1. On-Shell Methods in QCD and N=4 Super-Yang-Mills Theory
Lance Dixon (CERN & SLAC)
DESY Theory Workshop
21 Sept. 2010

2. L. Dixon On-Shell Methods
The S matrix reloaded
Almost everything we know experimentally about gauge theory is based on scattering processes with asymptotic, on-shell states, evaluated in perturbation theory.
Nonperturbative, off-shell information very useful, but in QCD it is often more qualitative (except for lattice).
All perturbative scattering amplitudes can be computed with Feynman diagrams – but that is not necessarily the best way, especially if there is hidden simplicity.
N=4 super-Yang-Mills theory has lots of simplicity, both manifest and hidden. A particularly beautiful application of on-shell methods
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

3. On-shell methods in QCD
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

4. LHC is a multi-jet environment
Every process also comes with one more jet at ~ 1/5 the rate
Understand not only SM production of X but also of
X + n jets
where
X = W, Z, tt, WW,
H, …
n = 1,2,3,…
7 TeV
Need precise
understanding
of “old physics”
that looks like
new physics
new physics?
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

5. Backgrounds to Supersymmetry at LHC
Cascade from gluino to neutralino
(dark matter, escapes detector)
Signal: missing energy + 4 jets
SM background from Z + 4 jets,
Z  neutrinos
Current state of art for Z + 4 jets based on LO tree amplitudes (matched to parton showers)
 normalization still quite uncertain c n
Motivates goal of
 n n
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

6. One-loop QCD amplitudes via Feynman diagrams
For V + n jets (maximum number of external gluons only)
# of jets
# 1-loop Feynman diagrams
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

7. Remembering a Simpler Time...
In the 1960s there was no QCD,
no Lagrangian or Feynman rules
for the strong interactions
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

8. L. Dixon On-Shell Methods
The Analytic S-Matrix
Bootstrap program for strong interactions: Reconstruct scattering amplitudes directly from analytic properties (on-shell information):
Poles
Chew, Mandelstam;
Eden, Landshoff,
Olive, Polkinghorne;
Veneziano;
Virasoro, Shapiro;
… (1960s)
Branch cuts
Analyticity fell out of favor in 1970s with the rise of QCD & Feynman rules
Now resurrected for computing amplitudes for perturbative QCD – as alternative to Feynman diagrams! Important: perturbative information now assists analyticity.
Works even better in theories with lots of SUSY, like N=4 SYM
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

9. Generalized unitarity
Ordinary unitarity:
Im T = T†T
put 2 particles on shell
Generalized unitarity:
put 3 or 4 particles on shell
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

10. One-loop amplitudes reduced to trees
When all external momenta are in D = 4, loop momenta in D = 4-2e
(dimensional regularization), one can write:
Bern, LD, Dunbar, Kosower (1994)
coefficients are all rational functions – determine algebraically
from products of trees using (generalized) unitarity
known scalar one-loop integrals,
same for all amplitudes
rational part
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

11. Generalized Unitarity for Box Coefficients di
Britto, Cachazo, Feng, hep-th/
No. of dimensions = 4 = no. of constraints  discrete solutions (2, labeled by ±)
Easy to code, numerically very stable
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

12. Box coefficients di (cont.)
Solutions simplify (and are more
stable numerically) when all
internal lines massless, at least one
external line (K1) massless:
BH, ; Risager
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

13. – numerical implementation
Unitarity method
– numerical implementation
Each box coefficient uniquely isolated by a “quadruple cut” given simply by a product of 4 tree amplitudes
Britto, Cachazo, Feng,
hep-th/
bubble coefficients come from ordinary double cuts, after removing contributions of boxes and triangles
triangle coefficients come from triple cuts, product of 3 tree amplitudes, but these are also “contaminated” by boxes
Ossola, Papadopolous, Pittau, hep-ph/ ;
Mastrolia, hep-th/ ; Forde, ;
Ellis, Giele, Kunszt, ; Berger et al., ;…
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

