and calculating an amplitude becomes intractable for a
person and very time consuming for even a computer.
In this chapter we will present two commonly used techniques that significantly re-
duce the complexity of the problem. These are the helicity and color-ordered amplitudes.
In these we create gauge invariant subamplitudes, which can then be added incoherently
to compute the full amplitude, to reduce the number of diagrams and terms we need to
look at. We will see that in the helicity representation amplitudes can take on particu-
larly simple and elegant forms in terms of the so-called spinor products.
We begin by presenting the helicity amplitudes and introduce spinor products and
their properties. We will see that even the gluon polarization vectors are written in terms
of spinor products. Next, we present color-ordered amplitudes and their properties.
Following this we use these techniques to calculate the Parke-Taylor amplitudes [1] for
low number of legs. Finally, we present the Berends-Giele recursion [10], which utilizes
these methods to create a recursion for an off-shell current that can later be used to
compute on-shell amplitudes for an arbitrary number of legs.
Even though our interest is computing amplitudes using light-front methods, we will
7
still be able to make use of these techniques. In fact, the spinor products that we’ll
discuss in this chapter emerge naturally in the light-front formalism due to the Galilean
structure of the light-front dynamics. To be precise, we shall see that the spinor products
are proportional to the variables which can be interpreted as the relative light-front
’velocities’ of the emitted particles in the scattering amplitude.
Finally, we do not aim to perform here a complete review of the helicity and color-
ordering methods, but rather we shall recall only basic properties which will be pertinent
for the later derivation of the tree-level amplitudes using the light-front perturbation
theory (LFPT). There are many excellent references and reviews of the helicity methods
in gauge theories, see for example [54, 52, 53, 55]. There are also many reviews, which
discuss the methods for the computations of loop QCD amplitudes, [56, 22, 57, 58]
and in N = 4 super-Yang-Mills theory [59, 60]. We also note the existence of the
Britto-Cachazo-Feng-Witten recursion relations [12, 11] which use decomposition of the
amplitudes into on-shell subamplitudes with shifted complex momenta. For a review see
[55, 13, 14].
2.1 Helicity amplitudes
The helicity formalism was first introduced in [61, 62] to study the process of trident
production of e+e− pairs in QED. It was later developed further in [63, 64, 65]. Currently,
it is because of this formalism that we have compact expressions for tree level and loop
processes in QCD [53]. The main idea in this approach is that we calculate matrix
elements for external massless particles of definite helicity. The primary reasons why
this simplifies our calculations are: 1) the fact that amplitudes for different helicity
configurations do not interfere with each other and, therefore, their squares can be added
incoherently and 2) as we will see, gauge invariance allows one to choose polarization
vectors that most simplify a problem. In fact, in Sec. 2.3 we will present a few examples
that showcase how choosing appropriate polarization vectors can sometimes give an
immediate answer.
To represent our massless particles of definite helicity we use Dirac spinors. Thus,
let us begin with the massless Dirac equation and its solutions for positive and negative
energies.
/ku(k) = /kv(k) = 0 . (2.1)
Recall the notation /k = γµkµ, where γµ are the Dirac matrices. The helicity eigenstates
coincide with the eigenspinors of γ5 and, furthermore, the helicity for negative solutions
8
is negative of the γ5 eigenvalue. Therefore, projecting u and v unto states of definite
helicity we get
u±(k) = P±u(k) =1
2(1± γ5)u(k), v±(k) = P∓v(k) =
1
2(1∓ γ5)v(k). (2.2)
u±(k) = v∓(k), u±(k) = v∓(k) , (2.3)
where P± are the projection operators and u = u†γ0, v = v†γ0. In addition, we will use
the following normalization convention
uλ(k)†uλ′(k) = 2k0δλ,λ′ = vλ(k)†vλ′(k), (2.4)
and we use charge conjugation to get
u±(k) = C (v±(k))T , (2.5)
where C represents the charge conjugation operator. We have the following properties
for C,
C−1(γµ)TC = −γµ, (2.6)
−C = C−1 = CT = C†. (2.7)
For the scattering amplitudes with many gluons, there are many external momenta,
we shall label them ki where i = 1, . . . , N , where N is the number of external legs, and
introduce the following notation:
|i〉 = |ki〉 = u+(ki) = v−(ki), |i] = |ki] = u−(ki) = v+(ki)
〈i| = 〈ki| = u−(ki) = v+(ki), [i| = [ki| = u+(ki) = v−(ki) (2.8)
Now we can construct spinor products
〈ij〉 = u−(ki)u+(kj), [ij] = u+(ki)u−(kj), (2.9)
which, using Dirac algebra properties, can be proven to be antisymmetric
〈ij〉 = −〈ji〉, [ij] = −[ji] . (2.10)
Furthermore, we note that using the projector operator P± actually creates two objects
9
|i〉 and |i] that live in distinct representations of the Lorentz group. Mathematically,
P∓P± = 0 and
[ij〉 = 0 = 〈ij] . (2.11)
We, additionally, have the following properties, which can be derived using the properties
of Dirac algebra,
• Gordon identity and the projection operators:
[i| γµ |i〉 = 〈i| γµ |i] = 2kµi , |i〉 [i| = P+/ki, |i] 〈i| = P−/ki (2.12)
• Fierz rearrangement
[i| γµ|j〉 [k| γµ|l〉 = 2[ik]〈lj〉 (2.13)
• Charge conjugation of the current
[i| γµ|j〉 = 〈j|γµ |i] (2.14)
• Schouten identity:
|i〉〈jk〉+ |j〉〈ki〉+ |k〉〈ij〉 = 0 (2.15)
Many times we will encounter terms like (ki+kj)2 = sij . Using the Fierz rearrangement
along with the Gordon identity we find
〈ij〉 [ji] = 2ki · kj = sij . (2.16)
This implies that the above spinor products are complex square roots of the Lorentz
product sij . Finally, from momentum conservation for N particles, assuming all mo-
menta are outgoing, we have∑N
i=1 ki = 0, and
N∑
i=1i 6=j,k
[ji]〈ik〉 = 0 . (2.17)
All of these properties prove to be extremely valuable when performing computations of
amplitudes.
If we want explicit expressions for the spinors then we need to choose a representation
10
for the γ matrices. Following [53] we use the Dirac representation,
γ0 =
(1 0
0 −1
), γi =
(0 σi
−σi 0
), γ5 =
(0 1
1 0
). (2.18)
With this choice the solutions for u and v are
u+(k) = v−(k) =1√2
√k+
√k−eiφk√k+
√k−eiφk
, u−(k) = v+(k) =
1√2
√k−e−iφk
−√k+
−√k−e−iφk√k+
, (2.19)
where
k± = k0 ± k3, e±iφk =k(1) ± ik(2)
√k+k−
. (2.20)
It is interesting to note that the light-front variables k+, k− appear in these expressions,
as in the following chapters we will use them extensively.
Now we can use our explicit expressions for the spinors to find the spinor products.
〈ij〉 =√k−i k
+j e
iφki −√k+i k−j e
iφkj =√|sij |eiφij , (2.21)
and
[ij] = −√k−i k
+j e−iφki +
√k+i k−j e−iφkj =
√|sij |e−iφij−iπ , (2.22)
where the angle φij is defined through
cosφij =k
(1)i k+
j − k(1)j k+
i√|sij |k+
i k+j
, sinφij =k
(2)i k+
j − k(2)j k+
i√|sij |k+
i k+j
. (2.23)
In the next chapter we will see that these spinor products appear in the context of
light-front perturbation theory.
One final important note is that these expressions are valid for the case when both
4-momenta have positive energy. If one or more of the momenta in 〈ij〉 have negative
energy, φij is calculated with the minus momenta , which have negative energy. Addi-
tionally, a phase lπ/2 is added to φij where l is the number of the negative momenta in
the spinor product.
11
2.1.1 Polarization vectors
As mentioned before, we want to represent everything in terms of Dirac spinors, and
that includes the polarization vectors for the vector bosons. Luckily, there is a way
to do this since the physical Hilbert space of a massless vector is isomorphic to the
physical Hilbert space of a massless spinor [52]. This isomorphism is realized through
the following transformation:
ε+µ (k, q) = A〈q|γµ |k] , ε−µ (k, q) = A∗ [q| γµ|k〉 , (2.24)
where A is a complex normalization (to be determined) and where we have used the
condition that the complex conjugation reverse the helicity
ε−µ (k, q) = (ε+µ (k, q))∗ , (2.25)
which resulted in the complex conjugate A∗ in the second term in Eq. (2.24). Here,
q is an arbitrary vector and is usually referred to as the reference momentum for the
polarization vector. In this isomorphism, gauge invariance is represented by the freedom
to select q. Also note that εµ is transverse to k regardless of q as
ε+(k, q) · k ∝ /k |k] = 0 ,
ε−(k, q) · k ∝ /k |k〉 = 0 (2.26)
To find A we just use the normalization condition
ε+ · (ε+)∗ = ε+ · ε− = −1 = −2|A|2q · k, (2.27)
which leads to
ε+µ (k, q) = eiθ(k,q)
〈q|γµ |k]√2〈qk〉
, (2.28)
ε−µ (k, q) = e−iθ(k,q)[q| γµ|k〉√
2[kq]. (2.29)
This still contains a phase which can depend on both the momentum of the particle k
and the reference momentum q. Nevertheless, in standard conventions [52, 53, 55] it is
set to zero.
Finally, let’s look at how changing the reference momentum affects the polarization
12
vectors. Changing q to q′ we get
ε+µ (k, q′)− ε+
µ (k, q) =〈q′|γµ |k]√
2〈q′k〉− 〈q|γµ |k]√
2〈qk〉
=〈q′|γµ |k] 〈qk〉 − 〈q|γµ |k] 〈q′k〉√
2〈q′k〉〈qk〉= −
√2〈q′q〉
〈q′k〉〈qk〉kµ . (2.30)
We arrive at this by using the identities (2.14), (2.15) and (2.12). Since the result is
proportional to kµ, any change induced in the polarization vector due to a change in
reference momentum vanishes in a Ward identity calculation. This confirms that we
have freedom in choosing the reference momentum, which can be exploited to make
the computations of the amplitudes easier. One could even choose different reference
momenta for different gauge invariant parts of the full amplitude.
With this in mind we list some identities which will help us simplify our calculations:
ki · ε±i (q) = q · ε±i (q) = 0, (2.31)
ε±i (q) · ε∓i(q′)
= ε±i (q) · ε±∗i(q′)
= −1. (2.32)
ε±i (q) · ε±i(q′)
= ε±i (q) · ε±j (q) = ε±i (q) · ε∓j (ki) = 0, (2.33)
/ε+i (kj)|j〉 = /ε−i (kj)|j] = 0 ,
[j|/ε−i (kj) = 〈j|/ε+i (kj) = 0 , (2.34)
where we have introduced short-cut notation ε(ki, q) ≡ εi(q). Here kj is a momentum of
another gluon and q′ is some other arbitrary null reference four vector. These suggest
that it is useful to choose the reference momenta of like-helicity gluons to be the same
and to be equal to the external momentum of one of the gluons from the opposite helicity
set. A choice like this might very well lead to the vanishing of many terms.
The final property that we’ll note is the completeness relation. We have
∑
λ=±ελµ(k, q) (ελν (k, q))∗ = −ηµν +
kµqν + kνqµk · q , (2.35)
which shows that gauges generated by this choice of polarization vectors are equivalent
to axial gauges.
13
2.2 Color ordering
The helicity amplitudes presented in the previous section simplify our calculations greatly,
yet it is not good enough. QCD particles possess color and with this comes a rapid in-
crease in the number of Feynman diagrams and number of terms. Luckily, there are
methods [66, 67, 68, 69, 70] that take advantage of the color structure to create gauge
invariant subsets of the full amplitude at tree level. These subsets are also invariant
under cyclic permutations of external gluons.
The main result is the following: at tree-level, the amplitude AN of any process
with N external gluons can be decomposed into terms containing single traces of color
matrices. Mathematically,
AN =∑
{1,2,...,n}′Tr(T a1T a2 . . . T aN )M(1λ12λ2 . . . nλN ) , (2.36)
where the sum is over all non-cyclic permutations of 1, 2, . . . , n, λi’s are polarizations of
the gluons and T a’s are the generators of the SU(Nc) group with Nc = 3. The generators
are normalized as
[T a, T b] = ifabcT c, Tr(T aT b) = δab, (2.37)
with fabc being the group structure constant. This color decomposition can be gener-
alized beyond tree level to the loop calculation [71], the difference being that the de-
composition becomes more complicated since it involves single and double trace terms.
Now, let’s quickly examine how we end up with this decomposition. If we consider a
tree level gluon diagram, each three gluon vertex will contain a factor fabc and the four
gluon vertex will contain the color factor fabef cde. Most of these f ’s will contract with
each other, but using (2.37) and
fabc = −iTr(T aT bT c − T cT bT a),
fabef cde = −Tr([T a, T b
] [T c, T d
]), (2.38)
which can also be derived from (2.37), we can write all these f factors in terms of traces
of the generators.
In (2.36), the partial amplitudes M, also called color-ordered amplitudes, depend
only on the kinematics. To find them we could use Feynman rules along with the helicity
methods described in the previous section, making sure to not include color factors. To
find, for example, M(1, 2, 3, 4) one would have to label the external legs in the same
14
cyclic order (1, 2, 3, 4). In our convention we label the particles following a clockwise
direction. Furthermore, we can only have planar diagrams. This means that diagrams
in which legs cross, such as the one depicted in Fig. 2.1a, cannot appear. Instead, this
diagram is in a sense taken into account by the diagram in Fig. 2.1b, which contributes
to the amplitude M(1, 2, 4, 3).
k1
k2 k3
k4
(a) Non-planar graph
k1
k2 k4
k3
(b) Planar graph
Figure 2.1: The figures show a non-planar graph and its planar equivalent. An attempt tohave Fig. 2.1a contribute toM(1, 2, 3, 4) fails, as the graph has two legs crossing, whichis not allowed in color-ordered amplitudes. Instead we can have Fig. 2.1b contribute tothe M(1, 2, 4, 3) amplitude.
The subamplitudes M(1, 2, 3, . . . , n) also possess a number of important properties
[52] :
• M(1, 2, 3, . . . , n) is gauge invariant
• M(1, 2, 3, . . . , n) is invariant under cyclic permutations of 1, 2, . . . , n
• M(n, n− 1, . . . , 2, 1) = (−1)nM(1, 2, . . . , n)
• They satisfy Dual Ward Identity or photon decoupling equation
M(1, 2, . . . , n) +M(2, 1, . . . , n) +M(2, 3, 1, . . . , n) + · · ·+M(2, 3, . . . , 1, n) = 0
• Factorization of M(1, 2, . . . n) on multi-gluon poles
• Incoherence to leading order in the number of colors
∑
colors
|An|2 = N2c (N2
c − 1)∑
{1,2,...,n}′
[|M(1, 2, . . . , n)|2 +O(1/N2
c )
].
The last property comes from the fact that traces of color matrices are approximately
orthogonal:
15
N2c−1∑
ai=1,bi=1
Tr (T a1T a1 . . . T aN )[Tr(T b1T b2 . . . T bN
)]∗
= NN−2c (N2
c − 1)(δ{a}{b} +O(N−2
c )). (2.39)
Looking at these properties it is easy to see why this decomposition, coupled with helicity
methods, is so useful. The gauge invariance ofM means that we can choose appropriate
reference vectors which will cause many of the contributing terms in an amplitude to
vanish, leaving only relatively few to deal with. Other properties decrease the amount
of color-ordered amplitudes we have to explicitly compute.
As an explicit example, suppose we want to calculate∑
colors |An|2 for the helicity con-
figuration (1−, 2−, 3+, 4+). The incoherent addition is over all non-cyclic permutations.
Thus, for this case there are six of these, but using cyclic permutations and reflections
we get only three independent partial amplitudes: M(1−, 2−, 3+, 4+),M(1−, 3+, 4+, 2−)
and M(1−, 4+, 2−, 3+). Now we explicitly calculate M(1−, 2−, 3+, 4+) by using Feyn-
man rules and an appropriate reference momentum vector. Using cyclic permutations
we can immediately findM(1−, 3+, 4+, 2−). AsM(1−, 4+, 2−, 3+) does not have the ad-
jacent negative helicities, it will be slightly less easy to find. This time using the photon
decoupling equation we get
M(1−, 4+, 2−, 3+) = −M(4+, 1−, 2−, 3+)−M(4+, 2−, 1−, 3+). (2.40)
Just like that we have found all the partial amplitudes involved in this incoherent sum
and we only had to explicitly compute one of them.
From now on, every amplitude we calculate will always be a color-ordered helicity
amplitude. Therefore, for brevity we will, henceforth, refer to them as amplitudes.
2.3 Tree level examples: Parke-Taylor amplitudes
In 1986 Parke and Taylor [1] made the following conjecture:
M(1+, . . . , N+) = 0, (2.41)
M(1−, 2+, . . . , N+) = 0, (2.42)
M(1−, 2−, 3+, . . . , N+) = igN−2 〈12〉4〈12〉〈23〉 . . . 〈N1〉 . (2.43)
16
The amplitudes on the last line are referred to as the Maximally Helicity Violating (MHV)
amplitudes and the set of all three are called the Parke-Taylor amplitudes. Parke and
Taylor arrived at these through some educated guesswork and challenged the theorists to
prove them. A few years later, Berends and Giele [10] rose to the challenge and proved
these expressions through a recursive method which made use of off-shell currents. This
will be the topic for the next section.
In the present day, the Parke-Taylor amplitudes serve as a base test for any new
hopeful methods for calculating amplitudes. For example, in their paper [11] outlining
the now-called Britto-Cachazo-Feng-Witten (BCFW) recursion, the authors used their
new, groundbreaking method to derive the Parke-Taylor amplitudes. Similarly, in the
following chapters, we will look to derive these amplitudes using light-front methods.
For now, we just want to look at the M(1, 2, 3, 4) processes as practical examples of
how to use the helicity and color-ordered methods we just learned. We should remark
that for these calculations the assumption is that the particles are massless and on-shell,
and all the momenta are defined as outgoing, as is standard in the literature [52]. Let’s
begin withM(1+, 2+, 3+, 4+). For this process we notice that all terms are proportional
to ε+i (qi) · ε+
j (qj). Thus, looking at (2.34) we see that if we choose the same reference
momenta q for all polarizations vectors we get ε+i (q) · ε+
j (q) = 0 and immediately
M(1+, 2+, 3+, 4+) = 0. (2.44)
This simple argument works even for arbitrary N number of gluons with same helicities.
We can show that M(1−, 2+, 3+, 4+) = 0 with a similar idea. This time we choose
the following reference vectors:
q1 = k4, q2 = q3 = q4 = k1 , (2.45)
which give
ε−1 (kn) · ε+i 6=1(k1) = 0, ε+
i (k1) · ε+j (k1) = 0 for i, j 6= 1. (2.46)
Once again, this argument follows through for arbitrary N number of legs, except N = 3.
For 3 legs the amplitude for does not vanish. This is because one cannot choose the
reference momenta to be equal to the external momenta since the polarization vectors
would become singular due to the energy-momentum conservation condition which gives
ki · kj = 0.
17
The vanishing properties of the scattering amplitudes in these particular helicity
configurations can be also proven using supersymmetry relations [72, 73].
Finally, let’s look at the MHV amplitude M4(1−, 2−, 3+, 4+), which should be non-
zero. For this problem we choose q1 = q2 = k4 and q3 = q4 = k1. It is easy to
convince ourselves that the only non-vanishing contribution comes from the contraction
ε−2 · ε+3 6= 0. There are three diagrams, but only one contributes here with this choice
of the polarization vectors and it is the diagram with the gluon exchanged in the s12
channel. The contribution from this graph is therefore
M4(1−, 2−, 3+, 4+) = −2ig2
s12(ε−2 · ε+
3 )(ε−1 · k2)(ε+4 · k3) = −ig2 〈12〉[34]2
[12]〈14〉[14]. (2.47)
Using various relations, like antisymmetry of spinor products, momentum conservation
one can finally cast the result into a very compact and appealing form
M4(1−, 2−, 3+, 4+) = ig2 〈12〉4〈12〉〈23〉〈34〉〈41〉 . (2.48)
As demonstrated at the end of the previous section, now that we haveM4(1−, 2−, 3+, 4+)
we could, if we wanted, find all other 4 gluon amplitudes with two negative and two
positive helicities.
Our results in this section have confirmed the Parke-Taylor results (2.43) for 4 gluons.
In the next section we will use Berends-Giele’s recursion to extend the result to N legs.
2.4 Berends-Giele recursion relations for off-shell currents
As we saw above, the proper use of helicity and color-ordered methods simplified our
problems and led us to quick solutions. However, the approach is still not good enough.
We still had to look at each color-ordered diagram, find each contribution and add
them–though for the case we looked at most contributions were zero. For large number
of external legs this becomes impractical. A better method developed by Berends and
Gieles [10] involves using recursive techniques .
The idea for the new method is that one defines a building block, an off-shell cur-
rent with one particle off-shell while the rest are on-shell. We note that momentum
conservation is still satisfied for this object. Once this is defined, one can construct the
current with higher number of legs using the currents with smaller number of legs and
18
appropriate vertices. We will denote our new off-shell current by
Jµ(1, 2, . . . , n) , (2.49)
where 1, 2, . . . , n are on-shell legs. The uncontracted leg is denoted by µ and the off-shell
propagator is included in Jµ. For the pure gluon case we can depict it as in Fig. 2.2. The
current could also be constructed for off-shell quark currents, see [10]. Once we have the
current, the on-shell subamplitude can then be obtained by multiplying by the inverse
propagator, allowing the momentum to be on-shell and contracting with the appropriate
polarization vector. This can be expressed in the following way:
J
µ
1
2
n
Figure 2.2: Color-ordered gluonic off-shell current Jµ(1, . . . , n). The red colored gluonleg labeled by µ is off-shell, the rest of the legs (in black) denoted by 1 to n are on-shell.
Mn+1(1, 2, . . . , n+ 1) =[ik2
1...nεµ(kn+1)Jµ(1, . . . , n)
]∣∣k1...n=−kn+1
, (2.50)
where we have defined handy notation which will be used later on extensively
k1...n ≡n∑
i=1
kn . (2.51)
Also note that since these currents are off-shell they are also (in principle) gauge depen-
dent. Therefore they depend on the reference momenta for the on-shell gluons which
must be kept fixed in the calculation. Nevertheless, the gauge invariance is restored
after the amplitude is extracted and the remaining off-shell gluon is put on-shell. Re-
gardless, these currents can be used as building blocks to obtain gauge invariant on-shell
amplitudes.
To obtain Jµ one can in principle sum over the color ordered Feynman graphs. We
19
can, instead, use this fact to write a recursion for Jµ. First, one considers an off-shell
gluonic current with one off-shell leg and N on-mass shell legs. From Feynman rules we
then know that the off-shell leg has to be connected to the rest of the diagram either
through the 3-gluon or 4-gluon vertex. Furthermore, these vertices have other legs that
need to be connected to other gluonic currents of fewer number of legs. This can be
diagrammatically illustrated in Fig. 2.3. The expression for this recursion is given by
µ
1
2
n
=
J
J
J +∑n−2
i=1∑n−1
j=i+1∑n
i=1
J
J
J
1 1
n ni + 1
i + 1
ii
j + 1
j
Figure 2.3: Diagrammatic expression for the Berends-Giele recursion relation Eq. (2.52)for the off-shell current Jµ. Red lines indicate off-shell gluons and black lines on-shellfinal state gluons.
Jµ(1, . . . , n) =−ik2
1...n
[n−1∑
i=1
V µνρ3 (k1...i, ki+1...n)Jν(1, . . . , i)Jρ(i+ 1, . . . , n)
+
n−2∑
i=1
n−1∑
j=i+1
V µνρσ4 Jν(1, . . . , i)Jρ(i+ 1, . . . , j)Jσ(j + 1, . . . , n)
, (2.52)
where V µνρ3 and V µνρσ
4 are three- and four-gluon color ordered vertices.
Note that this expression is pretty general, as we did not input any information
about helicities. For complicated helicity configurations these recursion relations can be
implemented into a computer code to obtain the amplitudes up to the desired number
of external legs. However, there are some configurations for which we can solve this
recursion and the result is surprisingly compact. In the case when all of the outgoing
helicities are the same, the current has the form [10]
Jµ(1+, 2+, . . . , N+) = gN−1 〈q|γµ/k1...n|q〉√2〈q1〉〈12〉 . . . 〈N − 1N〉〈Nq〉
, (2.53)
20
where q is the common reference momentum for all the gluons. For the case when one
gluon has a negative helicity the expression is little bit more complicated but also quite
compact, see for example [53]
Jµ(1−, 2+, . . . , N+) = gN−1 〈1|γµ/k2...n|1〉√2〈12〉 . . . 〈N − 1N〉〈N1〉
N∑
m=3
〈1|/km/k1...m|1〉k2
1...m−1k21...m
. (2.54)
The Parke-Taylor amplitude of Eq. (2.43) can be obtained by contracting the current
Jµ with the appropriate polarization vector, amputating the propagator and putting the
leg on-shell. These recursion relations were also formulated in the light-cone gauge [74]
and used to compute the amplitude for the (−,−,−,+, · · ·+) configuration of helicities.
Also note these additional properties that currents satisfy:
• The decoupling relation
Jµ(1, 2, . . . , N) + Jµ(2, 1, 3, . . . , N) + · · ·+ Jµ(2, 3, . . . , N, 1) = 0 ,
• The reflection identity
Jµ(1, 2, 3, . . . , N) = (−1)N+1Jµ(N, . . . , 3, 2, 1) ,
• Current conservation
kµ1...nJµ(1, 2, . . . , N) .
A few things we want to mention before we leave this section is that later in Chapters
4 and 6 we will encounter an object called a fragmentation function. Even though it is
defined for the light-front, it obeys a recursion relation practically identical to (2.52).
As such, we consider it to be the light-front analog of Jµ. Additionally, even though
the off-shellness of Jµ might seem like a drawback, it might not be. As we will see in
Chapter 7 one can formulate gauge invariant extensions of the off-shell amplitudes and
find the recurrence relations for them as well.
Chapter 3Introduction to light-front
In this chapter we introduce the light-front formalism, which is at the core of the work
done in this dissertation. This will be a quick overview, but more details can be found
in [75, 41, 76, 77]. For a review see [78].
We begin by defining the coordinates and then take a look at the algebra of the
Poincare group in the light-front. Of particular is interest is that there are only three
dynamical generators in the light-front, as opposed to the instant time, in which there
are four [36, 75, 39, 37]. In addition, there is an isomorphism between a subgroup of the
Poincare group in the front-form and the Galilean symmetry group in two dimensions
[38, 75, 39].
