1. Multi-Loop Amplitudes with Maximal Supersymmetry
Lance Dixon (SLAC)
work with Z. Bern, J.J Carrasco, H. Johansson, R. Roiban,
to appear
International workshop on gauge and string amplitudes
 IPPP Durham, 30 March 2009

2. Why maximal supersymmetry?
Two maximally supersymmetric theories in D=4:
N=4 super-Yang-Mills theory
N=8 supergravity
Each of interest in its own right:
N=4 conformal, related to gravity by AdS/CFT
N=8 amazingly well-behaved in ultraviolet, for a point-particle theory of gravity
The two theories are very closely linked:
Tree-level KLT (and more recent) relations
KLT leads to relations between multi-loop amplitudes
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

3. L. Dixon Multi-loop Amp's with Max. SUSY
vs spectra
28 = 256 massless states, ~ expansion of (x+y)8
SUSY
24 = 16 states
~ expansion of (x+y)4
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

4. Why multi-loop amplitudes?
N=4 super-Yang-Mills theory:
AdS/CFT connection – especially strong in planar (large Nc) limit  classical strings/gravity
Integrability (connection to amplitudes could be strengthened)
Infrared singularities resemble QCD, give insight into IR structure there
Interesting patterns emerge
pseudoconformality of (planar) loop integrals
dual superconformal invariance (in planar limit)
homogeneous maximal transcendentality in D = 4 – 2e;
still not well understood; also extends to nonplanar terms
color structure of UV poles
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

5. Why multi-loop amplitudes?
N=8 supergravity:
Possibility of ultraviolet finiteness
Study infrared singularities in gravity
Interpret what “ gravity ~ (gauge theory)2 ” means
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

6. Gravity ~ (gauge theory)2 at 1 and 2 Loops
Green, Schwarz, Brink;
Grisaru, Siegel (1981)
“color dresses kinematics”
Bern, Rozowsky, Yan (1997); Bern, LD, Dunbar, Perelstein, Rozowsky (1998)
2 loops:
N=8 supergravity: just remove color, square prefactors!
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

7. Kawai-Lewellen-Tye relations
KLT, 1986
Derive from relation between
open & closed string amplitudes.
Low-energy limit gives N=8 supergravity amplitudes as quadratic combinations of N=4 SYM amplitudes ,
consistent with product structure of Fock space,
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

8. More recent tree relations
Britto, Cachazo, Feng,
hep-th/ ;
BCFW, hep-th/
Elvang, Freedman,
Drummond, Spradlin, Volovich, Wen,
Derived from BCFW recursion relations.
Also gives N=8 supergravity amplitudes as
quadratic combinations of (pieces of) N=4 SYM amplitudes,
consistent with
Dual superconformal
invariants – e.g.
Drummond, Henn,
Dual superconformal
invariants –
Drummond, Henn,
where N=4 trees are
Dual superconformal
invariants –
Drummond, Henn,
Dual superconformal
invariants –
Drummond, Henn,
Dual superconformal
invariants –
Drummond, Henn,
Dual superconformal
invariants –
Drummond, Henn,
Dual superconformal
invariants –
Drummond, Henn,
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

9. Comparison of KLT and DSVW
Both give N=8 supergravity amplitudes as
quadratic combinations of (pieces of) N=4 SYM amplitudes.
Both respect
(very useful for exploiting at higher loops)
KLT has different permutations appearing,
[EF ] DSVW always has same permutation in a given term
KLT has “gravity factors” built out of Lorentz dot products
DSVW gravity factors are built from spinor strings, such as
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

10. Multi-loop generalized unitarity
Bern, LD, Kosower, hep-ph/ ; Bern, Czakon, LD, Kosower, Smirnov hep-th/ ;
Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/ ; BCJK, ;
Cachazo, Skinner, ; Cachazo, ; Cachazo, Spradlin, Volovich,
Ordinary cuts of multi-loop amplitudes contain loop amplitudes.
But it is very convenient to work with tree amplitudes only.
For example, at 3 loops, one encounters the product of a
5-point tree and a 5-point one-loop amplitude:
Cut 5-point loop amplitude further,
into (4-point tree) x (5-point tree),
in all 3 inequivalent ways:
cut conditions
satisfied by
real momenta
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