14. Triangle coefficients
Forde, ; BH,
Triple cut solution depends on one complex parameter, t
Solves
for suitable definitions of
Box-subtracted
triple cut has poles
only at t = 0, ∞
Triangle coefficient c0
plus all other coefficients cj
obtained by discrete Fourier
projection, sampling at
(2p+1)th roots of unity
Bubble similar
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

15. L. Dixon On-Shell Methods
Several Recent Implementations of On-Shell Methods for 1-Loop Amplitudes
Method for
Rational part:
CutTools: Ossola, Papadopolous, Pittau,
NLO WWW, WWZ, Binoth+OPP,
NLO ttbb, tt + 2 jets Bevilacqua, Czakon,
Papadopoulos, Pittau, Worek, ;
specialized
Feynman
rules
_ _ _
D-dim’l
unitarity
Rocket: Giele, Zanderighi,
Ellis, Giele, Kunszt, Melnikov, Zanderighi,
NLO W + 3 jets in large Nc approx./extrapolation
EMZ, , ; Melnikov, Zanderighi,
D-dim’l
unitarity
+ on-shell
recursion
SAMURAI: Mastrolia, Ossola, Reiter, Tramontano,
Blackhat: Berger, Bern, LD, Febres Cordero, Forde, H. Ita,
D. Kosower, D. Maître; T. Gleisberg,
, , , ,  
+ Sherpa  NLO production of W,Z + 3 (4) jets
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

16. L. Dixon On-Shell Methods
Virtual Corrections
Divide into leading-color terms, such as:
and subleading-color terms, such as:
The latter include many more terms, and are much more time-consuming for computer to evaluate. But they are much smaller (~ 1/30 of total cross section) so evaluate them much less often.
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

17. Recent analytic application: One-loop amplitudes
for a Higgs boson + 4 partons
Unitarity for cut parts, on-shell recursion for rational parts (mostly)
H = f + f†
Badger, Glover, Risager,
Glover, Mastrolia, Williams,
Badger, Glover, Mastrolia, Williams,
Badger, Glover, hep-ph/
LD, Sofianatos,
Badger, Campbell, Ellis, Williams,
by parity
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

18. 5-point – still analytic
BGMW DS
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

19. Besides virtual corrections, also need real emission
Infrared
singularities
cancel
General subtraction methods for integrating real-emission contributions developed in mid-1990s
Frixione, Kunszt, Signer, hep-ph/ ;
Catani, Seymour, hep-ph/ , hep-ph/
Recently automated by several groups
Gleisberg, Krauss, ;
Seymour, Tevlin, ;
Hasegawa, Moch, Uwer, ;
Frederix, Gehrmann, Greiner, ;
Czakon, Papadopoulos, Worek, ;
Frederix, Frixione, Maltoni, Stelzer,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

20. Les Houches Experimenters’ Wish List
Feynman
diagram
methods
now joined by
on-shell
BCDEGMRSW; Campbell, Ellis, Williams
Berger
table courtesy of
C. Berger
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

21. L. Dixon On-Shell Methods
W + n jets Data
Tevatron
CDF, [hep-ex]
n = 1
NLO parton level
(MCFM)
LO matched to parton shower MC with different
schemes
n = 2
n = 3
only LO
available
in 2007
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

22. W + 3 jets at NLO at Tevatron
Ellis, Melnikov, Zanderighi,
Berger et al.,
Rocket
Leading-color adjustment procedure
Exact treatment of color
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

23. L. Dixon On-Shell Methods
W + 3 jets at LHC
LHC has much greater dynamic range
Many events with jet ETs >> MW
Must carefully choose appropriate renormalization + factorization scale
Scale we used at the Tevatron,
also used in several other LO studies,
is not a good choice:
NLO cross section can even dive negative!
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

