Multi-loop scattering amplitudes in maximally supersymmetric gauge and gravity theories.

1
Multi-loop scattering amplitudes in maximally
supersymmetric gauge and gravity theories.

• Twistors, Strings and Scattering Amplitudes
• DurhamAugust 24, 2007
• Zvi Bern, UCLA

C. Anastasiou, ZB, L. Dixon, and D. Kosower,
hep-th/0309040 ZB, L. Dixon and V. Smirnov,
hep-th/0505205 ZB, M. Czakon, L. Dixon, D.
Kosower and V. Smirnov, hep-th/0610248 ZB, N.E.J.
Bjerrum-Bohr, D. Dunbar, hep-th/0501137 ZB, L.
Dixon , R. Roiban, hep-th/0611086 ZB, J.J.
Carrasco, L. Dixon, H. Johansson, D. Kosower, R.
Roiban, hep-th/0702112 ZB, J.J. Carrasco, H.
Johansson , D. Kosower arXiv0705.1864 hep-th
ZB, J.J. Carrasco, D. Forde, H. Ita and H.
Johansson, arXiv0707.1035 hep-th
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Outline
Since the twistor revolution of a few years ago
we have seen a very significant advance in our
ability to compute scattering amplitudes.

• QCD multi-parton scattering for LHC not
  discussed here.
• Supersymmetric gauge theory resummation of
  planar N 4 super-Yang-Mills scattering
  amplitudes to all loop orders.
• Quantum gravity reexamination of standard wisdom
  on ultraviolet properties of quantum gravity.

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Maximal Supersymmetry
In this talk we will discuss high loop orders of
scattering amplitudes in maximally supersymmetric
gauge and gravity theories.

• N 4 super-Yang-Mills theory is most promising
• D 4 gauge theory that we will likely be
  able to solve completely.
• Maximally supersymmetric gravity theory is the
  most promising theory which may be UV finite.
• Scattering amplitudes provide a powerful way to
  explore and confirm the AdS/CFT correspondence.

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Twistors Expose Amazing Simplicity
Penrose twistor transform
Early work from Nair
Witten conjectured that in twistorspace gauge
theory amplitudes have delta-function support on
curves of degree
Leads to MHV rules
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.
This simplicity gives us good reason to believe
that there is much more structure to uncover at
higher loops.
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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.
Complete Amplitudes?? Higher loops??
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Onwards to Loops Unitarity Method
Bern, Dixon, Dunbar and Kosower
Two-particle cut
Three- particle cut
Generalized unitarity
Bern, Dixon and Kosower
Generalized cut interpreted as cut propagators
not canceling.
A number of recent improvements to method
Bern, Dixon and Kosower Britto, Buchbinder,
Cachazo and Feng Berger, Bern, Dixon, Forde and
Kosower Britto, Feng and Mastrolia
Anastasiou, Britto, Feng Kunszt, Mastrolia ZB,
Carasco, Johanson, Kosower Forde
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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.
• Need on-shell formalism.

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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 of solving
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.
As we heard at this conference we are well on our
way to achieving this goal.
Talks from Alday, Volovich and Travaglini
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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.
Confirmation directly on integrands.
Cachazo, Spradlin and Volovich
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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
Confirmed by direct computation at five points!
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-Leg All-Loop Generalization
Why not be bold and guess scattering amplitudes
for all loop and all legs (at least for MHV
amplitudes)?

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

One-loop
constant
All loops

• Soft anomalous dimension
• Or leading twist high spin anomalous dimension
• Or cusp anomalous dimension

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All-loop Resummation in N 4 Super-YM Theory
ZB, Dixon, Smirnov
For MHV amplitudes
constant independent of kinematics.

IR divergences
cusp anomalous dimension
finite part of one-loop amplitude
all-loop resummed amplitude
Wilson loop
Gives a definite prediction for all values of
coupling given the Beisert, Eden, Staudacher
integral equation for the cusp anomalous
dimension.
See Roibans talk
In a beautiful paper Alday and Maldacena
confirmed this conjecture at strong coupling
from an AdS string computation.
See Aldays and Volovochs talks
Very suggestive link to Wilson loops even at weak
coupling
Drummond, Korchemsky, Sokatchev Brandhuber,
Heslop, and Travaglini
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Some Open Problems

• What is the multi-loop structure of non-MHV
  amplitudes?
• Very likely there is an iteration and
  exponentiation. Input from
• strong coupling would be very helpful.
• What is precice multi-loop twistor-space
  structure? Knowing
• this would be extremely helpful.
• What is the twistor-space structure at strong
  coupling?
• Is there an iteration and exponentiation for
  subleading color?
• What happens for CFT cases with less susy?
• Can we extract the spectrum of physical states
  from the
• exponentiated scattering amplitudes?

