of Maximal Supergravity

taupeselectionΜηχανική

14 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

182 εμφανίσεις


Henrik

Johansson

CERN


March 26, 2013


BUDS workshop

INFN
Frascati



1201.5366, 1207.6666, 1210.7709





Based on work with:
Zvi

Bern, John Joseph Carrasco, Lance Dixon,






Michael Douglas,
Radu

Roiban
, Matt von
Hippel

Towards
Determining the UV Behavior
of Maximal
Supergravity

H. Johansson, Frascati 2013

SUGRA status in one slide

After 35
years of
supergravity
, we can only now make very
precise statements about the
D
=4 ultraviolet structure.

No
D
=4 divergence of pure SG has been found to date.

Susy

forbids 1,2 loop div.,
R
2
,
R
3

c.t
. incompatible with
susy

Pure gravity 1
-
loop finite, 2
-
loop divergent
Goroff

&
Sagnotti

With matter: 1
-
loop divergent

‘t
Hooft

&
Veltman

Naively
susy

allows 3
-
loop div.
R
4

N
=8 SG and
N
=4 SG 3
-
loop finite!

N
=8
SG: no divergence before 7 loops

7
-
loop div. in
D
=4 implies a 5
-
loop
div.



in
D
=24/5

--

calculation in progress!

U
Finite
?

N=8 SG

H. Johansson, Frascati 2013

Why is it interesting ?

If
N
=8 SG is
perturbatively

finite, why is it interesting ?

It better be finite for a good reason!

H
idden new symmetry, for example

Understanding the mechanism might open a host of
possibilities

Any indication of hidden structures yet?

Gravity is a double copy of gauge theories

Color
-
Kinematics: kinematics = Lie algebra


Constraints from E
-
M
duality
Kallosh
,….

Hidden
superconformal

N=4 SUGRA
?

Symmetry?

Gravity

Bern, Carrasco,
HJ

Ferrara,
Kallosh
, Van
Proeyen

Henrik Johansson

G
auge Theory Analogy

Gauge theory in
D
>4 have same problem as
D
=4 gravity

N
on
-
renormalizable

due to
dimensionful

coupling

However,
D
=5 SYM has a UV completion: (2,0) theory in
D
=6

Is
D
=5 SYM
perturbatively

UV finite ?
Douglas; Lambert et al.

If yes, how does it work ?

If no, what do we need to add ?

Solitons
, KK modes ?
Douglas; Lambert et al
.



Understanding
D
=5 SYM might (or might not)


give clues to how to understand
D
=4 gravity.

(2,0)
theory ?

D=5

SYM

H. Johansson, Frascati 2013

Review UV status
N
=8
SUGRA
and

N
=4 SYM


4pt amplitudes and
UV divergences

3,4
-
loop
N
=8
SUGRA & N=4 SYM

5
-
loop
nonplanar

SYM

6
-
loop planar
D
=5 SYM



5pt
amplitudes and UV
divergences

1,2,3
-
loop
N
=8 SUGRA & N=4
SYM


Current 5
-
loop progress


Conclusion



Outline

H. Johansson, Frascati 2013

UV properties
N

=8 SG



N
=8 SG: conventional
superspace

power counting forbids
L
=1,2


divergences
Deser
, Kay,
Stelle
;
Howe and
Lindström
; Green, Schwarz, Brink;




Howe,
Stelle
; Marcus,
Sagnotti



Three
-
loop divergence ruled out by calculation:

Bern, Carrasco, Dixon, HJ,


Kosower
,
Roiban
, (2007),
Bern, Carrasco, Dixon, HJ,
Roiban

(2008
)


L
<7 loop divergences ruled out by
counterterm

analysis, using
E
7(7)

symmetry and other methods, but a
L
=7 divergence is still possible

Beisert
,
Elvang
, Freedman,
Kiermaier
, Morales,
Stieberger
;
Björnsson
, Green,
Bossard
, Howe,
Stelle
,
Vanhove
,
Kallosh
,
Ramond
,
Lindström
,
Berkovits
,
Grisaru
, Siegel, Russo, and more….






In
D
=4 dimensions
:

In D>4 dimensions
:

Through four loops
N
=8 SG and
N
=4 SYM diverge in exactly the

same dimension:

Marcus and
Sagnotti
; Bern, Dixon, Dunbar,
Perelstein
,
Rozowsky
;

Bern, Carrasco, Dixon, HJ,
Kosower
,
Roiban

H. Johansson, Frascati 2013

UV divergence trend

Plot of critical dimensions of
N

= 8 SUGRA

and
N

= 4 SYM

Known bound for
N

= 4

Bern, Dixon, Dunbar,
Rozowsky
,
Perelstein
; Howe,
Stelle

current trend for
N

= 8

If
N

= 8 div. at
L
=7

calculations:

L
= 7

lowest loop order for possible
D

= 4

divergence

Beisert
,
Elvang
, Freedman,
Kiermaier
, Morales,
Stieberger
;

Björnsson
, Green,
Bossard
, Howe,
Stelle
,
Vanhove

Kallosh
,
Ramond
,
Lindström
,
Berkovits
,
Grisaru
,
Siegel, Russo, and more….


1
-
2 loops:
Green, Schwarz, Brink; Marcus and
Sagnotti


3
-
5 loops:
Bern, Carrasco, Dixon, HJ,
Kosower
,
Roiban

6 loops:
Bern, Carrasco, Dixon, Douglas, HJ, von
Hippel

Finite

?