Since our main interest lies in calculating amplitudes, we need to know the relevant
light-front rules. In the spirit of Kogut and Soper [75], here we will derive them by
looking at a gluon scattering example. This method allows us to detect a potential
pitfall when calculating things like the Ward identity, and how to resolve it.
Finally, we look at one consequence of the rules, which is that the fragmentation
functions can be factorized. We will study fragmentation functions in much more detail in
the next chapters, but this particular property can be readily derived from the knowledge
within this chapter.
3.1 Light-front coordinates
We begin with the definition of the variables that are used on the light front [75]. Let
xµ be a contravariant space-time coordinate four-vector,
xµ = (x0, x1, x2, x3) , (3.1)
22
with the metric tensor in the standard convention g00 = 1, g11 = g22 = g33 = −1. One
can then perform the transformation to a new coordinate basis and introduce a new
time-like component, referred to as the light-front time,
x+ = x0 + x3 , (3.2)
and a corresponding space-like component
x− = x0 − x3 . (3.3)
Therefore in the new basis four-vectors are written as xµ = (x+, x1, x2, x−) and the
scalar product of two four-vectors is defined as
x · p = xµpµ =1
2(x+p− + x−p+)− ~x⊥ · ~p⊥ , (3.4)
where xµ⊥ = (0, x1, x2, 0) = (0, ~x⊥, 0) and ~x⊥ = (x1, x2). In particular we have
x2 = x+x− − ~x 2⊥ . (3.5)
The covariant four-vectors are obtained with the use of the metric tensor, which in the
light-front basis is defined as
gµν =
0 0 0 12
0 −1 0 0
0 0 −1 012 0 0 0
, gµν =
0 0 0 2
0 −1 0 0
0 0 −1 0
2 0 0 0
. (3.6)
Note that, with the above definition of the metric tensor we have x+ = 12x−, x− = 1
2x+.
In general to change from the standard basis to the light-front basis one needs to perform
the following transformation
xµ → Lµνxν , Aµν → LµαA
αβLνβ , (3.7)
23
with the transformation matrix
Lµν =
1 0 0 1
0 1 0 0
0 0 1 0
1 0 0 −1
. (3.8)
3.2 Poincare group
The standard generators of the infinitesimal Lorentz transformations form Poincare al-
gebra, for which the conventional relations hold in the standard basis
[Pµ, Pν ] = 0 , (3.9)
[Mµν , Pρ] = i(gνρPµ − gµρPν) , (3.10)
[Mµν ,Mρσ] = i(gµσMνρ + gνρMµσ − gµρMνσ − gνσMµρ) . (3.11)
The generators for the boosts are Mi0 = Ki and for the rotations Mij = εijkJk, where
i, j = 1, 2, 3 and εijk is the antisymmetric tensor.
In the light-front basis the generators can be obtained by performing the transfor-
mation given by (3.7) and (3.8), in which case they have the following form
Pµ = (P+, P 1, P 2, P−) , (3.12)
where
P+ = P 0 + P 3, P− = P 0 − P 3, P⊥ = (P1, P2), (3.13)
and
Mµν =
0 −S1 −S212K3
S1 0 J3 B1
S2 −J3 0 B2
−12K3 −B1 −B2 0
, (3.14)
with the new generators defined as
S1 =1
2(K1 − J2) , (3.15)
S2 =1
2(K2 + J1) , (3.16)
B1 =1
2(K1 + J2) , (3.17)
24
B2 =1
2(K2 − J1) . (3.18)
The relations among the operators P−, P 1, P 2, P+, J3, B1, B2 are the same as the rela-
tions between the symmetry operators of the non-relativistic Galilean dynamics in two
dimensions [75, 39]. We thus have
[P+, P⊥] = [P−, P⊥] = [P+, P−] = [J3, P−] = [J3, P
+] = [B1,2, P+] = 0 , (3.19)
[J3, Pk] = iεklP
l , (3.20)
[J3, Bk] = iεklB
l , (3.21)
[Bk, P−] = −iP k , (3.22)
[Bk, Pl] = −iδklP+ , (3.23)
with ε12 = −ε21 = 1 and ε11 = ε22. We can thus identify the different operators
with the Galilean dynamics if we make the correspondence P− → H, P+ → mass,
~P⊥ → 2-dim. momentum, J3 → angular momentum and B1 and B2 the generators
of the Galilean boosts in transverse directions 1 and 2 respectively. Thus there is an
isomorphism between the subgroup of the Poincare group in the front-form and the
Galilean symmetry group in two dimensions [38, 75, 39].
An example of this analogy can be illustrated in terms of the mass shell condition
for the single particle
m2 = pµpµ = p+p− − ~p 2⊥ , (3.24)
which, upon solving for p−, gives the Hamiltonian for the free particle in the form
p− = H =~p 2⊥p+
+ V0 , (3.25)
where V0 = m2
p+ is the constant potential. In the Lorentz group one can identify a
kinematical subgroup for each of the reference frames. This subgroup leaves the space-
like hyper surface of the constant time invariant. In the instant time formulation the
kinematical subgroup is formed by the translations and rotations and thus there are
6 kinematical generators. The remaining generators are referred to as the dynamical
generators.
In the case of the light-front formulation, the kinematical generators are
P 1, P 2, P+, J3,K3, B1, B2 , (3.26)
25
and note that there are 7 of these in contrast with 6 for the instant time [36, 75, 39, 37].
The remaining ones are S1, S2, P− which are the dynamical ones. Finally, we shall
mention that the operator K3 has the property that it rescales the other operators in
the following manner
exp(iωK3) η exp(−iωK3) = eωη ,
exp(iωK3) ~P⊥ exp(−iωK3) = ~P⊥ ,
exp(iωK3) P− exp(−iωK3) = e−ωP− ,
exp(iωK3) J3 exp(−iωK3) = J3 ,
exp(iωK3) ~B exp(−iωK3) = eω ~B ,
exp(iωK3) ~S exp(−iωK3) = e−ω ~S . (3.27)
3.3 Light-front rules for the gauge theory
In this section we will derive the pertinent light-front rules. These can be found in the
literature derived in different ways. One can start with writing the Hamiltonian in the
new variables and proceeding to find the commutation relations and the corresponding
rules for perturbation theory from there [39, 75, 79, 80, 78]. Another method is to use
old-fashioned perturbation theory and boost to the infinite momentum frame. From here
we get an expansion that gives us the rules [41, 39, 75, 80]. Here we will follow [75]. We
will analyze a scattering graph and see what results come out. The starting point is the
Feynman gluon propagator. We will transform our coordinates to light-front variables
and attempt to get a light-front time x+ ordered series. We should note that fermions
will not be covered in this derivation.
3.3.1 Conventions
We work in the light-cone gauge, A+ = 0 which is defined by η · A = 0, with vector η
which is light like and equal to
η = (0, 0, 0, 2) . (3.28)
This choice of the gauge defines the polarization four-vectors of the gluon with four-
momentum k
ε±µi (η) ≡ ε±µi = ε±µ⊥ +~ε ±⊥ · ~ki⊥ki · η
ηµ , (3.29)
26
with
ε±µ⊥ =1√2
(0, 1,±i, 0) . (3.30)
If one sets the reference momenta q = η in (2.29), we get the same polarization vectors
as above. One warning is warranted however. For historical reasons, the polarizations
we use in the light-front are opposite of the polarizations vectors from Chapter 2. In
other words, our plus helicity is their negative helicity and vice-versa. It is easy to check
that the polarization vectors satisfy the following properties
ε±i · (ε±i )∗ = −1, ε±i · (ε∓i ) = −1, ε±i · (ε±i ) = ε±i · (ε±j ) = 0 , (ε±i )∗ = ε∓i . (3.31)
With these definitions of the gauge and polarization vectors one can find the following
decomposition of the projector
dµν ≡∑
λ=±εµ(k, λ)(εν(k, λ))∗ = −gµν +
ηµkν + ηνkµ
η · k − ηµην
(η · k)2k2 . (3.32)
It is straightforward to verify that this projector is doubly transverse, i.e. it is transverse
both to the vector η as well as to the momentum k
dµνην = dµνkν = 0 . (3.33)
Also very important is that dµν is independent of k−.
3.3.2 Gluon propagator in the light-front theory
The Feynman propagator collects both retarded and advanced propagators into one
single expression. Since we are looking for x+ ordered diagrams, we want the gluon
propagator Dµν to take the following form:
Dµν = Dµν+ Θ(x+) +Dµν
− Θ(−x+) . (3.34)
In light-cone gauge, the propagator is given by
Dµν(x) =1
(2π)4
∫d4p exp(−ipαxα)
−gµν + ηµpν+ηνpµ
η·ppλpλ + iε
. (3.35)
In the following we will be performing the integral over p−. Notice that the numerator
of the propagator has an explicit dependance on p−. Therefore, we can use the projector
27
to write this as
Dµν(x) =1
(2π)4
∫d4p e−ip·x
dµν
p2 + iε+
1
(2π)4ηµην
∫d4p e−ip·x
1
(p+)2
p2
p2 + iε. (3.36)
In this expression, the numerator of the first term is independent of p−. The second
numerator is dependent on p−, but we will see that it is still very easy to compute that
integral. Writing the integral explicitly in terms of p+, p− and ~p⊥ we can rewrite the
propagator as
Dµν(x) = Dµν1 (x) +Dµν
2 (x) , (3.37)
where the two terms are
Dµν1 (x) =
1
2(2π)4
∫d2~p⊥dp
+dp−e−i2p+x−e−
i2p−x+
ei~p⊥·x⊥dµν
p+p− − ~p 2⊥ + iε
,
Dµν2 (x) =
1
(2π)4ηµην
∫d4pe−ip·x
1
(p+)2
p2
p2 + iε. (3.38)
Let’s work withDµν1 first. We can do this integral using contour integration. However,
notice that the sign of p+ will affect whether the pole is on the upper or lower half plane.
Thus, let’s make p+ be always positive.
Dµν1 (x) =
1
2(2π)4
∫d2p⊥e
i~p⊥·~x⊥∫ ∞
0dp+
∫ ∞
−∞dp−
e−i2
(p+x−+p−x+)dµν
p+p− − ~p 2⊥ + iε
+1
2(2π)4
∫d2p⊥e
i~p⊥·~x⊥∫ 0
−∞dp+
∫ ∞
−∞dp−
e−i2
(p+x−+p−x+)dµν
p+p− − ~p 2⊥ + iε
=1
2(2π)4
∫d2p⊥e
i~p⊥·~x⊥∫ ∞
0dp+
∫ ∞
−∞dp−
e−i2
(p+x−+p−x+)dµν
p+p− − ~p 2⊥ + iε
+1
2(2π)4
∫d2p⊥e
−i~p⊥·~x⊥∫ ∞
0dp+
∫ ∞
−∞dp−
e−i2
(−p+x−+p−x+)dµν
−p+p− − ~p 2⊥ + iε
. (3.39)
To get the last line we have changed variables, from p+ → −p+, ~p⊥ → −~p⊥. Note that
dµν , which is a function of p+ and ~p⊥, is invariant under this change. Continuing,
Dµν1 (x) =
1
2(2π)4
∫d2p⊥e
i~p⊥·~x⊥∫ ∞
0
dp+
p+
∫ ∞
−∞dp−
e−i2
(p+x−+p−x+)dµν
p− − ~p 2⊥/p+ + iε
− 1
2(2π)4
∫d2p⊥e
−i~p⊥·~x⊥∫ ∞
0
dp+
p+
∫ ∞
−∞dp−
e−i2
(−p+x−+p−x+)dµν
p− + ~p 2⊥/p+ − iε . (3.40)
28
The pole for the first term is on the lower half plane. For this contour we need x+ > 0.
For the second term the pole is in the upper half plane, so we need x+ < 0. Then,
defining p− = ~p 2⊥/p+, which we recognize as the value p− would take if p were on-shell,
Dµν1 (x) =
−2πi
2(2π)4
∫d2p⊥e
i~p⊥·~x⊥∫ ∞
0
dp+
p+
∫ ∞
−∞dp−dµνe−
i2
(p+x−+p−x+)Θ(x+)
− 2πi
2(2π)4
∫d2p⊥e
−i~p⊥·~x⊥∫ ∞
0
dp+
p+
∫ ∞
−∞dp−dµνe−
i2
(p+x−−p−x+)Θ(−x+)
=−2πi
2(2π)4
∫d2p⊥
∫ ∞
0
dp+
p+dµν
[e−ip·xΘ(x+) + eip·xΘ(−x+)
], (3.41)
where p = (p+, ~p⊥, p−).
For Dµν2 , the integrand is actually finite since
p2
p2 + iε→ 1, ε→ 0 . (3.42)
Therefore it can be evaluated readily evaluated as
Dµν2 (x) =
1
(2π)4ηµην
∫d4pe−ip·x
1
(p+)2=
ηµην
(2π)3δ(x+)
∫d2p⊥
∫ ∞
−∞
dp+
(p+)2e−i
12x−p++i~x⊥~p⊥ . (3.43)
We see that this part is proportional to the Dirac delta in the light-front time, and
therefore this is the instantaneous term of the propagator. This is reminiscent of the
Coulomb interaction which does appear when quantizing electrodynamics in the Coulomb
gauge.
Thus, we managed to get the propagator into the desired form (3.34), but now have
an additional instantaneous term. Therefore we can write the propagator in terms of
three contributions,
Dµν = Dµν+ +Dµν
− +Dµν2 , (3.44)
where we’ve absorbed the Θ functions into the Dµν ’s. As we will see in the next section,
this implies that a Feynman diagram with one propagator is equivalent to three time-
ordered diagrams in the light-front.
29
Figure 3.1: One of the Feynman diagrams for the 2 → 2 gluon scattering amplitude.The greek letters are the vertex indices and the x’s are the locations in position spaceof the vertices. The k’s are the momenta of the gluons. k1 and k2 are incoming and k3
and k4 are outgoing.
3.3.3 Analysis of a gluon scattering diagram
Suppose we have the Feynman diagram in Fig. 3.1. Let’s examine what contributions we
get in the light-front. Assuming that k1 and k2 are incoming and k3 and k4 are outgoing,
we would have the following,
I = −1
2
∫dx1dx2e
−ix1·k2ε∗µ12 eix1·k3εµ2
3 Vµ1µ2µ3(k2, k3, p)Dµ3ν3(x1 − x2)
× Vν1ν3ν2(k1, p, k4)e−ix2·k1ε∗ ν11 eix2·k4εν2
4 . (3.45)
Notice that there are no color factors above as we are going to assume we are working
with color-ordered helicity amplitudes (see Chapter 2). Also, the polarization vectors
should have a helicity index, but for brevity we’ve suppressed them. There is one subtlety
we need to address. We began this derivation by calculating the gluon propagator in the
light-front, which involved integrating over the p− component. Looking at the integral
above, we see that this was done prematurely, as the vertex factors can include some
p− components. Luckily, our chosen helicity vectors (3.29) project only on plus and
transverse components and, thus, everything is fine. In section 3.4 we will look at what
happens when the vertex does produce a p−, as is the case for a Ward identity calculation.
Let’s look at the contribution from Dµν+ first. The light-front rules will come out of
doing the x integrals, so we will group together everything that has no dependence on x
into one variable,
B(k1, k2, k3, p) = ε∗µ12 εµ2
3 Vµ1µ2µ3(k2, k3, p)dµ3ν3Vν1ν3ν2(k1, p, k4)ε∗ ν1
1 εν24 . (3.46)
30
Now we have,
I+ =2πi
4(2π)4
∫d2p⊥
∫ ∞
0
dp+
p+B
∫dx1dx2e
−ix1·(k2−k3+p)e−ix2·(k1−k4−p)Θ(x+1 − x+
2 ).
(3.47)
We can immediately do the x− and x⊥ integrals. These give conservation of longitudinal
and transverse momentum at every vertex. In addition, since p+ has to be positive, we
see that it must flow from x2 to x1. Let’s work on the x+ integrals. The term is
I+ = P∫dx+
1 dx+2 e− i
2x+
1 ·(k−2 −k
−3 +p−)e−
i2x+
2 ·(k−1 −k
−4 −p−)Θ(x+
1 − x+2 ), (3.48)
where
P =2πi
4(2π)4
∫d2p⊥
∫ ∞
0
dp+
p+B(2π)3δ(k+
2 − k+3 + p+)δ2(~k3 − ~k2 − ~p)
× (2π)3δ(k+1 − k+
4 − p+)δ2(~k4 + ~p− ~k1). (3.49)
We can make the following change of variables
T0 = x+2 , T1 = x+
1 − x+2 , (3.50)
to get
I+ = P∫dT0 dT1e
− i2T0(k−1 +k−2 −k
−3 −k
−4 )e−
i2T1(k−2 −k
−3 +p−)Θ(T1) (3.51)
= P∫dT0
∫ ∞
0dT1e
− i2T0(Ei−Ef )e−
i2T1(E1−Ef )Θ(T1). (3.52)
The result above effectively identifies three different states in light-front time. We have
the initial state, with energy Ei, which consists of the two incoming gluons. Then we
have the intermediate state, with energy E1. Gluon 1 has split into a virtual gluon and
the outgoing k4. Thus, this state is composed of k2, p and k4. At the end we have
the final state, with energy Ef , which is composed of the two outgoing gluons. This
decomposition means that we will be able to represent this contribution by the diagram
in Fig. 3.2a, where light-front time increases to the right and all momenta are also moving
to the right. The energies are defined as
Ei = k−1 + k−2 , E1 = k−2 + p− + k−4 , Ef = k−3 + k−4 , (3.53)
31
where each of the particles has taken its on-shell value, i.e. k−i = ~k2i⊥/k+
i . Then we get
I+ = P (2π)δ(Ei − Ef )4i
Ef − E1 + iε(3.54)
To get this result we used ∫ ∞
0dxeipx =
i
p+ iε. (3.55)
Finally, we can do the p integrals using the δ functions.
I+ = i(2π)4δ4(k1 + k2 − k3 − k4)Bi
p+(Ef − E1 + iε). (3.56)
The contribution to the scattering amplitude would be
M+ =∑
λ=±ε∗µ1
2 εµ23
(ελ ∗p
)µ3
Vµ1µ2µ3(k2, k3, p)i
p+(Ef − E1)Vν1ν3ν2(k1, p, k4)ελ ν3
p ε∗ ν11 εν2
4 ,
(3.57)
recalling that p+ is a positive quantity that flows from x2 to x1. In other words,
k+1 − k+
4 = p+ = k+3 − k+
2 , (3.58)
~k1⊥ − ~k4⊥ = ~p⊥ = ~k3⊥ − ~k2⊥. (3.59)
x+
k3
k4
k2
k1
p
(a)
x+
k3
k4
k2
k1
p
(b)
Figure 3.2: Two different light-front time ordered diagrams. Both came out of a singleFeynman diagram. In the figures, light-front time increases to the right, and all particlesmust move to right.
Following the same procedure for I− we get similar results. We get longitudinal and
transverse momentum conserved at every vertex and overall energy conservation, but
32
this time p+ flows from x1 to x2. From the x+ integrals we get
2πδ(Ei − Ef )4i
Ef − E2 + iε, E2 = k−3 + p− + k−1 , (3.60)
from which we have the following contribution to the scattering amplitude,
M− =∑
λ=±ε∗µ1
2 εµ23 ελµ3
p Vµ1µ2µ3(k2, k3, p)i
p+(Ef − E2)Vν1ν3ν2(k1, p, k4)
(ελ ∗p
)ν3
ε∗ ν11 εν2
4 ,
(3.61)
k+4 − k+
1 = p+ = k+2 − k+
3 , (3.62)
~k4⊥ − ~k1⊥ = ~p⊥ = ~k2⊥ − ~k3⊥. (3.63)
The light-front diagram that gives this contribution is depicted in Fig. 3.2b.
Let’s now examine the contribution from Dµν2 .
I2 = − 1
2(2π)3
∫d2p⊥
∫ ∞
−∞
dp+
(p+)2B
∫dx1dx2e
− i2x−1 (k+
2 −k+3 +p+)e−
i2x−2 (k+
1 −k+4 −p+)
× ei~x1(~k2−~k3+~p)ei~x2(~k1−~k4−~p)e−i2x+
1 (k−2 −k−3 )e−
i2x+
2 (k−1 −k−4 )δ(x+
1 − x+2 ), (3.64)
now with
B(k1, k2, k3, p) = ε∗µ12 εµ2
3 Vµ1µ2µ3(k2, k3, p)ηµ3ην3Vν1ν3ν2(k1, p, k4)ε∗ ν1
1 εν24 . (3.65)
Instead of a Θ function, Eq. (3.64) has a δ function. This tells us that the interac-
tion between the two vertices happens instantaneously. This is represented in Fig. 3.3.
Performing the x integrals we get
x+
k3
k4
k2
k1
p
Figure 3.3: In this figure the interaction occurs at the same light-front time.This repre-sents an instantaneous transfer of momentum and is denoted by the cross.
33
I2 = − 1
(2π)3
∫d2p⊥
∫ ∞
−∞
dp+
(p+)2B(2π)7δ(k+
2 − k+3 + p+)δ2(~k3 − ~k2 − ~p)
× δ(k+1 − k+
4 − p+)δ2(~k4 + ~p− ~k1)δ(Ei − Ef )
= i(2π)4δ4(k1 + k2 − k3 − k4) iB. (3.66)
Therefore, the contribution to the scattering amplitude from this term is
M 2 = ε∗µ12 εµ2
3 Vµ1µ2µ3(k2, k3, p)iηµ3ην3
(p+)2Vν1ν3ν2(k1, p, k4)ε∗ ν1
1 εν24 , (3.67)
k+1 − k+
4 = p+ = k+3 − k+
2 , (3.68)
~k1⊥ − ~k4⊥ = ~p⊥ = ~k3⊥ − ~k2⊥. (3.69)
We relate this to an instantaneous interaction, therefore p+ can be positive or negative.
3.3.4 Formal light-front rules
In the following we will use the longitudinal momentum fraction z instead of the longi-
tudinal momentum. Therefore, we will be redefining a few things. zi is defined by
k+i = ziP
+, (3.70)
where P+ is a total incoming longitudinal momentum. When all the factors are taken
into account, P+ cancel out, so we should not have any P+’s floating around.
Now, from our results and complemented by the literature [75, 41, 76, 77, 78] we can
write down a formal set of rules:
1. Write down all possible light-front time ordered diagrams for a given order of
perturbation theory. Do not draw the diagrams in which the momenta are flowing
backwards in light-front time. Also, make to sure to add graphs per possible helicity
configuration.
2. Each line will have 2-dimensional transverse momentum ~k⊥ and the longitudinal
momentum fraction zk > 0. Also, k2 = m2, i.e. is on mass-shell. Therefore
k− =~k2⊥zk
.
3. For each vertex include the conservation of the longitudinal and transverse com-
34
ponents
δ(z1 − z2 − z3)δ(2)(~k1⊥ − ~k2⊥ − ~k3⊥).
4. For each internal line include the phase space factor
d2~k⊥2(2π)3
dz
zΘ(z).
Thus all the particles are moving forward, and this eliminates large class of vacuum
diagrams.
5. For each vertex, again, assign an appropriate factor, the 3-gluon and 4-gluon ver-
tices are listed in Table A.1. For an incoming gluon use polarization vector ε∗. For
outgoing use ε.
6. For every intermediate state include the following energy denominator
i
D=
i∑i,initialEi −
∑i,intEi + iε
,
where
Ej ≡∑
j
~k2j⊥ +m2
j
zj
where the sum goes over the different particles in the given state. For our gluons
we have m = 0.
7. The gluon propagator has an instantaneous term ηµην/k+2 which we have to in-
clude as a separate graph. These instantaneous interactions are listed in Table A.1.
For an incoming gluon use polarization vector ε∗. For outgoing use ε. Alternatively
one can redefine the energy denominator to include the instantaneous interaction
inside. This is done by defining auxiliary momentum [76]
kµ =(k+,
∑
i,initial
E−i −∑
i,int′
E−i ,~k⊥), (3.71)
where the sum∑
int′ means that one sums over all the particles in the given inter-
mediate state except the particle of interest. This new momentum replaces k in
the polarization sum and in the triple-gluon vertex.
8. Integrate over all the internal momenta.
35
9. Sum over all the time orderings for a given topology of diagram and finally sum
over the diagrams.
10. Finally, one can impose energy conservation between the initial and final state.
3.3.4.1 Example
As an example, let’s use the light-front (LF) rules to study Fig. 3.1. The first we have
to do is to draw all possible time orderings, including the instantaneous graph. The
resulting graphs would be Figs. 3.2 and 3.3.
Let’s compute the contribution of Fig. 3.2a to the amplitude M under the assump-
tion the helicity configuration is (++→ ++). Immediately we see that the internal line
can have either a plus or negative helicity. Therefore, this one graph suddenly became
two, one for each internal helicity. Now, practically speaking, when calculating an am-
plitude all we care about are three things: vertex factors, internal line factors and energy
denominators. Nevertheless, the conservation of longitudinal and transverse momentum
is always in the mind, and we use them freely to simplify things. For positive internal
helicity we will get
M+ = V(+→++)
3 (k1, p, k4)i
zpD1V
(++→+)3 (p, k2, k3)Θ(zp), (3.72)
zp = z1 − z4, (3.73)
D1 = k−1 + k−2 − k−2 − p− − k−4 =~k 2
1⊥z1− (~k1⊥ − ~k4⊥)2
z1 − z4−~k 2
4⊥z4
. (3.74)
The vertices are listed in Table A.1, and one can write the energy denominator in terms
of v’s to help simplify things. We’ve also utilized longitudinal and transverse momentum
conservation to write everything in terms of external particles. Finally, the Θ function
is there to ensure that the internal line is positive.
For negative helicity we get a similar result:
M− = V(+→−+)
3 (k1, p, k4)i
zpD1V
(−+→+)3 (p, k2, k3)Θ(zp). (3.75)
3.4 Modified light-front rules
In Chapter 7 we will calculate Ward identities for our amplitudes. For this, we replace
a polarization vector by its momentum. Unfortunately, as was noted in [51], there is a
problem with that. As mentioned in Sec. 3.3.3, dotting a momentum with a vertex will
36
produce a p− component (here p is the momentum of the internal line). This means
that we cannot use our regular rules, as these were derived by integrating out the p−
component under the assumption that the numerator of the amplitude was independent
of p−.
