The SMatrix Reloaded: Twistors, Unitarity, Gauge Theories and Gravity

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The S-Matrix ReloadedTwistors, Unitarity, Gauge
Theories and Gravity

• Niels Bohr Summer Institute Conference 2006
• Zvi Bern, UCLA

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Outline
The past two years have seen a very significant
advance in our ability to compute scattering
amplitudes.

• The call of the LHC multi-parton scattering at
  loop level.
• Resummation of planar N 4 super-Yang-Mills
  scattering amplitudes to all loop orders.
• The structure of perturbative quantum gravity.
  Reexamine standard wisdom on quantum gravity.

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LHC Physics
The LHC will start operations in 2007.
We will have lots of multi-particle processes.
Want reliable predictions.
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Example Susy Search
Gianatti and Mangano
ALPGEN vs PYTHIA
Early ATLAS TDR studies using PYTHIA overly
optimistic.

• ALPGEN is based on LO
• matrix elements and much
• better at modeling hard jets.
• What will disagreement between
• ALPGEN and data mean? Hard
• to tell. Need NLO.

Such a calculation is well beyond anything that
has been done using Feynman diagrams
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What needs to be done at NLO?
Experimenters to theorists
Please calculate the following at NLO
A key theoretical problem for LHC is NLO
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More Realistic NLO Wishlist
Les Houches 2005

• Bold action required!

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State-of-the-Art NLO QCD
Five point is still state-of-the art for QCD
cross-sections
Typical examples
Brute force calculations give GB expressions
numerical stability? Amusing numbers 6g 10,860
diagrams, 7g 168,925 diagrams Much worse
difficulty integral reduction generates nasty
dets.
Grim determinant
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Consider an integral
Evaluate this integral via Passarino-Veltman
reduction. Result is
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Result of performing the integration
Numerical stability is a key issue. Clearly,
there should be a better way
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Tree-level example Five gluons
Consider the five-gluon amplitude
If you evaluate this you find
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Result of evaluation (actually only a small part
of it)
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Why are Feynman diagrams clumsy for loop or
high-multiplicity processes?

• Vertices and propagators involve
• gauge-dependent off-shell states.
• Origin of the complexity.
• To get at root cause of the trouble we must
  rewrite perturbative quantum field theory.

• All steps should be in terms of gauge invariant
• on-shell states. On shell
  formalism.
• Radical rewrite of gauge theory needed.

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• One of the most remarkable discoveries
• in elementary particle physics has been
• that of the existence of the complex
• plane.
• J. Schwinger in Particles, Sources and
  Fields Vol 1

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On-shell Formalisms
With on-shell formalisms we can exploit analytic
properties

• Curiously, a practical on-shell formalism was
  constructed at loop level prior to tree level
  unitarity method.
  (1994)
• Solution at tree-level had to await Wittens
  twistor inspiration.
  (2004)
• -- MHV vertices
• -- On-shell recursion
• Combining unitarity method with on-shell
  recursion gives loop-level on-shell bootstrap.
  (2006)

Bern, Dixon, Dunbar, Kosower
Cachazo, Svrcek Witten Brandhuber, Spence,
Travaglini
Britto, Cachazo, Feng, Witten
Berger, Bern, Dixon, Forde, Kosower
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Spinors
Xu, Zhang and Chang Berends, Kleis and
Causmaeker Gastmans and Wu Gunion and Kunszt
many others
Spinor helicity for gluon polarization vectors
More sophisticated version of circular
polarization
All required properties of circular polarization
satisfied
Changes in reference momentum q equivalent to
on-shell gauge transformations
Graviton polarization tensors are squares of
these
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Reconsider Five Gluon Tree
With spinor helicity
These are color stripped amplitudes
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Twistors
In a remarkable paper Ed Witten demonstrated that
twistor space reveals a hidden structure in
scattering amplitudes.
Link is for N 4 super-Yang-Mills theory, but
at tree level hardly any difference from QCD.
Early work from Nair
Penrose twistor transform
Wittens remarkable twistor-space link
Witten Roiban, Spradlin and Volovich
QCD scattering amplitudes
Topological String Theory
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Amazing Simplicity
Witten conjectured that in twistorspace gauge
theory amplitudes have delta-function support on
curves of degree
Connected picture
Disconnected picture
Witten Roiban, Spradlin and Volovich Cachazo,
Svrcek and Witten Gukov, Motl and Neitzke Bena
Bern and Kosower
Structures imply an amazing simplicity in the
scattering amplitudes. Amplitudes are much much
simpler than anyone imagined.
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A Remarkable Twistor String Formula
The following formula encapsulates the entire
tree-level S-matrix of N 4 super-Yang-Mills
Witten Roiban, Spradlin and Volovich
Integral over the Moduli and curves
Degree d polynomial in the moduli
Strange formula from Feynman diagram viewpoint.
But its true impressive checks by Roiban,
Spradlin and Volovich
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MHV Amplitudes
Parke and Taylor (1984)
At tree level Parke and Taylor conjectured a
very simple form for n-gluon scattering.