Divergent

H. Johansson, Frascati 2013

UV divergence trend

Plot of critical dimensions of
N

= 8 SUGRA

and
N

= 4 SYM

Known bound for
N

= 4

Bern, Dixon, Dunbar,
Rozowsky
,
Perelstein
; Howe,
Stelle

current trend for
N

= 8

If
N

= 8 div. at
L
=7

calculations:

L
= 7

lowest loop order for possible
D

= 4

divergence

Beisert
,
Elvang
, Freedman,
Kiermaier
, Morales,
Stieberger
;

Björnsson
, Green,
Bossard
, Howe,
Stelle
,
Vanhove

Kallosh
,
Ramond
,
Lindström
,
Berkovits
,
Grisaru
,
Siegel, Russo, and more….


1
-
2 loops:
Green, Schwarz, Brink; Marcus and
Sagnotti


3
-
5 loops:
Bern, Carrasco, Dixon, HJ,
Kosower
,
Roiban

6 loops:
Bern, Carrasco, Dixon, Douglas, HJ, von
Hippel

26/5 or 24/5 ?

Finite

?

Divergent

H. Johansson, Frascati 2013

N

=8 Amplitude and Counter Term Structure

4pt amplitude form

(any dimension)

divergence
occurs in

Counter term

D

= 8

D

= 6

D

= 7

D

= 5.5

Loop

order

1

2

3

4

If amplitude for
L


4

has at least 8
derivatives then by dimensional analysis:
no divergence before
L
= 7

!

D

= 24/5 ?

5

?

?

H. Johansson, Frascati 2013

N

=8 Amplitude and Counter Term Structure

4pt amplitude form

(any dimension)

divergence
occurs in

Counter term

D

= 8

D

= 6

D

= 7

D

= 5.5

Loop

order

1

2

3

4

If amplitude for
L


5

has at least 10
derivatives then by dimensional analysis:
no divergence before
L
= 8

!

D

= 26/5 ?

5

?

?

H. Johansson, Frascati 2013

Earliest appearance of
N

= 8 Divergence

3 loops

Conventional
superspace

power counting

Green, Schwarz, Brink (1982)

Howe and
Stelle

(1989)

Marcus and
Sagnotti

(1985)

5 loops

Partial analysis of unitarity cuts;
If

N

= 6 harmonic
superspace exists;
algebraic renormalisation

Bern, Dixon, Dunbar,

Perelstein
,
Rozowsky

(1998)

Howe and
Stelle

(2003,
2009
)

6 loops

If

N

= 7 harmonic superspace exists

Howe and
Stelle

(2003)

7 loops

If

N

= 8 harmonic
superspace

exists;

string theory U
-
duality
analysis;

lightcone

gauge locality arguments;

E
7(7)
analysis, unique 1/8 BPS candidate

Grisaru

and Siegel (1982);

Green
, Russo,
Vanhove
;
Kallosh
;
Beisert
,
Elvang
, Freedman,
Kiermaier
, Morales,
Stieberger
;
Bossard
, Howe,
Stelle
,
Vanhove

8 loops

Explicit identification of potential susy invariant
counterterm with full non
-
linear susy

Howe
and
Lindström
;

Kallosh

(1981)

9 loops

Assume Berkovits


superstring non
-
renormalization
theorems can be carried over to
N

= 8 supergravity

Green, Russo,
Vanhove

(2006)

Finite

Identified cancellations in
multiloop

amplitudes;
lightcone

gauge locality and E
7(7
)
,

inherited from hidden N=4 SC gravity

Bern, Dixon
,
Roiban

(2006),

Kallosh

(2009

12),

Ferrara,
Kallosh
, Van
Proeyen

(2012)

H. Johansson, Frascati 2013

3, 4, 5, 6
-
Loop Amplitudes

3
-
loop
N

=8 SG &
N

=4 SYM

Color
-
kinematics dual form:

Bern
, Carrasco, HJ

UV divergent in
D
=6:

Bern, Carrasco,
Dixon
, HJ,
Roiban

4
-
loops: 85 integral types

(8
0
)
2
1
3
4
(8
3
)
4
2
1
3
(8
5
)
4
2
1
3
(8
4
)
4
2
1
3
(8
2
)
2
1
3
4
(8
1
)
2
1
3
4
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
(8
0
)
2
1
3
4
(8
3
)
4
2
1
3
(8
5
)
4
2
1
3
(8
4
)
4
2
1
3
(8
2
)
2
1
3
4
(8
1
)
2
1
3
4
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
(8
0
)
2
1
3
4
(8
3
)
4
2
1
3
(8
5
)
4
2
1
3
(8
4
)
4
2
1
3
(8
2
)
2
1
3
4
(8
1
)
2
1
3
4
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
(8
0
)
2
1
3
4
(8
3
)
4
2
1
3
(8
5
)
4
2
1
3
(8
4
)
4
2
1
3
(8
2
)
2
1
3
4
(8
1
)
2
1
3
4
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
(8
0
)
2
1
3
4
(8
3
)
4
2
1
3
(8
5
)
4
2
1
3
(8
4
)
4
2
1
3
(8
2
)
2
1
3
4
(8
1
)
2
1
3
4
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
(8
0
)
2
1
3
4
(8
3
)
4
2
1
3
(8
5
)
4
2
1
3
(8
4
)
4
2
1
3
(8
2
)
2
1
3
4
(8
1
)
2
1
3
4
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
7
5
8
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
(6
8
)
2
1
4
3
2
1
(6
7
)
3
4
(7
0
)
2
1
3
4
(7
1
)
2
1
3
4
(6
9
)
2
1
3
4
6
8
5
5
6
5
7
8
5
7
8
5
6
2
1
(6
6
)
3
5
6
7
8
6
6
8
(7
2
)
2
4
1
3
(7
3
)
4
1
3
2
(7
4
)
4
3
2
1
5
7
6
8
7
6
8
5
7
6
8
5
7
7
8
7
4
7
5
8
6
(7
7
)
3
4
2
1
7
6
8
5
(7
6
)
4
2
1
3
(7
5
)
4
3
1
2
5
7
8
6
(7
9
)
3
2
1
4
7
6
8
5
8
5
(7
8
)
4
3
1
2
7
6
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
4
3
2
1
(5
3
)
2
(5
2
)
1
3
4
(5
5
)
4
1
2
3
2
1
(5
9
)
3
4
2
1
(5
8
)
4
3
(5
7
)
1
2
3
4
2
1
(6
2
)
3
4
2
1
(6
1
)
3
4
2
1
(6
0
)
3
4
2
1
(6
5
)
4
3
2
1
(6
4
)
4
3
2
1
(6
3
)
4
3
2
(5
1
)
4
1
3
6
7
5
8
6
7
5
8
7
8
6
5
6
7
8
5
7
8
5
7
5
8
6
7
5
8
6
5
7
6
(5
4
)
3
4
2
1
8
(5
6
)
1
2
3
4
6
7
8
6
5
6
7
8
5
7
5
8
6
6
8
5
7
7
5 8
6
7
5
6
8
7
8
6
5
H. Johansson, QMUL 2013