Therefore, as was done in [51], let’s start at the beginning and see what happens when
the vertex includes a p− component. Suppose the Ward identity produces a 12k
+i p− term,
the contribution of the diagram would now be
I = −1
2
∫dx1dx2e
−ix1·k2eix1·k3Dµ3ν3(x1 − x2)k+i p−
2e−ix2·k1eix2·k4B, (3.76)
where, as before, B includes all the non-x dependencies that are not crucial to our
current calculation. Notice that we can arrive at the same thing by using
Dµν(x)k+i p−
2=
k+i
(2π)4
∫d4p
p−
2exp(−ipαxα)
−gµν + ηµpν+ηνpµ
η·ppλpλ + iε
=k+i
(2π)4i∂
∂x+
∫d4p exp(−ipαxα)
−gµν + ηµpν+ηνpµ
η·ppλpλ + iε
= ik+i
∂
∂x+Dµν(x) (3.77)
As before, let’s examine the contribution from Dµν+ . Since the derivative is with
respect to x+, we only need to look at the x+ integrals. In the following we suppress all
the irrelevant variables, including the exponentials with x− and ~x⊥ in the second line.
∫dx+
1 dx+2 e− i
2x+
1 (k−2 −k−3 )e−
i2x+
2 (k−1 −k−4 )ik+
i
∂
∂(x+1 − x+
2 )Dµν
+ (x+1 − x+
2 )
=
∫dx+
1 dx+2 e− i
2x+
1 (k−2 −k−3 )e−
i2x+
2 (k−1 −k−4 )k+
i
×[p−
2e−
i2p−(x+
1 −x+2 )Θ(x+
1 − x+2 ) + ie−ip
−(x+1 −x
+2 ) ∂
∂(x+1 − x+
2 )Θ(x+
1 − x+2 )
]. (3.78)
We can do the same change of variables as before to get
∫dT0 dT1e
− i2T0·(Ei−Ef )e−
i2T1·(k−2 −k
−3 )k+
i
[p−
2e−
i2T1p−Θ(T1) + ie−
i2T1p− ∂
∂T1Θ(T1)
].
(3.79)
Now, let’s recall the following property about distributions. If we have a distribution h
37
and a function f , the derivative of the distribution is given by the following
∫h′(x)f(x)dx = −
∫h(x)f ′(x)dx . (3.80)
Thus, we get
∫dT0e
− i2T0·(Ei−Ef )
∫dT1k
+i Θ(T1)
[p−
2e−
i2T1·(k−2 −k
−3 +p−) − i ∂
∂T1e−
i2T1·(k−2 −k
−3 +p−)
]
=
∫dT0e
− i2T0·(Ei−Ef )
∫dT1k
+i Θ(T1)e−
i2T1·(k−2 −k
−3 +p−)
[p−
2− 1
2(k−2 − k−3 + p−)
]
=
∫dT0e
− i2T0·(Ei−Ef )
∫dT1Θ(T1)e−
i2T1·(k−2 −k
−3 +p−)k
+i (k−3 − k−2 )
2(3.81)
We began our calculation withk+i p−
2 in the numerator of I+ before integration of the
p− component. A term like this could come from any ki · p we may encounter. At the
end we see that p− gets replaced by k−3 − k−2 , which is the value p− would have taken if
energy conservation had been imposed.
It is easy to see that we should arrive at the same result when considering Dµν− .
For Dµν2 , getting this result is pretty straightforward. Once again, let’s look at the x+
integrals,
∫dx+
1 dx+2 e− i
2x+
1 (k−2 −k−3 )e−
i2x+
2 (k−1 −k−4 )ik+
i
∂
∂(x+1 − x+
2 )δ(x+
1 − x+2 )
=
∫dx+
1 dx+2 e− i
2x+
1 (k−2 −k−3 )e−
i2x+
2 (k−1 −k−4 )δ(x+
1 − x+2 )k+i (k−3 − k−2 )
2. (3.82)
What we’ve just shown is that if a linear p− component comes into the numerator
of our calculations we should not be discouraged nor abandon the LF rules. The LF
rules stay in place, but there is an added assumption that energy is conserved in the
numerator at the vertex that produced the term in the first place. Our calculation was
with Ward identities in mind, yet the only assumption here was that there was a linear
p− term in the numerator, which could presumably arise in other types of calculations.
What is very interesting is that, for an amplitude calculation, it is irrelevant what p−
is in the numerator, since polarization vectors eliminate all p− dependencies. Based on
the results of this section, can we then assume that energy is conserved in the numerator
regardless of the vertex factors?
38
3.5 Factorization of fragmentation tree amplitudes
As a consequence of the rules described, it turns out that an object that we will encounter
later–fragmentation functions–possesses the property of factorization. The fragmenta-
tion function, TNin→Nf is an object that describes the Nin → Nf process. Its initial state
is treated as an intermediate state–it could be part of a larger tree for example–and its
final state is an external “physical” state . What the factorization property refers to is
that a tree with Nin final gluons can be described as a set of smaller subtrees. Concretely,
TNin→Nf = Tl1→k1Tl2→k2 . . . Tli→ki ,i∑
j=1
li = Nin,i∑
j=1
kj = Nf . (3.83)
The fragmentation functions are calculated using LF rules. Therefore, to show the
factorization property let’s begin there. The rules specify to we need to include all
possible vertex orderings. This is a problem at the outset as the energy denominator is
non-local and mixes objects at different states. As for the vertex factors, and internal
lines, they don’t pose a problem as for they are fixed for a specific topology, regardless
of the vertex ordering. Thus, we only need to work with the denominators.
For the proof, it will suffice to show that a fragmentation function is composed of two
subtrees, each with fixed internal topology and vertex ordering. We will denote the m
denominators of subtree α in the absence of the other as Ai, where 1 ≤ i ≤ m. For the
other subtree β, it’s denominators will be Bi, where 1 ≤ i ≤ n. We represent by Tαi βj
the fragmentation function–in reality, only the denominators–with i A denominators and
j B denominators. To prove the factorization of the fragmentation function we need to
show that the denominators factorize. In other words,
Tαn βm =1
An . . . A1
1
Bm . . . B1,
Tα0 βm =1
Bm . . . B1,
Tαn β0 =1
An . . . A1. (3.84)
We should note that the lowest indexes represent the rightmost denominators.
Let’s begin with Tα1 β1 . This case is pretty simple and we get
Tα1 β1 =1
A1 +B1
1
A1+
1
A1 +B1
1
B1=
1
A1B1(3.85)
39
Now, let’s extend that to m B denominators. If we assume we have Tα1 βm−1 , then
the recursion would be
Tα1 βm =1
(A1 +Bm)Tα1 βm−1 +
1
(A1 +Bm)
1
BmTα0 βm−1 . (3.86)
The first term takes into account when Bm is to the left of A1. In this case, Bm does not
add any possible vertex ordering than the ones already present in Tα1 βm−1 and all we
need to do is add the new denominator. The second term takes into account the when
Bm is to the right of A1. In this case, we cannot use Tα1 βm−1 anymore as the presence of
Bm interrupted the addition. However, we notice that now A1 is the leftmost particle.
Furthermore, all possible vertex orderings that can happen to the right of Bm have been
taken into account in Tα0 βm−1 . Therefore, the presence of Bm introduces an extra two
denominators: 1/(A1 +Bm) and 1/Bm. Using (3.84) we now get
Tα1 βm =1
(A1 +Bm)
1
A1
1
Bm−1 . . . B1+
1
(A1 +Bm)
1
Bm
1
Bm−1 . . . B1(3.87)
=1
A1
1
Bm . . . B1(3.88)
This idea can be generalized to get T ′αn βm . Here the prime will denote that we don’t yet
know the form of T ′αn βm ; our job is to find it. It will be useful to define B0 = 0.
T ′αn βm =m∑
i=0
Tαn−1 βi
m∏
j=i
1
An +Bj
=m∑
i=1
Tαn−1 βi
m∏
j=i
1
An +Bj+ Tαn−1 β0
m∏
j=0
1
An +Bj
=
m∑
i=1
Tαn−1 βi
m∏
j=i
1
An +Bj+ Tαn β0
m∏
j=1
1
An +Bj. (3.89)
However, let’s note the following property
Tαn−1 βj + Tαn βj−1= Tαn βj (An +Bj) (3.90)
This relationship is fairly simple to derive. It then gives
Tαn−1 βi
m∏
j=i
1
An +Bj+ Tαn βi−1
m∏
j=i
1
An +Bj= Tαn βi
m∏
j=i+1
1
An +Bj(3.91)
40
We can solve for the first term of this expression and plug into (3.89).
T ′αn βm =m∑
i=1
Tαn βi
m∏
j=i+1
1
An +Bj−
m∑
i=1
Tαn βi−1
m∏
j=i
1
An +Bj+Tαn β0
m∏
j=1
1
An +Bj(3.92)
Everything cancels out in this expression except for i = m in the first term, leaving us
with
T ′αn βm =1
An . . . A1
1
Bm . . . B1(3.93)
Chapter 4Light-front wavefunctions and
fragmentation functions
In this chapter we will introduce the light-front wavefunction and fragmentation function
as discussed in [48] by Motyka and Stasto. In this work, the authors studied the gluon
cascade at tree level using light-front perturbation theory with the purpose of improving
the evolution equations at high energy by using exact kinematics. Their focus was on
the (+ → + . . .+)1 helicity configuration, as it provided a simple case study because
it involves only 3-gluon interactions. The authors also established a recursion relation
for the wavefunction and fragmentation function, and, with appropriate assumptions,
managed to get a closed form solution for these.
In addition to the recursive nature of the wavefunction and fragmentation function,
these are also off-shell objects. This makes them ideal building blocks for the amplitudes
we are interested in calculating. For this reason, in this chapter we present how they
were derived in [48]. The general procedure is to, first, find the underlying recursion
relation. Then, the lowest order functions are found and from this an educated guess to
the solution is made. Finally, the solution is shown to be correct via induction.
Afterwards, we also discuss the fragmentation function for the (− → − + . . .+)
helicity configuration, which will be important for calculating amplitudes for more com-
plicated configurations. The results from this gives us some insight into the structure of
the fragmentation functions and allows us to make a possible generalization.
We end the chapter by showing the relationship between the functions, and the off-
1Recall that in light-front perturbation theory the initial particle needs to be incoming, while the finalparticles are outgoing.
42
shell amplitude. This simple relationship allows us to easily find the amplitude once we
know either the wavefunction or the fragmentation function. As an example, we will
then use this to show that the on-shell amplitude M(+→+...+) = 0, as expected.
Before we continue let’s make one remark. LFPT dictates that each particle is on-
shell [78]. If that’s the case, what are we referring to when we say “off-shell”? We refer
to the energy of the state. In LFPT, energy is not conserved state by state; there is
only an overall conservation between the initial and final state. As such, we call any
intermediate state off-shell. Thus, an off-shell object is one that has an initial and/or
final state that is treated like an intermediate state, and energy conservation cannot be
applied.
4.1 Conventions and notations
Throughout this section we study processes where one initial incoming gluon splits or
fragments into N outgoing gluons. We will refer to the wavefunction and fragmentation
function generally as ΨN and TN respectively. More specifically,
Ψ(λ(1...N)→λ1...λN )(1, . . . , N) ≡ Ψ(λ(1...N)→λ1...λN )(k1, . . . , kN ), (4.1)
represents the wavefunction with outgoing gluons 1, . . . , N with momenta k1, . . . kN . The
incoming momenta is implicit but will be labeled k1...N (recall the momentum conserva-
tion rules from Sec. 3.3.4). Finally, λi is the helicity of gluon i. Also, recall that we are
using color-ordering, so color is suppressed. For fragmentation functions we would have
T (λ(1...N)→λ1...λN )(1, . . . , N) ≡ T (λ(1...N)→λ1...λN )(k1, . . . , kN ). (4.2)
Sometimes we will suppress the helicities when it is clear which helicity configuration we
are working with.
In the following calculations of wavefunctions and fragmentation functions, we will
use different conventions than for the amplitude. Specifically, we will have a different
normalization for the vertices: now for every vertex in a graph we will have a contribution
of V ,
V 3(ka, kb, kc) =1√zazbzc
V3(ka, kb, kc). (4.3)
This is for a 3-gluon vertex; a similar expression would apply for the 4-gluon interaction.
We should note that with this convention, the internal line factors are already taken
43
into account in V . In fact, this convention introduces some extra z factors that we will
remove later when calculating the amplitude.
Finally, for a wavefunction and fragmentation function calculation, the energy de-
nominators defined in Sec. 3.3.4 will, respectively, take the following forms:
Dwf(1,...,N) =
~k 21...N ⊥z1...N
−N∑
i=1
~k 2i ⊥zi
, (4.4)
Dff(1,...,N) =
N∑
i=1
~k 2i ⊥zi−~k 2
1...N ⊥z1...N
. (4.5)
In addition, we will often encounter the following scalar products,
ε±i · kj =~ε ±⊥ · ~ki⊥k+i
k+j − ~ε ±⊥ · ~kj⊥ = zj ~ε
±⊥ ·
(~ki⊥zi−~kj⊥zj
)(4.6)
Here we have used ki = ziP+, where P+ is a total incoming momentum. Finally, we
define
vij = ~ε −⊥ ·(~ki⊥zi−~kj⊥zj
), (4.7)
v∗ij = ~ε +⊥ ·
(~ki⊥zi−~kj⊥zj
), (4.8)
ξij =zizjzi + zj
. (4.9)
Here v is like a relative velocity and ξ is like a reduced mass. The variables vij are related
to the spinor products that are frequently used to express the helicity amplitudes in the
literature, see Chapter 2
[ji] =√
2zizj ε−⊥ ·(~ki⊥zi−~kj⊥zj
)=√
2zizjvij (4.10)
〈ij〉 =√
2zizj ε+⊥ ·(~ki⊥zi−~kj⊥zj
)=√
2zizjv∗ij . (4.11)
An important note is that our definitions of positive and negative helicities here and
in Chapter 3 are reversed from those in Chapter 2 and most of literature [52, 53, 14].
In other words, our positive helicity vector is actually Mangano and Parke’s negative
helicity vector, and vice-versa.
44
4.2 Wavefunction
k1
k2
kN+1
k0
k3
k4
k1
k2
kN+1
k0
k3
k4
×
ΨN
Figure 4.1: This figure shows how an amplitude graph can be decomposed into a wave-function times an extra vertex. Light-front time x+ increases to the right and the arrowsrepresent the longitudinal momentum direction. The blue lines indicate the intermediatestates, which would contribute energy denominators. Notice that the wavefunction hasan on-shell initial state, but an off-shell final state, to which other components can beattached. N is the number of final state gluons present in the wavefunction ΨN .
The wavefunction is defined as having its initial state on-shell and its final state off-
shell. In this sense, the final state is treated as an “intermediate” state which brings with
it an energy denominator. We could then have the decomposition shown in Fig. 4.1.
Here a graph which contributes to the 1 to N + 1 amplitude can be written in terms of a
wavefunction of N gluons times an extra vertex. More specifically, the graph in Fig. 4.1
is only one out of many which contribute to the wavefunction. The wavefunction actually
includes all possible processes that start with 1 particle and end with N . Therefore, a
better representation would be
ΨN
k1...N
=
k1
k2
kN
(4.12)
where the blob includes the vertices and denominators for this kind of process.
Now we proceed to find the wavefunction for the (+→ + . . .+) helicity configuration
45
by implementing the procedure outlined before. To be clear, the initial incoming gluon
and the final outgoing gluons all have positive helicity. Looking at Table A.1 we see
that this choice eliminates the 4-gluon and instantaneous interactions. We also want to
remark on the importance of the initial particle being on-shell: if the initial gluon had
a non-zero virtuality, obtaining a closed-form solution to the wavefunction seems very
difficult if not impossible.
4.2.1 Recursion relation
+
k12
k3
k4
kN+1
k1...N+1
×k1
k2
k1
k23
k4
kN+1
k1...N+1
×k2
k3
+
k1
k2
k3
kN N+1
k1...N+1+
×kN
kN+1
Figure 4.2: The diagrams represent the wavefunction recursion relation for N + 1 finalgluons. Note that we need to add over all the possibilities for the final splitting.
The wavefunction can be calculated with the light-front rules, see Sec. 3.3.4, and,
thus, could be schematically written as
ΨN ∼∏ V
D, (4.13)
46
where we have N − 1 factors in the product, one for each vertex present. Furthermore,
it should include all possible vertex arrangements for N particles. Thus, if we know ΨN
but are interested in ΨN+1, we need to figure out how to incorporate the additional V /D
term.
For this purpose, let’s imagine the process of arriving at N gluons as follows. We
begin with one gluon which splits into two. Then one of these two splits into two and we
end up with three gluons. Afterwards, one of these three splits and we end up with four.
This splitting continues until we reach N gluons. Hence, to arrive at N + 1 gluons one
of the N legs of ΨN must split. The splitting can occur at any of the legs and, therefore,
to obtain the full wavefunction one needs to sum over all the different possibilities. This
idea is illustrated in Fig. 4.2 and gives the following result
Ψ(+→+...+)(k1, k2, . . . , kN+1)
=i
Dwf(1,...,N+1)
N∑
i=1
V+→++3 (−ki i+1, ki, ki+1)Ψ+→+...+(k1, . . . , ki i+1, . . . , kN+1)
=2g
Dwf(1,...,N+1)
N∑
i=1
v∗i i+1√ξi i+1
Ψ+→+...+(k1, . . . , ki i+1, . . . , kN+1) . (4.14)
Here, Ψ(+→+...+)(k1, k2, . . . , ki i+1, . . . , kN+1) (with ki i+1 ≡ ki + ki+1) is the N gluon
wavefunction in momentum space before the splitting of gluon with momentum ki i+1
and Dwf(1,...,N+1) is the denominator for the last state with N+1 gluons. Also note that the
selected helicity configuration requires that all internal lines have only positive helicities,
as a negative helicity for one of these would give a zero contribution.
We note that all dependence on momenta of daughter gluons i and i + 1 is now
encoded in two variables: ξi i+1 and v∗i i+1. In addition, this recurrence formula is written
for the special choice of the helicities but it can be readily generalized for the other case
of helicities. Eq. (4.14), for the current has the explicit form
Dwf(1,...,N+1) Ψ(+→+...+)(1, 2, . . . , N + 1) = 2g
v∗12√ξ12
Ψ(+→+...+)(12, 3, . . . , N + 1) +
2gv∗23√ξ23
Ψ(+→+...+)(1, 23, . . . , N+1) + . . . +2gv∗nN+1√ξN N+1
Ψ(+→+...+)(1, 2, . . . , N N+1) .
(4.15)
Since we are working with only the (+ → + . . .+) helicity configuration in this
section, from now on we will suppress the helicities when writing the wavefunction.
47
4.2.2 Pattern
The wavefunction for the incoming state has the normalization Ψ(1) = 1. After the first
splitting one gets
Ψ(1, 2) = 2g1√ξ12
v∗12
−2ξ12v12v∗12
= −g 1√ξ12
1
ξ12v12, (4.16)
where we have used z12Dwf(1,2) = −2z1z2v12v
∗12. According to (4.15), the next splitting
leads from Ψ(1, 2) to Ψ(1, 2, 3) with the graphs depicted in Fig. 4.3 and the result is
k0
k1
k2
k3
(a)
k0
k1
k2
k3
(b)
Figure 4.3: Graphs contributing to the gluon wavefunction with 3 gluons.
D(1,2,3)Ψ(1, 2, 3) = 2g
[v∗12√ξ12
Ψ(12, 3) +v∗23√ξ23
Ψ(1, 23)
](4.17)
= − 2g2
[v∗12√ξ12ξ(12)3
1
ξ(12)3 v(12)3+
v∗23√ξ23ξ1(23)
1
ξ1(23) v1(23)
](4.18)
= −2g2
√z0√
z1z2z3
v∗12ξ1(23)v1(23) + v∗23ξ(12)3v(12)3
ξ(12)3ξ1(23)v(12)3 v1(23), (4.19)
where we used ξ12ξ(12)3 = ξ23ξ1(23) = z1z2z3z1+z2+z3
= z1z2z3z0
. We can further simplify the
numerator of this expression to obtain,
Ψ(1, 2, 3) = g2
√z0√
z1z2z3
1
ξ(12)3ξ1(23)
1
v(12)3 v1(23). (4.20)
48
Note that, the energy denominator Dwf(1,2,3) disappeared from the equation as it has can-
celed with the numerator when finding the common denominator for expression (4.19).
This is actually quite an important property and leads to the very simple expression
for the resummed gluon wavefunction. It is also important in recovering some crucial
properties of the helicity amplitudes as we shall show later.
A pattern has emerged from these two examples and the resummed wavefunction
with N gluon components seems to be given by the following compact expression [48]
Ψ(1, 2, . . . , N) = (−g)N−1
√z0√
z1z2 . . . zn
1
ξ(12...n−1)n ξ(12...n−2)(n−1n) . . . ξ1(2...n)
× 1
v(12...n−1)n v(12...n−2)(n−1n) . . . v1(2...n). (4.21)
4.2.3 Proof
The formula (4.21) can be proven by induction. Assuming the formula is correct, the
wavefunction for N + 1 particles would then take the following form:
Dwf(1,...,N+1)Ψ(1, . . . , N + 1) = (−1)N−12gN
√z0
N∑
i=1
v∗i+1 i√ξi i+1
1√z1z2 . . . (zi + zi+1) . . . zN
× 1
(ξ(12...N)N+1 ξ(12...N−1)(N N+1) . . . ξ1(2...N+1))′1
(v(12...N)N+1 v(12...N−1)(N N+1) . . . v1(2...N+1))′,
(4.22)
where the prime indicates that the term ξ(1...i)(i+1...N+1)v(1...i)(i+1...N+1) does not appear
in the denominator. Finding the common denominator we get
Dwf(1,...,N+1)Ψ(1, . . . , N + 1)
= (−1)N−1 2gN√z0√
z1z2 . . . zN
1
ξ(12...N)N+1 ξ(12...N−1)(N N+1) . . . ξ1(2...N+1)
× 1
v(12...N)N+1 v(12...N−1)(N N+1) . . . v1(2...N+1)
N∑
i=1
ξ(1...i)(i+1...N+1)v(1...i)(i+1...N+1)v∗i+1 i,
(4.23)
49
and with the help of the following identity
v(1...i)(i+1...N)ξ(1...i)(i+1...N) =i∑
j=1
~kj ⊥ −z(1...i)
z(1...N)
N∑
j=1
~kj ⊥ = ~k(1...i) ⊥ −z(1...i)
z(1...N)
~k(1...N) ⊥,
(4.24)
where the momentum conservation conditions have been utilized. The above identity
can be used to show that
N∑
i=1
v∗i i+1ξ(1...i)(i+1...N+1)v(1...i)(i+1...N+1) =1
2
(N+1∑
i=1
~k 2i ⊥zi−~k 2
1...N+1 ⊥z1...N+1
)=
− 1
2Dwf
(1,...,N+1) . (4.25)
The above expression is what appears in the numerator after combining the different
terms in the recursion formula to a common denominator. It, therefore, cancels with an
overall denominator for the wavefunction, leading to the simple form of the wavefunction
(4.21). The example of that property was used to simplify Eq. (4.19).
4.3 Fragmentation function
Using the techniques developed for the wavefunctions one can similarly resum the graphs
in a kinematical situation when the initial state is an off-shell gluon, and then it fragments
to give the final N -gluons which are now on-shell. The fragmentation function, TN can
be represented by Fig. 4.4 and we can think of it as being a subtree of a larger graph.
There are interesting similarities between the wavefunctions and fragmentation func-
tions. This can be expected as the topology of the graphs is exactly the same. There are
some important differences, however, which are related to the properties of factorization.
Namely, the fragmentation functions can be shown to factorize (see Sec. 3.5), whereas
the gluon wavefunctions do not exhibit this property (at least not in momentum space).
This stems from the fact that, in the latter case, one still has the energy denominator
corresponding to the N gluons, see Eq. (4.12). This means that the N gluons of the
wavefunction are in the “intermediate” state and still have to interact. Therefore they
do not necessarily form disconnected trees and, therefore, can be correlated and cannot
be factorized.
Next we proceed to calculate the fragmentation function for the (+ → + . . .+)
helicity configuration by following what is, basically, the same procedure as for the
wavefunction, albeit with some minor differences.
50
k1
k2
kn
k(1...n)
Figure 4.4: Pictorial representation of the fragmentation amplitude T (1, . . . , N) for asingle off-shell initial gluon. N is the number of final state gluons.
4.3.1 Recursion relation
The factorization property can be used then to write down the explicit recursion formula
for the fragmentation amplitudes. Namely, the fragmentation into N + 1 gluons denoted
by T (1, . . . , N + 1) can be represented by lower fragmentation factors T (1, . . . , i) and
T (i+ 1, . . . , N + 1) and by summing over the splitting combinations. To be precise one
has
T (+→+...+)(1, 2, . . . , N + 1) =2g
Dff(1,...,N+1)
N∑
i=1
{v∗(1...i)(i+1...N+1)√ξ(1...i)(i+1...N+1)
× T (+→+...+)(1, . . . , i ) T (+→+...+)(i+ 1, . . . , N + 1)
}. (4.26)
This expression is the final state analog of formula (4.14) for the iteration of the wave-
function and it is schematically depicted in Fig. 4.5.
k1
k2
ki
ki+1
ki+2
kn+1
k(12...n+1)Σi
k(i+1...n+1)
k(1...i)T
T
V3
Figure 4.5: Pictorial representation of the factorization property represented inEq. (4.26), a light-front analog of the Berends-Giele recursion relation. The helicities ofthe outgoing gluons are chosen to be the same in this particular case. Dashed verticalline indicates the energy denominator Dff
(1,...,N+1).
The above defined fragmentation functions T can be related to the gluonic currents
which play an important role in the Berends-Giele recursion relations. As we saw in Sec.
51
2.4, these recursive relations utilize the (gauge-dependent) current Jµ, which is obtained
from the amplitudes by taking one of the gluons off-shell. In the case of the light-front
calculation, the light-front current can also be defined and related to the fragmentation
function in the following way
T (1, . . . , N) ≡ εµ(12 . . . N)Jµ(1, 2, . . . , N)
where by εµ(12 . . . N) we denote the polarization vector of the incoming (off-shell) gluon
in the fragmentation function. With such a definition the factorization property for the
fragmentation function (4.26) is a light-front analog of the Berends-Giele [10] recursion
formula (2.52). The simpler form of (4.26) (as compared to (2.52)), which only includes
the 3-gluon vertex, stems from the fact that it has been written for a particular config-
uration of helicities. It is, nevertheless, possible to write down a general factorization
(recursion) relation for the fragmentation function which will include the 4-gluon vertex
as well as the instantaneous term. In fact, this will be necessary to reach our objective
in Chapter 6.
