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Multi-loop scattering amplitudes in maximally supersymmetric gauge and gravity theories. Twistors, Strings and Scattering Amplitudes Durham August 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

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multi loop scattering amplitudes in maximally supersymmetric gauge and gravity theories

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 arXiv:0705.1864 [hep-th]

ZB, J.J. Carrasco, D. Forde, H. Ita and H. Johansson, arXiv:0707.1035 [hep-th]


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.
maximal supersymmetry
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.
twistors expose amazing simplicity
Twistors Expose Amazing Simplicity

Penrose twistor transform:

Earlywork from Nair

Witten conjectured that in twistor–space gauge theory

amplitudes have delta-function support on curves of degree:

Leads to

MHV rules

Connected picture

Disconnected picture


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.

twistor structure at one loop
Twistor Structure at One Loop

At one-loop the coefficients of all integral functions

have beautiful twistor space interpretations

Box integral

Twistor space support

Three negative


Bern, Dixon and Kosower

Britto, Cachazo and Feng

Four negative


The existence of such twistor structures connected with loop-level simplicity.

Complete Amplitudes?? Higher loops??

onwards to loops unitarity method

Bern, Dixon, Dunbar and Kosower

Onwards to Loops: Unitarity Method

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

why are feynman diagrams clumsy for loop or high multiplicity processes
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.
n 4 super yang mills to all loops
N = 4 Super-Yang-Mills to All Loops

Since ‘t Hooft’s 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

n 4 multi loop amplitude
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!

loop iteration of the amplitude
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

loop iteration of the amplitude1
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 four–point

planar amplitude as iteration of one-loop amplitude.

Confirmation directly on integrands.

Cachazo, Spradlin and Volovich

generalization to n points

D = 4 – 2

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

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.

all leg all loop generalization
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.



All loops

  • Soft anomalous dimension
  • Or leading twist high spin anomalous dimension
  • Or cusp anomalous dimension
all loop resummation in n 4 super ym theory
All-loop Resummation in N = 4 Super-YM Theory

For MHV amplitudes:

ZB, Dixon, Smirnov

constant independent

of kinematics.

IR divergences

finite part of

one-loop amplitude

cusp anomalous


all-loop resummed


Wilson loop

Gives a definite prediction for all values of coupling

given the Beisert, Eden, Staudacher integral

equation for the cusp anomalous dimension.

See Roiban’s talk

In a beautiful paper Alday and Maldacena confirmed this

conjecture at strong coupling from an AdS string computation.

See Alday’s and Volovoch’s talks

Very suggestive link to Wilson loops even at weak coupling

Drummond, Korchemsky, Sokatchev ; Brandhuber, Heslop, and Travaglini


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?

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 R’s or more derivatives

Dimensionful coupling


Any supergravity:

is not a valid supersymmetric counterterm.

Produces a helicity amplitude forbidden by susy.

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.

why n 8 supergravity
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 1980’s.

where are the n 8 divergences
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.

reasons to reexamine this
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 Green’s talk

Witten; ZB, Bjerrum-Bohr, Dunbar

gravity feynman rules
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

feynman diagrams for gravity
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 hasn’t 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.

basic strategy

N = 8


Loop Amplitudes

ZB, Dixon, Dunbar, Perelstein

and Rozowsky (1998)

Basic Strategy

N = 8


Tree Amplitudes


N = 4


Tree Amplitudes



  • 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.
klt relations
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


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


In D = 4 diverges for L 5.

L is number of loops.

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


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


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)


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.
l loop observation






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!


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:




N = 8 supergravity cuts are sums of products of

N = 4 super-Yang-Mills cuts


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.


Cancellation of Leading Behavior

To check leading UV behavior we can expand in external momenta

keeping only leading term.

Get vacuum type diagrams:



Violates NTH

Does not violate NTH

but bad power counting

After combining contributions:

The leading UV behavior cancels!!


Finiteness Conditions

Through L = 3 loops the correct finiteness condition is (L > 1):


in D = 4

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

notthe weaker result from iteratedtwo-particle cuts:


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.


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?


How does behave as


Proof of BCFW recursion requires

Tree Cancellations in Pure Gravity

Unitarity method implies all loop cancellations come from tree amplitudes. Can we find tree cancellations?

You don’t need to look far: proof of BCFW tree-level on-shell

recursion relations in gravity relies on the existence such


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:




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 generic

to Einstein gravity

Cancellation from N = 8 susy

Proposal: This continues to higher loops, so that most of the observed N = 8 multi-loop cancellations are notdue to susy but in fact are generic to gravity theories!


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.


  • 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, establishedN = 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.
non trivial confirmation
Non-trivial confirmation

Method is designed to give same results as Feynman diagrams.


  • 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 Beisert’s 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 Alday’s talk

ZB, Czakon, Dixon,

Kosower, Smirnov


Opinions from the 80’s

We have not shown that a three-loop counterterm is not present

in N > 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

comments on higher loops
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.

what s new
What’s New?
  • In the 1960’s 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 , ...)


Other technical difficulties in the 60’s:

  • 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.