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Quantum Gravity at High Loop Orders
A key unsolved question is whether a finite
point-like quantum gravity theory is possible.

• Gravity is non-renormalizable by power counting.
  
• Every loop gains mass
  dimension - 2.
• At each loop order potential counterterm gains
  extra
• As loop order increases potential counterterms
  must have
• either more Rs or more derivatives

Dimensionful coupling
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Divergences in Gravity
Vanish on shell
One loop Pure gravity 1-loop finite (but not
with matter)
vanishes by Gauss-Bonnet theorem
t Hooft, Veltman (1974)
Two loop Pure gravity counterterm has non-zero
coefficient
Goroff, Sagnotti (1986) van de Ven (1992)
Grisaru (1977) Tomboulis (1977)
The first divergence in any supergravity theory
can be no earlier than three loops.
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Why N 8 Supergravity?

• UV finiteness of N 8 supergravity would imply
  a new
• symmetry or non-trivial dynamical mechanism.
• The discovery of either would have a fundamental
  impact on
• our understanding of gravity.
• High degree of supersymmetry makes this the most
  promising
• theory to investigate.
• By N 8 we mean ungauged Cremmer-Julia
  supergravity.

No known superspace or supersymmetry argument
prevents divergences from appearing at some loop
order.
Potential counterterm predicted by susy power
counting
Deser, Kay, Stelle (1977) Kaku, Townsend, van
Nieuwenhuizen (1977) Deser and Lindtrom(1980)
Kallosh (1981) Howe, Stelle, Townsend (1981).
A three loop divergence was the widely accepted
wisdom coming from the 1980s.
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Where are the N 8 Divergences?
Depends on who you ask and when you ask.
Howe and Lindstrom (1981) Green, Schwarz and
Brink (1982) Howe and Stelle (1989) Marcus and
Sagnotti (1985)
3 loops Conventional superspace power counting.
5 loops Partial analysis of unitarity cuts.
If harmonic superspace with N 6
susy manifest exists 6 loops If harmonic
superspace with N 7 susy manifest exists 7
loops If a superspace with N 8 susy manifest
were to exist. 8 loops Explicit identification
of potential susy invariant counterterm
with full non-linear susy. 9 loops Assume
Berkovits superstring non-renormalization
theorems can be naively carried over to N
8 supergravity. Naïve
extrapolation from 6 loops needed.
ZB, Dixon, Dunbar, Perelstein, and Rozowsky
(1998)
Howe and Stelle (2003)
Howe and Stelle (2003)
Grisaru and Siegel (1982)
Kallosh Howe and Lindstrom (1981)
Green, Vanhove, Russo (2006)
Note none of these are based on demonstrating a
divergence. They are based on arguing susy
protection runs out after some point.
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Reasons to Reexamine This

• The number of established counterterms in any
  supergravity
• theory is zero.
• 2) Discovery of remarkable cancellations at 1
  loop
• the no-triangle hypothesis. ZB, Dixon,
  Perelstein, Rozowsky
• ZB, Bjerrum-Bohr and Dunbar
  Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager
• 3) Every explicit loop calculation to date finds
  N 8 supergravity
• has identical power counting as in N 4
  super-Yang-Mills theory,
• which is UV finite. Green, Schwarz and
  Brink ZB, Dixon, Dunbar, Perelstein, Rozowsky
• Bjerrum-Bohr, Dunbar, Ita, PerkinsRisager
  ZB, Carrasco, Dixon, Johanson, Kosower, Roiban.
• 4) Very interesting hint from string dualities.
  Chalmers Green, Vanhove, Russo
• Dualities restrict form of effective
  action. May prevent
• divergences from appearing in D 4
  supergravity.
• Difficulties with decoupling of towers
  of massive states.
• 5) Gravity twistor-space structure similar to
  gravity.
• Derivative of delta function support

