FEATURE ARTICLE
THE GEOMETRY OF TREES
FREDDY CACHAZO
BY
THE S-MATRIX
n 1937, J.A. Wheeler introduced the concept of the
Scattering or S-matrix and in 1942 Heisenberg proposed to use it as a way to describe particle physics.
The Scattering matrix encodes the information needed to compute the probability of a certain outcome given a
particular set of incoming particles. One of the beautiful
properties of quantum mechanical systems is that the
S-matrix is computed using complex numbers. In particle
physics, where the initial conditions are determined in
terms of the four-momenta of the particles, one is naturally led to an object that is an analytic function of the initial
and final data. Of course, in practice one is only interested in four-momenta which are real and thus correspond to
particles used in accelerators. Each matrix element of the
S-matrix is called a scattering amplitude. New methods of
calculating scattering amplitudes, and what they could
imply for our understanding of the underlying physics, are
the subject of this article.
I
From the time of Rutherford scattering alpha particles
against gold atoms to our modern powerful accelerators
which smash subatomic particles at high energies the basic
theoretical idea has been the same: to use the measurements of the outcoming particles to infer properties of
short distance physics. With the Large Hadron Collider
(LHC) starting operations at CERN we expect to have
access to physics at distances on the order of 10n17cm.
The LHC will smash protons at 14 TeV [1].
Protons are made out of quarks and gluons. The most
spectacular collisions will happen when two gluons
within the protons, each carrying a large fraction of the
proton’s total energy, collide. Previously unseen particles
are expected to be produced which will then decay into
known particles. The huge CMS and ATLAS detectors
will be capable of tracking and determining the properties
of the final products. From these one can trace back and
determine the properties of the new particles. The reader
might wonder how is one supposed to know whether the
interaction was the result of new physics. The answer is
that one has to compute the predictions of the Standard
Model [1] of particle physics, for which Glashow, Salam
and Weinberg won the 1979 Nobel prize, and then find the
discrepancy between the observed phenomena and the theory.
Computing the predictions, also called the Standard
Model background, is one motivation for finding new and
efficient ways of computing scattering amplitudes. The
textbook procedure for computing scattering amplitudes is
very clear. One has to add up complex numbers associated with all ways of producing the final particles from the
incoming ones. The rules for the allowed interactions are
determined by the underlying theory, in this case the
Standard Model. These are called the Feynman rules 1 of
the theory and can be represented using diagrams called
Feynman diagrams. The rules also give the complex number associated to each diagram. At leading order in perturbation theory, the diagrams are called Tree diagrams
(see Figure 1).
As mentioned, the scattering of two gluons into several
gluons is of particular interest for the LHC. Gluons are
massless particles which come in 8 H 2 = 16 different
types. In the previous formula, 8 represents the so-called
color charge while the 2 represents the helicity, which can
be (±1). Unlike photons, which do not carry electric
charge, gluons can interact among themselves because of
their color charge! The part of the Standard Model which
governs the interactions of gluons among themselves is
called Quantum chromodynamics (or QCD) [2].
As usual in physics, computations and often the final
answer can be simplified depending on the variables chosen. To encode the data of the initial and final gluons, the
Fig. 1
SUMMARY
Collisions of gluons, happening at the core
of hadron colliders, have beautiful descriptions in terms of abstract mathematical
spaces called Grassmannians.
F. Cachazo
<fcachazo@
perimeterinstitute.ca>,
Perimeter Institute for
Theoretical Physics,
31 Caroline St. N.,
Waterloo ON
N2L 2Y5
The Feynman rules for gluons tell us that they can only
interact through cubic and quartic vertices. At leading
order in perturbation theory, the diagrams are called
Tree diagrams. Some of them are shown in the figure.
1. These rules can be formally derived by using Quantum Field Theory,
but this is beyond the scope of this article.
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THE GEOMETRY OF TREES (CACHAZO)
textbook recipe uses momenta pμ, which are null vectors for
gluons, and polarization vectors εμ for each particle. With
these variables the scattering amplitudes can be very lengthy
and quickly get out of hand even with modern computers.
A simple transformation, done using the three Pauli matrices
and the identity matrix to form a four-vector σμ, which converts pμ into a 2 H 2 matrix pμσμ gives rise to a dramatic simplification. Each particle is now described by a pair of Weyl
spinors which are nothing but two-component objects whose
entries are complex numbers. All amplitudes become functions
of the Lorentz invariant inner products of such spinors which
are denoted by +i, j, for particles i and j. In the 1980’s, Parke
and Taylor [3] wrote down a shockingly simple proposal for the
scattering amplitude of two (+1) gluons, say particles 1 and 2,
into any number of (+1) gluons, particles 3, . . ., n. The answer
is given by a sum of terms of the form
3
1, 2
2, 3 3, 4 4, 5 5, 6 6, 7 ... n − 1, n n,1
(1)
weighted by a factor that accounts for the color charge of the
gluons participating in the interaction. This proposal was later
proven by Berends and Giele [4].