15

4
-
loops
N
=4 SYM and
N
=8 SG

Bern, Carrasco, Dixon, HJ,
Roiban

1201.5366


85 diagrams


Power
counting manifest both
N
=4 and
N
=8


Both diverge in
D
=11/2

V
1
V
2
V
8
FI
G.
1
7
:
T
h
e
b
a
s
i
c
f
o
u
r
-l
o
o
p
v
a
c
u
u
m
i
n
t e
g
r
a
l
s
V
1
,
V
2
an
d
V
8
,t
o
w
h
i
c
h
a
l
l o
t
h
e
r
s
c
a
n
b
e
r
e
d
u
c
e
d
.
r
e
v
e
a
ls
t
h
a
t
t
h
e
le
a
d
in
g
U
V
b
e
h
a
v
io
r
c
o
m
e
s
s
o
le
ly
f
r
o
m
in
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
.
T
h
e
s
e
s
ix
in
t
e
g
r
a
ls
h
a
v
e
1
1
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
,
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
e
p
e
n
d
e
n
t
o
f
t
h
e
lo
o
p
m
o
m
e
n
t
u
m
.
T
h
e
r
e
f
o
r
e
t
h
e
y
d
iv
e
r
g
e

r
s
t
in
D
c
= 1
1
/
2
,
w
h
ic
h
m
a
t
c
h
e
s
t
h
e
e
x
p
e
c
t
e
d
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
c
= 4
+
6
/L
w
it
h
L
= 4
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
1
P
I
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
h
a
v
e
1
3
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
T
h
e
ir
n
u
-
m
e
r
a
t
o
r
s
w
o
u
ld
h
a
v
e
t
o
b
e
q
u
a
r
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
f
o
r
t
h
e
m
t
o
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
w
e
n
o
t
e
f
r
o
m
e
q
.
(
3
.1
4
)
t
h
a
t
t
h
e
m
a
s
t
e
r
n
u
m
e
r
a
t
o
r
s
N
18
an
d
N
28
a
r
e
q
u
a
d
r
a
t
ic
in
t h
e
τ
i j
,
a
n
d
h
e
n
c
e
m
e
r
e
ly
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
T
h
e
J
a
c
o
b
i
r
e
la
t
io
ns
(
A4
)
a
nd
(
A
5
)
p
r
e
s
e
r
v
e
t
h
is
q
u
a
d
r
a
t
ic
b
e
h
a
v
io
r
f
o
r
a
ll
n
u
m
e
r
a
t
o
r
s
.
T
h
e
r
e
f
o
r
e
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
a
r
e

n
it
e
in
D
c
= 1
1
/
2
.
T
h
e
1
P
R
in
t
e
g
r
a
ls
I
53
t h
ro
u
g
h
I
79
h
a
v
e
1
2
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
I
f
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
w
e
r
e
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
,
t
h
e
n
t
h
e
y
w
o
u
ld
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
it
is
e
a
s
y
t
o
s
e
e
f
r
o
m
e
q
s
.
(
A
5
)
a
n
d
(
B
1
)
t
h
a
t
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
a
r
e
a
ll
lin
e
a
r
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
I
n
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
r
e
d
u
c
e
e
a
s
ily
t
o
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
t
h
e
lim
it
t
h
a
t
t
h
e
e
x
t
e
r
n
a
l
m
o
m
e
n
t
a
v
a
n
is
h
.
T
h
e
p
la
n
a
r
in
t
e
g
r
a
ls
I
80
an
d
I
83
r
e
d
u
c
e
t
o
t
h
e
v
a
c
u
u
m
in
t
e
g
r
a
l
V
1
d
e
p
ic
t
e
d
in