24. L. Dixon On-Shell Methods
Better Scale Choices
Q: What’s going on?
A: Powerful jets and wimpy Ws
If (a) dominates, then
is OK
But if (b) dominates, then the scale ETW is too low.
Looking at large ET for the 2nd jet forces configuration (b).
Better: total (partonic) transverse energy
(or fixed fraction of it, or sum in quadrature?); gets large properly for both (a) and (b)
Bauer, Lange  
Another reasonable scale is invariant mass of the n jets
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

25. Compare Two Scale Choices
logs not properly
cancelled for
large jet ET
– LO/NLO quite flat,
also for many other observables
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

26. Total Transverse Energy HT at LHC
often used in supersymmetry searches  
flat LO/NLO ratio
due to good
choice of
scale m = HT
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

27. NLO pp  W + 4 jets now available
C. Berger
et al.,
Virtual terms: leading-color (including quark loops); omitted terms only ~ few %
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

28. One indicator of NLO progress
pp  W + 0 jet 1978 Altarelli, Ellis, Martinelli pp  W + 1 jet 1989 Arnold, Ellis, Reno pp  W + 2 jets 2002 Arnold, Ellis pp  W + 3 jets 2009 BH+Sherpa; EMZ pp  W + 4 jets 2010 BH+Sherpa
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

29. NLO Parton-Level vs. Shower MCs
Recent advances on Les Houches NLO Wish List all at parton level: no parton shower, no hadronization, no underlying event.
Methods for matching NLO parton-level results to parton showers, maintaining NLO accuracy
Frixione, Webber (2002), ...
POWHEG Nason (2004); Frixione, Nason, Oleari (2007); ...
POWHEG in SHERPA Höche, Krauss, Schönherr, Siegert,
GenEvA Bauer, Tackmann, Thaler (2008)
However, none is yet implemented for final states with multiple light-quark & gluon jets
NLO parton-level predictions generally give best normalizations for total cross sections (unless NNLO available!), and distributions away from shower-dominated regions.
Right kinds of ratios will be considerably less sensitive to shower + nonperturbative effects
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

30. On-shell methods in N=4 SYM
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

31. L. Dixon On-Shell Methods
Why N=4 SYM?
Dual to gravity/string theory on AdS5 x S5
Very similar in IR to QCD talk by Magnea
Planar (large Nc) theory is integrable talk by Beisert
Strong-coupling limit a minimal area problem (Wilson loop) Alday, Maldacena
Planar amplitudes possess dual conformal invariance Drummond, Henn, Korchemsky, Sokatchev
Some planar amplitudes “known” to all orders in coupling Bern, LD, Smirnov + AM + DHKS
More planar amplitudes “equal” to expectation values of light-like Wilson loops talk by Spradlin
N=8 supergravity closely linked by tree-level Kawai-Lewellen-Tye relation and more recent “duality” relations Bern, Carrasco, Johansson
More recent Grassmannian developments Arkani-Hamed et al.
Excellent arena for testing on-shell & related methods
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

32. L. Dixon On-Shell Methods
N=4 SYM “states”
all states in adjoint representation, all linked by N=4 supersymmetry
Interactions uniquely specified by gauge group, say SU(Nc), 1 coupling g
Exactly scale-invariant (conformal) field theory: b(g) = 0 for all g
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

33. Planar N=4 SYM and AdS/CFT
In the ’t Hooft limit,
fixed, planar diagrams dominate
AdS/CFT duality
suggests that weak-coupling perturbation series in l for large-Nc (planar) N=4 SYM should have special properties, because
large l limit  weakly-coupled gravity/string theory
on AdS5 x S5
Maldacena; Gubser, Klebanov, Polyakov; Witten
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

34. L. Dixon On-Shell Methods
AdS/CFT in one picture
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

35. Scattering at strong coupling
Alday, Maldacena, [hep-th]
Use AdS/CFT to compute an appropriate scattering amplitude
High energy scattering in string theory is semi-classical
Gross, Mende (1987,1988) r
Evaluated on the classical solution, action is imaginary
 exponentially suppressed tunnelling configuration
Can also do with dimensional regularization instead of
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

