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Multi-loop scattering Multi-loop scattering amplitudes in maximally amplitudes in maximally

supersymmetric gauge and supersymmetric gauge and gravity theories.gravity theories.

Twistors, Strings and Scattering AmplitudesTwistors, Strings and Scattering AmplitudesDurhamDurham

August 24, 2007August 24, 2007Zvi Bern, UCLAZvi 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/0610248ZB, 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/0702112ZB, 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]

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OutlineOutline

• 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.

Since the “twistor revolution” of a few years ago we have seen a very significant advance in our ability to compute scattering amplitudes.

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Maximal SupersymmetryMaximal Supersymmetry

• 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.

In this talk we will discuss high loop orders of scatteringamplitudes in maximally supersymmetric gauge and gravity theories.

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Twistors Expose Amazing SimplicityTwistors Expose Amazing Simplicity

Witten conjectured that in twistor–space gauge theoryamplitudes have delta-function support on curves of degree:

Connected picture Disconnected picture

Structures imply an amazing simplicity in the scattering amplitudes.

WittenRoiban, Spradlin and VolovichCachazo, Svrcek and WittenGukov, Motl and NeitzkeBena Bern and Kosower

This simplicity gives us good reason to believe that there is much more structure to uncover at higher loops.

Leads to MHV rules

Penrose twistor transform:

Early work from Nair

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Twistor Structure at One Loop

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

At one-loop the coefficients of all integral functionshave beautiful twistor space interpretations

Twistor space supportBox integral

Bern, Dixon and KosowerBritto, Cachazo and Feng

Three negativehelicities

Four negativehelicities

Complete Amplitudes?? Higher loops??

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Onwards to Loops: Unitarity MethodOnwards to Loops: Unitarity Method

Two-particle cut:

Generalized unitarity:

Three- particle cut:

Generalized cut interpreted as cut propagators not canceling.A number of recent improvements to method

Bern, Dixon, Dunbar and Kosower

Bern, Dixon and Kosower

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 N = 4 Super-Yang-Mills to All Loops= 4 Super-Yang-Mills to All Loops

Can we solve planar N = 4 super-Yang-Mills theory?

Initial Goal: Resum amplitudes to all loop orders.

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.

Talks from Alday, Volovich and Travaglini

As we heard at this conference we are well on our wayto achieving this goal.

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N N = 4 Multi-loop Amplitude= 4 Multi-loop Amplitude

Consider one-loop in N = 4:

The basic D-dimensional two-particle sewing equation

Bern, Rozowsky and Yan

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.

Obtained via unitarity method.

Integrals known and involve 4th order polylogarithms.

Planar contributions.

V. Smirnov

Bern, Rozowsky, Yan

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Loop Iteration of the AmplitudeThe 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

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Generalization to n PointsAnastasiou, Bern, Dixon, Kosower

Can we guess the n-point result? Expect simple structure.Trick: use collinear behavior for guess

Have calculated two-loop splitting amplitudes.Following ansatz satisfies all collinear constraints

Valid for planarMHV amplitudes

D = 4 – 2 Confirmed by direct computation at five points!

Bern, Dixon, Kosower

Cachazo, Spradlin and VolovichBern, Czakon, Kosower, Roiban, Smirnov

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Three-loopThree-loop GeneralizationGeneralizationFrom unitarity method we get three-loop planar integrand:

Bern, Rozowsky, Yan

Use Mellin-Barnes integration technology and applyhundreds of harmonic polylog identities:

V. Smirnov

Bern, Dixon, Smirnov

Answer actually does not actually depend on c1 and c2. Five-point calculation would determine these.

Vermaseren and Remiddi

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All-Leg All-Loop GeneralizationAll-Leg All-Loop GeneralizationWhy 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.

constant

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

One-loop

All loops

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All-loop Resummation in All-loop Resummation in NN = 4 Super-YM Theory = 4 Super-YM Theory

ZB, Dixon, Smirnov

In a beautiful paper Alday and Maldacena confirmed this conjecture at strong coupling from an AdS string computation.