g
.
1
7
.
W
h
ile
in
t
e
g
r
a
ls
I
81
an
d
I
84
a
r
e
n
o
n
p
la
n
a
r
a
s
f
o
u
r
-
p
o
in
t
g
r
a
p
h
s
,
in
t
h
e
v
a
c
u
u
m
lim
it
t
h
e
y
r
e
d
u
c
e
t
o
t
h
e
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
2
.
F
in
a
lly
,
in
t
e
g
r
a
ls
I
82
an
d
I
85
re
d
u
c
e
t o
t
h
e
n
o
n
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
8
.
A
s
w
a
s
t
h
e
c
a
s
e
a
t
t
h
r
e
e
lo
o
p
s
,
t
h
e
c
o
lo
r
f
a
c
t
o
r
s
f
o
r
t
h
e
le
a
d
in
g
U
V
g
r
a
p
h
s
a
r
e
r
e
la
t
e
d
b
y
c
o
lo
r
J
a
c
o
b
i
id
e
n
t
it
ie
s
.
I
n
t
h
is
c
a
s
e
w
e
c
a
n
s
u
b
t
r
a
c
t
,
f
o
r
e
x
a
m
p
le
,
C
81
fr
o
m
C
80
,a
n
d
u
s
e
a
J
a
c
o
b
i
id
e
n
t
it
y
o
p
e
r
a
t
in
g
o
n
t
h
e
b
o
x
a
t
t
h
e
t
o
p
c
e
n
t
e
r
o
f
t
h
e
g
r
a
p
h
s
in

g
.
1
1
.
T
h
e
n
w
e
r
e
d
u
c
e
t
h
e
r
e
s
u
lt
in
g
t
r
ia
n
g
le
s
u
b
g
r
a
p
h
s
it
e
r
a
t
iv
e
ly
,
t
o

n
d
C
80
,
83

2
N
4
c
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
=
C
81
=
C
82
=
C
84
=
C
85
.
(
4
.1
3
)
A
g
a
in
,
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
p
a
r
t
s
o
f
a
ll
c
o
n
t
r
ib
u
t
in
g
c
o
lo
r
f
a
c
t
o
r
s
a
r
e
e
q
u
a
l.
Fr
o
m
e
q
.
(
A
5
)
,
t
h
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
o
b
e
y
N
80
=
N
81
=
N
82
an
d
N
83
=
N
84
=
N
85
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
t
h
e
c
o
m
b
in
a
t
o
r
ia
l
f
a
c
t
o
r
o
f
I
81
in
e
q
.
(
3
.3
)
is
t
w
ic
e
a
s
la
r
g
e
a
s
t
h
o
s
e
fo
r
I
80
an
d
I
82
,
a
n
d
s
im
ila
r
ly
f
o
r
I
84
w
it
h
r
e
s
p
e
c
t
t
o
I
83
an
d
I
85
.
T
a
k
in
g
in
t
o
a
c
c
o
u
n
t
b
o
t
h
c
o
m
b
in
a
t
o
r
ia
l
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
,
t
h
e
c
o
n
t
r
ib
u
t
io
n
o
f
I
83
is

9
8
t
im
e
s
t
h
a
t
o
f
I
80
,a
n
d
s
im
ila
r
ly
f
o
r
t
h
e
o
t
h
e
r
t
w
o
p
a
ir
s
o
f
g
r
a
p
h
s
.
C
o
m
b
in
in
g
a
ll
t
e
r
m
s
a
n
d
s
w
it
c
h
in
g
t
o
t
h
e
c
o
lo
r
-
t
r
a
c
e
b
a
s
is
,
w
e

n
d
t
h
a
t
t
h
e
U
V
d
iv
e
r
g
e
n
c
e
in
t
h
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
D
= 1
1
/
2
is
g
iv
e
n
by
,
A
(4
)
4
(1
,
2
,
3
,
4)
SU
(
N
c
)
po
l
e
=

6
g
10
K
N
2
c
N
2
c
V
1
+ 1
2
(
V
1
+ 2
V
2
+
V
8
)
(
4
.1
4
)
×
s
( T
r
1324
+ T
r
1423
) +
t
( T
r
1243
+ T
r
1342
) +
u
( T
r
1234
+ T
r
1432
)
,
33
V
1
V
2
V
8
FI
G.
1
7
:
T
h
e
b
a
s
i
c
f
o
u
r
-l
o
o
p
v
a
c
u
u
m
i
n
t e
g
r
a
l
s
V
1
,
V
2
an
d
V
8
,t
o
w
h
i
c
h
a
l
l o
t
h
e
r
s
c
a
n
b
e
r
e
d
u
c
e
d
.
r
e
v
e
a
ls
t
h
a
t
t
h
e
le
a
d
in
g
U
V
b
e
h
a
v
io
r
c
o
m
e
s
s
o
le
ly
f
r
o
m
in
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
.
T
h
e
s
e
s
ix
in
t
e
g
r
a
ls
h
a
v
e
1
1
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
,
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
e
p
e
n
d
e
n
t
o
f
t
h
e
lo
o
p
m
o
m
e
n
t
u
m
.
T
h
e
r
e
f
o
r
e
t
h
e
y
d
iv
e
r
g
e

r
s
t
in
D
c
= 1
1
/
2
,
w
h
ic
h
m
a
t
c
h
e
s
t
h
e
e
x
p
e
c
t
e
d
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
c
= 4
+
6
/L
w
it
h
L
= 4
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
1
P
I
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
h
a
v
e
1
3
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
T
h
e
ir
n
u
-
m
e
r
a
t
o
r
s
w
o
u
ld
h
a
v
e
t
o
b
e
q
u
a
r
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
f
o
r
t
h
e
m
t
o
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
w
e
n
o
t
e
f
r
o
m
e
q
.
(
3
.1
4
)
t
h
a
t
t
h
e
m
a
s
t
e
r
n
u
m
e
r
a
t
o
r
s
N
18
an
d
N
28
a
r
e
q
u
a
d
r
a
t
ic
in
t h
e
τ
i j
,
a
n
d
h
e
n
c
e
m
e
r
e
ly
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
T
h
e
J
a
c
o
b
i
r
e
la
t
io
ns
(
A4
)
a
nd
(
A
5
)
p
r
e
s
e
r
v
e
t
h
is
q
u
a
d
r
a
t
ic
b
e
h
a
v
io
r
f
o
r
a
ll
n
u
m
e
r
a
t
o
r
s
.
T
h
e
r
e
f
o
r
e
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
a
r
e