36. Dual variables and strong coupling
T-dual momentum variables introduced by Alday, Maldacena
Boundary values for world-sheet
are light-like segments in :
for gluon with momentum
For example,
for gg  gg 90-degree scattering,
s = t = -u/2, the boundary looks like:
Corners (cusps) are located at
– same dual momentum variables
appear at weak coupling (in planar theory)
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

37. Generalized unitarity for N=4 SYM
Found long ago that one-loop N=4 amplitudes contain only boxes, due to SUSY cancellations of loop momenta in numerator: Bern, LD, Dunbar, Kosower (1994)
More recently, L-loop generalization of this property conjectured: All (important) terms determined by “leading-singularities” – imposing 4L cuts on the L loop momenta in D= Cachazo, Skinner, ; Arkani-Hamed, Cachazo, Kaplan,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

38. Multi-loop generalized unitarity at work
Allowing for complex cut momenta, one can chop an amplitude entirely into 3-point trees
 maximal cuts or ~ leading singularities
Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/ ; Bern, Carrasco, Johansson, Kosower,
These cuts are maximally simple, yet give an excellent starting point for constructing the full answer. (No conjectures required.)
In planar (leading in Nc) N=4 SYM, maximal cuts find all terms in the complete answer for 1, 2 and 3 loops
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

39. L. Dixon On-Shell Methods
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,
Maximal cut method is completely systematic
not restricted to N=4 SYM
not restricted to planar contributions
Recent supersum advances to evaluate more complicated cuts
Drummond, Henn, Korchemsky, Sokatchev, ; Arkani-Hamed, Cachazo, Kaplan, ; Elvang, Freedman, Kiermaier, ; Bern, Carrasco, Ita, Johansson, Roiban, 2009
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

40. 4-gluon amplitude in N=4 SYM at 1 and 2 Loops
Green, Schwarz, Brink;
Grisaru, Siegel (1981)
Bern, Rozowsky, Yan (1997); Bern, LD, Dunbar, Perelstein, Rozowsky (1998)
2 loops:
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

41. Dual Conformal Invariance
Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/
A conformal symmetry acting in momentum space, on dual (sector) variables xi
First seen in N=4 SYM planar amplitudes in the loop integrals x5 x1 x2 x3 x4 k
invariant under inversion:
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

42. Dual conformal invariance at 4 loops
Simple graphical rules:
4 (net) lines into inner xi
1 (net) line into outer xi
Dotted lines are for
numerator factors
4 loop planar integrals
all of this form
BCDKS,
hep-th/
also true at 5 loops
BCJK,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

43. Insight from string theory
As a property of full (planar) amplitudes, rather than integrals,
dual conformal invariance follows, at strong coupling, from bosonic T duality symmetry of AdS5 x S5.
Also, strong-coupling calculation ~ equivalent to computation of Wilson line for n-sided polygon with vertices at xi Alday, Maldacena,
Wilson line blind to helicity formalism
– doesn’t know MHV from non-MHV.
Some recent attempts to go beyond this
Alday, Eden, Maldacena, Korchemsky, Sokatchev,
; Eden, Korchemsky, Sokatchev, ,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

44. L. Dixon On-Shell Methods
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 On-Shell Methods
DESY Workshop Sept 2010

45. L. Dixon On-Shell Methods
3 loop cubic graphs
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 On-Shell Methods
DESY Workshop Sept 2010

46. L. Dixon On-Shell Methods
N=4 numerators at 3 loops
Omit overall
manifestly quadratic in loop momentum
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

47. Four loops: full color N=4 SYM as input for N=8 SUGRA
Bern, Carrasco, LD, Johansson, Roiban,
BCDJR,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

48. 4 loop 4 point amplitude in N=4 SYM
Number of cubic 4-point graphs with nonvanishing
Coefficients and various topological properties
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

49. L. Dixon On-Shell Methods
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 On-Shell Methods
DESY Workshop Sept 2010

50. L. Dixon On-Shell Methods
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 On-Shell Methods
DESY Workshop Sept 2010

