1. Trees in N=8 SUGRA and Loops in N=4 SYM
Congkao Wen
based in part on
J.Drummond, M.Spradlin, and A.Volovich,
, ,
Durham University
April 2nd, 2009 1

2. Outline N=8 Supergravity: N=4 Super Yang-Mills:
Review of tree level S-matrix in N=4 SYM.
Drummond, Henn (2008)
Tree level S-matrix in N=8 SUGRA.
Drummond, Spradlin, Volovich, CW (2009)
New understanding for MHV gravity amplitude.
Spradlin, Volovich, CW(2008)
N=4 Super Yang-Mills:
All loop BDS ansatz and leading singularity method.
Anastasiou,Bern,Dixon,Kosower(2003), Bern,Dixon,Smirnov(2005)
Cachazo, Skinner(2008), Cachazo(2008), Cachazo, Spradlin, Volovich(2008)
Three-loop five-point amplitude. Spradlin, Volovich, CW (2008) 2

3. N=8 super gravity
We try to solve the simplest problem of the “simplest quantum field theory”.
With two definitions:
The simplest, but very nontrivial, problem of a quantum
field theory is the all tree level scattering amplitude.
The simplest quantum field theory is N=8 SUGRA.
Arkani-Hamed, Cachazo, Kaplan(2008) 3

4. The simplest problem is not really simple:
Look at the literature:
There are a few different formulas for MHV amplitude. The first one was conjectured by Berend, Giele and Kuijf 20 years ago, which is far more complicated than Parke-Taylor formula of N=4 SYM.
Beyond MHV, the situation is of course worse.
There is no general formula.
In principle, KLT relation may be used to find the desired amplitude. 4

5. Basic tool  BCFW recursion relations
Britto, Cachazo, Feng & Witten An
Ak+1
An-k+1
Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs,
and with momenta shifted by a complex amount.
Having a recursion is great, having a solution is even better!

6. All tree-level amplitudes In N=4 SYM
Drummond, Henn
Use supersymmetric version of BCFW recursion relations and dual superconformal symmetry as guide.
All tree-level solution:
is a dual conformal invariant factor.
Drummond,Henn, Korchemsky,Sokatchev (2008)
Split helicity gluon amplitude ( ) was solved by Britto, Feng, Roiban, Spradlin, Volovich(2005)

7. Definition of where the chiral spinor is given by
For the dual coordinates we use 7

8. Examples
NMHV:
NNMHV:
NNNMHV:

9. The tree-level tree

10. Tree level in N=8 SUGRA Our results for N=8 SUGRA:
Drummond,Spradlin,Volovich,CW,0901,2363
Now we want to solve the BCFW recursion relation for N=8 SUGRA.
Our results for N=8 SUGRA:
‘Gravity factor’ G is not dual conformal invariant.
We find the “KLT relation” from BCFW.

11. Basic Ideas Introduce an ordered ‘gravity subamplitude’ Difficulties:
Gravity amplitudes are not color ordered. Actually there is no such
a thing called color.
There is no dual superconformal invariant to guide us.
Introduce an ordered ‘gravity subamplitude’
Then we find the physic amplitude
via
(Supersymmetry is important here)

12. A simple proof(assuming it’s true for n-1)12

13. The subamplitude can be calculated by BCFW relations!
The extra is the reason for gravity factor G.
Now, the major job is to find the gravity factor G.

14. Examples
Actually, the MHV amplitude in this form was found by Elvang & Freedman:
(Elvang, Freedman (2007))
5-point NMHV: 14

15. For NMHV, we found in 0901.2363 that
with
where
and

16. General formulas for , which will be important beyond NMHV level.
NNMHV: 16

17. with:
and
Finally

18. Numerical check on 6-point NMHV amplitude
We have checked numerically our expression agrees with other representations in literatures, see for example:
Cachazo, Svrcek (2005), Bianchi, Elvang, & Freedman(2008) 18

19. Summary We find a solution to the recursion of the form
and a specific procedure for writing down
any gravity factor G.
The expressions we have found can certainly be used in loop
supergravity amplitudes by applying generalized unitarity technique
in a manifestly supersymmetric way.
Bern,Dixon,Dunbar,Kosower(1994)
Arkani-Hamed,Cachazo,Kaplan(2008)
Brandhuber, Spence, & Travaglini(2008)
Drummond,Henn,Korchemsky,Sokatchev(2008)
Elvang,Freedman,Kiermaier(2008)