One important detail is that the Berends-Giele relations can be written on the level
of individual diagrams, whereas for the derivation of the analogous recursion relations
on the light-front (4.26), the summation over the time-ordering is necessary to decouple
the disconnected fragmentation trees.
4.3.2 Pattern
With T1 = 1 and suppressing the helicity configuration, the case of the splitting of an
off-shell gluon, denoted by (12), into 2 on-shell gluons (denoted respectively by 1 and 2)
the fragmentation amplitude can be expressed as
T (1, 2) =2g
Dff(1,2)
v∗12√ξ12
= 2gv∗12
2ξ3/212 v12v∗12
= g ξ−3/212
1
v12, (4.27)
where z12Dff(1,2) = 2z1z2v12v
∗12 for this case. For the next step in the recursion, we have
the graphs depicted in Fig. 4.3. However, this time there is no energy denominator from
the final state. Instead, we have an energy denominator from the initial gluon. Using
(4.26), we get
T (1, 2, 3) =2g
Dff(1,2,3)
[v∗1(23)√ξ1(23)
T (1)T (2, 3) +v∗(12)3√ξ(12)3
T (1, 2)T (3)
]
52
=2g2
Dff(1,2,3)
v∗1(23)
ξ1/21(23) ξ
3/223 v23
+v∗(12)3
ξ1/2(12)3 ξ
3/212 v12
=2g2
Dff(1,2,3)
(z123
z1z2z3
)3/2[ξ1(23)v
∗1(23)v12 + ξ(12)3v
∗(12)3v23
v12v23
]
= g2
(z123
z1z2z3
)3/2 1
v12v23, (4.28)
where we have used ξ1(23)v∗1(23)v12 + ξ(12)3v
∗(12)3v23 = Dff
(1,2,3)/2 on the last line.
This time the pattern suggests that the general expression for the fragmentation part
of the amplitude for 1 to N gluons at tree-level is given by [48]
T (1, 2, . . . , N) = gN−1
(z(12...N)
z1z2 . . . zN
)3/2 1
v12v23 . . . vN−1N. (4.29)
4.3.3 Proof
The above formula can be again proven by mathematical induction using analogous re-
lations to the (4.24) and (4.25). On top of that, one crucial assumption is that of the
factorization property of the fragmentation functions, which are formed from topologi-
cally disconnected trees and originate from different off-shell parents. This was discussed
in Sec. 3.5.
Assuming (4.29) to be correct, we would then have
T (1, . . . , N+1) =2gN
D(1,...,N+1)
N∑
i=1
{v∗(1...i)(i+1...N+1)√ξ(1...i)(i+1...N+1)
(z(12...i)
z1z2 . . . zi
)3/2( z(i+1...N+1)
zi+1 . . . zN+1
)3/2
× 1
v12v23 . . . vi−1 i
1
vi+1 i+2 vi+2 i+3 . . . vN N+1
}
=2gN
D(1,...,N+1)
(z(1...N+1)
z1 . . . zN+1
)3/2 1
v12 . . . vN N+1
N∑
i=1
ξ(1...i)(i+1...N+1)v∗(1...i)(i+1...N+1)vi i+1
= 2gN(z(1...N+1)
z1 . . . zN+1
)3/2 1
v12 . . . vN N+1, (4.30)
which is of the same form as (4.29). In getting this result we have, once again, used the
fragmentation function analog of (4.25) and
z(12...i)z(i+1...N+1)
ξ(1...i)(i+1...N+1)z1z2 . . . zizi+1 . . . zN+1=
z(1...N+1)
z1 . . . zN+1. (4.31)
53
4.3.4 T (−→−+...+)
In this section we will show how the fragmentation function T (−→−+...+)(1, . . . , N) is
derived. We will follow the same procedure used to calculate T (+→+...+)(1, . . . , N).
We begin with N = 2. It can easily be seen that the fragmentation function for this
case is
T (−→−+)(1, 2) = 2gz1√
z12z1z2
v∗(12)2
Dff(1,2)
= 2gz1√
z12z1z2
v∗(12)2
2ξ12v∗12v12
= g
(z1
z12
)2( z12
z1z2
)3/2 1
v12. (4.32)
One should note that this is the same as that for T (+→++)(1, 2) multiplied by an extra
factor of (z1/z12)2. To get T (−→−++)(1, 2, 3) recursively, it is slightly more complicated.
This is because it includes a combination of T (−→−+) and T (+→++).
T (−→−++)(1, 2, 3)
=2g2
Dff(1,2,3)
[z1√
z123z1z23
(z23
z2z3
)3/2 v∗(123)(23)
v23+
z12√z123z12z3
(z1
z12
)2( z12
z1z2
)3/2 v∗(123)3
v12
]
=2g2
Dff(1,2,3)
[(z1
z123
)2( z23
z2z3
)3/2 v∗1(23)√ξ1(23)v23
+
(z12
z123
)2( z1
z12
)2( z12
z1z2
)3/2 v∗(12)3√ξ(12)3v12
]
=2g2
Dff(1,2,3)
(z1
z123
)2 v∗1(23)
ξ3/223 ξ
1/21(23)v23
+v∗(23)3
ξ3/212 ξ
1/2(12)3v12
, (4.33)
where the first and second terms inside the brackets in the first line come from T (+→++)
and T (−→−+) respectively. We, once again, recognize that this is the expression given in
(4.28) for T (+→++)(1, 2, 3) multiplied by (z1/z123)2. Thus, the final result for N = 3 is
T (−→−++)(1, 2, 3) = g2
(z1
z123
)2( z123
z1z2z3
)3/2 1
v12v23. (4.34)
This pattern implies that for general N the fragmentation function should be of the form
T (−→−+...+)(1, . . . , N) =
(z1
z1...N
)2
T (+→+...+)(1, . . . , N). (4.35)
We show that this expression is indeed correct by substituting (4.35) into the recursion
54
relation for T (−→−+...+)(1, . . . , N + 1) given below.
T (−→−+...+)(1, . . . , N + 1) =2g
Dff(1,...,N+1)
N∑
i=1
{(z1...i
z1...N+1
)2 v∗(1...i)(i+1...N+1)√ξ(1...i)(i+1...N+1)
× T (−→−+...+)(1, . . . , i ) T (+→+...+)(i+ 1, . . . , N + 1)
}. (4.36)
We see that T (−→−+...+)(1, 2 . . . , i) includes a factor of (z1/z1...i)2 which can be combined
with the (z1...i/z1...N+1)2 in (4.36) to give
T (−→−+...+)(1, . . . , N + 1) =2g
D(1,...,N+1)
(z1
z1...N+1
)2 n∑
i=1
{v∗(1...i)(i+1...N+1)√ξ(1...i)(i+1...N+1)
× T (+→+...+)(1, . . . , i ) T (+→+...+)(i+ 1, . . . , N + 1)
}. (4.37)
This is simply the recursion relation for T (+→+...+)(1, . . . , N+1) given in (4.26) multiplied
by (z1/z1...N+1)2. Thus, combining (4.26), (4.35) and (4.36) we get
T (−→−+...+)(1, . . . , N + 1) =
(z1
z1...N+1
)2
T (+→+...+)(1, . . . , N + 1), (4.38)
which proves (4.35) to be correct.
Before leaving this section we want to discuss a possible generalization that arises
due to the results from this section. For a moment let us think of our gluons as all
outgoing. In this case, for the case of T (−→−+...+), the incoming gluon with negative
helicity would actually be an outgoing gluon with positive helicity. Hence, only line 1
would have a negative helicity. What we want to note is that it is gluon 1 which appears
at the numerator of the prefactor in (4.35). What about T (+→+...+)? For this case we
can imagine it having a prefactor of 1 = (z1...N/z1...N )2. This time we notice that line
(1 . . . N) is the only one with negative helicity. This seems to imply that the numerator
of the prefactor will be determined by which gluon is the only one that has a negative
helicity. Thus, we can make a further generalization for the fragmentation functions:
T (−→+...+−+...+)(1, . . . , i, . . . N) =
(zi
z1...N
)2
T (+→+...+)(1, . . . , N), (4.39)
where the negative helicity belongs to the i-th gluon.
A similar relation appears also for the wavefunctions. Finally, let us note that these
55
are educated guesses and we currently do not have a formal proof for these.
4.4 Connection to amplitudes
Ultimately our goal is to find amplitudes. Now that we have an idea of how to calculate
wavefunctions and fragmentation functions, at least for simple helicity configurations,
we must figure out how to reuse these to find amplitudes. This becomes very simple
once we make the following observation. The wavefunctions, fragmentation functions
and amplitudes for a particular helicity configuration are all calculated from the same
set of diagrams and using the same LF rules. The only difference is the treatment of
the initial and final states. Compared to the amplitude, both the wavefunction and the
fragmentation function have an extra energy denominator. Therefore, all we need to
do is remove this extra factor and we obtain the gluon amplitudes, except for one more
detail.
We now need to take into account the vertex conventions (4.3). This convention
appropriately took into account all the internal line factors from the LF rules, but it in-
troduced extra√z factors. Concretely, it introduced an extra factor of 1/
√z1...Nz1 . . . zN .
Taking all of this into account,
M(λ(1...N)→λ1...λN )(k1, . . . , kN )
= −iDwf(1,...,N)
√z(1...N)z1 . . . zN Ψ(λ(1...N)→λ1...λN )(k1, . . . , kN ), (4.40)
M(λ(1...N)→λ1...λN )(k1, . . . , kN )
= −iDff(1,...,N)
√z(1...N)z1 . . . zN T (λ(1...N)→λ1...λN )(k1, . . . , kN ), (4.41)
where the −i comes with the energy denominator. The amplitudeMN is, in its current
state, off-shell. I.e., either its initial or final state is off-shell depending on which of the
above relations we use. To make it on-shell, we impose D(1,...,N) = 0 and evaluate the
amplitude. I.e., MN |D(1,...,N)=0 gives the on-shell amplitude.
4.4.1 M(+→+...+)N
As an example, we will quickly calculate the on-shell tree-level gluon amplitudeM(+→+...+)N .
This corresponds to the (− + . . .+) helicity configuration when all lines are outgoing.
Hence, it is expected to be 0 (see Chapter 2).
56
Using either (4.40) or (4.41) we get that M(+→+...+)N ∝ D(1,...,N). This is so because
the denominators in neither of the solutions (of the wavefunction and fragmentation
function) can cancel the D(1,...,N). Therefore, when going to the on-shell limit
M(+→+...+)N =M(+→+...+)
N
∣∣∣D(1,...,N)→0
= 0
as expected.
We should note that we could have also used a recursion relation similar to (4.14) or
(4.26) to obtain the on-shell M(+→+...+)N . We arrive at the same result but the process
is slightly lengthier, not as convenient and, definitely, not what we want to use as we
move forward towards more complicated calculations.
Chapter 5Crossing symmetry on the light-front
In the previous chapter we saw how to calculate wavefunctions and fragmentation func-
tions, and how to obtain transition amplitudes from these objects. However, as was
discussed in previous chapters, the process that is of more interest to experimentalists
is that of gluon scattering, i.e. (gg → g . . . g). Why then have we focused so much–and
will continue to do so in the following chapters–on the transition amplitude? Simply
because these are easier to compute. Since LFPT [41, 76, 77, 78, 75] tells us that we
need to sum over all possible light-front time orderings, calculating the scattering am-
plitude requires many more graphs than if we were calculating the associated transition
amplitude. Finally, because of crossing symmetry, if we know the expression for the
transition amplitude, we can analytically continue an outgoing particle to an incoming
one to find the scattering amplitude.
The fact that crossing symmetry holds on the light-front is, however, not a given. In
standard perturbation theory, this symmetry is intuitive as there is no time ordering, and
the graphs for both processes are topologically equivalent. Furthermore, each light-front
graph includes energy denominators which entangle the momenta of the particle, and
these, in principle, change when a line goes from outgoing to incoming. Nevertheless, it
is possible to show that crossing symmetry does hold on the light-front [49, 81] and the
proof of this is the subject of this chapter.
We will begin with a simple example, (+− → ++), that showcases crossing symmetry
on the light-front. Specifically, we will show that the expression for this scattering
amplitude is the same, modulo a phase1, as the expression we derived in Chapter 4 for
1The phase depends on you the choice of polarization vectors. For our choice it is equal to 1. If oneutilizes the conventions in [78] the phase would be -1.
58
the (+→ + . . .+) transition amplitude.
Following this, we will then provide a perturbative proof for crossing symmetry on
the light-front. With the aid of another example, we will establish that the scattering
graphs can all be obtained from the transition graphs. Then, we just need to examine
how changing a gluon from outgoing to incoming changes the contribution of a graph.
The end result is that the expressions for the scattering and transition amplitudes are
the same up to a phase. This will also allow us to compare our results from the next
chapter with the MHV amplitudes [1, 53, 52] discussed in Chapter 2.
There is, however, one caveat to our proof for crossing symmetry: it requires that
energy be conserved between the initial and final states. In other words, the initial and
final states need to be on-shell. This, unfortunately, means that we do not appear to have
crossing symmetry for off-shell amplitudes. We can still calculate scattering amplitudes
using LF rules, but we cannot make use of our results for off-shell transition amplitudes.
Because of this, this chapter will focus on on-shell amplitudes. Therefore, let us
introduce a notation to represent these:
M 2→N =M(λ0λ1→λ2...λN+1)(k0, k1; k2, . . . , kN+1), (5.1)
MN+1 =M(λ0→λ1...λN+1)(k1, . . . , kN+1), (5.2)
where M 2→N and MN+1 are the 2 → N on-shell scattering and 1 → N + 1 on-shell
transition amplitudes, respectively, that we are trying to relate. In (5.1) k0 and k1 are
defined as incoming and k2 to kN+1 as outgoing. In (5.2) k1 to kN+1 are outgoing and
k0, which is incoming, is left to be implicit.
5.1 Crossing symmetry example
In order to prepare the ground for the derivation of general relations between the gluon
transition amplitudes derived earlier and scattering amplitudes, we shall consider at first
a lowest order example of the 2 → 2 amplitude using the rules of the LFPT and the
notation introduced in the previous sections. The goal of this simple example is to
illustrate the fact that the resulting amplitude has a very similar form to the 1 → 3
transition amplitude.
To calculate the amplitude for 2 → 2 gluon scattering, M2→2 one needs to compute
and add up all the graphs shown in Fig. 5.1. Let us focus on the situation where
the helicities for this amplitude are (+− → ++), the momenta of the gluons 0 and 1
59
k0
k1 k2
k3
k03
(a)
k0
k1 k2
k3
k03
(b)
k0
k1 k2
k3
k01
(c)
k0
k1 k2
k3
(d)
k0
k1 k2
k3
(e)
Figure 5.1: Basic graphs which contribute to the 2 → 2 amplitude in LFPT. Dashedlines indicate the energy denominators for the intermediate states. The cross on thevertical line in graph (e) indicates the instantaneous gluon exchange.
are incoming and the momenta of the gluons 2 and 3 are outgoing. It is well known
that the amplitude for such helicity configuration has to vanish, see [52] and discussion
in Chapter 2. Thus, it is the simplest example. Nevertheless it is instructive to see
how that happens on the light-front. Graphs d and e give contributions which vanish
identically for this helicity configuration. However, the contributions from graphs a, b, c
do not vanish separately. The amplitude Ga for the graph in Fig. 5.1a is given by
Ga =i
z12
V1V2
D1, (5.3)
where the vertices and the energy denominator are given by
V1 = V +→++3 (−k0, k(12), k3) = −2igz0v
∗(12)3 , (5.4)
V2 = V +−→+3 (−k(12),−k1, k2) = −2igz12v
∗12 , (5.5)
D1 =~k2
0 ⊥z0−~k2
(12) ⊥
z12
−~k2
3 ⊥z3
, (5.6)
to give
Ga = −4ig2z0
v∗(12)3
v∗12
~k20 ⊥z0−
~k2(12) ⊥z12
− ~k23 ⊥z3
. (5.7)
60
Here we used notation 1 to denote the change in the sign of the corresponding transverse
and longitudinal momenta, i.e. zij ≡ zi − zj and ~k(i j)⊥ ≡ ~ki⊥ − ~kj⊥. Similarly, the
amplitudes Gb and Gc for the graphs in Figs. 5.1b and 5.1c, respectively, are given by
Gb = 4ig2z0
v∗(12)3
v∗12
~k21⊥z1−
~k2(12)⊥z12
− ~k22⊥z2
, (5.8)
Gc = −4ig2z0
v∗1(23)
v∗23
~k20⊥z0
+~k2
1⊥z1−
~k2(23)⊥z23
. (5.9)
We have to take into account that the internal lines need to have positive fractions
of longitudinal momenta in Figs. 5.1a and 5.1b. This means we need to include the
corresponding step functions which enforce the ordering of the momenta. Hence,M2→2
resulting from these three graphs is given by
M2→2 = Θ(z0 − z3)Ga + Θ(z3 − z0)Gb +Gc . (5.10)
It is trivial to show that Ga and Gb are actually the same and can be combined into
one expression. Using~k2
0⊥z0
+~k2
1⊥z1
=~k2
2⊥z2
+~k2
3⊥z3
we can write the denominator in (5.8) as
−~k20⊥z0−
~k2(12)⊥z12
+~k2
3⊥z3
. The following identities also hold
v(12)3 = −v3(21), z(12) = −z21 . (5.11)
Using them in (5.7) and (5.8) we realize that the expressions for Ga and Gb are the same
which gives
M2→2 = Ga +Gc , (5.12)
where the theta functions are absent (only global momentum conservation conditions
on external longitudinal momenta are present but they are implicit). The fact that the
graphs a and b give the same contributions, can be shown to hold for more complicated
graphs. This is discussed in detail in the next section and it will be used to prove the
relationships between transition and scattering amplitudes. In fact, this is where the
caveat that we discussed in the introduction comes in. We used energy conservation
to rewrite the denominator in (5.8), so as to get rid of the theta functions, allowing
us to arrive at the simplified expression (5.17) in the following paragraphs . If this
were an off-shell amplitude, we would not be able to do this and we would not be
able to significantly simplify (5.10). The best we could do would be to multiply Gc by
61
1 = Θ(z0 − z3) + Θ(z3 − z0) to get
M2→2 = Θ(z0 − z3)(Ga +Gc) + Θ(z3 − z0)(Gb +Gc) . (5.13)
Even then, the usefulness of this expression is unclear.
Now, having reduced the expression for the sum of both graphs to (5.12), we proceed
to show that amplitude M 2→2 in (5.12) has identical structure to the form of the tran-
sition amplitude M 3 arrived at by combining Eqs. (4.19) and (4.40) (modulo a phase).
To demonstrate this one can use the relation D1 = −2ξ(12)3v(12)3v∗(12)3
to obtain
Ga = 2ig2z0v∗12
ξ(12)3v(12)3
, (5.14)
Similarly, we can rewrite the energy denominator of graph (c)
~k20⊥z0
+~k2
1⊥z1−~k2
(23)⊥
z23= −2ξ1(23)v1(23)v
∗1(23) ,
to get
Gc = 2ig2z0v∗23
ξ1(23)v1(23)
. (5.15)
Putting both expressions together we obtain,
M 2→2 = 2ig2z0
(v∗12
ξ(12)3v(12)3
+v∗23
ξ1(23)v1(23)
)(5.16)
= 2ig2z0
(v∗12ξ1(23)v1(23) + v∗23ξ(12)3v(12)3
ξ(12)3ξ1(23)v(12)3v1(23)
). (5.17)
We notice that the structure of M2→2 now has exactly the same structure as the tran-
sition amplitude forM 3 in (4.19) (combining with (4.40)), when replacing 1→ 1. Most
importantly, just as in (4.19), the numerator in (5.17) is proportional to the difference
of the energies from the initial to the last state in the problem. Since in the case con-
sidered (i.e. on-shell amplitude) the last state is on-shell state, the numerator, and thus
M2→2, will be zero. Therefore we see that the relatively compact form of the transition
amplitude derived in Chapter 4 and the recursion relations for the wavefunctions are
closely related to the properties of the amplitudes. In this particular case the simplicity
of the wavefunctions are related to the fact that the amplitudes with a single helicity
different from the rest are vanishing. The result derived for lowest order scattering can
62
be generalized to arbitrary number of the final state particles (at the tree-level). It is
important to note though that the wave functions derived in Sec. 4.2 themselves are
non-zero due to the off-shellness of the last state.
Before leaving this section, there are two more observations we want to make. First,
we saw that the sum Ga+Gc has the same structure asM(+→+++)(k1, k2, k3). Therefore,
it could be expected that Gb + Gc would have the same structure as some other kind
of 1→3 transition amplitude. Specifically, we expect it to have the same structure
as M(−→++−)(k2, kN+1, k0), with k1 being the sole incoming gluon. This transition
amplitude could also be analytically continued to obtain M 2→2.
The second observation is that M2→2 has the same structure as the wavefunctions
derived in the previous chapter. Why is that? Why don’t they have same structure as
fragmentation functions, Eq. (4.28)? The reason lies in the denominators. The denomi-
nators in Eqs. (5.7) and (5.8) are written in the same form as the energy denominators
for the wavefunctions: energy of initial state minus energy of intermediate state. In fact,
the denominator in (5.7) can be written as Dwf((12),3)
. If, instead, we had written them
as energy of final state minus energy of intermediate state, like the energy denominators
of fragmentation functions, we would have gotten the structure of M 2→2 to be that of
fragmentation functions.
5.2 Crossing symmetry proof
Given the result of the explicit example shown in the previous section we proceed to
investigate the relation between the amplitude for a general number of final state gluons
M 2→2 and the transition amplitude MN+1 (via the wavefunction) which were derived
previously, see Ref. [48] and Sec. 4.2. That such relation should exist is quite intuitive as
the two objects correspond to the graphs which are topologically equivalent with only the
one external leg changing the momentum from the final to initial state. As mentioned in
the introduction, in the case of the LFPT [41, 76, 77, 78, 75] it is however not an entirely
obvious relation as the energy denominators, which entangle the momenta of the particles
in a given graph, in principle change via such operation and, moreover, some graphs are
vanishing. For example, for the case of the M 3 we have only two graphs depicted in
Fig. 4.3 which contain the 3-gluon vertices and give non-trivial contributions, whereas for
the 2→ 2 amplitude there are three graphs (with 3-gluon vertices) whose contributions
are non-vanishing, those shown in Fig. 5.1a, 5.1b, 5.1c. Therefore by changing the
momentum line from outgoing to incoming in Fig. 4.3, one needs to perform additional
63
summation over the time-ordering of some of the vertices, so that in our example the
contribution from graph (a) in Fig. 4.3 has to give two separate contributions in Fig. 5.1a
and 5.1b. We shall show how to identify and sum over the time-ordering for the case of
2→ 3 and then we perform a proof for the general case and show that the amplitude from
theM 2→N graphs can be obtained through analytical continuation from the light-front
transition amplitude MN+1 which contain in general a smaller number of graphs.
We will restrict ourselves to the case of only 3-gluon vertices which will limit the
possible helicity states. However, the methods developed are much more general and we
will utilize them later to find amplitudes with more complicated helicity states.
The following proof follows [49].
5.2.1 Guiding example
Let us consider the case of 2 → 3 gluon scattering. We want to find the M 2→3 graphs
from the graphs which describe the 1 → 4 gluon transition in the + → + + · · ·+configuration. This transition contains (4− 1)! = 6 graphs.
k0
k1 k2
k3
k4
k0
k1
k2
k3
k4
Figure 5.2: By changing the outgoing to incoming momentum one can recover the graphsof the scattering amplitudes from the transition amplitudes.
The way to obtain graphs for 2 → 3 scattering is by changing the k1 line from
outgoing into incoming from the gluon transition, as shown in Fig. 5.2. Naively, one
recovers only one graph from the transition, which is shown in this figure. However, since
our calculation is being done in the LFPT, we need to find all possible time-orderings.
Thus, from the sole graph depicted on the left hand side of Fig. 5.2 we obtain three
graphs of the same topology, which are depicted in Fig. 5.3. This continuation and
summation needs to the performed with all the graphs in theM 4 transition to arrive at
all the corresponding graphs needed to calculate M 2→3, depicted in Figs. 5.3-5.6. Here
we have excluded all the vacuum diagrams since they vanish on the light-front.
Figs. 5.3-5.6 group together all the graphs with the same topology. These groups are
in turn composed of subgroups where the directions of the lines are taken into account.
64
k0
k1 k2
k3
k4
(a) Subgroup G1,1
k0
k1 k2
k3
k4 k0
k1 k2
k3
k4
(b) Subgroup G1,2
Figure 5.3: A group of topologically equivalent graphs for 2 → 3 amplitude obtainedfrom the continuation of the k1 momentum, as shown in Fig. 5.2.
k0
k1 k2
k4
k3
(a) Subgroup G2,1
k0
k1 k2
k4
k3
(b) Subgroup G2,2
k0
k1 k2
k4
k3
k0
k1 k2
k4
k3
(c) Subgroup G2,3
Figure 5.4: A group of topologically equivalent graphs for 2→ 3 amplitude with differenttime-orderings.
65
k0
k1 k2
k4
k3
k0
k1 k2
k4
k3
(a) Subgroup G3,1
k0
k1 k2
k4
k3
(b) Subgroup G3,2
Figure 5.5: A group of topologically equivalent graphs for 2→ 3 amplitude with differenttime-orderings.
k0
k1
k2
k3
k4 k0
k1
k2
k3
k4
Figure 5.6: s - channel graph contributions to 2→ 3 scattering. Groups G4 and G5 areshown on the left and right respectively. Both consist of only one graph.
We know that for a graph to have a non-zero value all its lines must have a positive
longitudinal momentum. Thus, different kinematical conditions which are satisfied by
the longitudinal momenta of the external lines apply for the different subgroups inside a
group. To make this statement more clear we can introduce the following classification
for the graphs. Let us call Gi the group of graphs which have the same topology, with
G2 being the group depicted in Fig. 5.4. We denote by Gi,j the subgroup of graphs in
group Gi, which are distinguished by different kinematical conditions on the external
momenta. In our example G2,j denote the subgroups listed in plots a-c in Fig. 5.4.
Let
s2,j ≡ sum of graphs in subgroup G2,j , (5.18)
66
S2 ≡ sum of graphs in group G2 . (5.19)
We can then write
S2 = s2,1Θ(z0−z4) Θ(z2−z1)+s2,2Θ(z4−z0) Θ(z1−z2)+s2,3Θ(z0−z4) Θ(z1−z2) . (5.20)
However, it can be demonstrated that contributions from all the graphs are actually the
same, i.e. s2 ≡ si,1 = s2,2 = s2,3 (this is discussed in Sec. 5.2.3 for a general case). Thus,
S2 = s2 . (5.21)
This sum contains all physical arrangements of the kinematical conditions on the external
lines, so we can remove the Θ functions.