11. L. Dixon Multi-loop Amp's with Max. SUSY
Chop further
Allowing for complex cut momenta, one can chop an amplitude entirely into 3-point trees
 maximal cuts or ~ leading singularities
Advantage is that these cuts are maximally simple, yet give an excellent starting point for constructing the full answer.
For example, in planar (leading in Nc) N=4 SYM
they find all terms in the complete answer for 1, 2 and 3 loops
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

12. L. Dixon Multi-loop Amp's with Max. SUSY
Finding missing terms
Maximal cut method:
Allowing one or two propagators
to collapse from each maximal cut,
one obtains near-maximal cuts
These near-maximal cuts are very
useful for analyzing
N=4 SYM (including nonplanar)
and N=8 SUGRA at 3 loops
BCDJKR, BCJK (2007);
Bern, Carrasco, LD, Johansson, Roiban,
Leading singularity method:
Uses consistent behavior with
respect to “hidden singularities”
Cachazo, Skinner; Cachazo (2008)
Recent supersum advances for more complicated cuts
Drummond, Henn, Korchemsky, Sokatchev, ; Arkani-Hamed, Cachazo, Kaplan, ; Elvang, Freedman, Kiermaier, ; Bern, Carrasco, Ita, Johansson, Roiban, to appear
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

13. Multi-loop “KLT copying”
Bern, LD, Dunbar, Perelstein, Rozowsky (1998)
Suppose we know an N=4 SYM amplitude at some loop order – both planar and nonplanar terms.
Then we have “simple” forms for all of its generalized cuts, i.e. products of N=4 SYM trees, already summed over all internal states
The KLT relations let us write the N=8 SUGRA cuts, which are products of N=8 SUGRA trees, summed over all internal states, very simply in terms of
sums of products of two copies of the N=4 SYM cuts.
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

14. Example of KLT copying at 3 loops
Using
it is easy to see that
N=8 SUGRA
N=4 SYM
N=4 SYM
rational function of Lorentz products
of external and cut momenta;
all state sums already performed
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

15. L. Dixon Multi-loop Amp's with Max. SUSY
The rung rule
Many higher-loop contributions to gg  gg scattering
deduced from a simple property of the 2-particle cuts at one loop
Bern, Rozowsky, Yan (1997)
Leads to “rung rule” for easily computing all contributions which can be built by iterating 2-particle cuts
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

16. L. Dixon Multi-loop Amp's with Max. SUSY
Rung rule for gravity
“KLT copying” for 4-point amplitudes is very simple, leads to simple rung rule for N=8 supergravity as well.
BDDPR (1998)
N=4 SYM rung rule
N=8 SUGRA rung rule 2
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

17. L. Dixon Multi-loop Amp's with Max. SUSY
3 loop amplitude
Nine basic integral
topologies
Seven (a-g) were
already known
(2-particle cuts
 rung rule)
BDDPR (1998)
Two new ones (h,i)
have no 2-particle cuts
BCDJKR (2007); BCDJR (2008)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

18. L. Dixon Multi-loop Amp's with Max. SUSY
N=4 numerators at 3 loops
Omit overall
manifestly quadratic in loop momentum
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

19. L. Dixon Multi-loop Amp's with Max. SUSY
N=8 numerators at 3 loops
Omit overall
also manifestly quadratic in loop momentum
BCDJR (2008)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

20. Applications of the (full color) three-loop amplitudes
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

21. L. Dixon Multi-loop Amp's with Max. SUSY
N=8 SUGRA in the UV
N=8 supergravity in higher dimension D still has same critical dimension for UV divergence
as N=4 SYM: Dc = L = 1
Dc = 4 + 6/L L = 2,3
If this persists to all loop orders then
N=8 supergravity would be perturbatively finite.
We can also evaluate the precise 3-loop divergence in Dc = 6 – 2e (mainly to verify that it’s nonzero):
Note homogeneous transcendentality (z3, no 1)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