Proven by Berends and Giele
Amazingly, this simplicity continues to loops and
to general helicities.
Bern, Dixon, Dunbar, Kosower Cachazo, Svrcek,
Witten Bern, Dixon, Kosower Brandhuber, Spence
and Travaglini
These MHV amplitudes can be thought of as
vertices for building new amplitudes.
Cachazo, Svrcek and Witten
Arbitrary null momentum
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Six gluon example
220 Feynman diagrams
A perfect calculation
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MHV Vertices at Loop Level
There were various arguments why it cant work at
loop level

• Vertices are non-local so unitarity would be
  messed up.
• At loop level the twistor string is polluted by
  conformal supergravity.
• The prescription is unclear on some
  denominators.
• The twistor structure of known loop amplitudes
  seemed confused and complicated. (Confusion due
  to "holomorphic anomaly.)

Berkovits and Witten
Happily these arguments are all wrong.
Bradhuber, Spence and Travaglini had sufficient
faith
Explicit one-loop calculation in N 4, N 1
and QCD demonstrate the you get right answers!
Agreement with BDDK.
Now there is even a Lagrangian derivation which
demystifies it.
Mansfield Gorsky and Rosly
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Twistor Structure at One Loop
At one-loop the coefficients of all integral
functions have beautiful twistor space
interpretations
Twistor space support
Box integral
Three negative helicities
Bern, Dixon and Kosower Britto, Cachazo and Feng
Four negative helicities
The existence of such twistor structures
connected with loop-level simplicity.
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Unitarity Method
Bern, Dixon, Dunbar and Kosower
Two-particle cut
Three- particle cut
Generalized unitarity
As observed by Britto, Cachazo and Feng quadruple
cut freezes integral Coefficients of box
integrals always easy.
Bern, Dixon and Kosower
Generalized cut interpreted as cut propagators
not canceling.
Recent improvements for bubble and triangle
contributions
Britto, Buchbinder, Cachazo and Feng Britto,
Feng and Mastrolia
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Early On-Shell Bootstrap
Bern, Dixon, Kosower (1997)
Early Approach

• Use Unitarity Method with D 4 helicity states.
  Efficient means
• for obtaining logs and polylogs.
• Use factorization properties to find rational
  function contributions.

Key problems preventing widespread applications

• Difficult to find rational functions with
  desired factorization properties.
• Systematization unclear key problem.

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Tree-Level On-Shell Recursion
New representations of tree amplitudes from IR
consistency of one-loop amplitudes in N 4
super-Yang-Mills theory.
Bern, Del Duca, Dixon, Kosower Roiban, Spradlin,
Volovich
Using intuition from twistors and generalized
unitarity

Britto, Cachazo, Feng BCF Witten
An
On-shell conditions maintained by shift.

• Proof relies on so little. Power comes from
  generality
• Cauchys theorem
• Basic field theory factorization properties
• Applies as well to massive theories

Britto, Cachazo, Feng and Witten
Badger, Glover, Khoze and Svrcek
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Loop-Level On-Shell Bootstrap
Berger, Bern, Dixon, Forde and Kosower
Shifted amplitude function of a complex parameter
Shift maintains on-shellness and momentum
conservation
Use unitarity method (in special cases on-shell
recursion)
Use on-shell recursion
Use auxiliary on-shell recursion in another
variable
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Loop-Level Recursion
z

• New Features
• Presence of branch cuts.
• Unreal poles poles appear with complex
  momenta.
• Double poles.
• Spurious singularities that cancel only against
  polylogs.
• Double count between cuts and recursion.