n
it
e
in
D
c
= 1
1
/
2
.
T
h
e
1
P
R
in
t
e
g
r
a
ls
I
53
t h
ro
u
g
h
I
79
h
a
v
e
1
2
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
I
f
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
w
e
r
e
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
,
t
h
e
n
t
h
e
y
w
o
u
ld
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
it
is
e
a
s
y
t
o
s
e
e
f
r
o
m
e
q
s
.
(
A
5
)
a
n
d
(
B
1
)
t
h
a
t
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
a
r
e
a
ll
lin
e
a
r
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
I
n
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
r
e
d
u
c
e
e
a
s
ily
t
o
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
t
h
e
lim
it
t
h
a
t
t
h
e
e
x
t
e
r
n
a
l
m
o
m
e
n
t
a
v
a
n
is
h
.
T
h
e
p
la
n
a
r
in
t
e
g
r
a
ls
I
80
an
d
I
83
r
e
d
u
c
e
t
o
t
h
e
v
a
c
u
u
m
in
t
e
g
r
a
l
V
1
d
e
p
ic
t
e
d
in

g
.
1
7
.
W
h
ile
in
t
e
g
r
a
ls
I
81
an
d
I
84
a
r
e
n
o
n
p
la
n
a
r
a
s
f
o
u
r
-
p
o
in
t
g
r
a
p
h
s
,
in
t
h
e
v
a
c
u
u
m
lim
it
t
h
e
y
r
e
d
u
c
e
t
o
t
h
e
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
2
.
F
in
a
lly
,
in
t
e
g
r
a
ls
I
82
an
d
I
85
re
d
u
c
e
t o
t
h
e
n
o
n
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
8
.
A
s
w
a
s
t
h
e
c
a
s
e
a
t
t
h
r
e
e
lo
o
p
s
,
t
h
e
c
o
lo
r
f
a
c
t
o
r
s
f
o
r
t
h
e
le
a
d
in
g
U
V
g
r
a
p
h
s
a
r
e
r
e
la
t
e
d
b
y
c
o
lo
r
J
a
c
o
b
i
id
e
n
t
it
ie
s
.
I
n
t
h
is
c
a
s
e
w
e
c
a
n
s
u
b
t
r
a
c
t
,
f
o
r
e
x
a
m
p
le
,
C
81
fr
o
m
C
80
,a
n
d
u
s
e
a
J
a
c
o
b
i
id
e
n
t
it
y
o
p
e
r
a
t
in
g
o
n
t
h
e
b
o
x
a
t
t
h
e
t
o
p
c
e
n
t
e
r
o
f
t
h
e
g
r
a
p
h
s
in

g
.
1
1
.
T
h
e
n
w
e
r
e
d
u
c
e
t
h
e
r
e
s
u
lt
in
g
t
r
ia
n
g
le
s
u
b
g
r
a
p
h
s
it
e
r
a
t
iv
e
ly
,
t
o

n
d
C
80
,
83

2
N
4
c
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
=
C
81
=
C
82
=
C
84
=
C
85
.
(
4
.1
3
)
A
g
a
in
,
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
p
a
r
t
s
o
f
a
ll
c
o
n
t
r
ib
u
t
in
g
c
o
lo
r
f
a
c
t
o
r
s
a
r
e
e
q
u
a
l.
Fr
o
m
e
q
.
(
A
5
)
,
t
h
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
o
b
e
y
N
80
=
N
81
=
N
82
an
d
N
83
=
N
84
=
N
85
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
t
h
e
c
o
m
b
in
a
t
o
r
ia
l
f
a
c
t
o
r
o
f
I
81
in
e
q
.
(
3
.3
)
is
t
w
ic
e
a
s
la
r
g
e
a
s
t
h
o
s
e
fo
r
I
80
an
d
I
82
,
a
n
d
s
im
ila
r
ly
f
o
r
I
84
w
it
h
r
e
s
p
e
c
t
t
o
I
83
an
d
I
85
.
T
a
k
in
g
in
t
o
a
c
c
o
u
n
t
b
o
t
h
c
o
m
b
in
a
t
o
r
ia
l
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
,
t
h
e
c
o
n
t
r
ib
u
t
io
n
o
f
I
83
is

9
8
t
im
e
s
t
h
a
t
o
f
I
80
,a
n
d
s
im
ila
r
ly
f
o
r
t
h
e
o
t
h
e
r
t
w
o
p
a
ir
s
o
f
g
r
a
p
h
s
.
C
o
m
b
in
in
g
a
ll
t
e
r
m
s
a
n
d
s
w
it
c
h
in
g
t
o
t
h
e
c
o
lo
r
-
t
r
a
c
e
b
a
s
is
,
w
e

n
d
t
h
a
t
t
h
e
U
V
d
iv
e
r
g
e
n
c
e
in
t
h
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
D
= 1
1
/
2
is
g
iv
e
n
by
,
A
(4
)
4
(1
,
2
,
3
,
4)
SU
(
N
c
)
po
l
e
=