22. L. Dixon Multi-loop Amp's with Max. SUSY
N=4 SYM in the UV
Determine the full color dependence of the UV divergences in N=4 SYM in the critical dimension.
Bern, Carrasco, LD, Johansson, Roiban, to appear
Dc = (L = 1)
Dc = 4 + 6/L (L = 2,3)
Of interest in light of recent investigations of potential counterterms Bossard, Howe, Stelle,
We have results for general gauge group G. Here we just give the case G = SU(Nc),
in terms of the color structures:
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

23. L. Dixon Multi-loop Amp's with Max. SUSY
N=4 SYM in UV at one loop
Box integral in Dc = 8 - 2e with color factor 
where
Corresponds to counterterms such as
and (no extra derivatives)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

24. L. Dixon Multi-loop Amp's with Max. SUSY
N=4 SYM in UV at two loops
Planar and nonplanar double box integrals in Dc = 7 - 2e easy [BDDPR 1998] with color factors 
“20” terms linked to each other by group theory
Corresponds to counterterms such as
and (two extra derivatives)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

25. N=4 SYM in UV at three loops
UV poles of integrals (e), (f), (g), (i) in Dc = 6 - 2e [BCDJR 2008] with color factors 
Corresponds to type counterterms.
Absence of double-trace terms of form
at L = 3 calls out for explanation.
Curiously, full divergence does not have uniform transcendentality – but each coefficient of Nck does!
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

26. N=4 SYM in IR at three loops
Full color dependence of IR singularities in gauge amplitudes in D = 4 - 2e is dictated by two constants in collinear jet functions,
plus the soft anomalous dimension matrix
Akhoury (1979); Mueller (1979);
Collins (1980); Sen (1981);
Sterman (1987);
Botts, Sterman (1989);
Catani, Trentadue (1989);
Korchemsky (1989);
Magnea, Sterman (1990);
Korchemsky, Marchesini (1992);
Catani (1998);
Sterman, Tejeda-Yeomans (2002)
See talk by Lorenzo Magnea
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

27. N=4 SYM in IR at three loops (cont.)
Constants in jet functions already known in N=4 SYM at three loops, where they are purely planar
Bern, LD, Smirnov, hep-th/
Also equal to the maximal transcendentality part
of the QCD constants!
Kotikov, Lipatov, Onishschenko, Velizhanin, hep-th/
Moch, Vermaseren, Vogt, hep-ph/ , hep-ph/ , hep-ph/
transcendentality (weight): n for pn
n for zn
What’s not completely known at 3 loops (although there are many constraints & conjectures) is the soft anomalous dimension matrix – see talk by Magnea
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

28. N=4 SYM in IR at three loops (cont.)
What is known at three loops is the
matter-dependent part of LD,
The difference between QCD and N=4 SYM is “just matter”
So a 3-loop quantity needed for QCD resummation can be extracted by evaluating, through 1/e, integrals from an N=4 SYM amplitude!
The 2 planar integrals are known, but unfortunately the 7 non-planar integrals are still intractable…
Recent progress though with one fewer leg:
Heinrich, Huber, Kosower, Smirnov, ;
Baikov, Chetyrkin, Smirnov2, Steinhauser,
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

29. N=8 SUGRA in IR at three loops
Similarly, once the (related) integrals are evaluated (even numerically), one could test the exponentiation of IR divergences in gravity, as observed so far at two loops,
Naculich, Nastase, Schnitzer, ;
Brandhuber, Heslop, Nasti, Spence, Travaglini,
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

30. On to four loops (full color N=4 SYM)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

31. The 4 loop 4 point amplitude in N=4 SYM
Scope of the problem illustrated by the number of
cubic 4-point graphs with nonvanishing coefficients
and various topological properties
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

32. Vacuum cubic graphs at 4 loops
To decorate with 4 external legs
cannot generate
a nonvanishing
(no-triangle)
cubic 4-point
graph
only generate
rung rule
topologies
the most complex
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

33. Planar terms all known
Bern, Czakon, LD, Kosower, Smirnov, hep-th/
Planar result has pseudoconformal
invariance, acting on dual or sector
variables xi
No such guide for the nonplanar terms
Drummond, Henn, Smirnov,
Sokatchev, hep-th/
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