See Greens talk
Witten ZB, Bjerrum-Bohr, Dunbar
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Gravity Feynman Rules
Propagator in de Donder gauge
Three vertex has about 100 terms
An infinite number of other messy vertices
Gravity looks to be a hopeless mess
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Feynman Diagrams for Gravity
Suppose we want to put an end to the speculations
by explicitly calculating to see what is true and
what is false
Suppose we wanted to check superspace claims with
Feynman diagrams
If we attack this directly get terms in diagram.
There is a reason why this hasnt been evaluated
previously.
In 1998 we suggested that five loops is where
the divergence is
This single diagram has terms prior
to evaluating any integrals. More terms than
atoms in your brain.
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Basic Strategy
ZB, Dixon, Dunbar, Perelstein and Rozowsky (1998)
N 8 Supergravity Tree Amplitudes
Unitarity
N 4 Super-Yang-Mills Tree Amplitudes
KLT
Divergences

• Kawai-Lewellen-Tye relations sum of products of
  gauge
• theory tree amplitudes gives gravity tree
  amplitudes.
• Unitarity method efficient formalism for
  perturbatively
• quantizing gauge and gravity theories. Loop
  amplitudes
• from tree amplitudes.

ZB, Dixon, Dunbar, Kosower (1994)
Key features of this approach

• Gravity calculations mapped into much simpler
  gauge
• theory calculations.
• Only on-shell states appear.

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KLT Relations
At tree level Kawai, Lewellen and Tye 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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N 8 Power Counting To All Loop Orders
From 98 paper

• Assumed iterated 2 particle cuts give
• the generic UV behavior.
• Assumed no cancellations with other
• uncalculated terms.

No evidence was found that more than 12 powers
of loop momenta come out of the integrals.
Result from 98 paper
Elementary power counting gave finiteness
condition
counterterm was expected in D 4, for
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Additional Cancellations at One Loop
Crucial hint of additional cancellation comes
from one loop.
Surprising cancellations not explained by any
known susy mechanism are found beyond four points
One derivative coupling
Two derivative coupling
One loop
Two derivative coupling means N 8 should have a
worse power counting relative to N 4
super-Yang-Mills theory.

• Cancellations observed in MHV amplitudes.
• No-triangle hypothesis cancellations in all
  other amplitudes.
• Confirmed by explicit calculations at 6,7 points.

ZB, Dixon, Perelstein Rozowsky (1999)
ZB, Bjerrum-Bohr and Dunbar (2006)
Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager
(2006)
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No-Triangle Hypothesis
One-loop D 4 theorem Any one loop massless
amplitude is a linear combination of scalar box,
triangle and bubble integrals with rational
coefficients

• In N 4 Yang-Mills only box integrals appear.
  No
• triangle integrals and no bubble integrals.
• The no-triangle hypothesis is the statement
  that
• same holds in N 8 supergravity.

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L-Loop Observation
ZB, Dixon, Roiban
numerator factor
From 2 particle cut
1 in N 4 YM
Using generalized unitarity and no-triangle
hypothesis all one-loop subamplitudes should
have power counting of N 4 Yang-Mills
numerator factor
From L-particle cut
Above numerator violates no-triangle hypothesis.
Too many powers of loop momentum.
There must be additional cancellation with other
contributions!
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Complete Three-Loop Calculation
ZB, Carrasco, Dixon, Johansson, Kosower,
Roiban
Besides iterated two-particle cuts need following
cuts
reduces everything to product of tree amplitudes
For first cut have
Use KLT
supergravity
super-Yang-Mills
N 8 supergravity cuts are sums of products of
N 4 super-Yang-Mills cuts
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Complete three -loop result
ZB, Carrasco, Dixon, Johansson, Kosower,
Roiban hep-th/0702112
All obtainable from iterated two-particle cuts,
except (h), (i), which are new.
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Cancellation of Leading Behavior
To check leading UV behavior we can expand in
external momenta keeping only leading term.
Get vacuum type diagrams
Doubled propagator
Violates NTH
Does not violate NTH but bad power counting
After combining contributions
The leading UV behavior cancels!!
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Finiteness Conditions
Through L 3 loops the correct finiteness
condition is (L gt 1)
superfinite in D 4