6
g
10
K
N
2
c
N
2
c
V
1
+ 1
2
(
V
1
+ 2
V
2
+
V
8
)
(
4
.1
4
)
×
s
( T
r
1324
+ T
r
1423
) +
t
( T
r
1243
+ T
r
1342
) +
u
( T
r
1234
+ T
r
1432
)
,
33
V
1
V
2
V
8
FI
G.
1
7
:
T
h
e
b
a
s
i
c
f
o
u
r
-l
o
o
p
v
a
c
u
u
m
i
n
t e
g
r
a
l
s
V
1
,
V
2
an
d
V
8
,t
o
w
h
i
c
h
a
l
l o
t
h
e
r
s
c
a
n
b
e
r
e
d
u
c
e
d
.
r
e
v
e
a
ls
t
h
a
t
t
h
e
le
a
d
in
g
U
V
b
e
h
a
v
io
r
c
o
m
e
s
s
o
le
ly
f
r
o
m
in
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
.
T
h
e
s
e
s
ix
in
t
e
g
r
a
ls
h
a
v
e
1
1
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
,
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
e
p
e
n
d
e
n
t
o
f
t
h
e
lo
o
p
m
o
m
e
n
t
u
m
.
T
h
e
r
e
f
o
r
e
t
h
e
y
d
iv
e
r
g
e

r
s
t
in
D
c
= 1
1
/
2
,
w
h
ic
h
m
a
t
c
h
e
s
t
h
e
e
x
p
e
c
t
e
d
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
c
= 4
+
6
/L
w
it
h
L
= 4
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
1
P
I
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
h
a
v
e
1
3
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
T
h
e
ir
n
u
-
m
e
r
a
t
o
r
s
w
o
u
ld
h
a
v
e
t
o
b
e
q
u
a
r
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
f
o
r
t
h
e
m
t
o
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
w
e
n
o
t
e
f
r
o
m
e
q
.
(
3
.1
4
)
t
h
a
t
t
h
e
m
a
s
t
e
r
n
u
m
e
r
a
t
o
r
s
N
18
an
d
N
28
a
r
e
q
u
a
d
r
a
t
ic
in
t h
e
τ
i j
,
a
n
d
h
e
n
c
e
m
e
r
e
ly
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
T
h
e
J
a
c
o
b
i
r
e
la
t
io
ns
(
A4
)
a
nd
(
A
5
)
p
r
e
s
e
r
v
e
t
h
is
q
u
a
d
r
a
t
ic
b
e
h
a
v
io
r
f
o
r
a
ll
n
u
m
e
r
a
t
o
r
s
.
T
h
e
r
e
f
o
r
e
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
a
r
e

n
it
e
in
D
c
= 1
1
/
2
.
T
h
e
1
P
R
in
t
e
g
r
a
ls
I
53
t h
ro
u
g
h
I
79
h
a
v
e
1
2
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
I
f
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
w
e
r
e
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
,
t
h
e
n
t
h
e
y
w
o
u
ld
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
it
is
e
a
s
y
t
o
s
e
e
f
r
o
m
e
q
s
.
(
A
5
)
a
n
d
(
B
1
)
t
h
a
t
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
a
r
e
a
ll
lin
e
a
r
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
I
n
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
r
e
d
u
c
e
e
a
s
ily
t
o
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
t
h
e
lim
it
t
h
a
t
t
h
e
e
x
t
e
r
n
a
l
m
o
m
e
n
t
a
v
a
n
is
h
.
T
h
e
p
la
n
a
r
in
t
e
g
r
a
ls
I
80
an
d
I
83
r
e
d
u
c
e
t
o
t
h
e
v
a
c
u
u
m
in
t
e
g
r
a
l
V
1
d
e
p
ic
t
e
d
in

g
.
1
7
.
W
h
ile
in
t
e
g
r
a
ls
I
81
an
d
I
84
a
r
e
n
o
n
p
la
n
a
r
a
s
f
o
u
r
-
p
o
in
t
g
r
a
p
h
s
,
in
t
h
e
v
a
c
u
u
m
lim
it
t
h
e
y
r
e
d
u
c
e
t
o
t
h
e
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
2
.
F
in
a
lly
,
in
t
e
g
r
a
ls
I
82
an
d
I
85
re
d
u
c
e
t o
t
h
e
n
o
n
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
8
.
A
s
w
a
s
t
h
e
c
a
s
e
a
t
t
h
r
e
e
lo
o
p
s
,
t
h
e
c
o
lo
r
f
a
c
t
o
r
s
f
o
r
t
h
e
le
a
d
in
g
U
V
g
r
a
p
h
s
a
r
e
r
e
la
t
e
d
b
y
c
o
lo
r
J
a
c
o
b
i
id
e
n
t
it
ie
s
.
I
n
t
h
is
c
a
s
e
w
e
c
a
n
s
u
b
t
r
a
c
t
,
f
o
r
e
x
a
m
p
le
,
C
81
fr
o
m
C
80
,a
n
d
u
s
e
a
J
a
c
o
b
i
id
e
n
t
it
y
o
p
e
r
a
t
in
g
o
n
t
h
e
b
o
x
a
t
t
h
e
t
o
p
c
e
n
t
e
r
o
f
t
h
e
g
r
a
p
h
s
in

g
.
1
1
.
T
h
e
n
w
e
r
e
d
u
c
e
t
h
e
r
e
s
u
lt
in
g
t
r
ia
n
g
le
s
u
b
g
r
a
p
h
s
it
e
r
a
t
iv
e
ly
,
t
o