34. L. Dixon Multi-loop Amp's with Max. SUSY
Box cut
Bern, Carrasco, Johansson, Kosower,
If the diagram contains a box subdiagram, can use the simplicity of the 1-loop 4-point amplitude to compute the numerator very simply
Planar example:
Only five 4-loop cubic topologies
do not have box subdiagrams.
But there are also “contact terms”
to determine.
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

35. L. Dixon Multi-loop Amp's with Max. SUSY
Twist identity
If the diagram contains a four-point tree subdiagram, can use a Jacobi-like identity to relate it to other diagrams.
Bern, Carrasco, Johansson,
Relate non-planar topologies to planar, etc.
For example, at 3 loops, (i) = (e) – (e)T [ + contact terms ] 2 3 1 4 - =
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

36. Checks on the final result
Lots of different products of MHV tree amplitudes.
NMHV7 * anti-NMHV7 and MHV5 * NMHV6 * anti-MHV5
– evaluated by Elvang, Freedman, Kiermaier,
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

37. Simple (rung rule) diagrams
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

38. (mostly) box cut diagrams
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

39. L. Dixon Multi-loop Amp's with Max. SUSY
Most complex diagrams
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

40. L. Dixon Multi-loop Amp's with Max. SUSY
N=4 in UV at 4 loops
Need UV poles of
4-loop vacuum graphs
(doubled propagators
represented by
blue dots).
By injecting external momentum in right place,
can rewrite as 4-loop propagator integrals
that factorize into product of
-1-loop propagator integral with UV pole
- finite 3-loop propagator integral
Do this in multiple ways
 Either “gluing relations” or cross-check.
Only 3 vacuum
integrals required
Dc = 4 + 6/4 = 11/2
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

41. L. Dixon Multi-loop Amp's with Max. SUSY
N=4 in UV at 4 loops (cont.)
Reduce 3-loop propagator integrals to master integrals using integration by parts (IBP), a la MINCER.
Chetyrkin, Tkachov (1981)
Most nontrivial integral is nonplanar master integral, for which we only have numerical results (obtained using Gegenbauer polynomial x-space technique
Chetyrkin, Tkachov (1981); Bekavac, hep-ph/
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

42. N=4 SYM in UV at four loops
Combining UV poles of integrals with color factors 
Again corresponds to type counterterms.
Absence of double-trace terms at L = 4 again calls out for explanation.
Related to better UV behavior of abelian theories?
Transcendentality = 3 ??
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

43. L. Dixon Multi-loop Amp's with Max. SUSY
Conclusions
Full color 4 point 4 loop amplitude has been computed in N=4 super-Yang-Mills theory, using a combination of rung rule, box cut, twist identity, and ansatze.
UV divergences have been extracted, and compared with results for L = 1,2,3.
Double-trace terms drop out after L = Why?
Next task (in progress) is to compute the 4-point loop amplitude in N=8 supergravity, and inspect its UV behavior. Will it continue to be as well-behaved as N=4 super-Yang-Mills theory?
Wealth of IR information in gauge theory & gravity is also available … once technology is developed for doing non-planar 4-point integrals (even numerically) in D = 4 – 2e at L = 3,4.
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

44. L. Dixon Multi-loop Amp's with Max. SUSY
Extra Slides
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

45. L. Dixon Multi-loop Amp's with Max. SUSY
Vacuum integrals at four loops (before cancellations between different topologies)
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

46. Integrals for planar amplitude at 5 loops
Bern, Carrasco, Johansson, Kosower,
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009

47. Leading transcendentality relation between QCD and N=4 SYM
KLOV (Kotikov, Lipatov, Onishschenko, Velizhanin, hep-th/ )
noticed (at 2 loops) a remarkable relation between kernels for
BFKL evolution (strong rapidity ordering)
DGLAP evolution (pdf evolution = strong collinear ordering)
 includes cusp anomalous dimension
in QCD and N=4 SYM:
Set fermionic color factor CF = CA in the QCD result and
keep only the “leading transcendentality” terms. They coincide with the full N=4 SYM result (even though theories differ by scalars)
Conversely, N=4 SYM results predict pieces of the QCD result
transcendentality (weight): n for pn
n for zn
Similar counting for HPLs and
for related harmonic sums
used to describe DGLAP kernels
at finite j
L. Dixon Multi-loop Amp's with Max. SUSY
Amp's Durham March 2009