Unfortunately, the simplicity observed by Parke and Taylor
was not generalizable to other amplitudes and this particular set
of amplitudes was considered to be very special. This point of
view changed radically with the discovery of new methods,
some of them inspired by a string theory living in twistor
space constructed by Edward Witten in 2003 [5]. Some of these
methods are the Cachazo-Svrcek-Witten (CSW) diagram
expansion of amplitudes, where all scattering amplitudes are
built using Parke-Taylor amplitudes as building blocks, and the
Britto-Cachazo-Feng-Witten (BCFW) recursion relations [5],
where scattering amplitudes of many particles are built out of
those of a smaller number of particles. These new techniques,
together with powerful methods developed in the past two
decades, have been implemented in a very efficient computer
code called BlackHat [6] which has been used to compute
interesting Standard Model backgrounds for hadron colliders
like the Tevatron at Fermilab and the LHC.
In physics, it is often the case that when the results of computations are much simpler than expected then there is some
principle or symmetry that has been missed. At the very least
there is usually an alternative formulation of the theory where
the simplicity is more manifest. Putting together all the clues
coming from the different techniques in order to find the
new formulation is a very fascinating challenge. Finding new
formulations of the same theory is useful because one such
formulation might be the springboard for the next breakthrough
in our understanding of nature. A classic example is the
Hamiltonian or Lagrangian formulation and the least action
principle of Newton’s equations.
TWISTOR SPACE
One could say that this might have been the motivation for
Roger Penrose to introduce Twistor space in 1967 as a new
arena for the formulation of physical theories meant to replace
space-time [7]. Twistor space takes as fundamental objects the
paths followed by massless particles in space-time. Each such
path is represented by a point in twistor space. The idea was
that perhaps quantization in this new space would be the natural springboard for a theory of quantum gravity. Penrose was
also very driven by the power of complex analysis. A problem
with this was that the space of null rays is five dimensional
(called PN) and thus does not admit a complex variable
description (which requires an even number of real dimensions). The solution Penrose gave was beautifully physical:
null rays are the trajectories of massless particles of helicity
zero. If particles with non-zero helicity, say ±1/2, are considered then the solution to the wave equations can be nicely
encoded into a six dimensional space which is chopped into
two halves by PN. The two halves are called PT+ and PTn.
Depending on the sign of the helicity, the particle is described
by a point in PT+ or PTn. It turns out that the total space
PNcPT+cPTn is a very familiar space for mathematicians: it
is CP 3, a complex projective space, i.e., the space of complex
lines in C4.
As part of the twistor programme, scattering amplitudes were
modeled using a construction called twistor diagrams. These
diagrams were meant to encode the scattering information as
multidimensional contour integrals. A puzzling fact was that,
somehow, twistor diagrams were encoding the information in a
different way than Feynman diagrams do. Very few people kept
working on twistor diagrams for this reason and also because
of their mathematical complexity. One who did was Andrew
Hodges, who, after working on the subject since the 70’s, proposed a surprising connection in 2004 with the BCFW construction. In 2008, the connection was made precise. It turns
out that the terms in BCFW recursion relations are, in fact,
twistor diagrams!
TWISTOR STRING THEORY
One of the well known facts about string theory is that it lives
in a space-time of 9+1 dimensions. How can one even start to
think about a string theory in twistor space which only has
6 real dimensions? The answer is that a special string theory
called the topological B model can live on any space which satisfies a mathematical condition called the Calabi-Yau condition. It turns out that twistor space itself does not satisfy this
condition but if the space is made maximally supersymmetric [5] then it meets the requirement 2. The precise construction
is delicate and beyond the scope of this article. However, the
implications of the theory are remarkably simple to state.
2. Supersymmetry is usually known as a symmetry that relates bosons and fermions. Likewise, if one thinks about the usual coordinates used in geometry as “bosonic” then one can add “fermionic” coordinates in order to have a superspace!
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THE GEOMETRY OF TREES (CACHAZO)AA
development worth mentioning for how unexpected it is. By
using the AdS/CFT correspondence [2,8] and by perturbation
theory arguments as well, it was found that scattering amplitudes of gluons possess a hidden symmetry! In addition to
invariance under conformal transformations 3, the new symmetry is the so-called dual conformal invariance [8], which acts
just like conformal transformations but on an auxiliary spacetime where the momenta of the particles are written as the difference of two “coordinates”: p μi = x μi n x μi+1 .
Is there a framework where all these developments can be seen
as different faces of the same object? Could such a framework
be an interesting reformulation of space-time physics?
Fig. 2
CSW and connected maps into twistor space from the
twistor string worldsheet. The physical amplitude is
obtained by mapping the twistor space objects into space
time! The map in the figure is of degree 2.