n
d
C
80
,
83

2
N
4
c
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
=
C
81
=
C
82
=
C
84
=
C
85
.
(
4
.1
3
)
A
g
a
in
,
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
p
a
r
t
s
o
f
a
ll
c
o
n
t
r
ib
u
t
in
g
c
o
lo
r
f
a
c
t
o
r
s
a
r
e
e
q
u
a
l.
Fr
o
m
e
q
.
(
A
5
)
,
t
h
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
o
b
e
y
N
80
=
N
81
=
N
82
an
d
N
83
=
N
84
=
N
85
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
t
h
e
c
o
m
b
in
a
t
o
r
ia
l
f
a
c
t
o
r
o
f
I
81
in
e
q
.
(
3
.3
)
is
t
w
ic
e
a
s
la
r
g
e
a
s
t
h
o
s
e
fo
r
I
80
an
d
I
82
,
a
n
d
s
im
ila
r
ly
f
o
r
I
84
w
it
h
r
e
s
p
e
c
t
t
o
I
83
an
d
I
85
.
T
a
k
in
g
in
t
o
a
c
c
o
u
n
t
b
o
t
h
c
o
m
b
in
a
t
o
r
ia
l
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
,
t
h
e
c
o
n
t
r
ib
u
t
io
n
o
f
I
83
is

9
8
t
im
e
s
t
h
a
t
o
f
I
80
,a
n
d
s
im
ila
r
ly
f
o
r
t
h
e
o
t
h
e
r
t
w
o
p
a
ir
s
o
f
g
r
a
p
h
s
.
C
o
m
b
in
in
g
a
ll
t
e
r
m
s
a
n
d
s
w
it
c
h
in
g
t
o
t
h
e
c
o
lo
r
-
t
r
a
c
e
b
a
s
is
,
w
e

n
d
t
h
a
t
t
h
e
U
V
d
iv
e
r
g
e
n
c
e
in
t
h
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
D
= 1
1
/
2
is
g
iv
e
n
by
,
A
(4
)
4
(1
,
2
,
3
,
4)
SU
(
N
c
)
po
l
e
=

6
g
10
K
N
2
c
N
2
c
V
1
+ 1
2
(
V
1
+ 2
V
2
+
V
8
)
(
4
.1
4
)
×
s
( T
r
1324
+ T
r
1423
) +
t
( T
r
1243
+ T
r
1342
) +
u
( T
r
1234
+ T
r
1432
)
,
33
up to overall factor,

divergence

same as for
N
=
4 SYM
part

H. Johansson, Frascati 2013

N

=4 SYM 5
-
loop Amplitude

N
=4 SYM important stepping stone to
N
=8 SG


1207.6666 [
hep
-
th
]

Bern, Carrasco, HJ,
Roiban


2
(410)
(404)
(335)
(184)
(284)
1
2
4
3
10
18
20
13
(1)
1
2
4
3
(370)
(12)
4
5
1
2
3
FI
G
.
1
:
S
a
m
p
l
e
g
ra
p
h
s
f
o
r
t
h
e

v
e
-
l
o
o
p
f
o
u
r-
p
o
i
n
t
N
= 4
sY
M
a
m
p
l
i
t
u
d
e
.
T
h
e
g
r
a
p
h
l
a
b
e
l
s
c
o
r
r
e
sp
o
n
d
t
o
t
h
e
o
n
e
s
us
e
d
i
n
t
he
a
nc
i
l
l
a
r
y
fil
e
[
2
3
]
.
t
h
r
e
e
-
lo
o
p
c
o
u
n
t
e
r
t
e
r
m
[2
0
];
t
h
e
c
o
e
f
f
i
c
ie
n
t
o
f
t
h
is
c
o
u
n
-
t
e
r
t
e
r
m
h
a
s
r
e
c
e
n
t
ly
b
e
e
n
e
x
p
lic
it
ly
s
h
o
w
n
t
o
v
a
n
is
h
[
1
2
].
(
S
e
e
r
e
f
.
[2
1
]
f
o
r
a
s
t
r
in
g
-
b
a
s
e
d
a
r
g
u
m
e
n
t
.)
T
h
is
e
x
h
ib
it
s
b
e
t
t
e
r
b
e
h
a
v
io
r
t
h
a
n
im
p
lie
d
b
y
k
n
o
w
n
s
y
m
m
e
t
r
y
c
o
n
-
s
id
e
r
a
t
io
n
s
a
n
d
is
in
lin
e
w
it
h
c
a
n
c
e
lla
t
io
n
s
s
u
g
g
e
s
t
e
d
b
y
u
n
it
a
r
it
y
a
r
g
u
m
e
n
t
s
[2
2
].
I
n
p
a
r
t
ic
u
la
r
,
it
e
m
p
h
a
s
iz
e
s
t
h
e
im
p
o
r
t
a
n
c
e
o
f
d
ir
e
c
t
ly
c
h
e
c
k
in
g
t
h
e
a
m
p
lit
u
d
e
s
w
h
e
t
h
e
r
e
q
.
(
1
)
h
o
ld
s
f
o
r
N
=
8
s
u
p
e
r
g
r
a
v
it
y
a
t
L
= 5
.
O
u
r
c
o
n
s
t
r
u
c
t
io
n
o
f
t
h
e