Gives a definite prediction for all values of couplinggiven the Beisert, Eden, Staudacher integral equation for the cusp anomalous dimension.

all-loop resummed amplitude

IR divergences cusp anomalous dimension

finite part of one-loop amplitude

constant independent of kinematics.

See Alday’s and Volovoch’s talks

See Roiban’s talk

For MHV amplitudes:

Wilson loop

Very suggestive link to Wilson loops even at weak couplingDrummond, Korchemsky, Sokatchev ; Brandhuber, Heslop, and Travaglini

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Some Open ProblemsSome 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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• 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

Quantum Gravity at High Loop OrdersQuantum Gravity at High Loop Orders

A key unsolved question is whether a finite point-like quantum gravity theory is possible.

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Grisaru (1977); Tomboulis (1977)

Divergences in GravityDivergences in Gravity

Any supergravity:

is not a valid supersymmetric counterterm.Produces a helicity amplitude forbidden by susy.

Two loop: Pure gravity counterterm has non-zero coefficient:

Goroff, Sagnotti (1986); van de Ven (1992)

One loop:

Pure gravity 1-loop finite (but not with matter)

The first divergence in any supergravity theory can be no earlier than three loops.

Vanish on shell

vanishes by Gauss-Bonnet theorem

‘t Hooft, Veltman (1974)

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Why Why N N = 8 Supergravity?= 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.

A three loop divergence was the widely accepted wisdom coming from the 1980’s.

Potential countertermpredicted by susypower counting

Deser, Kay, Stelle (1977); Kaku, Townsend, van Nieuwenhuizen (1977);Deser and Lindtrom(1980); Kallosh (1981); Howe, Stelle, Townsend (1981).

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Where are the Where are the N N = 8 Divergences? = 8 Divergences?

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.

Depends on who you ask and when you ask.

Note: none of these are based on demonstrating a divergence. They are based on arguing susy protection runs out after some point.

Green, Vanhove, Russo (2006)

Kallosh; Howe and Lindstrom (1981)

ZB, Dixon, Dunbar, Perelstein, and Rozowsky (1998)

Howe and Lindstrom (1981)Green, Schwarz and Brink (1982)Howe and Stelle (1989)Marcus and Sagnotti (1985)

Howe and Stelle (2003)

Howe and Stelle (2003)

Grisaru and Siegel (1982)

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Reasons to Reexamine ThisReasons to Reexamine This1) 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

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Gravity Feynman Rules

An infinite number of other messy vertices

Propagator in de Donder gauge:

Three vertex has about 100 terms:

Gravity looks to be a hopeless mess

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Feynman Diagrams for GravityFeynman Diagrams for Gravity

Suppose we wanted to check superspace claims with Feynman diagrams:

This single diagram has termsprior to evaluating any integrals.More terms than atoms in your brain.

Suppose we want to put an end to the speculations by explicitlycalculating to see what is true and what is false:

In 1998 we suggested that five loops is where the divergence is:

If we attack this directly get terms in diagram. There is a reason why this hasn’t been evaluated previously.

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Basic StrategyBasic Strategy

N = 4 Super-Yang-Mills Tree Amplitudes

KLTN = 8

Supergravity Tree Amplitudes

Unitarity N = 8Supergravity

Loop Amplitudes

ZB, Dixon, Dunbar, Perelsteinand Rozowsky (1998)

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.

Key features of this approach:

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

ZB, Dixon, Dunbar, Kosower (1994)

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KLT RelationsKLT 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

where we have stripped all coupling constants Color stripped gauge theory amplitude

Full gauge theory amplitude

Gravityamplitude

Holds for any external states.See review: gr-qc/0206071

Progress in gauge theory can be importedinto gravity theories

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NN = 8 Power Counting To All Loop Orders = 8 Power Counting To All Loop Orders

No evidence was found that more than 12 powers of loop momenta come out of the integrals.

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

Elementary power counting gave finiteness condition:

From ’98 paper:

counterterm was expected in D = 4, for

Result from ’98 paper

In D = 4 diverges for L 5. L is number of loops.

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Additional Cancellations at One LoopAdditional Cancellations at One Loop

Surprising cancellations not explained by any known susy mechanism are found beyond four points

Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager (2006)

Two derivative coupling means N = 8 should have a worse power counting relative to N = 4 super-Yang-Mills theory.