We should note that some groups consist of only a single subgroup. These will
be composed of only s-channel graphs and the internal lines are guaranteed to have a
positive longitudinal momentum regardless of the external lines. These are graphs shown
in Fig. 5.6.
What we have just seen is that, for each group, we can select one subgroup to
represent the whole group. In other words, the amplitude of this subgroup will give us
the contribution of the entire group. Then, it turns out that, when calculating M 2→N ,
it is possible to select a set of subgroups such that each of the m graphs in the set comes
from a direct 1 to 1 mapping from the m graphs in the transition amplitude MN+1.
To summarize, the point we are trying to make is that since the value of all subgroups
inside a group are the same, in order to calculate M 2→N all we need are the N ! graphs
that can be obtained by simply changing the direction of the k1 line in each of the N !
graphs in the 1 to N +1 gluon transitionMN+1. We implement this below for arbitrary
number of gluons.
5.2.2 General proof
In this subsection we shall show that the expression for the value of a graph G is equal
to the negative value of the expression of the 1 to N + 1 graph H from which G was
obtained. In general we know, Chapter 3, that a graph will be equal to (modulo overall
factors)VZD
, (5.22)
67
2 to N 1 to N + 1
~k0⊥ + ~k1⊥ =∑N+1
i=2~ki⊥ ~k0⊥ =
∑N+1i=1
~ki⊥
z0 + z1 =∑N+1
i=2 zi z0 =∑N+1
i=1 zi~k2
0⊥z0
+~k2
1⊥z1
=∑N+1
i=2
~k2i⊥zi
~k20⊥z0
=∑N+1
i=1
~k2i⊥zi
Table 5.1: Physical conditions for gluon scattering for the case of the 2 → N and1→ N + 1 amplitudes.
with
V ≡∏
all vertices
Vk , (5.23)
Z ≡∏
internal lines
zi , (5.24)
D ≡∏
intermediate states
Di , (5.25)
Thus, we need to compare Z, V and D for G and H.
It is important to set up the physical conditions for each of the cases. These conditions
are listed in Table 5.1. Note that we have included the energy condition, which we have
arrived at by setting the initial and final states on-shell.
We shall introduce two variables ~kA⊥ and zA. These will be defined differently for
the two cases we are working on. For the 2→ N case ~kA⊥ ≡ −~k1⊥, zA ≡ −z1 whereas
for 1→ N + 1 case ~kA⊥ ≡ ~k1⊥, zA ≡ z1. We can now see that these definitions allow us
to use the same mathematical expression for both cases. To be precise, for both 2 to N
and 1 to N + 1 the physical conditions are
~k0⊥ = ~kA +
N+1∑
i=2
~ki⊥
z0 = zA +
N+1∑
i=2
zi
~k20⊥z0
=~k2A⊥zA
+
N+1∑
i=2
~k2i⊥zi. (5.26)
Now we can proceed to compare factors Z. Let’s take as an example the graphs in
Fig. 5.2. Looking at these it is clear that, using ~kA⊥ and zA, we can write, for both
68
cases,
~kl⊥ = ~k0⊥ − ~k4⊥ = ~kA⊥ + ~k2⊥ + ~k3⊥, zl = z0 − z4 = zA + z2 + z3, (5.27)
~kq⊥ = ~k0⊥ − ~k4⊥ − ~kA⊥ = ~k2⊥ + ~k3⊥, zq = z0 − z4 − zA = z2 + z3, (5.28)
where l and q refer to the internal lines of the graphs. Once again we see that we can
use the same expression for both cases. This is true for any internal line one considers.
Thus, Z does not change from graphs 2→ N and 1→ N + 1.
Let’s consider the vertex factors V. By taking a graph and changing the direction of
the k1 line only one vertex is changed, the one with the k1 line (V1). The other vertices
remain the same and, because the expressions for the internal lines do not change, neither
do the expression for these vertices. Looking at Table A.1 we see that the expression for
V1 does not change. We have V1, 2→N = V1, 1→N+1. Therefore V2→N = V1→N+1. As it
turns out, however, this vertex factor can introduce a phase depending on the choice of
helicity vectors. For example, here we get a phase of +1, but in [49], where the authors
used the Brodsky-Lepage convention [78], the phase was -1.
Finally, let us examine the energy denominators D. Let us consider a general 2→ N
graph with arbitrary number of vertices and internal lines. The graph can also contain
loops and is schematically depicted in Fig. 5.7. Let Q denote the point where gluon
1 interacts with the rest of a graph. Using it as a reference, we split the graph into
a left sub-graph and a right sub-graph. Next, let denote by Da and Db the energy
denominators of the intermediate states a and b which are present just next to the
vertex, we thus assume that there are more vertices to the left and right of the vertex
Q, such that a and b are the intermediate states. For Fig. 5.7a we can write
Da = Ea −~k2
0⊥z0
= Ea +~k2
1⊥z1−
N∑
i=2
~k2i⊥zi
= Ea −~k2A⊥zA−
N∑
i=2
~k2i⊥zi
, (5.29)
and
Db = Eb −~k2
0⊥z0−~k2
1⊥z1
= Eb −~k2
0⊥z0
+~k2A⊥zA
= Eb −N∑
i=2
~k2i⊥zi
, (5.30)
where Ea and Eb are the sums of the energies in intermediate states a and b respectively.
Here, we’ve used (5.26) and explicitly excluded the energy of gluon 1 from Ea. Similarly,
69
for Fig. 5.7b we can write
Da = Ea −~k2
0⊥z0
= Ea −~k2
1⊥z1−
N∑
i=2
~k2i⊥zi
= Ea −~k2A⊥zA−
N∑
i=2
~k2i⊥zi
, (5.31)
Db = Eb −~k2
0⊥z0
+~k2
1⊥z1
= Eb −~k2
0⊥z0
+~k2A⊥zA
= Eb −N∑
i=2
~k2i⊥zi
. (5.32)
This time we’ve excluded the energy of the gluon 1 from Eb. Looking at Da and Db for
both cases once again one sees that, using ~kA⊥ and zA, the expressions are the same,
though in each of these cases ~kA⊥ and zA are defined differently. Hence, the contribution
from energy denominators D also does not change. This result is true for all graphs in
light-front theory with arbitrary topology.
These results essentially demonstrate that crossing symmetry holds for on-shell am-
plitudes. We can therefore analytically continue whatever expressions we obtain for the
1→ N + 1 transition MN+1 to obtain the 2→ N scattering amplitudes M 2→N :
...
k2
kn
k1
k0
Da Db
Q
(a)
...
k2
kn
k1
k0
Da Db
Q
(b)
Figure 5.7: Schematic representation of the crossing of the momentum of gluon 1 torelate the graphs for 1 → N + 1 with graphs 2 → N . Point Q denotes the vertex atwhich gluon 1 attaches to the graph. Da and Db denote the energy denominators for theintermediate states.
M(λ0λ1→λ2...λN+1)(k0, k1; k2, . . . , kN+1)
= eiθM(λ0→(−λ1)λ2...λN+1)(kA, k2, . . . , kN+1)∣∣∣~kA⊥→−~k1⊥, zA→−z1
, (5.33)
where θ = 0 in our conventions. Furthermore, since the physical conditions for both
cases are the same when written in terms of kA and zA, we can use them to simplify
70
M 2→N by simplifying M1→N+1 first as much as possible. For example, we know that
M(+→+...+)1→N+1 = 0, which immediately reveals that
M(+−→+...+)2→N = 0, (5.34)
as we expect.
5.2.3 Equal contributions from all Gi,j for fixed i
k1
k0Q
kc
P
...
...
k2
ki
ki+1
kn
(a)
k1
−kc
P
k0Q
...
...
k2
ki
ki+1
kn
(b)
Figure 5.8: Generic 2 to N tree-level graphs. In (a) gluon c departs from the lowersubgraph at Q and arrives at the upper subgraph at P. For (b) the opposite is true.
In this sub-section we shall investigate a general structure of the tree-level amplitudes
in light-front theory. In particular we shall demonstrate that the typical tree-level graphs
can be written in terms of the factorizable contributions originating from initial and final
state emissions and non-factorizable contribution which typically comes from the energy
denominator of the intermediate state exchanged between two subsets of graphs. On
top of that we shall show for a given group of topologically identical graphs, all the
subgroups which differ by the orientation of the exchanged particle within such group
have the same value. This is true with with the exception of those subgraphs for which
the physical conditions constrain the graphs to have a value of zero (i.e. vacuum graphs).
We will show this by demonstrating that graphs within the given subgroup sum up to the
expression which can be analytically continued to obtain the graphs from the different
subgroup. Generic tree-level graphs are shown in Fig. 5.8a and Fig. 5.8b. In these
figures, the top and bottom box both have a set of vertices and thus form subgraphs
with a definite topology. The graph represents the sum of all the possible time-orderings
between the upper and lower parts of the diagrams.
Points P and Q denote the vertices at which gluon labeled by c attaches to both
71
subgraphs. We need also to perform the sum over the relative orderings of these ver-
tices with respect to the orderings in the lower and upper subgraphs. For instance, for
situation depicted in Fig.5.8a we need to perform the sum over all the orderings of the
splittings in the upper box which occur before the vertex P with respect to the position
of vertex Q. One can call them initial state splittings of the subgraph U, as they occur
before the interaction P. Similarly, we have to sum over all the orderings of the splittings
of lower subgraph L with respect to P which occur after interaction depicted by the
vertex Q. These splittings one can refer to as the final state splittings of the subgraph
L. A similar situation is valid for a graph shown in Fig. 5.8b.
k0
k1
−kc
...
ki+1
kn
...
k2
ki
P
Q
Figure 5.9: 2 to n tree-level graph with only final state splittings.
Let us consider a graph with only final state splittings, that is with the splittings
which occur in the upper part occur only after vertex P and splittings in the lower
subgraph occur only after vertex Q. Such graph is depicted in Fig. 5.9. Let us label by
Ai the energy denominators of the upper subgraph U in the absence of lower subgraph
and gluon c, and by Bj the energy denominators in the lower subgraph in the absence of
the upper part as well as the gluon c. In particular Ai(Bj) is the energy denominator of
the intermediate state which is on the i(j) position when we count from the right hand
side of the corresponding subgraph. We denote by C the energy of the intermediate
gluon c. Consider a class of subset of graphs in which all the splittings in L and U occur
after vertex P. It was shown in [48] (see Chapter 3) that in such case the fragmentation
trees factorize. Using this property for such configuration we can write the contribution
from the energy denominators in the form
1
Am − C + B
1
A1A2 . . . AmB1 . . . Bn, (5.35)
where by B we denoted the difference of the energy denominator of the gluon 0,i.e.
B = B0 − B0 where B0 is the energy of the incoming gluon into the lower subgraph L
and B0 is the sum of the energies of the final states in the same subgraph L.
72
The next step is to include the remaining graphs of the same topology, but this time
with the splittings in the lower subgraphs occurring earlier than the point P but later
than Q. Summing this subset of diagrams together with the diagrams which result in
(5.35) one arrives at the factorized expression. Again, the upper tree factorizes from the
lower part and the sum of all the time-orderings for the graphs in Fig. 5.9 is
1
−C + B
1
A1A2 . . . AmB1 . . . Bn,
k0
k1
...ki+1
kn
...k2
ki
−kc
P
Q
Figure 5.10: 2 to N tree-level graph with only initial state splittings.
The first factor 1−C+B
is just the energy denominator for the simple 2 → 2 graph,
without any splittings in the final state, with the only assignment that the final energy
of the lower leg in this graph is given by B0. This denominator in turn multiplies the
factorized contributions to the fragmentation trees originating from subgraphs L and U.
The same procedure can be repeated for the general graph depicted in Fig. 5.10 with only
the initial state radiation. Finally one needs to consider the most general case in which
the splittings can occur both before and after P and Q on both the upper and lower
subgraphs simultaneously. In this case the contribution from the energy denominators
is given by1
A1A2 . . . AkB1B2 . . . Bl
1
−C + B
1
A1A2 . . . AmB1 . . . Bn. (5.36)
A similar result can be obtained for the graph Fig. 5.8a for which the contribution from
the energy denominators reads
1
A1A2 . . . AkB1B2 . . . Bl
1
−C + A
1
A1A2 . . . AmB1 . . . Bn.
where now A is the difference between the energy of the incoming gluon 1 in the upper
subgraph U and the sum of the final energies of the states in the same subgraph U. Using
73
global light-front energy conservation A = −B and performing substitution C = −C we
see that the contribution from the energy denominators for both graphs is the same up
to a global sign. One needs to take into account also the additional sign change between
graphs 5.8b and 5.8a coming from the contributions from the longitudinal fractions.
Thus the general expression for the graph (either 5.8a or 5.8b) has a compact (partially)
factorizable form with the energy denominators given by expression (5.36).
Notice that this proof relies on having energy conservation, A = −B. Thus, it can
only work for on-shell amplitudes.
Chapter 6Off-shell (+→ − + . . .+) amplitudes
on the light-front
From the previous chapters we now have the tools and background necessary to work on
calculating light-front off-shell amplitudes for more complicated helicity configurations
than we have seen. In this chapter we will follow the work done in [50] to calculate
M(+→−+...+)N . In terms of all outgoing particles, this is equivalent to the (− − + . . .+)
helicity configuration and is better known as a Maximally Helicity Violating amplitude [1,
53, 52], as we first saw it in Chapter 2. This is the the simplest, non-trivial amplitude and
provides a test case for any new method of calculating amplitudes. For example, when
Berends and Giele [9, 66], and the creators of the BCFW technique [11, 12] each wrote
down their respective recursion relations, they immediately tested them by computing
this amplitude. We will do the same thing. After computing the off-shell amplitude, we
will make sure it is correct by making sure it reduces to the well known result [1]
M(+→−+···+)(k1, . . . , kN ) ≡ −2igN−1 z1...N z1
z2z3 . . . zN
v3(1...N)1
v12v23 . . . vN−1 NvN(1...N)
= −i(√
2g)N−1 [(1 . . . N)1]4
[(1 . . . N)N ] [N(N − 1)] . . . [21] [1(1 . . . N)],
(6.1)
under the right conditions. Recall that to obtain an on-shell amplitude we need to set
D(1,...,N) = 0. I.e., M(+→−+···+)(k1, . . . , kN ) =M(+→−+···+)(k1, . . . , kN )∣∣D(1,...,N)=0
.
Our result for the off-shell amplitude turns out to have an interesting structure.
75
Namely, it can be written as a sum of lower order, off-shell, MHV-like amplitudes.
Furthermore, most of the terms in this expression are multiplied by D(1,...,N), such that
when the on-shell limit is taken, only the MHV amplitude is left. In Chapter 7 we will see
that our solution can also be obtained through matrix elements of Wilson line operators
[34] and that our MHV-like amplitudes are, in fact, gauge invariant objects [51].
Note that we’ve been talking about M(+→−+...+)N as if it were a unique object. In
fact, in the light-front we actually have two distinctM(+→−+...+)N ’s: one with initial state
off-shell, the other with final state off-shell. Thus, we will calculate both by using the
wavefunction and the fragmentation function [48]. Nevertheless, looking at (4.29), we
can see that the fragmentation functions already have the correct Parke-Taylor structure
built in. Thus, we will obtain a general solution only with the fragmentation functions.
For the wavefunction, we will limit ourselves to M(+→−++)3, wf .
Throughout this, we will follow the same procedure of Chapter 4. We will first write
down the recursion relation that holds for this helicity configuration. As before, this
recursion is the light-front analog of the Berends-Giele recursion and, this time, the
similarity is more apparent as 4-gluon and instantaneous interactions are present in this
case. Afterwards, we will find a few of the lowest order examples to establish a pattern
from which we will then guess a solution and prove it via induction.
6.1 M(+→−++)3, wf
The first thing we need to establish for our calculation of M(+→−++)3, wf is the recursion
relation that holds for the helicity configuration (+→ −+. . .+). In writing down (4.14),
only the vertex V(+→++)
3 was necessary. For the current case, the vertices depicted in
Fig. 6.1 are not zero (see Table A.1) and must be included. Thus, the recursion relation
appears as shown in Fig. 6.2. This can be written mathematically in the following way:
M(+→−+...+)wf (k1, k2, . . . , kN ) =
Ψ(+→+...+)(123, 4, . . . , N)×√z0z4 . . . zN
z123
(ig2 + ig2 (z123 + z3)(z2 − z1)
(z3 − z123)2
)
+ Ψ(+→+...+)(12, 3, . . . , N)×√z0z3 . . . zN
z12
(2ig z2 v(12)1
)
+ Ψ(+→−+...+)(12, 3, . . . , N)×√z0z3 . . . zN
z12
(2ig z1 v
∗2(12)
)
76
+N−1∑
i=2
Ψ(+→−+...+)(1, 2, . . . , i i+ 1, . . . , N)×√
z0z1 . . . zNzi (i+1)zizi+1
(2ig zi (i+1) v
∗(i+1) i
),
(6.2)
where the terms in parenthesis in the first three lines come from the vertices in Figs.
6.1a - 6.1d respectively.
k1, −
k2, +
k3, +
k0, +
(a)
k0, +
k1, −
k2, +
k3, +
(b)
k1, −
k2, +
k0, +
(c)
k1, −
k2, +
k0, −
(d)
Figure 6.1: Vertices that are included in recursion relation for the wavefunctions (6.2).The gluons on the left hand side of the vertices are treated as incoming.
1, +
2, +
n − 2, +
−
+
+
×
(a)
1, +
2, +
n − 1, +
−
+
×
(b)
1, −
2, +
n − 1, +
−
+
×
(c)
1, −
2, +
n − 1, +
+
+
∑i
i, +×
(d)
Figure 6.2: A schematic representation of four terms on the right hand side of recursion(6.2). The blob in the four gluon vertex in graph (a) denotes the combined 4-gluonand instantaneous interaction. Vertical dashed lines indicate that the states are takenoff-shell and the energy denominators are taken into account.
There are two observations that we would like to make about our recursion relation.
77
First, it is composed of two different types of wavefunctions. Ψ(+→+...+), which we
know from (4.21), and Ψ(+→−+...+). This latter one we can connect to M(+→−+...+)N,wf
using (4.40), which definitely establishes (6.2) as a recursion relation for M(+→−+...+)N,wf .
Secondly, the recursion is composed of wavefunctions of different number of legs due to
the presence of the 4 gluon vertex and the instantaneous term. This makes searching
for the solution to this recursion for general N much more involved than in the previous
case, i.e. Ψ(+→+...+)N , and is yet another reason why we focus on the 1 → 3 case for the
present calculation with wavefunctions.
Now, for the (+→ -++) configuration we have
M(+→−++)wf (k1, k2, k3) = Ψ(+→+)(123)×
√z0
z123
2ig2
(z0 − z3)2(z0z2 − z1z3)
+ Ψ(+→++)(12, 3)×√z0z3
z12
(2ig z2 v(12)1
)+ Ψ(+→−+)(12, 3)×
√z0z3
z12
(2ig z1 v
∗2(12)
)
+ Ψ(+→−+)(1, 23)×√z0z1
z23(2ig z23 v
∗32) . (6.3)
The Ψ(+→+)1 and Ψ
(+→++)2 wave functions have already been computed and their ex-
pressions are given by Eq. (4.21). The new object Ψ(+→−+)2 can be found by multiplying
the vertex in Fig. 6.1c by i√z0z1z2D
wf(1,2)
:
Ψ(+→−+)(1, 2) = g1√ξ(21)1
1
ξ(21)1v∗(21)1
. (6.4)
In addition, we can combine the following terms
Ψ(+→+)(123)× 2ig2
(z0 − z3)2z0z2 + Ψ(+→++)(12, 3)×
√z0z3
z12
(2ig z2 v(12)1
)=
Ψ(+→++)(12, 3)×√z0z3
z12(2ig z2 v01) , (6.5)
Ψ(+→+)(123)× 2ig2
(z0 − z3)2(−z1z3) + Ψ(+→−+)(12, 3)×
√z0z3
z12
(2ig z1 v
∗2(12)
)=
Ψ(+→−+)(12, 3)×√z0z3
z12(2ig z1 v
∗23) . (6.6)
Putting in the explicit expressions for Ψ’s the following expression is obtained
78
M(+→−++)wf (k1, k2, k3) = 2ig2
[z0z2
z3z12
v01
v30+
z1z3
z0z12
v∗23
v∗30
+z2
23
z0z1
v∗32
v∗01
]
=2ig2
z0z1z3z12
1
v30v∗30v∗01
[z1z2z
20 v01v
∗01v∗30 + (z1z3)2 v∗23v30v
∗01 + z2
23z12z3 v∗32v30v
∗30
],
(6.7)
where we used z12v3(12) = z0v30 and z12v∗0(12) = z3v
∗30 to get the first line. We can now
do some manipulations to the third term of the second line:
z223z12z3 v
∗32v30v
∗30 = z23z12z3z2v
∗32v∗32v30 + z23z12z3z1v
∗32v∗01v30
= z1z2z3z12v∗32v∗32v01 + z2
2z3z12v∗32v∗32v32 + (z1z3)2v∗32v
∗01v30 + z0z1z2z3v
∗32v∗01v30
= z1z2z3z12v∗32v∗32v01 + z2
2z3z0v∗32v∗30v32 + z2
2z3z1v∗32v∗12v32 + (z1z3)2v∗32v
∗01v30
+ z0z1z2z3v∗32v∗01v30 (6.8)
where we have used the following relations z23v30 = z1v01 + z2v32, z12v∗32 = z0v
∗30 + z1v
∗12
and z23z12 = z1z3 + z0z2 in order to rewrite the expressions. Thus,
M(+→−++)(k1, k2.k3) =2ig2
z0z1z3z12
1
v30v∗30v∗01
[z1z2z3z12v
∗32v∗32v01
+(z1z2z
20 v01v
∗01v∗30 + z2
2z3z0v∗32v∗30v32
)+(z2
2z3z1v∗32v∗12v32 + z0z1z2z3v
∗32v∗01v30
) ]
(6.9)
Finally, we can group some terms which are proportional to denominator Dwf(1,2,3) by
using the following identities:
z1z0 v01v∗01 + z2z3v
∗32v32 = −1
2z23D
wf(1,2,3), (6.10)
z2v∗12v32 + z0v
∗01v30 = z2v12v
∗32 + z0v01v
∗30 = −1
2Dwf
(1,2,3). (6.11)
Therefore,
M(+→−++)(k1, k2, k3)
=2ig2
z0z1z3z12
1
v30v∗30v∗01
[z1z2z3z12v
∗32v∗32v01 −
1
2z2D
wf(1,2,3) (z1z3v
∗32 + z0z23v
∗30)
]
= −2ig2 (z0z1)2
z0z1z2z3
v401
v01v12v23v30− 1
2Dwf
(1,2,3)
2ig2
z0z1z2z3z12
1
v12v23v30v∗30v∗01
79
×(z0z1z
212v
201v∗01 − z1z2z12z23v01v12v
∗32 + z1z
22z3v12v23v
∗32 + z0z
22z23v12v23v
∗30
)(6.12)
The on-shell energy condition Dwf(1,2,3) = 0 can now be applied to the above expression as
required for the on-shell amplitudes. This leaves only the first term in Eq. (6.12) which
is finite and it matches the MHV amplitude.
Thus by utilizing the recursion relation for the wave functions we are able to recon-
struct the the on-shell MHV amplitude. The full proof would of course involve solving
exactly the recursion relation. An interesting structure though emerges from this ex-
ample, namely the fact that the off-shell amplitude MN can be written as a sum of an
MHV-like amplitude MN1 plus the term proportional to the energy denominator which
vanishes if one takes the on-shell condition Dwf(1,2,3) → 0. Thus we can expect that the
general structure for arbitrary number of final particles in the LFPT has the following
form
MN = MN +O(D(1,...,N)) , (6.13)
where the term which is related to the off-shellness of the amplitude vanishes upon taking
the physical condition Ein = Efin, i.e. it is at least linear in the energy denominator
D(1,...,N).
6.2 General tree-level off-shell amplitudes:
M(+→−+...+)N, ff and MHV
In this section we will calculate the general, tree-level, off-shell amplitude for a process
with one incoming gluon with positive helicity fragmenting into N outgoing gluons with
helicities (−+ . . .+). Looking back at Chapter 2 and at our discussion in the previous
chapter, we would expect the on-shell limit to be that of the MHV amplitudes, with
structure as in Eq. (6.21). This structure is what compels us to use fragmentation
functions as building blocks towards a general solution. The reason being that if we look
at (4.29), the structure of the MHV seem to be already built-in into the fragmentation
functions. Thus, we should remark that we are then calculating an off-shell amplitude
with initial off-shell state and final on-shell state.2
1MN is really an off-shell object, i.e. off-shell kinematics, that has an on-shell structure. In this casethe structure is that of an MHV amplitude.
2As previously mentioned, this amplitude shares similarities with the off-shell amplitude with initialon-shell state and final off-shell state. They are both calculated from the same graphs and their on-shelllimit has to be the same. However, these two off-shell amplitudes are fundamentally different from eachother.
80
Another point that we want to make is that there is an infinite number of ways in
which we can write down our amplitude M(+→−+...+)N, ff . We decided on trying to arrive
at an expression with the structure that Eq. (6.13) hinted at. And, indeed, we cast our
solution in terms of an MHV-like amplitude plus terms proportional to D(1...,N).
6.2.1 Recursion relations for (+→ −+ . . .+)
Just as for the wavefunction recursion relation in Sec. 6.1, 4-gluon and instantaneous
interactions will be present because of our current choice of helicities. Due to the factor-
ization of fragmentation functions, we are once again able to use them as building blocks
and, thus, have the recursion depicted in Fig. 6.3. Mathematically, we write this as,
Σnj=2 k(1...j),+
k(j+1...n+1),+ kj+1,+
kj+2,+
kn+1,+
T
k1,−k2,+
kj,+
T
k(1...n+1),+
(a)
Σnj=1 k(1...j),−
k(j+1...n+1),+ kj+1,+
kj+2,+
kn+1,+
T
k1,−k2,+
kj,+
T
k(1...n+1),+
(b)
k(1...i),−
k(j+1...n+1),+
k1,−k2,+
ki,+
T
ki+1,+ki+2,+
kj,+
T
kj+1,+kj+2,+
kn+1,+
T
k(i+1...j),+
Σnj=2Σ
j−1i=1
k(1...n+1),+
(c)
Σnj=2 Σ
j−1i=1
k1,−k2,+
ki,+
T
ki+1,+ki+2,+
kj,+
T
kj+1,+kj+2,+
kn+1,+
T
k(1...n+1),+ k(j+1...n+1),+
k(i+1...j),+
k(1...i),−
(d)
Figure 6.3: Graphs involved in the fragmentation of a single off-shell gluon into N + 1on-shell gluons. The initial and final helicities are specified in the figures. We denotethe 3-gluon vertex in Figs. 6.3a and 6.3b as V+ and V− respectively, the 4-gluon vertexin Fig. 6.3c as V4, and the instantaneous interaction in Fig. 6.3d as VInst. Verticallines denote the energy denominators that need to be taken, they are implicit in allintermediate states denoted by blobs. There are no energy denominators in the finalstate.
