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
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]
1
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.
2
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.
3
Twistors Expose Amazing Simplicity
Penrose twistor transform:
Early work 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
Structures imply an amazing simplicity
in the scattering amplitudes.
Witten
Roiban, Spradlin and Volovich
Cachazo, Svrcek and Witten
Gukov, Motl and Neitzke
Bena Bern and Kosower
This simplicity gives us good reason to believe that there
is much more structure to uncover at higher loops. 4
Twistor Structure at One Loop
At one-loop the coefficients of all integral functions
have beautiful twistor space interpretations
Box integral
Three negative
helicities
Twistor space support
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??
5
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;
6
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?
• 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.
7
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.
8
Talks from Alday, Volovich and Travaglini
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!
9
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
10
Loop Iteration of the Amplitude
The planar four-point two-loop amplitude undergoes
Anastasiou, Bern, Dixon, Kosower
fantastic simplification.
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
11
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!
D=4–2
Cachazo, Spradlin and Volovich
12
Bern, 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
V. Smirnov
Vermaseren and Remiddi
hundreds of harmonic polylog identities:
Bern, Dixon, Smirnov
Answer actually does not actually depend on c1 and c2. Five-point calculation would
determine these.
13
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
All loops
constant
• Soft anomalous dimension
• Or leading twist high spin anomalous dimension
• Or cusp anomalous dimension
14
All-loop Resummation in N = 4 Super-YM Theory
ZB, Dixon, Smirnov
For MHV amplitudes:
constant independent
of kinematics.
all-loop resummed
amplitude
IR divergences
cusp anomalous
dimension
finite part of
one-loop 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 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 15
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?
16
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.
Dimensionful coupling
• 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
17
Divergences in Gravity
One loop:
Vanish on shell
vanishes by Gauss-Bonnet theorem
Pure gravity 1-loop finite (but not with matter)
‘t Hooft, Veltman (1974)
Two loop: Pure gravity counterterm has non-zero coefficient:
Goroff, Sagnotti (1986); van de Ven (1992)
Any supergravity:
is not a valid supersymmetric counterterm.
Produces a helicity amplitude
forbidden by susy.
Grisaru (1977); Tomboulis (1977)
The first divergence in any supergravity theory
can be no earlier than three loops.
18
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
19
from the 1980’s.
Where are the N = 8 Divergences?
Depends on who you ask and when you ask.
3 loops: Conventional superspace power counting.
Howe and Lindstrom (1981)
Green, Schwarz and Brink (1982)
Howe and Stelle (1989)
Marcus and Sagnotti (1985)
Dixon, Dunbar, Perelstein,
5 loops: Partial analysis of unitarity cuts. ZB,
and Rozowsky (1998)
If harmonic superspace with N = 6 susy manifest exists
Howe and Stelle (2003)
6 loops: If harmonic superspace with N = 7 susy manifest exists
Howe and Stelle (2003)
7 loops: If a superspace with N = 8 susy manifest were to exist.
Grisaru and Siegel (1982)
8 loops: Explicit identification of potential susy invariant counterterm
Kallosh; Howe and Lindstrom (1981)
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. 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. 20
Reasons to Reexamine This
1) 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.
See Green’s talk
5) Gravity twistor-space structure similar to gravity.
Derivative of delta function support
Witten; ZB, Bjerrum-Bohr, Dunbar
21
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
22
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.
23
Basic Strategy
N=4
Super-Yang-Mills
Tree Amplitudes
KLT
ZB, Dixon, Dunbar, Perelstein
and Rozowsky (1998)
N =8
N =8
Unitarity
Supergravity
Supergravity
Loop Amplitudes
Tree Amplitudes
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
ZB, Dixon, Dunbar, Kosower (1994)
from tree amplitudes.
Key features of this approach:
• Gravity calculations mapped into much simpler gauge
theory calculations.
• Only on-shell states appear.
24
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
Gravity
amplitude
where we have stripped all coupling constants
Full gauge theory
amplitude
Color stripped gauge
theory amplitude
Holds for any external states.
See review: gr-qc/0206071
Progress in gauge
theory can be imported
into gravity theories
25
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:
In D = 4 diverges for L
L is number of loops.
counterterm was expected in D = 4, for
5.
26
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.
ZB, Dixon, Perelstein Rozowsky (1999)
• “No-triangle hypothesis” — cancellations in all other amplitudes.
ZB, Bjerrum-Bohr and Dunbar (2006)
• Confirmed by explicit calculations at 6,7 points.
Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager (2006) 27
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.
28
L-Loop Observation
2
ZB, Dixon, Roiban
3
..
numerator factor
1
4
From 2 particle cut:
1 in N = 4 YM
numerator factor
From L-particle cut:
Using generalized unitarity and
no-triangle hypothesis all one-loop
subamplitudes should have power
counting of N = 4 Yang-Mills
Above numerator violates no-triangle
hypothesis. Too many powers of loop
momentum.
There must be additional cancellation with other contributions!
29
Complete Three-Loop Calculation
ZB, Carrasco, Dixon,
Johansson, Kosower, Roiban
Besides iterated two-particle cuts need following cuts:
For first cut have:
reduces everything to
product of tree amplitudes
Use KLT
supergravity
super-Yang-Mills
N = 8 supergravity cuts are sums of products of
N = 4 super-Yang-Mills cuts
30
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.
31
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
Doubled
propagator
Does not violate NTH
but bad power counting
After combining contributions:
The leading UV behavior cancels!!
32
Finiteness Conditions
Through L = 3 loops the correct finiteness condition is (L > 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.
33
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?
34
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;
cancellations!
Susy not required
Bedford, Brandhuber, Spence and Travaglini
Cachazo and Svrcek; Benincasa, Boucher-Veronneau and Cachazo
Consider the shifted tree amplitude:
How does
?
behave as
Proof of BCFW recursion requires
35
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.
n
legs
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 not due to susy but
in fact are generic to gravity theories!
36
What needs to be done?
• N = 8 four-loop computation. Can we demonstrate that fourloop 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?
• Link to a twistor string description of N = 8?
Bern, Bjerrum-Bohr, Dunbar
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.
37
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.
38
Extra Transparencies
39
Non-trivial confirmation
Method is designed to give same results as Feynman diagrams.
Examples:
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
ZB, Czakon, Dixon,
Kosower, Smirnov
– 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
40
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
Green, Schwarz, Brink (1982)
starting at three loops.
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
41
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.
42
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.
A(s,t )
Mandelstam representation
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
A(s1 , s2 , s3 , ...)
Unitarity method builds
loops from tree ampitudes.43
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
44
imaginary parts.