Pure phase for real momenta
On-shell bootstrap is designed to deal with these
issues
Berger, Bern, Dixon, Forde and Kosower
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One-Loop Five-Point Example
The most challenging part was rational function
terms at the end of chain of integral
reductions. Assume we already have log terms
computed from D 4 cuts.
Only one non-vanishing recursive diagram
Tree-like calculations
Only two double-count diagrams
These are computed by taking residues
Rational function terms obtained from tree-like
calculation!
No integral reductions. No unwanted Grim
dets.
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Six-Point Example
Rational function parts of scalar loops were by
far most difficult to calculate.
Using on-shell bootstrap rational parts are
given by tree-like calculations. No integral
reductions.
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Numerical Results for n Gluons
Choose specific points in phase-space see
hep-ph/0604195
Scalar loop contributions
Naive diagram count
6 points
7 points
8 points
3,017,489 other diagrams
Obtained from numerically evaluating analytic
expressions
Modest growth in complexity as number of legs
increases
At 6 points agreement with numerical results of
Ellis, Giele and Zanderighi and analytic results
of Xiao, Yang and Zhu (XYZ)
More work needed to construct program for
cross-sections.
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N 4 Super-Yang-Mills to All Loops
Since t Hoofts paper thirty years ago on the
planar limit of QCD we have dreamed that we can
solve QCD in this limit. This is too hard. N
4 sYM is much more promising.

• Heuristically, we expect magical simplicity in
  the
• scattering amplitude especially in planar limit
  with large
• t Hooft coupling dual to weakly coupled
  gravity in AdS
• space.

Can we solve planar N 4 super-Yang-Mills
theory? Initial Goal Resum amplitudes to all
loop orders.
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N 4 Multi-loop Amplitude
Bern, Rozowsky and Yan
Consider one-loop in N 4
The basic D-dimensional two-particle sewing
equation
Applying this at one-loop gives
Agrees with known result of Green, Schwarz and
Brink.
The two-particle cuts algebra recycles to all
loop orders!
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Loop Iteration of the Amplitude
Four-point one-loop , N 4
amplitude
To check for iteration use evaluation of two-loop
integrals.
Planar contributions
Obtained via unitarity method
Bern, Rozowsky, Yan
Integrals known and involve 4th order
polylogarithms.
V. Smirnov
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Loop Iteration of the Amplitude
The planar four-point two-loop amplitude
undergoes fantastic simplification.
Anastasiou, Bern, Dixon, Kosower
is universal function related to IR singularities
Thus we have succeeded to express two-loop
fourpoint planar amplitude as iteration of
one-loop amplitude.
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Generalization to n Points
Anastasiou, Bern, Dixon, Kosower
Can we guess the n-point result? Expect simple
structure.
Trick use collinear behavior for guess
Bern, Dixon, Kosower
Have calculated two-loop splitting
amplitudes. Following ansatz satisfies all
collinear constraints
Valid for planar MHV amplitudes
Has correct analytic properties. Explicitly
checked for n5
Cachazo, Spradlin and VolovichBern, Czakon,
Kosower, Roiban, Smirnov
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Three-loop Generalization
From unitarity method we get three-loop planar
integrand
Bern, Rozowsky, Yan
Use Mellin-Barnes integration technology and
apply hundreds of harmonic polylog identities
V. Smirnov
Vermaseren and Remiddi
Bern, Dixon, Smirnov
Answer actually does not actually depend on c1
and c2. Five-point calculation would determine
these.
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All legs and All Loop Generalization
Why not be bold and guess scattering amplitudes
for all loops and legs (at least for MHV
amplitudes)?

• Remarkable formula from Magnea and Sterman tells
  us
• IR singularities to all loop orders. Guides
  construction.
• Collinear limits gives us the key analytic
  information, at
• least for MHV amplitudes.

After subtracting IR singularities finite
remainder is
All loop resummation of finite remainder
An unnamed constant
One- loop finite remainder. Complicated function
of kinematic variables

• Soft anomalous dimension
• Or leading twist high spin
• anomalous dimension
• Or cusp anomalous dimension
• Or high moment limit of
• Altarelli-Parisi splitting kernel

Same anomalous dimension recently conjectured by
Eden and Staudacher to all loop orders using
integrability
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Finite Remainder

• All loop resummation of a one-loop amplitude in
  planar limit.
• In QCD this type of function contributes to
  physical quantities such as jet rates.
• IR divergences cancel against similar divergences
  from real emission diagrams.

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Gravity
Propagator in de Donder gauge
Three vertex
About 100 terms in three vertex An
infinite number of other messy vertices. Naïve
conclusion Gravity is a nasty mess.
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Connection of Gravity and Gauge Theory
At tree level Kawai, Lewellen and Tye have
presented a relationship between closed and open
string amplitudes. In field theory limit,
relationship is between gravity and gauge theory
Gravityamplitude
Color stripped gauge theory amplitude
where we have stripped all coupling constants
Full gauge theory amplitude
Holds for any external states. See review
gr-qc/0206071
Progress in gauge theory can be imported into
gravity theories
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Simplicity of Gravity Amplitudes
The KLT relations imply gravity is simpler than
you thought, but actually it is much simpler.