M(+→−+...+)(k1, k2, . . . , kN+1) =
81
N∑
j=2
V+
√z1z2 . . . zN+1
z1...jzj+1...N+1T (+→−+...+)(1, 2 . . . , j) T (+→+...+)(j + 1, . . . , N + 1)
+
N∑
j=1
V−√
z1z2 . . . zN+1
z1...jzj+1...N+1T (−→−+...+)(1, 2 . . . , j) T (+→+...+)(j + 1, . . . , N + 1)
+N∑
j=2
j−1∑
i=1
(V4 + VInst)
√z1z2 . . . zN+1
z1...izi+1...jzj+1...N+1T (−→−+...+)(1, 2 . . . , i)
× T (+→+...+)(i+ 1, . . . , j) T (+→+...+)(j + 1, . . . , N + 1) , (6.14)
which, once again, is a light-front analog to the Berends-Giele recursion (2.52). However,
the recursion above includes an extra instantaneous term that is not present in the
Berend-Giele recursion. The first, second and third line come from Fig. 6.3a, Fig. 6.3b
and Figs. 6.3c - 6.3d respectively. The V’s are the vertex factors and these are given by
V+ = V(+→++)
3 (−k1...N+1, k1...j , kj+1...N+1) = 2igz1...N+1v∗(j+1...N+1)(1...j) , (6.15)
V− = V(+→−+)
3 (−k1...N+1, k1...j , kj+1...N+1) = 2igzj+1...N+1v(1...N+1)(1...j) , (6.16)
V4 = V(+→−++)
4 (−k1...N+1, k1...i, ki+1...j , kj+1...N+1) = ig2 , (6.17)
VInst = V(+→−++)
Inst (−k1...N+1, k1...i, ki+1...j , kj+1...N+1)
= ig2 (z1...N+1 + zj+1...N+1)(zi+1...j − z1...i)
(z1...N+1 − zj+1...N+1)2. (6.18)
Inspecting formula (6.14) we see that the fragmentation functions involved in the
process correspond to three different helicity configurations. Two of them, T(+→+...+)i
and T(−→−+...+)i were discussed in Chapter 4 and their explicit expressions are given
by (4.29) and (4.35). The third fragmentation function, T(+→−+...+)i , however, remains
unknown, but is related to M(+→−+...+)i via (4.41). Thus, Eq. (6.14) which is depicted
in Fig. 6.3 turns out to be a recursion relation forM(+→−+...+)N+1 . In the next subsections
we will find a solution to this equation and prove it via the method of mathematical
induction.
6.2.2 Pattern and solution
Let us start with the initial conditions for the recursion formula. The normalization is
such that the initial fragmentation functions are set to T (+→−)(1) = 0, T (+→+)(1) =
82
T (−→−)(1) = 1. Finding M(+→−+)2 , (N = 1), is trivial,
M(+→−+)(k1, k2) = 2igz2v(12)1
√z1z2
z1z2= 2igz2v(12)1
z212v(12)1v(12)1
z212v(12)1v(12)1
= 2igz1z
212
z2
v3(12)1
z1v21v21= −2ig
z1z12
z2
v3(12)1
v12v2(12)= M (+→−+)(k1, k2) , (6.19)
where we defined
M (+→−+)(k1, k2) = −2igz1z12
z2
v3(12)1
v12v2(12). (6.20)
In general, we will define
M (+→−+...+)(k1...N ) ≡ −2igN−1 z1...N z1
z2z3 . . . zN
v3(1...N)1
v12v23 . . . vN−1 NvN(1...N), (6.21)
for arbitrary N number of outgoing particles. This object is off-shell, yet its structure
is identical to that of the on-shell MHV amplitudes. This object is not found directly
from any of the graphs, yet it will keep appearing throughout our calculations and, in
fact, M(+→−+...+)N will be written as a linear sum of these. Finally, it turns out that
M(+→−+...+)N is gauge invariant, but that is the subject of the next chapter.
FindingM(+→−++)3 , (N = 2), is much more complicated and we should remark that
the order in which terms will be added is the same as when we perform the proof via
induction. The order is important because it makes the structure of the solution easier
to see. To begin, it will be convenient for our calculations to add V4 and VInst to get
V4 + VInst = Vcomb, a + Vcomb, b , (6.22)
where
Vcomb, a = 2ig2 z1...N+1 zi+1...j
z21...j
, (6.23)
Vcomb, b = −2ig2 z1...i zj+1...N+1
z21...j
. (6.24)
Thus, we can replace Figs. 6.3c and 6.3d with Figs. 6.4a and 6.4b. The white and black
blobs represent the contributions from the vertices Vcomb, a and Vcomb, b respectively.
From recursion (6.14) we see that there are five different terms which contribute to
83
Σnj=2 Σ
j−1i=1
k(1...i),−
k(j+1...n+1),+
k1,−k2,+
ki,+
T
ki+1,+ki+2,+
kj,+
T
kj+1,+kj+2,+
kn+1,+
T
k(1...n+1),+ k(i+1...j),+
(a)
Σnj=2 Σ
j−1i=1
k(1...i),−
k(j+1...n+1),+
k1,−k2,+
ki,+
T
ki+1,+ki+2,+
kj,+
T
kj+1,+kj+2,+
kn+1,+
T
k(1...n+1),+ k(i+1...j),+
(b)
Figure 6.4: Graphs representing the contribution from the equivalent 4-gluon verticesVcomb, a and Vcomb, b to the fragmentation function.
M(+→−++)3 :
I = 2igz123v∗(3)(12)
√z1z2z3
z12z3
[1√
z12z1z2
i
D(1,2)M(+→−+)(k1, k2)
], (6.25)
II = 2igz23v(123)1
√z1z2z3
z1z23
[g
(z23
z2z3
)3/2 1
v23
]= 2ig2 z
223
z2z3
v(321)1
v23, (6.26)
III = 2igz3v(123)(12)
√z1z2z3
z12z3
[g
(z1
z12
)2( z12
z1z2
)3/2 1
v12
]
= 2ig2 z1z3
z12z2
v(321)(21)
v12, (6.27)
IV = 2ig2 z123 z2
z212
√z1z2z3
z1z2z3= −g z123z3
z212z3
M (+→−+)(k1, k2)
v(12)1, (6.28)
V = −2ig2 z1 z3
z212
√z1z2z3
z1z2z3. (6.29)
I, IV and V are obtained from Figs. 6.3a, 6.4a and 6.4b respectively. II and III are
obtained from the sum in Fig. 6.3b. Here we have already written IV in terms of
M(+→−+)2 as a simple example of how the term coming from Fig. 6.4a will be simplified
later on. Next, III is added to V to get
VI = III + V = −2ig2 z1z3
z12
[1
z12+
1
z2
v(321)(21)
v21
]= −2ig2 z1z3
z2z12
v(123)1
v21, (6.30)
84
which added to II gives
VI + II = VII = −2ig2 v(123)1
z2
[z2
23
z3
1
v32+z1z3
z12
1
v21
]. (6.31)
However, using the following relation
z123z2
z12+z1z
23
z12z2
v3(123)
v12+z2
23
z2
v3(123)
v23
=1
z12z2
1
v12v23
[z123z
22v12v23 + z1z
23v3(123)v23 + z12z
223v3(123)v12
]
=1
z12z2
1
v12v23
[z123z2v12
(z2v23 + z23v3(123)
)+ z1z3v3(123) (z3v23 + z23v12)
]
=z123z1
z12z2
v(123)1
v12v23
[−z12v(123)1
], (6.32)
we see that term VII can be written as
VII = v(123)1
{M1→3
v(123)1− 1
z3
z2123
z12z123
g
v3(123)
M1→2
v(12)1
}. (6.33)
Now that M(+→−++)3 = I + IV + VII is written completely in terms of MHV-like
amplitudes we can collect terms proportional to M(+→−+...+)j to get,
M(+→−++)(k123)
= M (+→−++)(k123)− g z123
z12M (+→−+)(k12)
1
z3
v(123)1
v3(123)v(12)1− 1
z12v(12)1+ 2
v∗3(12)
Dff(1,2)
= M (+→−++)(k123)− g 1
z3
z123
z212z123
1
v3(123)
M (+→−+)(k12)
v(12)1Dff(1,2)
×{Dff
(1,2)
(z123z12v(123)1 − z3z123v3(123)
)+ z12v(12)1
(2z3z123v
∗3(12)v3(123)
)}, (6.34)
where we have introduced the following shorthand notation
M(k1...i, ki+1, . . . , kj) ≡M(k(1...i)i+1...j) . (6.35)
Using
2z3z123v∗3(12)v3(123) = z3z12
(~k3⊥z3−~k12⊥z12
)2
= z123
(Dff
(1,2) −Dff(1,2,3)
), (6.36)
85
and
−z123z12v(123)1 + z3z123v3(123) = −z123z12v(123)1 − z12z123v(12)(123)
= −z123z12v(12)1 , (6.37)
our final result for M(+→−++)3 is then
M(+→−++)(k123) = M (+→−++)(k123)− g 1
z3
z123
z12
1
v3(123)
D(1,2,3)
D(1,2)M (+→−+)(k12) , (6.38)
where for the sake of brevity we’ve suppressed the ff marking. From now on it should
be implicit that the energy denominators are the fragmentation function denominators.
Unfortunately, (6.19) and (6.38) are not enough to establish a pattern. Therefore,
we need the next step in the iteration. M(+→−+++)4 can be found following the same
procedure, yet it is a much more tedious process. The result ends up being
M(+→−+++)(k1234) = M (+→−+++)(k1234)
− gD(1,2,3,4)
D(1,2,3)
z21234
z123z1234
1
z4
1
v4(1234)M (+→−++)(k123)
− g2D(1,2,3,4)
D(1,2)
z21234
z12z123
1
z3z4
1
v34
M (+→−+)(k12)
v3(123). (6.39)
Interestingly, the off-shell amplitude is expressed as a linear combination of the “MHV-
like” objects with the pre factors which are proportional to the energy denominators.
In particular we see that by putting the on-shell constraint D(1,2,3,4) = 0 we recover the
true on-shell amplitude.
Following the pattern found, one would then expect, for a general integer
N ≥ 2,
M(+→−+···+)(k1...N ) = M (+→−+···+)(k1...N )−
z21...ND(1,...,N)
N−1∑
i=2
1
z1...iz1...i+1
1
zi+1 . . . zN
gN−i
vi+1 i+2 . . . vN−1 N
× 1
vi+1(1...i+1)D(1,...,i)M (+→−+···+)(k1...i) . (6.40)
One interesting aspect to note about M(+→−+...+) is that it is written as a sum of
amplitudes M (+→−+...+) with different number of legs. Looking at Fig. 6.3, it is not
86
immediately obvious as to why M (+→−+...+) should dominate the expression since only
Fig. 6.3a has a substructure with helicity configuration (+ → − + . . .+). It turns out
that the other graphs, even though their substructures do not have the right helicity
configuration, do contribute terms proportional to M (+→−+...+). In the next subsection
we will write out the these terms explicitly and through some algebraic manipulation, it
will be shown that the following term emerges from Figs. 6.3b-6.3d:
k∑
j=1
z1z2j+1...k+1
z1...j
1
vk+1 k . . . vj+2 j+1vj j−1 . . . v21.
This term can be rewritten using the following identity (to be proven in Section 6.2.4)
− 2i(−g)kvk+1 (1...k+1)
z2z3 . . . zk
k∑
j=1
z1z2j+1...k+1
z1...j
1
vk+1 k . . . vj+2 j+1vj j−1 . . . v21
= zk+1vk+1(1...k+1)
{M (+→−+...+)(k1...k+1)
v(1...k+1)1
−k∑
j=2
1
zj+1 . . . zk+1
z21...k+1
z1...jz1...j+1
1
vj+1 j+2 . . . vk k+1
gk+1−j
vj+1(1...j+1)
M (+→−+...+)(k1...j)
v(1...j)1
},
(6.41)
showing where the M (+→−+...+)’s come from.
In addition, the recursion relation (6.40) can be rewritten in a more elegant way,
which demonstrates factorization into different subamplitudes. In order to do that let
us inspect the second term in (6.40). We shall show that it can be expressed as the sum
over the products
M(+→+···+)(k(1...i)i+1...N )i
z1...iD(1,...,i)M (+→−+···+)(k1...i) .
Let us start with the definition of the off-shell subamplitude for the helicity configuration
(+→ + · · ·+)
M(+→+···+)(k(1...i)i+1...N ) = −igN−i z21...N
z1...izi+1 . . . zN
D((1...i),i+1,...,N)
v(1...i)i+1vi+1 i+2 . . . vN−1 N, (6.42)
which comes from combining (4.29) and (4.41). The energy denominator in this expres-
87
kN
k1...N
k1...i
ki+1
z1...iDi
+
+
+
+
ki
k1
k2
−
+
+
i
Figure 6.5: Schematic representation of the second term in the recursion formula (6.44). Thedotted vertical line represents the energy denominator 1/Di, the graph on the left hand side ofthis line is the amplitude M(+→+···+)(k(1...i)i+1...N ) whereas the graph on the right hand side is
the amplitude M (+→−+···+)(k1...i). The double line around the blob in the latter graph indicatesthat this is M (with an explicit form of (6.21)) rather than M which are different objects asexplained in the text.
sion is equal to
D((1...i),i+1,...,N) = E1...i +N∑
j=i+1
Ej − E1...N =~k2
1...i⊥z1...i
+i∑
j=i+1
~k2j⊥zj−~k2
1...N⊥z1...N
. (6.43)
Comparing with (6.40), we see that the term inside the sum is very similar to the above
off-shell subamplitude with some additional prefactors.
Substituting (6.42) in (6.40) and using z1...j+1vj+1 (1...j+1) = z1...jvj+1 (1...j) we finally
obtain the following version of the recursion relation
M(+→−+···+)(k1...N ) = M (+→−+···+)(k1...N )
+
N−1∑
i=2
D(1,...,N)
D((1...i),i+1,...,N)M(+→+···+)(k(1...i)i+1...N )
i
z1...iD(1,...,i)M (+→−+···+)(k1...i) .
(6.44)
We see that the second term on the right hand side of this recursion has a nice
factorized form which can be recast diagrammatically as in Fig. 6.5. It consists of the
sum over the factorized products of amplitudes M and M. M, however, is evaluated
with a different denominator in the sense that the ratio D(1,...,N)/D((1...i),i+1,...,N) cancels
the energy denominator inside of M and replaces it with D(1,...,N). Later, in Chapter 7,
88
we will see that (6.40) can also be obtained from a matrix element of a straight infinite
Wilson line.
6.2.3 Proof
We shall now present the proof (6.40) using the method of mathematical induction.
Before we begin, the following are two relationships which we will use many times in the
rest of this section,
z1...j+1v(1...j+1)1 = z2...j+1v(2...j+1)1 =
j∑
i=1
zi+1...j+1vi+1 i , (6.45)
and
zi+1...j+1vj+1(i+1...j+1) = zi+1...jvj+1(i+1...j) =
j∑
l=i+1
zi+1...lvl+1 l . (6.46)
To perform the proof we assume (6.40) is true and then use it in (6.14) to find
M(+→−+...+)N+1 . At the end,M(+→−+...+)
N+1 should be of the form given by expression (6.40)
for N → N + 1. Let us remind that for the result one needs to add all the contributions
from Figs. 6.3a, 6.3b, 6.4a and 6.4b. We begin with Fig. 6.4a. For fixed j, the expression
for this graph reads
E1 =
j−1∑
i=1
(−1)N2igNz1...N+1z1zj+1...N+1z
2i+1...j
z1...iz21...j z2 . . . zN+1
× 1
vN+1 N . . . vj+2 j+1vj j−1 . . . vi+2 i+1vi i−1 . . . v21
= (−1)N2igNz1...N+1zj+1...N+1
z21...j z2 . . . zN+1
1
vN+1 N . . . vj+2 j+1
×j−1∑
i=1
z1z2i+1...j
z1...i
1
vj j−1 . . . vi+2 i+1vi i−1 . . . v21. (6.47)
Next, we add the expressions for the graphs presented in Figs. 6.3b and 6.4b for fixed j
(j 6= 1),
Aj = (−1)N−1 2igNz1z
2j+1...N+1
z1...j z2 . . . zN+1
v(1...N+1)1
vN+1 N . . . vj+2 j+1vj j−1 . . . v21, (6.48)
89
where we have used Eq. (6.45) . For j = 1 in Fig. 6.3b,
B = (−1)N−1 2gNz2
2...N+1
z2 . . . zN+1
v(1...N+1)1
vN+1 N . . . v32. (6.49)
The overall contribution from Figs. 6.3b and 6.4b would then be given by
E2 =
N∑
j=2
Aj +B
= (−1)N−1 2gNv(1...N+1)1
z2 . . . zN+1
N∑
j=1
z1z2j+1...N+1
z1...j
1
vN+1 N . . . vj+2 j+1vj j−1 . . . v21. (6.50)
For Fig. 6.3a, for fixed j, we get
E3 = (−1)N−1 2i j−1gN−j+1 z1...N+1zj+1...N+1
z1...jzj+1 . . . zN+1
×v∗(N+1...j+1)(j...1)
vN+1 N . . . vj+2 j+1
1
D(1,...,j)M(+→−+...+)(k1...j) . (6.51)
Now, we need to add the contributions from (6.47), (6.50) and (6.51) to get the final
expression for M(+→−+...+)N+1 . To simplify the calculations it is useful to rewrite the
expressions entirely in terms M(+→−+...+)j . We can use (6.41) to rewrite E1 in (6.47) and
E2 in (6.50),
E1 = (−1)N+j gN−j+1 z1...N+1zj+1...N+1
z21...j zj+1 . . . zN+1
1
vN+1 N . . . vj+2 j+1
{M (+→−+...+)(k1...j)
v(1...j)1
−j−1∑
i=2
1
zi+1 . . . zj
z21...j
z1...iz1...i+1
1
vi+1 i+2 . . . vj−1 j
gj−i
vi+1(1...i+1)
M (+→−+...+)(k1...i)
v(1...i)1
}, (6.52)
E2 = v(1...N+1)1
{M (+→−+...+)(k1...N+1)
v(1...N+1)1
−N∑
j=2
1
zj+1 . . . zN+1
z21...N+1
z1...jz1...j+1
1
vj+1 j+2 . . . vN N+1
gN−j+1
vj+1(1...j+1)
M (+→−+...+)(k1...j)
v(1...j)1
}.
(6.53)
We can now find M(+→−+...+)N+1 from the contributions of (6.51), (6.52) and (6.53),
90
where we must remember to sum over j in (6.51) and (6.52) from j = 2 to j = N and
collect terms proportional to M(+→−+...+)l , where 2 ≤ l ≤ N + 1. For l = N + 1 we get
only one term, which comes from the first term in (6.53),
M (+→−+...+)(k1...N+1) . (6.54)
For any other l, after simplifying and remembering to use (6.40) we get
− gN−l+1 1
zl+1 . . . zN+1
z1...N+1
z21...lz1...l+1
1
vl+1 l+2 . . . vN N+1
1
vl+1(1...l+1)
M (+→−+...+)(k1...l)
v(1...l)1D(1,...,l)
×{−D(1,...,l)C + z1...lv(1...l)1F
}, (6.55)
where
C = −z1...N+1z1...lv(1...N+1)1 + zl+1...N+1z1...l+1vl+1(1...l+1) −N∑
j=l+1
zj+1...N+1z1...lvj j+1 ,
(6.56)
F = 2zl+1...N+1z1...l+1v∗(N+1...l+1)(l...1)vl+1(1...l+1)
− 2
N∑
j=l+1
zj+1...N+1z1...jv∗(N+1...j+1)(j...1)vj j+1 . (6.57)
However, we can write C as
C = −z21...lv(1...l)1 −
z1...l
N+1∑
j=l+1
zjvj 1
+ z1...l
N+1∑
j=l+1
zj
vl+1(1...l) − z1...l
N∑
j=l+1
N+1∑
m=j+1
zmvj j+1 . (6.58)
After some little algebra and changing the order of summation in the last term, i.e.
replacing∑N
j=l+1
∑N+1m=j+1 with
∑N+1m=l+2
∑m−1j=l+1, we arrive at
C = −z1...lz1...N+1v(1...l)1 . (6.59)
91
Furthermore, we can write F as
F = 2N+1∑
j=l+1
zjz1...lv∗j(l...1)vl+1(1...l) + 2
N∑
j=l+1
N+1∑
m=j+1
zmz1...jv∗m(j...1)vj+1 j . (6.60)
Since j will always be greater than l we can write
z1...jv∗m(j...1) = z1...lv
∗m(l...1) +
j∑
j=l+1
ziv∗m i.
We use this to expand the second term in F . Changing the order of summation in these
two new terms we get
F = 2N+1∑
j=l+1
zjz1...lv∗j(l...1)vl+1(1...l)
+ 2z1...l
N+1∑
m=l+2
zmv∗m(l...1)vm l+1 + 2
N+1∑
m=l+2
m−1∑
i=l+1
zmziv∗m ivm i
= 2N+1∑
j=l+1
zjz1...lv∗j(l...1)vj(1...l) + 2
N+1∑
j=l+2
j−1∑
i=l+1
zjziv∗j ivj i . (6.61)
We can rewrite this in terms of k’s by using
v∗abvab =1
2
(~ka⊥za−~kb⊥zb
)2
. (6.62)
Manipulating the sums and simplifying the expression we end up with
F = z1...N+1
k
21...l
z1...l+
N+1∑
j=l+1
k2j
zj
−
k2
1...l +N+1∑
j=l+1
k2j
− 2kl+1...N+1 · k1...l −N+1∑
j=l+2
j−1∑
i=l+1
2kj · ki
= −z1...N+1
k
21...N+1
z1...N+1− k2
1...l
z1...l−
N+1∑
j=l+1
k2j
zj
= z1...N+1
(D(1,...,N+1) −D(1,...,l)
). (6.63)
92
Thus, (6.55) reduces to
− gN−l+1 1
zl+1 . . . zN+1
z21...N+1
z1...lz1...l+1
1
vl+1 l+2 . . . vN N+1
× 1
vl+1(1...l+1)
D(1,...,N+1)
D1,...,l)M (+→−+...+)(k1...l) . (6.64)
The final result is the sum of (6.54) and (6.64)
M (+→−+...+)(k1...N+1)−N∑
l=2
1
zl+1 . . . zN+1
z21...N+1
z1...lz1...l+1
1
vl+1 l+2 . . . vN N+1
× gN−l+1
vl+1(1...l+1)
D(1,...,N+1)
D1,...,l)M (+→−+...+)(k1...l) , (6.65)
which gives us M(+→−+...+)N+1 written as a linear combination of on-shell amplitudes.
Comparing these terms to (6.40) we see that, indeed, M(+→−+...+)N+1 is of the same form,
which completes our proof. It is important to note that if we now apply the condition
that the initial state be on-shell, i.e. D(1,...,N+1) → 0, M(+→−+...+)N+1 does reduce to the
known expression for the MHV amplitudes.
6.2.4 Proof of identity (6.41)
In this subsection we provide the proof for (6.41), which is done using induction. We
begin by rewriting the (6.41) as
k∑
j=1
z2j+1...k+1
z1...jvj+1 j −
k∑
j=2
z21...k+1
z1...j+1vj+1 j
v2(1...j)1
vj+1(1...j+1)vj(1...j)− z1...k+1
v2(1...k+1)1
vk+1(1...k+1)= 0.
(6.66)
To get this we have substituted (6.21) into (6.41) and taken out some overall factors.
We will label the left hand side of this equation as fk and assume that fk = 0 for k < k′,
where k′ is an arbitrary upper limit. If, under our assumption, fk′ = 0 for k′ = k + 1
then it will be true that fk = 0 for all k subject to f1 = 0 = f2.
Our current expression for fk is, however, too cumbersome to work with. Thus,
we will derive a simpler form. We can combine the first and third terms by using
(6.45). Then using v(1...k+1)1 = −vk+1(1...k+1) + v(k+1)1 and (6.46) we get, after some
manipulation,
93
k∑
j=1
z2j+1...k+1
z1...jvj+1 j − z1...k+1
v2(1...k+1)1
vk+1(1...k+1)
=
k∑
j=1
zj+1...k+1vj+1 j
(zj+1...k+1
z1...j−
v(1...k+1)1
vk+1(1...k+1)
)
= −k∑
j=2
j−1∑
l=1
zj+1...k+1zl+1...j
z1...j
vj+1 j vl+1 l
vk+1(1...k+1)+k−1∑
j=1
k∑
l=j+1
zj+1...k+1zj+1...l
z1...j
vj+1 j vl+1 l
vk+1(1...k+1).