Two derivative couplingOne derivative coupling

Crucial hint of additional cancellation comes from one loop.

• Cancellations observed in MHV amplitudes.

• “No-triangle hypothesis” — cancellations in all other amplitudes.

• Confirmed by explicit calculations at 6,7 points.ZB, Bjerrum-Bohr and Dunbar (2006)

ZB, Dixon, Perelstein Rozowsky (1999)

One loop

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No-Triangle HypothesisNo-Triangle Hypothesis

• 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.

One-loop D = 4 theorem: Any one loop massless amplitude is a linear combination of scalar box, triangle and bubble integrals with rational coefficients:

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LL-Loop Observation-Loop Observation

From 2 particle cut:

From L-particle cut:

There must be additional cancellation with other contributions!

Above numerator violates no-triangle hypothesis. Too many powers of loop momentum.

numerator factor

numerator factor1

2 3

4

..

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

ZB, Dixon, Roiban

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Complete Three-Loop CalculationComplete Three-Loop Calculation

Besides iterated two-particle cuts need following cuts:

For first cut have:

Use KLT

supergravity super-Yang-Mills

reduces everything to product of tree amplitudes

N = 8 supergravity cuts are sums of products of N = 4 super-Yang-Mills cuts

ZB, Carrasco, Dixon, Johansson, Kosower, Roiban

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Complete three -loop resultComplete three -loop result

All obtainable from iteratedtwo-particle cuts, except(h), (i), which are new.

ZB, Carrasco, Dixon, Johansson, Kosower, Roiban; hep-th/0702112

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Cancellation of Leading BehaviorCancellation of Leading Behavior

To check leading UV behavior we can expand in external momenta keeping only leading term.

Get vacuum type diagrams: Doubledpropagator

Violates NTH Does not violate NTHbut bad power counting

The leading UV behavior cancels!!

After combining contributions:

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Finiteness ConditionsFiniteness Conditions

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

not the weaker result from iterated two-particle cuts:

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

(’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.

“superfinite”in D = 4

finitein D = 4

for L = 3,4

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Origin of Cancellations?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 GravityTree Cancellations in Pure Gravity

You don’t need to look far: proof of BCFW tree-level on-shellrecursion relations in gravity relies on the existence such cancellations!

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

Consider the shifted tree amplitude:

Britto, Cachazo, Feng and Witten;Bedford, Brandhuber, Spence and TravagliniCachazo and Svrcek; Benincasa, Boucher-Veronneau and Cachazo

Proof of BCFW recursion requires

How does behave as ?

Susy not required

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Loop Cancellations in Pure GravityLoop Cancellations in Pure 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!

Powerful new one-loop integration method due to Forde makes it much easier to track the cancellations. Allows us to linkone-loop cancellations to tree-level cancellations.

Observation: Most of the one-loop cancellationsobserved in N = 8 supergravity leading to “no-triangle hypothesis” are already present in non-supersymmetric gravity. Susy cancellations are on top of these.

Cancellation from N = 8 susyCancellation generic to Einstein gravity

Maximum powers ofLoop momenta

nlegs

ZB, Carrasco, Forde, Ita, Johansson

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What needs to be done?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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SummarySummary

• 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 Non-trivial confirmation

1. 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

Method is designed to give same results as Feynman diagrams.Examples:

ZB, Czakon, Dixon, Kosower, Smirnov

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Opinions from the 80’sOpinions from the 80’s

If certain patterns that emerge should persist in the higherorders 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)

We have not shown that a three-loop counterterm is not presentin N > 4, although it is tempting to conjecture that this may be the case. Howe and Lindstrom (1981)

Widespread agreement that N = 8 supergravity should diverge at 3 loops

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Comments on Higher LoopsComments on Higher LoopsRule 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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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.

With unitarity method we can build arbitrary amplitudesat any loop order from tree amplitude.

A(s,t )

A(s1 , s2 , s3 , ...)

Mandelstam representationDouble dispersion relationOnly 2 to 2 processes.

Unitarity method builds loops from tree ampitudes.

Bern, Dixon, Dunbar and Kosower

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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.