• On-shell recursion works

Brandhuber, spence and Travaglini Cachazo and
Svrcek Bjerrum-Bohr, Dunbar, Ita, Perkins and
Risager

• Perturbative gravity follows solely from three
  vertex.
• Higher-point vertices are irrelevant for
  scattering amplitudes.

• Similar twistor structures exist in gravity as
  in gauge theory.

Witten Bern, Bjerrum-Bohr, Dunbar
Derivative of delta function support
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Divergences in Supergravity
Conventional wisdom states that it impossible to
construct a finite quantum field theory of
gravity.

• Flaw with all previous studies of divergences.
  Rely on powercounting, taking into account only
  supersymmetry.
• We now have a much deeper understanding hidden
  structures,
• dualities, twistors, connection to sYM via
  KLT.
• Perturbative N 8 supergravity inherits its
  property from N 4 sYM.

Conjecture N 8 supergravity is finite.
Suppose we wanted to check this with Feynman
diagrams
First potential divergence is at 5 loops
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UV divergence of Supergravity

• We may use KLT relations in conjunction with the
  unitarity method to check the divergence
  structure of gravity theories.
• Strategy already used to demonstrate that N 8
• sugra is less divergent than previously
  thought.
• First potential divergence will be at least 5
  loops
• not 3 loops as had been previously thought.
• Cancellations exists beyond those of susy.

Bern, Dixon, Dunbar, Rozowsky, and Perelstein
Howe and Stelle
Bern, Dixon, Rozowsky, and Perelstein Bern,
Bjerrum-Bohr and Dunbar
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Summary

• Three examples of recent studies of S matrices
• (a) LHC demands QCD loop calculations.
• (b) Resummation of planar N 4
  super-Yang-Mills amplitudes.
• (c) Conjecture that N 8 supergravity is
  finite, contrary to
• accepted wisdom. Demands explicit
  computations.
• Inspiration from twistor space amazing
  simplicity.
• Key advance on-shell methods unitarity and
  factorization.
• State-of-the art calculations in perturbative
  QCD.
• Explicit conjecture for resumming the planar MHV
  
• amplitudes of N 4 super-Yang-Mills theory to
  all loop orders.

There are a variety of exciting avenues for
further exploring the S-matrices of gauge and
gravity theories.
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Extra Transparancies
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N 4 Super-Yang-Mills and the S-Matrix
People often repeat the statement The S-matrix
of N 4 super-Yang-Mills does not exist
because it is a conformal field theory
While strictly speaking true, it is completely
misleading.

• The origin of this comment has to do with IR
  singularitites.
• However, perturbative QCD has similar issues.
• Use dimension regularization.
• Use Lee-Nauenberg Theorem to guide construction
  of
• IR safe quantities. Analogous to jets in QCD.
  Define
• in terms of a resolution parameter.

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Analytic Reconstruction of S-Matrix
Basic Claim In any massless theory, the on-shell
tree-level amplitudes in D dimensions contain
sufficient information for obtaining all loop
contributions to all orders. Proven technology
state of the art loop calculations, in QCD,
supersymmetric gauge and gravity theories. At
one-loop in susy gauge theories D 4 tree
amplitudes are sufficient called D 4 cut
constructibility. Here we need to use D
dimensional version since we want higher loop
orders
Bern, Dixon, Dunbar and Kosower Bern and Morgan
Bern, Dixon Kosower
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Whats New?

• In the 1960s unitarity and analyticity widely
  used.
• However, not understood how to use unitarity to
  
• reconstruct complete amplitudes with more than
  2
• kinematic variables.

Mandelstam representation Double dispersion
relation Only 2 to 2 processes.
A(s,t )
With unitarity method we can build arbitrary
amplitudes at any loop order from tree amplitude.
Bern, Dixon, Dunbar and Kosower
Unitarity method builds loops from tree
ampitudes.
A(s1 , s2 , s3 , ...)
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Other technical difficulties in the 60s

• Non-convergence of dispersion relations.
• Ambiguities or subtractions in the dispersion
  relations.
• Confusion when massless particles present.
• Inability to reconstruct rational functions with
  no
• branch cuts.

The unitarity method overcomes these difficulties
by (a) Using dimension regularization to make
everything well defined. (b) Bypassing
dispersion relations by writing down
Feynman representations giving both real and
imaginary parts.