(6.67)
Here we have changed the lower and upper limits of j appropriately. We now choose to
change the order in which the sums are performed in the second term of (6.67). I.e., we
replace∑k−1
j=1
∑kl=j+1 by
∑kl=2
∑l−1j=1. Furthermore, since j and l are dummy indices we
can exchange them so that we can now combine the two terms in (6.67) into a single
double sum. Thus, for the sum of the first and third terms of (6.66) we end up with
−k∑
j=2
j−1∑
l=1
zj+1...k+1zl+1...j
z1...j
vj+1 j vl+1 l
vk+1(1...k+1)+k−1∑
j=1
k∑
l=j+1
zj+1...k+1zj+1...l
z1...j
vj+1 j vl+1 l
vk+1(1...k+1)
=k∑
j=2
j−1∑
l=1
z2l+1...jz1...k+1
z1...l z1...j
vj+1 j vl+1 l
vk+1(1...k+1). (6.68)
We can also rewrite the second term in (6.66) in the following way,
−k∑
j=2
z21...k+1
z1...j+1vj+1 j
v2(1...j)1
vj+1(1...j+1)vj(1...j)=
−k∑
j=2
j−1∑
l=1
z21...k+1zl+1...j
z1...j+1z1...jvj+1 jvl+1 l
v(1...j)1
vj+1(1...j+1)vj(1...j)(6.69)
Finally, fk will be given by the sum of (6.68) and (6.69). Let us now define
gk ≡vk+1(1...k+1)
z1...k+1fk. (6.70)
For k′ = k + 1 we would then have
gk′ =vk+2(1...k+2)
z1...k+2fk+1
=
k+1∑
j=2
j−1∑
l=1
z2l+1...j
z1...l z1...jvj+1 j vl+1 l
94
−k+1∑
j=2
j−1∑
l=1
z1...k+2zl+1...j
z1...j+1z1...jvj+1 jvl+1 l
v(1...j)1vk+2(1...k+2)
vj+1(1...j+1)vj(1...j)
=vk+2 k+1
z1...k+1
(k∑
l=1
z2l+1...k+1
z1...lvl+1 l −
k∑
j=2
z21...k+1
z1...j+1vj+1 j
v2(1...j)1
vj+1(1...j+1)vj(1...j)
− z1...k+1
v2(1...k+1)1
vk+1(1...k+1)
)
+k∑
j=2
j−1∑
l=1
z2l+1...j
z1...l z1...jvj+1 j vl+1 l −
k∑
j=2
j−1∑
l=1
z1...k+1zl+1...j
z1...j+1z1...jvj+1 jvl+1 l
v(1...j)1vk+1(1...k+1)
vj+1(1...j+1)vj(1...j)
=vk+2 k+1
z1...k+1fk + gk (6.71)
To get the third line we have used (6.45) and z1...k+2vk+2(1...k+2) = z1...k+1vk+2(1...k+1) =
z1...k+1(vk+2 k+1 +vk+1(1...k+1)). Our assumption fk = 0 implies gk = 0, since vk+1(1...k+1)
is in general non-zero. Hence, this shows that fk+1 ∝ gk+1 = 0. It is fairly easy to show
that (6.66) is true for k = 1, 2. Thus, we have shown that (6.66) is true for all k ≥ 1.
Chapter 7Gauge Invariance of Off-Shell
Amplitudes
Through our discussions of off-shell amplitudes we’ve touched but haven’t gone in depth
into the issue of gauge invariance. The key idea is that, when calculating scattering
amplitudes for physical processes, our results should not depend on what gauge we
choose. In this sense, on-shell methods, if used correctly, have the advantage that they
are guaranteed to be gauge invariant. However, there is no such guarantee for off-shell
amplitudes. This is a problem since alternative methods [30, 31, 32] to compute cross
sections, which take into account kinematics more accurately, make use of off-shell matrix
elements. Nevertheless, there has been some progress recently [33, 34, 35] in constructing
off-shell amplitudes which do satisfy Ward identities and hence obey gauge invariance.
With this in mind, in this chapter we will follow [51] in studying the gauge dependence
of the off-shell, light-front amplitudes we’ve constructed. To get some insight, we begin
by calculating the Ward identity of the 1→ 3 processM(+→−+k3)1→3 , which is a simple yet
non-trivial example. Here, the k3 indicates that we have replace the polarization vector
of k3 for its momentum. The result will be that M is not gauge invariant. However,
it turns out that, using (6.44), we find that M1→3 is gauge invariant. This is very
interesting as this object was not calculated directly from any graphs. Instead, it just
appeared in our calculations. Furthermore, this object has the exact form of an MHV
amplitude [1, 53, 52], yet it is still off-shell.
As mentioned in the previous chapters, M turns out to be gauge invariant for an
arbitrary number of legs. To show this we rely on the Wilson line method of [34].
In this framework, the matrix element of an infinite straight line Wilson operator is
96
used to construct off-shell gauge invariant amplitudes. These amplitudes can be written
in a recursion relation which, remarkably, exactly matches our recursion relation for
M (6.40)1. This gives the physical origin of (6.40) as a direct consequence of gauge
invariance. Additionally, it will allow us to conclude that M is gauge invariant.
Given our reliance on the Wilson line method, we will also give a brief introduction
to this topic and then proceed with our derivations, which were first done in [51].
7.1 The Ward identity for light-front amplitudes
In this section we calculate the Ward identity for the (+→ −+ k3) process. We will use
light front graphs for our first calculation and see that it is proportional to the energy
denominator of the initial state. This indicates that the Ward identity holds in the
on-shell limit. Secondly, we shall perform the Ward identity check using the recursion
relation (6.40). It turns out that the second term in the r.h.s. of (6.44), which is a
sum of lower order amplitudes in this recursion, gives the expression which is exactly
equal to the term previously derived by the explicit calculation of the Ward identity from
diagrams. This means that the new amplitude M which appears in the recursion relation
is gauge invariant, i.e. the Ward identity gives exactly zero for this object despite the
fact that it is off-shell.
7.1.1 Example: the Ward identity check for the lowest order amplitude
Let us recall, that for a generic QCD amplitude M with external momenta ki on-shell
and corresponding polarization vectors εi, the Ward identities read
M∣∣εi→ki
= 0 for any i . (7.1)
For our N particle off-shell amplitude it is unknown whether this will hold. However,
we do know that in the on-shell limit (D(1,...,N) → 0) it must hold; thus, we can expect
that the Ward identity to, at least, be proportional to D(1,...,N). I.e.,
M∣∣εi→ki
∝ D(1,...,N) for any i not off-shell . (7.2)
Turns out that to get this result one must be careful; one cannot naively use the
regular LFPT rules [41, 76, 77, 78, 75] derived in Sec. 3.3.4. Instead, we have to use the
1This recursion was originally meant to be a recursion for M. Here we decide to use it as a recursionfor M
97
k123
k1
k2
k3
+
−
+ =
(A1)
(A3) (A4) (A5)
(A2)
Figure 7.1: Diagrams for the Ward identity check for 1→ 3 light-front amplitude with helicityconfiguration + → − + ±. The solid line with an arrow instead of a gluon line representsreplacement of a polarization vector with the corresponding momentum. The two right-mostdiagrams contain instantaneous interactions.
modified rules discussed in Sec. 3.4, which impose and effective conservation of full 4-
momentum in the numerator of a graph. The reason for this is that in the calculation of
Ward identities one injects into the vertices a minus light cone component when replacing
a gluon polarization vector by its momentum, while for regular LFPT rules the actual
minus light cone components flowing through the diagram are integrated out prior to
this replacement.
Now that we’re aware of this subtlety, we can proceed to calculate the Ward identity
for the 1 → 3 process with the helicity configuration (+→ −+ +) . Replacing the
polarization vector of the third outgoing particle by the corresponding momentum. We
have
M(+→−+k3)1→3 = A1 +A2 +A3 +A4 +A5, (7.3)
where A1-A5 are the contributions from the diagrams depicted in Fig. 7.1. Here, and
below we use a notation for replacement εi ↔ ki in the superscript, i.e. we replace the
helicity indication by the corresponding momentum. Using the modified LFPT rules and
color-ordered vertices we get
A1 = −4ig2 z2z3z12z123
z212D(1,2)
v(12)1v(123)3v∗(123)3 , (7.4)
A2 = −4ig2 z2z3z223
z223D(2,3)
v(123)1v23v∗23 , (7.5)
A3 = ig2 z3
(v(123)3 − 2v13
), (7.6)
98
A4 = ig2 z3 (z1 − z2)
z12v(123)3 , (7.7)
A5 = 0 . (7.8)
Adding the diagrams we get
M(+→−+k3)1→3 = −2ig2
[z3
z12
(z1v1(123) + z2v13
)+ z23v(123)1 −
z2z3z123
z1z12v∗12
v(123)3v∗(123)3
].
(7.9)
Finally, we can use
z2z1v∗12v12 = −z0z3v03v
∗03 +
1
2z12D(1,2,3), (7.10)
z12z23 = z1z3 + z2z123, (7.11)
z123v(123)1 = z2v21 + z3v31, (7.12)
to get the desired result:
M(+→−+k3)1→3 = ig2 z2
z1
D(1,2,3)
v∗12
. (7.13)
We can now conclude that, in general, M is not gauge invariant. However, it does have
the correct on-shell limit.
If instead we would have wrongly used the regular LFPT rules, the contributions
from A1 and A2 would have been different:
A1 = −2ig2 z2z3z123 (z123 + z12)
z212D(1,2)
v(12)1v(123)3v∗(123)3 , (7.14)
A2 = −2ig2 z2z3z23 (z2 + z23)
z223D(2,3)
v(123)1v23v∗23 . (7.15)
And the Ward identity calculation gives
M(+→−+k3)1→3 = ig2
[z3
z12
(z2
12v01 − z22v12
)+z2(z123 + z12)
z1z12
D(1,2,3)
2v∗12
], (7.16)
which does not vanish for an on-shell amplitude.
The above problem stems from the fact that (7.14) and (7.15) were obtained assuming
that there are no external minus components. Indeed, for an amplitude calculation,
the polarization vectors (3.29) project only on plus and transverse components. The
99
only minus components that flowed inside diagrams were integrated out giving energy
denominators and instantaneous terms. However, for the above Ward identity check, the
triple gluon vertex that appears, for example, in A2, reads
V +→+k33 (−k23, k2, k3) = ig
[− k3 · (k2 + k23)
(ε+
23
)∗ · ε+2 +
(ε+
23
)∗ · (k2 − k3) k3 · ε+2
+ ε+2 · (k23 + k3)
(ε+
23
)∗ · k3
]= ig
z2z3 (z23 + z2)
z23v23v
∗23. (7.17)
The problematic term is the first one in the square bracket. Formally, we cannot write
k3 · (k2 + k23) = 2k2 · k3 since in the regular LFPT rules we do not have full momentum
conservation. We have to consider k2 · k3 and k23 · k3 as different scalar products and
this causes the Ward identity to fail.
7.1.2 Ward identity and the recursion relation for the lowest order
amplitude
In the recursion relation (6.40) the new amplitude M that actually solves the recurrence,
has exactly the form of the MHV amplitude. Therefore, once we impose the on-shell
condition for the process, the result is equal to the MHV amplitude as expected. The
amplitude M , however, as it stands in the recursion relation, is an off-shell object. As
we shall see shortly, it has a remarkable property, namely, it turns out that it is gauge
invariant.
This can be explicitly illustrated by taking (6.44) for N = 3 and checking the Ward
identity. However, one needs to take care of the issues discussed in the preceding section.
We need to calculate
M(+→−+k3)1→3 =M(+→−+k3)
1→3 −D(1,2,3)
D(12,3)M(+→+k3)
1→2
i
z12D(1,2)M
(+→−+)1→2 . (7.18)
In the above expression we have replaced the polarization vector both in the 1 → 3
amplitude and 1→ 2 subamplitude. The second term can be simplified to
D(1,2,3)
D(12,3)
(2ig z3z123v(123)3v
∗(123)3
) i
z12D(1,2)
(2igz2v(12)1
)= ig2 z2
z1
D(1,2,3)
v∗12
(7.19)
where we have used the relations z12D(1,2) = 2z1z2v12v∗12, z12D(12,3) = 2z123z3v(123)3v
∗(123)3
and z1v(12)1 = −z2v12. We see that (7.19) precisely cancels the previously derived term
(7.13), leaving M(+→−+k3)1→3 equal to zero. Therefore M
(+→−++)1→3 is the gauge invariant
100
amplitude irrespectively whether the incoming leg is on-shell or off-shell.
It may be argued that the Ward identity for M is satisfied in general, for arbitrary
number of external legs
M(+→−+...ki···+)1→N = 0. (7.20)
We shall undertake this task in the next section.
7.2 Proof of gauge invariance of the amplitude M from
Wilson lines
In this and the previous chapter we mentioned that the off-shell amplitude M which
appears in the recurrence relation (6.40) is gauge invariant. One possible way to prove
this would be to show that the Ward identities (7.20) are satisfied for an arbitrary
number of legs by using some form of recursion relation. Here we take a different and
more interesting approach.
In [34] it was shown that a gauge invariant off-shell amplitude M could be obtained
from the matrix element of a certain Wilson line operator and that it could be written
in terms of a recursion relation. This relation is remarkably similar to (6.40). To prove
that M is gauge invariant, in the following sections (and was originally done in [51]) we
will, show that, for the helicity configuration (+ → − + . . .+), these two relations are,
in fact, identical. Furthermore, we will see that M corresponds to the gauge invariant
amplitude M and must, therefore, be itself gauge invariant.
In the following subsections we will quickly review the Wilson line approach from
[34] and, then, proceed to our proof.
7.2.1 Matrix elements with Wilson lines and off-shell amplitudes
Let us consider a tree level gluonic Green’s function in momentum space with external
momenta k1...N , k1, . . . , kN satisfying momentum conservation (we assume, as before,
that k1...N is incoming and the rest are outgoing). As such, the Green’s function is a
purely off-shell object, i.e. the external momenta have arbitrary virtuality; moreover,
the external gluon Lorentz indices are not contracted. In order to obtain a scattering
amplitude, we reduce the Green’s function by amputating the external propagators,
taking the on-shell limit for the external momenta, and contracting the external legs
with appropriate polarization vectors transverse to (on-shell) momenta. Here, we shall
consider the Green’s function where the legs k1, . . . , kN are on-shell and reduced as above,
101
while the leg k1...N is kept off-shell and is contracted with a vector e1...N . We shall call
this vector a “polarization” vector for the off-shell gluon. At this point, it is only assumed
that this vector is transverse to the off-shell momentum, e1...N · k1...N = 0. We will call
the Green’s function reduced in that manner an off-shell amplitude.
The off-shell amplitudeM constructed according to the above procedure is not gauge
invariant, i.e. it does not satisfy the Ward identities with respect to the on-shell legs
(for a general choice of e1,...,N and external polarization vectors). However, one can find
a gauge invariant extension of such off-shell amplitude. For example, in the analysis of
scattering at high-energy, i.e. in the Regge limit s � |t| within perturbative QCD, one
encounters similar objects. There, the e1...N is set to one of the light-cone components
n± (n2± = 0) of a hadron momentum and k1...N = xn±+ k⊥, so that k1...N ·n± = 0. The
gauge invariant vertices corresponding to transitions of such off-shell gluons to a set of
on-shell gluons can be derived from the so-called Lipatov’s effective action [82, 83].
In [84] an equivalent, but more computationally friendly, method to Lipatov’s effec-
tive vertices was derived. The idea was to add to M a gauge restoring amplitude W so
that
M =M+W, (7.21)
is gauge invariant. This is similar to how, in Sec. 7.1, (7.19) cancels (7.13) to show that
M in (7.18) is gauge invariant. Knowing the relationship between the Green’s function
and M, the authors then used the Slavnov-Taylor identities to find W:
W (ε1, . . . , εN ) = −∣∣∣~k⊥ 1...N
∣∣∣(−g√
2
)N−1
ε1 · pA . . . εN · pAk1 · pA (k1 − k2) · pA . . . (k1 − k2 − . . .− kN−1) · pA
, (7.22)
where pA is the gauge vector and is defined such that it is transverse to k1...N .
Later in [34] it was found that the gauge invariant extension of the off-shell amplitude
can actually be found by considering a matrix element of a straight infinite Wilson line
operator. More precisely, one defines an object
Ma1...Na1...aNe1...N
(k1...N ; k1, . . . , kN ) =
∫d4x eik1...N ·x
⟨0∣∣T{R a1...N
e1...N(x) eiSY-M
}∣∣ k1, λ1, a1; . . . ; kN , λN , aN⟩c, (7.23)
102
with
R a1...Ne1...N
(x) = Tr
[ta1...NP exp
(ig
∫ +∞
−∞dsAbµ (x+ s e1...N ) eµ1...N t
b
)], (7.24)
where T is the time-ordering, P is the path-ordering, SY-M is the Yang-Mills interaction
action, and, finally, |ki, λi, ai〉 are one-gluon on-shell states with momentum ki, helicity
λi and color ai. The color of the off-shell gluon is a1...N . The subscript c means that
we take only connected contributions. The infinite Wilson line operator R a1...Ne1...N sand-
wiched in the matrix element is explicitly gauge invariant with respect to small gauge
transformations. In [34], instead of a straight infinite path in (7.24), deformed paths
were considered in order to regularize the integrals and to show that they form certain
generalized functions. In fact it is proportional to the momentum conservation Dirac
delta and the delta assuring the Wilson line direction e1...N and the momentum k1...N
are mutually transverse. It can thus be written as
Me1...N (k1...N ) = δ4 (k1...N − k1 − . . .− kN ) δ (e1...N · k1...N )M(λ1...λN )e1...N (k1...N ) , (7.25)
where we have used the shorthand notation for momenta arguments as defined in Eq. (6.35).
The above relation defines the gauge invariant off-shell amplitude M with “polariza-
tion” vector e1...N for the off-shell gluon. If we use the expansion of (7.24), we can see
that this M is the same from (7.21). As an example, a calculation of M for the g∗gg
process gives [34]
M (ε12; ε1, ε2) = −g δ4 (k12 + k1 − k2) δ (k12 · ε12) fc12 c1c2ε12γ ελ1∗α1
(k1) ελ2α2
(k2){
1
k212
[gα1α2 (kγ1 + kγ2 ) + gγα1 (kα212 − kα2
1 ) + gα2γ (−kα12 − kα1
12 )]
− εα112 ε
α212
ε12 · k1
}, (7.26)
where for pA = ε12 we see that the first term in brackets matches the non gauge invariant
M and the second term matches the gauge restoring amplitudeW. M then satisfies the
Ward identities with respect to the external on-shell legs (but not with respect to e1...N ,
i.e. the Wilson line slope)
M(λ1...ki...λN )e1...N (k1...N ) = 0 for i = 1, . . . , N . (7.27)
103
Indeed, this can be shown using Slavnov-Taylor identities [34]. Let us stress that the
amplitude M is gauge invariant only when k1...N · e1...N = 0.
Since we are interested in relating M to the light-front amplitude M , let us now con-
sider a color-ordered version of the matrix element (7.23) with order (a1...N , a1, . . . , aN ).
Diagrammatically, the amplitude M can be written as
N
. . .= +
+ . . .
. . .
N − m − k
. . .
k
. . .
m
. . .
N
. . .
N − m
. . .
m
m = 1
N − 1
M =
k1kN
k1...N
k = 1
N − m − 1
+
m = 1
N − 2
(7.28)
The double line represents the Wilson line in momentum space. Each double line con-
necting two gluon attachments contributes the propagator i/p ·e1...N , with p being the
momentum flowing through the line. The gluons couple to the Wilson line via an igeµ1...N
vertex. More on the Feynman rules can be found in [34]. The shaded blobs represent
standard QCD contributions with the numbers indicating the number of external on-
shell legs. Note that the first contribution in (7.28) is the off-shell amplitude defined
at the beginning of this subsection (modulo ig factor due to a coupling with the gauge
link). In what follows we will denote this amplitude as M(e1...N→λ1...λN )(k1...N ). It will
contain the off-shell propagator and and a coupling to the Wilson line (we include an
additional i factor for further convenience)
M(e1...N→λ1...λN )(k1,...,N ) = ig
−ik2
1...N
iM(e1...N→λ1...λN ) (k1,...,N ) , (7.29)
where M is the standard QCD amplitude calculated from Feynman diagrams (with,
however, off-shell kinematics). Let us underline one more time that the amplitude M(or M) itself does not satisfy the Ward identities, but they are restored thanks to the
rest of the r.h.s of Eq. (7.28).
Let us look at a quick example for (7.28). If N = 4 we get that the gauge invariant
amplitude M4 would be given by the terms depicted in Fig. 7.2. We can write this in a
different manner by grouping terms with the same leftmost blob, such as in Fig. 7.3. This
allows to notice that the terms inside the brackets have follow (7.28) and are themselves
amplitudes M but for a lower number of legs. Thus, the decomposition (7.28) can be
104
M4
=+
k1
k2
k1
k2
k3
k4
+
k2
k3
k4
k1
k3
k4
+
k1
k2
k3
k4
+
k3
k4
k1
k2
+
k1
k2
k3
k4
+
k4
k2
k3
k1
+
k1
k2
k3
k4
Fig
ure
7.2:
Exp
an
sion
ofM
4b
ase
don
(7.2
8).
105
M4
=
k1
k2
k3
k4
+
k2
k3
k4
k1
+
k3
k4
+
M1
M2
M3
k1
k2
k2
k1
++
+
k4
k1
k2
k3
k2
k3
k1
+
k1
k2
k3
k1
k2
k3
Fig
ure
7.3:
Th
issh
ows
how
can
we
wri
teM
4in
term
sof
are
curs
ion
rela
tion
,sp
ecifi
call
y(7
.2)
106
written in a more compact form:
N
. . .=
k1kN
k1...N
. . .
N − m
m = 0
N − 1
kN km+1
. . .
m
k1km
(7.30)
which establishes a recursion relation for M with the following algebraic form:
M(λ1...λN )e1...N (k1...N ) =
N−1∑
m=0
M(λ1...λm)e1...N (k1...m)
1
k1...m · ε1...NM(e1...N→λm+1...λN )
(km+1,...,N ) ,
(7.31)
with
M(λi)e1...N (ki) = i2g e∗1...N · ελii , (7.32)
and M(λ1...λm)e1...N (k1...m) = 1 for m = 0. Here and in the following we should recall that
e1...N is the polarization vector for an incoming momentum, which is why we have the
star in (7.32).
Nevertheless, there is an important difference between the amplitudes M which ap-
pear on both sides of Eq. (7.31). The amplitude on the left hand side of (7.31) satisfies
property (7.27) and thus is gauge invariant. On the contrary, the amplitude M which
appears on the right hand side of (7.31) is not gauge invariant. This stems from the fact
that the replacement εi ↔ ki will lead to the non-vanishing result
M(λ1...ki...λm)e1...N (k1...m) 6= 0 for i = 1, . . . ,m . (7.33)
This is because the Wilson line slope defining M is not perpendicular to the off-shell
momentum, e1...N · k1...m 6= 0, as required by (7.27).
7.2.2 Light-front recursion relation from Wilson lines
We will now relate the recursion with Wilson lines (7.31) to the recursion (6.40) obtained
within the light-front formalism. To this end, we first have to choose the appropriate
“polarization” vector e1...N for the off-shell gluon. We choose, of course, the same vector
as in the formalism to obtain (6.40), i.e. we choose
eµ1...N = ε+µ1...N , (7.34)
107
where ε+1...N is defined by (3.29). Note that ε1...N · k1...N = 0, despite that k1...N is
off-shell and, thus, M(λ1...λN )
ε+1...N(k1...N ) is gauge invariant. Choosing helicities as λ1 = −,
λ2 = · · · = λN = + and the reference momenta for the polarization vectors to be η as in
(3.29), we can write (7.31) as
M(−+···+)
ε+1...N(k1...N ) =M(+→−···+)
(k1...N )
+
N−1∑
m=2
M(−+···+)
ε+1...N(k1...m)
−1
z1...mv(1...m)(1...N)M(+→+···+)
(km+1...N ) . (7.35)
Note that now the sum starts with the index m = 2. For m = 1 the term vanishes since
according to (7.32) it is proportional to
e∗1...N · ε−1 = ε+∗1...N · ε−1 = 0 . (7.36)
The recursion (7.35) relates two amplitudes M, each of them having different number
of external legs. That means that the object on the left hand side of this recursion can be
obtained from the amplitude with lower number of legs. This is similar to the recursion
(6.40) obtained within the light-front formalism and therefore we shall show that it is
possible to relate both recursions. It is tempting to identify immediately the amplitude
M on the right hand side of Eq. (7.31) with the amplitude M on the right-hand side
of (6.40). However, as mentioned earlier, the two objects which are on two sides of the
recursion (7.31) are actually different as the one is gauge invariant and the other isn’t.
Therefore we need to find the transformation which will allow us to compare directly
the two recursions. In other words, we will look to eliminate M(−+···+)
ε+1...N(k1...m) from the
r.h.s of (7.35) and replace it with M(−+···+)
ε+1...m(k1...m), which is gauge invariant.
Another issue which arises is that (7.35) involves two amplitudes for two different
sets of helicities. Therefore one first needs to compute the off-shell amplitude for the
(+ → + · · ·+) configuration and then this object can be used in the recursion above
to compute the other amplitude. Thus, in order to proceed, we have to find an explicit
expression for the off-shell amplitudeM(+→+···+). This is similar to how in Chapter 6 the
recursion relation for M(+→−+...+) (6.14) involved fragmentation functions for different
helicity configurations, T (+→+...+) and T (−→−+...+). These, in turn, had previously been
found through their own recursion relations, (4.26) and (4.36) respectively.
To summarize, to successfully compare (7.35) and (6.40) we will need to do three
things. First, we need to find a concrete expression for M(+→+···+). Secondly, we must
108
find a relationship between the non gauge invariant M(−+···+)
ε+1...N(k1...m) and the gauge
invariant M(−+···+)
ε+1...m(k1...m). Finally, we need to rewrite (7.35) entirely in terms of off-
shell, gauge invariant amplitudes. Specifically, we look to arrive at the following:
M(−+···+)
ε+1...N(k1...N ) =M(+→−···+)
(k1,...,N )
+
N−1∑
m=2
M(−+···+)
ε+1...m(k1...m) KmN M(+→+···+)
(km+1,...,N ) . (7.37)
The relationship we will find in the second step is not a simple one, so getting this result
will be non-trivial. In this equation, KmN is a kernel that we will see is precisely what
it needs to be so that (7.37) identically matches (6.40). An important note is that it is
not obvious that (7.35) can be rewritten as in (7.37). Nevertheless, the authors of [51],
where this was originally done, were encouraged by the light-front results and proceeded
to find that it was, indeed, possible. It, however, remains unclear whether such a kernel
exists for arbitrary helicity configuration.
In the following parts, we will do all three things that we mentioned and finalize our
proof that the light-front, off-shell, MHV-like amplitude M found in Chapter 6 is gauge
invariant.
7.2.2.1 Off-shell (+→ + . . .+) amplitude from Wilson lines
Let us consider the gauge invariant amplitude M(+···+)+ with the choice of the polariza-
tion vectors (3.29). As mentioned in Sec. 3.3.1 this corresponds to choosing η as the
reference momentum for all the polarization vectors. Since M(+···+)+ ≡ M(ε+1 (η)...ε+N (η))
ε+1...N (η)is
gauge invariant, we can freely change the reference momenta of the polarization vectors
ε+1 (η) , . . . , ε+
N (η). Let us thus use (2.30) and set k1...N as the reference momentum
M(ε+1 (η)...ε+N (η))ε+1...N (η)
= M(ε+1 (k1...N )...ε+N (k1...N ))ε+1...N (η)
. (7.38)
Note that the properties (2.31)-(2.33) still hold for εi (k1...N ) despite the fact that k1...N is
off-shell (actually, only the last relation of (2.33) is non-trivial to check). The amplitude
(7.38) is given by the expansion (7.28). Consider any blob
M(ε+1...N (η)→ε+i (k1...N )...ε+j (k1...N )), j > i ,
109
attached to the Wilson line. Such blob contains terms with at least one scalar product
of polarization vectors. This is due to the following standard argument (see e.g. [52])).
