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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
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
In this talk we will discuss high loop orders of scattering
amplitudes in maximally supersymmetric gauge and
D = 4 gauge theory that we will likely be able to solve completely.
Penrose twistor transform:
Earlywork from Nair
Witten conjectured that in twistor–space gauge theory
amplitudes have delta-function support on curves of degree:
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.
At one-loop the coefficients of all integral functions
have beautiful twistor space interpretations
Twistor space support
Bern, Dixon and Kosower
Britto, Cachazo and Feng
The existence of such twistor structures connected with loop-level simplicity.
Complete Amplitudes?? Higher loops??
Three- particle cut:
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
gauge-dependent off-shell states.
Origin of the complexity.
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.
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
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!
Four-point one-loop, N = 4 amplitude:
To check for iteration use evaluation of two-loop integrals.
Obtained via unitarity method.
Bern, Rozowsky, Yan
Integrals known and involve 4th order polylogarithms.
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
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
Confirmed by direct computation
at five points!
Cachazo, Spradlin and VolovichBern, Czakon, Kosower, Roiban, Smirnov
From unitarity method we get three-loop planar integrand:
Bern, Rozowsky, Yan
Use Mellin-Barnes integration technology and apply
hundreds of harmonic polylog identities:
Vermaseren and Remiddi
Bern, Dixon, Smirnov
Answer actually does not actually depend on c1 and c2. Five-point calculation would determine these.
Why not be bold and guess scattering amplitudes for all loop and all legs (at least for MHV amplitudes)?
For MHV amplitudes:
ZB, Dixon, Smirnov
finite part of
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
A key unsolved question is whether a finite point-like quantum gravity theory is possible.
is not a valid supersymmetric counterterm.
Produces a helicity amplitude forbidden by susy.
Divergences in Gravity
Vanish on shell
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.
No known superspace or supersymmetry argument prevents
divergences from appearing at some loop order.
predicted by susy
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.
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.
See Green’s talk
Witten; ZB, Bjerrum-Bohr, Dunbar
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
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.
ZB, Dixon, Dunbar, Perelstein
and Rozowsky (1998)Basic Strategy
N = 8
N = 4
ZB, Dixon, Dunbar, Kosower (1994)
Key features of this approach:
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
L is number of loops.
N = 8 Power Counting To All Loop Orders
From ’98 paper:
No evidence was found that more than 12 powers of
loop momenta come out of the integrals.
Elementary power counting gave finiteness condition:
counterterm was expected in D = 4, for
Crucial hint of additional cancellation comes from one loop.
Surprising cancellations not explained by any known susy
mechanism are found beyond four points
Two derivative coupling
Two derivative coupling means N = 8 should have a worse power counting relative to N = 4 super-Yang-Mills theory.
ZB, Dixon, Perelstein Rozowsky (1999)
ZB, Bjerrum-Bohr and Dunbar (2006)
Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager (2006)
One-loop D = 4 theorem: Any one loop massless amplitude is a linear combination of scalar box, triangle and bubble integrals with rational coefficients:
ZB, Dixon, Roiban
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
From L-particle cut:
Above numerator violates no-triangle hypothesis. Too many powers of loop momentum.
There must be additional cancellation with other contributions!
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
ZB, Carrasco, Dixon, Johansson,
Kosower, Roiban; hep-th/0702112
All obtainable from iterated
two-particle cuts, except
(h), (i), which are new.
To check leading UV behavior we can expand in external momenta
keeping only leading term.
Get vacuum type diagrams:
Does not violate NTH
but bad power counting
After combining contributions:
The leading UV behavior cancels!!
Through L = 3 loops the correct finiteness condition is (L > 1):
in D = 4
notthe weaker result from iteratedtwo-particle cuts:
in D = 4
for L = 3,4
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.
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?
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
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!
Method is designed to give same results as Feynman diagrams.
ZB, Czakon, Dixon,
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
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.
Double dispersion relation
Only 2 to 2 processes.
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 , ...)
The unitarity method overcomes these difficulties by
(a) Using dimension regularization to make everything
(b) Bypassing dispersion relations by writing down
Feynman representations giving both real and