20. MHV Revisited
Spradlin,Volovich,CW,
There are a few different MHV formulas in the literatures:
BGK formula conjectured by Berend, Giele and Kuijf 20 years ago.
BBST formula from BCFW relations.
Elvang & Freedman formula1 from BCFW relations,
EF formula2 conjectured by Elvang & Freedman, and equivalent
to one by Bern, Carrasco, Forde, Ita & Johansson.
5. Mason & Skinner formula from twistor string theory.
Berend, Giele, & Kuijf (2007)
Bedford, Brandhuber, Spence, & Travaglini(2005)
Elvang, Freedman(2007)
Bern, Carrasco, Forde, Ita & Johansson(2007)
Mason,Skinner(2008)

21. Divide all the formulas into two classes:
(n-2)!-term formulas:
BBST formula
Elvang & Freedman formula1 == ??
(n-3)!-term formulas:
BGK formula
Elvang & Freedman formula2
Mason & Skinner formula
Of course, we can check numerically that all these formulas agree,
but we would like an analytic understanding.

22. Bonus relation M(z) ～ 1/z^2 leads to one more relation:
Benincasa,Boucher-Veronneau,Cachazo(2007)
Arkani-Hamed, Kaplan(2008)
Arkani-Hamed, Cachazo, Kaplan(2008)
Besides BCFW recursion relation, we also have the bonus relation,
Use the second equation, we can delete M_3 in the first equation.
Apply this recursively, we can reduce (n-2)!-term formulas to
(n-3)!-term formulas.

23. The Results
Using the bonus relations, we analytically proved all five MHV formulas satisfy BCFW recursion relations, namely all of them are equivalent. As a byproduct, we also found a new MHV formula by simplifying (n-2)!-term BBST formula to a (n-3)!-term formula.

24. Loops in N=4 SYM
There are many remarkable properties of N=4 SYM scattering amplitude have been discovered.
One of the discoveries will be relevant to our discussion
is the BDS ansatz.
Anastasiou,Bern,Dixon,Kosower(2003), Bern,Dixon,Smirnov(2005)
The ansatz needs to be modified beyond five points.
Alday, Maldacena(2007),
Bern, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich(2008)
Drummond, Henn, Korchemsky, Sokachev(2008)

25. Three-loop five-point amplitude
Spradlin, Volovich, CW,
BDS relation:
Use leading singularity method to determine integral coefficients,
then check with BDS relation to determine N1 & N2 numerically. 25 25

26. Leading singularities
Cachazo, Skinner(2008)
Cachazo(2008)
Cachazo, Spradlin, Volovich(2008)
The leading singularity method:
a natural basis of integral is provided.
the coefficients are determined by solving simple linear equations.
and the linear equations are easy to write down by hand. (for MHV
at least)
Leading singularity method is refinement of
generalized unitarity (Bern, Dixon, Dunbar, Kosower), (Britto, Cachazo, Feng) and maximal cuts (Bern, Carrasco, Johansson, Kosower) Basic idea: Feynman diagrams possess singularity which must be
reproduced by any representation of the amplitude in terms of simpler
integrals. For the maximal supersymmetric theory, it is conjectured that
the whole amplitude is determined by the leading singularity. 26 26

27. All possible topologies27 27

28. Collapse and expansion rules
Collapse rule
Expansion rule

29. Reduce loops to tree Example:
Natural integrals related to this topology:
L’s are coefficients of integrals

30. Linear equations of the coefficients
Example:
Solutions of r :
We can simply solve it to get the coefficients.
Actually, there are more equations than variables.
strong consistency checks!

31. Then we numerically evaluate the following nine
dual conformal integrals:
and check with BDS relation, we find
Remark: We have only evaluated obstruction P, not full amplitude.
Cachazo, Spradlin, Volovich(2006)

32. Summary
An algorithm for writing down explicit formulas for all tree amplitudes in N=8 super gravity.
All gravity MHV formulas are equivalent and a new MHV formula.
Full coefficients, both parity even and parity odd parts, of three-loop five-point amplitude in N=4 SYM are found by using leading sigularity. 32

33. Thanks!