Since the are at most j−i−1 triple gluon vertices there may be at most j−i−1 momentum
vectors in the numerator. These vectors are contracted with j−i+1 polarization vectors,
which means that at least two polarization vectors must be contracted together. Due
to our choice of reference momenta all such scalar products vanish due to (2.33). This
happens for all the blobs, therefore
M(+···+)+ = 0. (7.39)
Of course, for the reference momenta set to η the blobs itself no longer vanish, but
different contributions get cancelled due to the gauge invariance.
Let us now look at the consequences of the above equation. Consider the recursion
(7.30) for N = 2 and the Wilson line slope set to a vector u defined by
uµ = ε+µ⊥ +
~ε +⊥ · ~p⊥p · η ηµ (7.40)
for certain momentum p (for example for p = k1...N we have u = ε1...N , but we want to
keep it more general here). We have
M(++)u (k12) =M(u→++)
(k12) + M(+)u (k1)
1
k1 · uM(u→+)
(k2) . (7.41)
If u = ε+12 the l.h.s vanishes according to (7.39) and we have
M(+→++)(k12) = −M(+)
+ (k1)1
k1 · ε12M(+→+)
(k2) (7.42)
or diagrammatically
= − (7.43)
Calculating the r.h.s we get (remember that by convention we include an additional i
factor, c.f. (7.29))
M(+→++)(k12) = −i2 (ig)2 ε−12 · ε+
1 ε−12 · ε+
2
k1 · ε−12
= g2 1
z1v1(12)= g2 z12
z1z2
1
v12. (7.44)
110
Similarly, we have
M(+)u (k1)
1
k1 · uM(u→+)
(k2) = −g2 1
z1v1p, (7.45)
where we have introduced
v1p = ~ε⊥ ·(~ki⊥zi− ~p⊥zp
), zp = p+ . (7.46)
Inserting this back to (7.41) we get
M(++)u (k12) =
g2
z1
(1
v1(12)− 1
v1p
)=g2
z1
v(12)p
v1(12)v1p, (7.47)
where we have utilized the fact that in the light-cone gauge
M(+→++)(k12) =M(u→++)
(k12) , (7.48)
as the propagator on the l.h.s of (7.43) always projects (7.40) to ε+12.
For N = 3 we have
M(+++)u (k123) =M(u→+++)
(k123) + M(+)u (k1)
1
k1 · uM(u→++)
(k23)
+ M(++)u (k12)
1
k12 · uM(u→+)
(k3) (7.49)
Inserting (7.47) we get
M(+++)u (k123) =M(u→+++)
(k123)− g3
(1
z1z2
1
v1p v2(23)− 1
z1z12
vp(12)
v1(12)v1pv(12)p
)
=M(u→+++)(k123)− g3 z123
z1z2z3
1
v12v23
v1(123)
v1p(7.50)
Setting u = ε+123 we eliminate l.h.s and thus
M(+→+++)(k123) = g3 z123
z1z2z3
1
v12v23. (7.51)
Again, one can calculate M(+++)u by inserting the above to (7.50).
The above results generalize. We have
M(+→+···+)(k1...N ) = gN
z1...N
z1 . . . zN
1
v12 v23 . . . v(N−1)N(7.52)
111
and thus
M(+···+)u (k1...N ) = gN
z1...N
z1 . . . zN
(1
v1(1...N)− 1
v1p
)v1(1...N)
v12 v23 . . . v(N−1)N
= gNz1...N
z1 . . . zN
v(1...N)p
v1p
1
v12 v23 . . . v(N−1)N(7.53)
The proof is by checking, that these expressions satisfy the recursion relation (7.30)
rewritten for the current helicity case and for u = ε+1...N . That is, we need to verify if
M(+→+···+)(k1,...,N ) =
N−1∑
m=1
M(+···+)
ε+1...N(k1...m)M(+→+···+)
(km+1,...,N )1
z1...mv(1...m)(1...N).
(7.54)
The r.h.s. with (7.52) and (7.53) reads
N−1∑
m=1
gmz1...m
z1 . . . zm
v(1...m)(1...N)
v1(1...N)
1
v12 v23 . . . v(m−1)m[gN−m
zm+1...N
zm+1 . . . zN
1
v(m+1)(m+2) . . . v(N−1)N
]1
z1...mv(1...m)(1...N)
= gNN−1∑
m=1
zm+1...N
z1 . . . zN
vm(m+1)
v1(1...N)
1
v12 v23 . . . v(N−1)N
= gN1
z1 . . . zN
1
v1(1...N)
1
v12 v23 . . . v(N−1)N
[N−1∑
m=1
zm+1...Nvm(m+1)
]
= gNz1...N
z1 . . . zN
1
v12 v23 . . . v(N−1)N, (7.55)
where we have usedN−1∑
i=m
zi+1...Nvi(i+1) = zm...Nvm(m...N)
to perform the sum. This indeed coincides with (7.52).
Remarkably, this expression is identical to the one obtained from the light-front
approach (6.42), even though the two are (gauge dependent) off-shell amplitudes obtained
within the two different frameworks. To show this we set i = 1 in (6.42), and use the
relation (7.29) with k21...N = z1...ND1...N . This result is encouraging but the recursion
(7.35) is still different then the one obtained within the light-front formulation. Indeed,
this recursion does not involve the same object on the l.h.s and r.h.s of the equation. In
the next section we will find a relation ship between these two objects.
112
7.2.2.2 Relationship between M(−+···+)
ε+1...N(k1...N ) and
M(−+···+)
ε+1...m(k1...m)
To find the relation between M(−+···+)
ε+1...N(k1...m) and M(−+···+)
ε+1...m(k1...m). we can observe
that the first term on the r.h.s of (7.35) does not depend on the Wilson line direction
(this is the consequence of the gauge we are using). Thus, we can write this equation
separately for M(−+···+)
ε+1...N(k1...m) and M(−+···+)
ε+1...m(k1...m) and subtract them. Then we get
M(−+···+)
ε+1...N(k1...m) = M(−+···+)
ε+1...m(k1...m)
+m−1∑
i=2
−M(−+···+)
ε+1...N(k1...i)
z1...i v(1...i)(1...N)+M(−+···+)
ε+1...m(k1...i)
z1...i v(1...i)(1...m)
M(+→+···+)
(ki+1,...,m)
= Mmm +m−1∑
i1=2
(MN i1C
Nmi1 − Mmi1C
mmi1
), (7.56)
for N ≥ m. For brevity we’ve defined
Mji = M(−+···+)
ε+1...j(k1...i) , Cjm i = −M
(+→+···+)(ki+1,...,m)
z1...i v(1...i)(1...j). (7.57)
We want to express the r.h.s of (7.56) solely in terms of objects Mii, which are gauge
invariant. This can be done by using (7.56) recursively. For the first two iterations we
get
MNm = Mmm +
m−1∑
i1=2
Mi1i1
(CNmi1 − Cmmi1
)
+m−1∑
i1=2
i1−1∑
i2=2
Mi2i2
[CNmi1
(CNi1 i2 − Ci1i1 i2
)− Cmmi1
(Cmi1 i2 − Ci1i1 i2
) ]+ . . . (7.58)
With the aid of a few examples this can be readily generalized to
MNm = Mmm +
m−1∑
i1=2
Mi1i1a(1)N mi1
+
m−1∑
i1=2
i1−1∑
i2=2
Mi2i2a(2)N mi1i2
+ . . .
= Mmm +
m−2∑
p=1
m−1∑
i1=2
i1−1∑
i2=2
· · ·ip−1−1∑
ip=2
Mipipa(p)N mi1...ip
, (7.59)
113
where
a(p)N mi1...ip
= CNmi1a(p−1)N i1...ip
− Cmmi1a(p−1)mi1...ip
(7.60)
with the initial condition
a(0)N m = 1. (7.61)
To find the solution for a(p)N mi1...ip
, let’s look at a few examples. For p = 1 we have
a(1)N mi1
= −M(+→+···+)
(ki1+1,...,m)
z1...i1
(1
v(1...i)(1...N)− 1
v(1...i)(1...m)
)
=M(+→+···+)
(ki1+1,...,m)
z1...i1
v(1...m)(1...N)
v(1...i1)(1...m)v(1...i1)(1...N). (7.62)
Similarly, for p = 2 we will have
a(2)N mi1i2
=M(+→+···+)
(ki2+1,...,i1)M(+→+···+)(ki1+1,...,m)
z1...i2 z1...i1v(1...i2)(1...i1)
v(1...m)(1...N)
v(1...i2)(1...m)v(1...i2)(1...N)
(7.63)
and so on. In fact, the solution to (7.60) reads
a(p)N mi1...ip
=M(+→+···+) (
kip+1,...,ip−1
). . .M(+→+···+)
(ki2+1,...,i1)M(+→+···+)(ki1+1,...,m)
z1...ip . . . z1...i1v(1...ip)(1...ip−1) . . . v(1...ip)(1...i1)
×v(1...m)(1...N)
v(1...ip)(1...m)v(1...ip)(1...N), (7.64)
for p > 1, as can be easily verified. Therefore, the equation (7.59) can be written as
M(−+···+)
ε+1...N(k1...m) = M(−+···+)
ε+1...m(k1...m)
+m−2∑
p=1
m−1∑
i1=2
i1−1∑
i2=2
· · ·ip−1−1∑
ip=2
M(−+···+)
ε+1...ip
(k1...ip
)
M(+→+···+) (kip+1,...,ip−1
). . .M(+→+···+)
(ki1+1,...,m)
1
z1...ip . . . z1...i1
1
v(1...ip)(1...ip−1) . . . v(1...ip)(1...i1)
v(1...m)(1...N)
v(1...ip)(1...m)v(1...ip)(1...N). (7.65)
7.2.2.3 Arriving at (7.37)
Thanks to our previous results, we will now be able to go from (7.35) to (7.37), where
all M are gauge invariant. This will allow a direct comparison with (6.40) which will
114
allow us to conclude that the MHV-like M obtained in Chapter 6 is gauge invariant.
First, we use (7.65) to rewrite (7.35) purely in terms of gauge invariant amplitudes.
The resulting equation is, however, very complicated. In order to simplify it we note the
following identity
M(+→+···+) (kip+1,...,ip−1
). . . M(+→+···+)
(ki1+1,...,m)
= zip+1...ip−1 . . . zi2+1...i1
zi1+1...m
zip+1...mvip−1(ip−1+1) . . . vi1(i1+1)
M(+→+···+) (kip+1,...,m
). (7.66)
It follows directly from (7.52). Using this we can write (7.35) as
M(−+···+)
ε+1...N(k1...N ) =M(+→−···+)
(k1,...,N )
+N−1∑
m=2
M(−+···+)
ε+1...m(k1...m)
−1
z1...mv(1...m)(1...N)M(+→+···+)
(km+1,...,N )
+N−1∑
m=2
m−2∑
p=1
m−1∑
i1=2
i1−1∑
i2=2
· · ·ip−1−1∑
ip=2
M(−+···+)
ε+1...ip
(k1...ip
)M(+→+···+) (
kip+1,...,N
)
zip+1...ip−1 . . . zi2+1...i1
z1...ip . . . z1...i1z1...m
zi1+1...m
zip+1...Nzm+1...N
vm(m+1)
v(1...N)(1...ip)v(1...ip)(1...m)
vi1(i1+1) . . . vip−1(ip−1+1)
v(1...ip)(1...i1) . . . v(1...ip)(1...ip−1). (7.67)
The amplitudes on the r.h.s. depend only on single summation variable ip, and it turns
out that one can perform the remaining sums. To this end, let us reorganize the sums
as follows
N−1∑
m=2
m−2∑
p=1
m−1∑
i1=2
i1−1∑
i2=2
· · ·ip−1−1∑
ip=2
=N−3∑
p=1
N−1∑
ip=2
N−1∑
ip−1=ip+1
· · ·N−1∑
i1=i2+1
N−1∑
m=i1+1
. (7.68)
Let us also rename variables m→ i0, ip → m. This allows to rewrite (7.67) as
M(−+···+)
ε+1...N(k1...N ) =M(+→−···+)
(k1,...,N )
+N−1∑
m=2
M(−+···+)
ε+1...m(k1...m)M(+→+···+)
(km+1,...,N )−1
z1...mzm+1...Nv(1...m)(1...N){zm+1...N +
N−4∑
p=0
N−1∑
ip=m+1
· · ·N−1∑
i0=i1+1
zm+1...ip . . . zi1+1...i0zi0+1...N
z1...i0 . . . z1...ip
115
vi0(i0+1) . . . vip(ip+1)
v(1...m)(1...i0) . . . v(1...m)(1...ip)
}. (7.69)
The tower of sums above can be utilized as follows. First, let us introduce
κmN = zm+1...N
+N−4∑
p=0
N−1∑
ip=m+1
· · ·N−1∑
i0=i1+1
zm+1...ip . . . zi1+1...i0zi0+1...N
z1...i0 . . . z1...ip
vi0(i0+1) . . . vip(ip+1)
v(1...m)(1...i0) . . . v(1...m)(1...ip)(7.70)
It can be rewritten as
κmN = zmN +
N−4∑
p=0
∑
ip
· · ·∑
i0
zmiphip (m) zipip−1 . . . hi0 (m) zi0N , (7.71)
where
zij = Θ (j − i− 1) z(i+1)...j (7.72)
with Θ (j − i) being the Heaviside step function, and
hi (m) =vi(i+1)
z1...iv(1...m)(1...i). (7.73)
We can consider zij as a “free propagator” and hi (m) as a “vertex”, and (7.71) as the
equation for the “full propagator”. Graphically it can be represented as
=m + 1 n m + 1m + 1 n
+ +n
+ . . .m + 1 ni0
+ . . .m + 1 i0 in−4 n
(7.74)
where the blob represents κmn, the black dots represent vertices hi, and the lines repre-
sent propagators zij . At each vertex there is a summation over the corresponding index.
It is easy to see that (7.71) has the following factorization property, graphically
=m + 1m + 1 n
+m + 1 n i n
(7.75)
or
κmn = zmn +∑
i
κmihi (m) z(i+1)n. (7.76)
116
We will prove, that the solution to this equation reads
κmn =z1...mz1...nv(1...m)(1...n)
z1...m+1v(1...m+1)(m+1). (7.77)
First, consider the sum on the r.h.s. of (7.76) inserting the above ansatz
∑
i
Θ (n− i− 1) Θ (i−m− 1)z1...mzi+1...nv(1...m)(1...i)
z1...m+1v(1...m+1)(m+1)
v(i+1)i
zi+1v(1...i)(1...m)
= − z1...m
z1...m+1v(1...m+1)(m+1)
n−1∑
i=m+1
zi+1...nv(i+1)i = −z1...mzm+1...nv(m+1...n)(m+1)
z1...m+1v(1...m+1)(m+1). (7.78)
The complete r.h.s. of (7.76) now reads
z(m+1)...n −z1...mzm+1...nv(m+1...n)(m+1)
z1...m+1v(1...m+1)(m+1)=zm+1...nv(1...m)(m+1...n)
v(1...m)(m+1)
=z1...mz1...nv(1...m)(1...n)
z1...m+1v(1...m+1)(m+1). (7.79)
We see that this is the same as (7.77), thus we have accomplished the proof.
It is now easy to read out the expression for KmN defined in (7.37). It follows from
comparison of (7.37) and (7.69) and simply reads
KmN =z1...N
z1...m+1zm+1...Nv(m+1)(1...m+1)(m+1). (7.80)
It is straightforward to check that (7.37) with (7.80) and (6.42) coincides exactly with
(6.40) obtained within the light-front approach. To this end one only needs to redefine
M and M in (6.40) to include the energy denominators with appropriate zi’s forming
in fact propagators. This means also, that the MHV off-shell amplitude M in (6.40) is
indeed gauge invariant since it corresponds to the gauge invariant M from the Wilson
line approach. Yet another confirmation of this result comes from the work [35] where
similar off-shell gauge invariant helicity amplitude was calculated and turned out to have
also the MHV form.
Chapter 8Conclusions
In this dissertation we took initial, important steps in understanding helicity ampli-
tudes in the light-front formalism. We studied the (− → + . . .+), (+ → + . . .+)
and (+ → − + . . .+) helicity configurations and obtained off-shell amplitudes which
reduce to the Parke-Taylor amplitudes [1] in the on-shell limit. For this purpose we
obtained recursion relations (6.14) for an off-shell amplitude, which turned out to be
the light-front analogs of the Berends-Giele equation (2.52), yet they contained an extra
instantaneous interaction term. Particularly interesting was the solution for the off-shell
amplitude(6.40,6.44). It turns out that it can be written entirely as a linear sum of
off-shell MHV-like objects. We were not looking for these, but they still appeared from
the algebra.
Trying to give a physical interpretation to our recursion relation we calculated the
Ward identity for a simple example and it turned out that our off-shell MHV-like am-
plitude seemed to be gauge invariant, see Sec. 7.1.2. For further proof we studied the
matrix elements of a straight infinite Wilson line operator [34], as these had been shown
to generate gauge invariant off-shell amplitudes. It turns out, that the same recursion
relation appears in both frameworks, and that our off-shell MHV-like amplitude matches
the gauge invariant amplitude generated from the Wilson line operator. This leads us to
conclude that our recursion relation has its physical origin in gauge invariance and that
our off-shell MHV-like amplitude is itself gauge invariant.
Throughout this work we also studied crossing symmetry on the light-front, see Chap-
ter 5. The fact that crossing symmetry should hold on the light-front is not obvious,
due to the presence of energy denominators. These are non-local and mix energies from
different states. Yet, we found an approach to relate transition amplitudes to scattering
118
amplitudes. The only requirement is that the amplitudes need to on-shell. Here we also
saw that the analytic continuation involved introduces a phase which depends on the
choice of polarization vectors.
We also saw that off-shell amplitudes can be obtained from either wavefunctions or
fragmentation functions, see Chapter 6. However, the fragmentation functions already
have the appropriate structure so they seem like the best choice.
Finally, we saw that for a Ward identity calculation we cannot naively use standard
light-front rules. Instead, a modification is needed where an instantaneous term appears.
The net effect of this term is to conserve energy on the numerator of the amplitude. See
Sec. 3.4.
As for the outlook to this work, it seems promising. For starters, there is thirty years
of amplitude research in instant-time from which we can learn. For example, Eq. (6.44)
and Fig. 6.5 seem reminiscent of a BCFW [14, 11, 13] type recursion, but in this case for
off-shell amplitudes. We can see this by looking at the term in Fig. 6.5. It is composed of
two amplitudes, with the correct selection legs, multiplied by a propagator. Furthermore,
one of those amplitudes is already in an on-shell form.
Some other questions include:
1. Can we calculate in an easy way amplitudes for other helicity configurations?
2. In [48] the authors talked about a duality between wavefunctions and fragmentation
functions for a simple helicity configuration. Is it possible this generalizes to more
complicated helicities?
3. Is there any way to calculate amplitudes for two incoming off-shell gluons based
on information we already have?
4. How do we calculate loops?
5. Can the recursion between off-shell and “on-shell” amplitudes be used for higher
order computations?
6. Our off-shell gauge invariant amplitude took on its on-shell form. Does this result
generalize?
In summary, there is still more work to be done and many more interesting questions
to be answered.
Appendix AVertex factors
This appendix provides an overview of how to calculate the vertex factors relevant to
this dissertation. A table with a summary of these is also provided for convenience. No
color factors are included as these will be used in color-ordered amplitudes
A.1 Polarization vector relations
The following relationships will make our lives a little easier when computing the vertex
factors. Recall from Chapter 3
aµ = (a+, a1, a2, a−), (A.1)
ε±µi (η) ≡ ε±µi = ε±µ⊥ +~ε ±⊥ · ~ki⊥ki · η
ηµ =
(0, ~ε ±⊥ ,
2 ~ε ±⊥ · ~ki⊥k+i
), (A.2)
ε±µ⊥ =1√2
(0, 1,±i, 0) =1√2
(0, ~ε ±⊥ , 0
), (A.3)
ε±i · (ε±j )∗ = −1, ε±i · (ε∓j ) = −1, ε±i · (ε±i ) = ε±i · (ε±j ) = 0 , (ε±i )∗ = ε∓i . (A.4)
Using (A.2) we also have
ε±i · kj =~ε ±⊥ · ~ki⊥k+i
k+j − ~ε ±⊥ · ~kj⊥ = zj ~ε
±⊥ ·
(~ki⊥zi−~kj⊥zj
)(A.5)
Here we have used ki = ziP+, where P+ is a total incoming momentum. Finally, we
120
k2, ±
k3, ±
k1, ± (−1)j−12igz1v±32
k3, ∓
k2, ∓
k1, ∓
k2, ∓
k3, ±
k1, ± (−1)j−12igz3v∓12
k3, ∓
k2, ±
k1, ∓
k2, ±
k3, ∓
k1, ± (−1)j−12igz2v∓31
k3, ±
k2, ∓
k1, ∓
k1, ± k3, ±
k2, ∓
k4, ±
ig2 k1, ± k3, ±
k2, ±
k4, ∓
k1, ± k3, ∓
k2, ±
k4, ±
−2ig2
k1, ± k4, ±
k2, ∓
k3, ±ig2 (z1+z4)(z3−z2)
(z4−z1)2
k1, ± k4, ±
k2, ±
k3, ∓
k3, ∓
k4, ±
k2, ±k1, ±
ig2 (z1+z2)(z3−z4)(z2−z1)2
k3, ±
k4, ∓
k2, ±k1, ±
Table A.1: Helicity vertex factors relevant to us. j is the number of incoming particles.Light-front time increases to the right and all particles are moving to the right. Weinclude 1→ 2, 2→ 1, and 1→ 4, vertices, but not 2→ 2. All other factors are zero.
121
define
vij = ~ε −⊥ ·(~ki⊥zi−~kj⊥zj
), (A.6)
v∗ij = ~ε +⊥ ·
(~ki⊥zi−~kj⊥zj
), (A.7)
v+ij = v∗ij , v−ij = vij . (A.8)
An important note here is that our definitions of positive and negative helicities here and
in Chapter 3 are reversed from those in Chapter 2 and most of literature [52, 53, 14].
A.2 3-gluon vertex
k2, µ2
k3, µ3
k1, µ1
Figure A.1: 3-gluon vertex
Assuming all momenta outgoing, the 3 gluon vertex (Fig. A.1) is given by
V3,µ1µ2µ3(k1, k2, k3) = ig[(k1 − k2)µ3
gµ1µ2 + (k2 − k3)µ1gµ2µ3 + (k3 − k1)µ2
gµ1µ3
].
(A.9)
Note that Mangano and Parke’s [52] vertex has an extra 1/√
2. Since we will be working
with helicity amplitude, we multiply by appropriate polarization vectors:
V λ1λ2λ33 (k1, k2, k3) = (ελ1
1 )µ1(ελ22 )µ2(ελ3
3 )µ3V3,µ1µ2µ3
= −ig[(z1v
λ331 − z2v
λ332 )δλ1
−λ2+ (z2v
λ112 − z3v
λ113 )δλ2
−λ3+ (z3v
λ223 − z1v
λ221 )δλ3
−λ1
]. (A.10)
This equation is for all outgoing momenta. For any incoming momentum, use ε∗ instead
and reverse the sign of the momentum in (A.9).
For specific helicities see Table A.1.
122
A.3 4-gluon vertex
k1, µ1
k4, µ4
k2, µ2
k3, µ3
Figure A.2: 4-gluon vertex
For the 4-vertex in Fig. A.2 we would have, assuming all momenta are outgoing,
V λ1λ2λ3λ44 (k1, k2, k3, k4)
= ig2(ελ11 )µ1(ελ2
2 )µ2(ελ33 )µ3(ελ4
4 )µ4(2gµ1µ3gµ2µ4 − gµ1µ4gµ2µ3 − gµ1µ2gµ3µ4). (A.11)
This time we are missing a factor of 1/2 compared to [52].
For any incoming momentum, use ε∗ instead.
For specific helicities see Table A.1.
A.4 Instantaneous interaction
k1, µ1 k4, µ4
k2, µ2 k3, µ3
Figure A.3: Instantaneous interaction
As we saw in Chapter 2, this interaction is not really a vertex. It is an instantaneous
interaction between two 3-gluon vertices. However, it is convenient to think of it as its
own vertex. For Fig. A.3 we have
V λ1λ2λ3λ4Inst (k1, k2, k3, k4)
123
= εµ22 εµ3
3 V3,µ2µ3ν1(k2, k3,−k2 − k3)iην1ην2
(k+1 + k+
4 )2V3,µ1ν2µ4(k1,−k1 − k4, k4)εµ1
1 εµ44
(A.12)
This equation is for all outgoing momenta. For any incoming momentum, use ε∗ instead
and reverse the sign of the momentum in (A.12).
For specific helicities see Table A.1.
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Vita
Christian A. Cruz Santiago
Education
Ph.D. in PhysicsThe Pennsylvania State University, University Park, PA, August 2015
B.S. in Computer EngineeringUniversity of Puerto Rico, Mayaguez, PR, June 2008
Awards
Gary McCartor Travel Award 2012
David C. Duncan Graduate Fellowship 2012
Climate and Diversity Award 2009
The Bunton-Waller Scholarship 2008-2010
UPRM ECE Computer Engineering Award 2008
Verizon Scholarship 2003-2008
Publications
Gluon cascades and amplitudes in light-front perturbation theoryC.A. Cruz-Santiago and A.M. Stasto.Nucl. Phys. B 869, 1 (2013). arXiv:1301.3075 [hep-ph].
Gluon Wavefunctions and Amplitudes on the Light-FrontC.A. Cruz-Santiago.Acta Phys. Polon. Supp. 6, 201 (2013). arXiv:1301.5598 [hep-ph].
Recursion relations and scattering amplitudes in the light-front formalismC.A. Cruz-Santiago and A.M. Stasto.Nucl. Phys. B 875, 368 (2013). arXiv:1308.1062 [hep-ph].
Recursion relations for multi-gluon off-shell amplitudes on the light-frontand Wilson linesC. Cruz-Santiago, P. Kotko, and A.M. Stasto.Nucl. Phys. B 895, 132 (2015). arXiv:1503.02066 [hep-ph].