The basic idea is that all scattering amplitudes are best
described with a two-dimensional worldsheet and are universal. In other words, the core of all scattering amplitudes is as
simple as that of the Parke-Taylor amplitudes! If this is true,
how can one get the complicated answers one expects when the
final gluons are not all of (+1) helicity? In fact, one expects
amplitudes to get more and more complicated the larger the
number of (n1) helicity gluons in the final state. Let us denote
the number of (n1) gluons in the final state by m. Witten conjectured that all the complication comes from the way the core
interaction is mapped into twistor space and furthermore in the
mapping from twistor space into spacetime. The ways to map a
sphere into twistor space are classified by their degree. For
example, if f (z) = z2 is a mapping from the complex numbers
to themselves we say that such a map has degree 2. The reason
is that a generic point on the image, yo, comes from 2 points in
the domain, for if f (zo ) is equal to yo so is f (nzo ). Witten
argued that scattering amplitudes with a given m are obtained
by using a map of degree m + 1. Clearly, the Parke-Taylor formula, with m = 0, corresponds to a map of degree 1 which
means that the amplitude is basically the same as the core one!
This striking formulation gave rise to the CSW expansion of
scattering amplitudes, where the degree m + 1, is taken to produce m + 1 degree 1 spheres in twistor space, each of them with
the simplicity of the original core or equivalently the ParkeTaylor one. Another form is obtained by taking the image to
be a single curve of high degree, e.g., for degree 2, a conic.
This led to the Roiban-Spradlin-Volovich (RSV) connected
formula.
THE GRASSMANNIAN UNIFICATION
The answer might come from the Arkani-Hamed-CachazoCheung-Kaplan (ACCK) Grassmannian formulation [9], where
the physics of scattering amplitudes in the sector containing the
scattering of two (+) gluons into k n 2 (n) gluons and the
remainder (+) gluons has been conjectured to be encoded in a
contour integral in the space of k-planes containing the origin
of Cn. This space has a name in the mathematical literature: it
is called the Grassmannian G(k,n). Here n is the total number
of particles.
A k-plane in Cn can be specified by a matrix made out of k vectors in Cn. Let such a matrix be denoted by C. Then it is proposed that all the information needed for computing the amplitude is contained in
Lk , n = ∫
k
d k × nCαa
δ4 4 ( CαaWa )
∏
n
∏ i =1 ( i, i + 1, ... , i + k − 1) α =1
where Wa are points in super-(dual)-twistor space 4 which
encode the data of the particles involved in the interaction and
Fig. 3
We have mentioned several different constructions developed
since 2004, which have led to ways of computing the amplitudes of gluons in simpler forms. There is one more recent
Unification of Formulations. Center: The Grassmannian
integral Lk,n . Top left: RSV. Top right: CSW. Middle left:
Twistor Diagrams. Middle right: Polygon of momenta for
Dual Conformal Invariance. Bottom: Feynman diagrams.
3. These are nothing but the familiar translations, rotations and Lorentz boosts together with dilations and inversions which act on space-time coordinates as xμ a
xμ and xμ xμ/x2, respectively.
4. Here we are using supersymmetry in the form of super-twistors simply as a bookkeeping device but at tree-level, which is the main focus of this article, everything
can be done completely within QCD!
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(i, i + 1, . . . , i + kn1) is the k H k minor of the matrix C made
out of columns i through i + k n 1.
there is a single relation among the residues of a given function, in the multidimensional case there are many.
Scattering amplitudes are determined by purely combinatorial
data. For example, for n = 7 and k = 3 one finds the amplitude
to be
These relations among the residues of Lk,n , which stem from
the topology of the Grassmannian, have been shown to be
equivalent to relations which follow from space-time locality.
An even more surprising application of the global residue
theorem, as it is called, is the “duality” between Lk,n and the
Grassmannian formulation of CSW and the RSV formulation.
Noting that the residues of Lk,n contain all possible objects constructed from BCFW recursion relations and hence twistor diagrams, we can conclude that Lk,n leads to a unification of all
known formulations!
(1)[(2) + (4) + (6)] + (3)[(4) + (6)] + (5)(6)
(2)
where (i)( j) implies that minors starting with i and j vanish.
This defines an algebraic variety in the Grassmannian and a
residue can be associated with it.
The form of Lk,n makes conformal invariance manifest. A
Fourier transform and a simple linear algebra argument show
that all residues computed using Lk,n are also dual conformal
invariant.
In physics we are familiar with the power of the one dimensional residue theorem or Cauchy’s theorem. It turns out that
the generalization to more complex variables is even more
powerful in the sense that while in the one-dimensional case
ACKNOWLEDGEMENTS
Research at Perimeter Institute is supported by the Government
of Canada through Industry Canada and by the Province of
Ontario through the Ministry of Research and Innovation. The
author also acknowledges further support provided by an
NSERC Discovery grant and by an Early Researcher Award
from the Province of Ontario.
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