• same as N 4 super-Yang-Mills
• bound saturated at L 3

not the weaker result from iterated two-particle
cuts
finite in D 4 for L 3,4
(98 prediction)
Beyond L 3, as already explained, from special
cuts we have good reason to believe that the
cancellations continue.
All one-loop subamplitudes should have same UV
power-counting as N 4 super-Yang-Mills theory.
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Origin of Cancellations?
There does not appear to be a supersymmetry
explanation for observed cancellations,
especially if they hold to all loop orders, as we
have argued.
If it is not supersymmetry what might it be?
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Tree Cancellations in Pure Gravity
Unitarity method implies all loop cancellations
come from tree amplitudes. Can we find tree
cancellations?
You dont need to look far proof of BCFW
tree-level on-shell recursion relations in
gravity relies on the existence such
cancellations!
Britto, Cachazo, Feng and Witten Bedford,
Brandhuber, Spence and Travaglini Cachazo and
Svrcek Benincasa, Boucher-Veronneau and Cachazo
Susy not required
Consider the shifted tree amplitude
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Loop Cancellations in Pure Gravity
ZB, Carrasco, Forde, Ita, Johansson
Powerful new one-loop integration method due to
Forde makes it much easier to track the
cancellations. Allows us to link one-loop
cancellations to tree-level cancellations.
Observation Most of the one-loop
cancellations observed in N 8 supergravity
leading to no-triangle hypothesis are already
present in non-supersymmetric gravity. Susy
cancellations are on top of these.
Maximum powers of Loop momenta
Cancellation from N 8 susy
Cancellation generic to Einstein gravity
Proposal This continues to higher loops, so
that most of the observed N 8 multi-loop
cancellations are not due to susy but in fact are
generic to gravity theories!
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What needs to be done?

• N 8 four-loop computation. Can we demonstrate
  that four-
• loop N 8 amplitude has the same UV power
  counting as
• N 4 super-Yang-Mills? Certainly feasible.
• Can we construct a proof of perturbative UV
  finiteness of N 8?
• Perhaps possible using unitarity method
  formalism is recursive.
• Investigate higher-loop pure gravity power
  counting to
• study cancellations. (It does diverge.)
  Goroff and Sagnotti van de Ven
• Twistor structure of gravity loop amplitudes?
  Bern, Bjerrum-Bohr, Dunbar
• Link to a twistor string description of N 8?
  Abou-Zeid, Hull, Mason
• Can we find other examples with less susy that
  may be finite?
• Guess N 6 supergravity theories will be
  perturbatively finite.

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Summary

• Twistor-space structures tell us gauge theory
  and gravity
• amplitudes are much simpler than previously
  anticipated.
• Unitarity method gives us a powerful means for
  constructing
• multi-loop amplitudes.
• Resummation of N 4 MHV sYM amplitudes match
  to
• strong coupling!
• At four points through three loops, established
  N 8 supergravity
• has same power counting as N 4 Yang-Mills.
• Proposed that most of the observed N 8
  cancellations are
• present in generic gravity theories, with
  susy cancellations
• on top of these.

• N 8 supergravity may be the first example of a
  unitary point-like
• perturbatively UV finite theory of quantum
  gravity in D 4.
• Proof is an open
  challenge.

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Extra Transparencies
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Non-trivial confirmation
Method is designed to give same results as
Feynman diagrams. Examples

• Two-loop-four gluon QCD amplitudes
  ZB, De Frietas, Dixon
• matches results of Glover, Oleari and
  Tejeda-Yeomans
• 2. Four-loop cusp anomalous dimension in N 4
  sYM
• Beisert, Eden and Staudacher equation
  matches result. see Beiserts talk
• 3. Resummation of 1,2,3 loop calculations to all
  loop order in
• N 4 super-Yang-Mills theory
  Anastasiou, ZB, Dixon, Kosower
• 
  
  ZB, Dixon, Smirnov
• matches the recent beautiful strong
  coupling construction of
• Alday and Maldacena.
  see Aldays
  talk

ZB, Czakon, Dixon, Kosower, Smirnov
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Opinions from the 80s
We have not shown that a three-loop counterterm
is not present in N gt 4, although it is tempting
to conjecture that this may be the case.

Howe and Lindstrom (1981)
If certain patterns that emerge should persist in
the higher orders of perturbation theory, then
N 8 supergravity in four dimensions would have
ultraviolet divergences starting at three loops.
Green, Schwarz, Brink (1982)
Unfortunately, in the absence of further
mechanisms for cancellation, the analogous N 8
D 4 supergravity theory would seem set to
diverge at the three-loop order.
Howe, Stelle (1984)
There are no miracles It is therefore very
likely that all supergravity theories will
diverge at three loops in four dimensions. The
final word on these issues may have to await
further explicit calculations.
Marcus, Sagnotti (1985)
Widespread agreement that N 8 supergravity
should diverge at 3 loops
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Comments on Higher Loops
Rule of thumb If we can compute N 4
Yang-Mills to a given order we can do the same
for N 8 supergravity.
We obtained the planar N 4 YM amplitude at 5
loops
ZB, Carrasco, Johansson, Kosower
Origin of claim that even 5 loop N 8
supergravity is feasible.
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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.