51. 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 On-Shell Methods
DESY Workshop Sept 2010

52. Planar terms well known
Bern, Czakon, LD, Kosower, Smirnov, hep-th/
Dual conformal (pseudoconformal)
invariance, acting on dual or sector
variables xi
Greatly limits the possible numerators
No such guide for the nonplanar terms
Drummond, Henn, Smirnov,
Sokatchev, hep-th/
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

53. Simplest (rung rule) graphs N=4 SYM numerators shown
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

54. L. Dixon On-Shell Methods
Most complex graphs N=4 SYM numerators shown [N=8 SUGRA numerators much larger]
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

55. Checks on final N=4 result
Lots of different products of MHV tree amplitudes.
NMHV7 * anti-NMHV7 and MHV5 * NMHV6 * anti-MHV5
– evaluated by Elvang, Freedman, Kiermaier,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

56. L. Dixon On-Shell Methods
N=4 SYM in the UV
Want the full color dependence of the UV divergences in N=4 SYM in the critical dimension.
BCDJR (April `09)
Dc = (L = 1)
Dc = 4 + 6/L (L = 2,3)
For G = SU(Nc), divergences organized in terms of color structures:
Found absence of double-trace terms,
later studied by Bossard, Howe, Stelle, , ;
Berkovits, Green, Russo, Vanhove, ; Bjornsson, Green,
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

57. L. Dixon On-Shell Methods
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 On-Shell Methods
DESY Workshop Sept 2010

58. UV behavior of N=8 at 4 loops
All 50 cubic graphs have numerator factors composed of terms
loop momenta l
external momenta k
Maximum value of m turns out to be 8 in every integral, vs. 4 for N=4 SYM
Integrals all have 13 propagators, so
In order to show that
need to show that
all
cancel
But they all do 
N=8 SUGRA still no worse than N=4 SYM in UV at 4 loops!
DESY Workshop Sept 2010
L. Dixon On-Shell Methods

59. L. Dixon On-Shell Methods
Conclusions
On-shell methods at one loop have many practical applications to LHC physics – analytically, but especially in numerical implementations
On-shell methods in (planar) N=4 SYM have led to BDS ansatz and information about its violation at 6 points.
Very recently used to construct the planar NMHV 2 loop 6 point amplitude Kosower, Roiban, Vergu,
And the full-color 4 point 4 loop amplitude in N=4 SYM
Latter result was used to construct the 4 point 4 loop amplitude in N=8 supergravity, which showed that it is still as well-behaved as N=4 super-Yang-Mills theory through this order
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 On-Shell Methods
DESY Workshop Sept 2010

60. L. Dixon On-Shell Methods
Extra Slides
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

61. L. Dixon On-Shell Methods
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 On-Shell Methods
DESY Workshop Sept 2010

62. L. Dixon On-Shell Methods
N=4 SYM in UV at two loops
Planar and nonplanar double box integrals in Dc = 7 - 2e [BDDPR 1998] with color factors 
Corresponds to counterterms such as
and (two extra derivatives)
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

63. 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 = 3 and 4.
L. Dixon On-Shell Methods
DESY Workshop Sept 2010

64. Cancellations between integrals
Cancellation of k4 l8 terms [vanishing of coefficient of ]
simple: just set external momenta ki  0,
collect coefficients of 2 resulting vacuum diagrams,
observe that the 2 coefficients cancel.
Cancellation of k5 l7 [and k7 l5] terms is trivial:
Lorentz invariance does not allow an odd-power divergence.
Cancellation of k6 l6 terms [vanishing of coefficient of ]
more intricate: Expand to second subleading order in limit ki  0,
generating 30 different vacuum integrals.
Evaluating UV poles for all 30 integrals (or alternatively deriving
consistency relations between them), we find that
UV pole cancels in D=5-2e
N=8 SUGRA still no worse than N=4 SYM in UV at 4 loops!
DESY Workshop Sept 2010
L. Dixon On-Shell Methods