v
e
-
lo
o
p
f
o
u
r
-
p
o
in
t
a
m
p
lit
u
d
e
of
N
=
4
s
Y
M
t
h
e
o
r
y
o
r
g
a
n
iz
e
s
it
in
t
h
e
f
o
r
m
,
A
(5
)
4
=
i g
12
st
A
t re
e
4
S
4
416
i
= 1
9
j
= 5
d
D
l
j
(2
π
)
D
1
S
i
C
i
N
i
20
m
= 5
l
2
m
,
(2
)
w
h
e
r
e
t
h
e
s
e
c
o
n
d
s
u
m
r
u
n
s
o
v
e
r
a
s
e
t
o
f
4
1
6
d
is
t
in
c
t
(
n
o
n
-
is
o
m
o
r
p
h
ic
)
g
r
a
p
h
s
w
it
h
o
n
ly
c
u
b
ic
(
t
r
iv
a
le
n
t
)
v
e
r
-
t
ic
e
s
.
S
o
m
e
s
a
m
p
le
g
r
a
p
h
s
a
r
e
s
h
o
w
n
in

g
.
1
.
T
h
e

r
s
t
s
u
m
r
u
n
s
o
v
e
r
a
ll
2
4
p
e
r
m
u
t
a
t
io
n
s
o
f
e
x
t
e
r
n
a
l
le
g
la
b
e
ls
in
d
ic
a
t
e
d
b
y
S
4
.T
h
e
s
y
m
m
e
t
r
y
f
a
c
t
o
r
s
S
i
re
m
o
v
e
o
v
e
r
c
o
u
n
t
s
,
in
c
lu
d
in
g
t
h
o
s
e
a
r
is
in
g
f
r
o
m
in
t
e
r
n
a
l
a
u
t
o
-
m
o
r
p
h
is
m
s
y
m
m
e
t
r
ie
s
w
it
h
e
x
t
e
r
n
a
l
le
g
s

x
e
d
.
H
e
r
e
w
e
a
b
s
o
r
b
a
ll
c
o
n
t
a
c
t
t
e
r
m
s
(
i.e
.
t
e
r
m
s
w
it
h
f
e
w
e
r
t
h
a
n
t
h
e
m
a
x
im
u
m
n
u
m
b
e
r
o
f
p
r
o
p
a
g
a
t
o
r
s
)
in
t
o
g
r
a
p
h
s
w
it
h
o
n
ly
c
u
b
ic
v
e
r
t
ic
e
s
,
b
y
m
u
lt
ip
ly
in
g
a
n
d
d
iv
id
in
g
b
y
a
p
p
r
o
p
r
i-
at
e
p
r
op
agat
or
s
.
W
e
d
e
n
ot
e
e
x
t
e
r
n
al
m
om
e
n
t
a
b
y
k
i
fo
r
i
= 1
,...
,
4
a
n
d
t
h
e

v
e
in
d
e
p
e
n
d
e
n
t
lo
o
p
m
o
m
e
n
t
a
b
y
l
j
fo
r
j
= 5
,...,
9
.
T
h
e
r
e
m
a
in
in
g
l
j
a
r
e
lin
e
a
r
c
o
m
b
i-
n
a
t
io
n
s
o
f
t
h
e
s
e
.
T
h
e
c
o
lo
r
f
a
c
t
o
r
s
C
i
o
f
a
ll
g
r
a
p
h
s
a
r
e
o
b
t
a
in
e
d
b
y
d
r
e
s
s
in
g
e
v
e
r
y
t
h
r
e
e
-
v
e
r
t
e
x
in
t
h
e
g
r
a
p
h
w
it
h
a f
a
c
t
o
r
o
f
˜
f
ab
c
= T
r
(
[
T
a
,T
b
]
T
c
),
w
h
e
r
e
t h
e
g
a
u
g
e
g
r
o
u
p
ge
n
e
r
at
or
s
T
a
a
r
e
n
o
r
m
a
liz
e
d
a
s
T
r
(
T
a
T
b
) =
δ
ab
.T
h
e
g
a
u
g
e
c
o
u
p
lin
g
is
g
a
n
d
t
h
e
c
r
o
s
s
in
g
s
y
m
m
e
t
r
ic
p
r
e
f
a
c
-
t o
r
st
A
t re
e
4
is
in
t
e
r
m
s
o
f
t
h
e
c
o
lo
r
-
o
r
d
e
r
e
d
D
-
d
im
e
n
s
io
n
a
l
t
r
e
e
a
m
p
lit
u
d
e
A
t re
e
4

A
t re
e
4
(1
,
2
,
3
,
4)
an
d
s
= (
k
1
+
k
2
)
2
an
d
t
= (
k
2
+
k
3
)
2
.
To
c
o
n
s
t
r
u
c
t
t
h
e
n
u
m
e
r
a
t
o
r
s
N
i
,w
e
u
s
e
t
h
e
m
e
t
h
o
d
o
f
m
a
x
im
a
l
c
u
t
s
[8
],
b
a
s
e
d
o
n
t
h
e
u
n
it
a
r
it
y
m
e
t
h
o
d
[
2
4
].
A
p
-
p
lic
a
t
io
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id
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r
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le
d
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t
a
il
in
r
e
f
.
[6
],
s
o
h
e
r
e
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e
g
iv
e
o
n
ly
a
b
r
ie
f
s
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m
m
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.
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h
e
m
e
t
h
o
d
w
o
r
k
s
in
D
d
i-
N
3
MC
NM
C
MC
N
2
MC
FI
G
.
2
:
S
a
m
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l
e
N
k
-m
a
x
i
m
a
l
c
u
t
s
f
o
r
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= 0
,
1
,
2
,
3.
T
h
e
e
x
-
po
s
e
d
l
i
n
e
s
a
r
e
a
l
l
c
u
t
.
(b
)
(a
)
=
s
57
s
36
s
67
s
38
(c
)
3
5
6
7
1
2
4
8
3
5
6
7
1
2
4
8
FI
G
.
3
:
E
x
a
m
p
l
e
s
o
f
s
i
m
p
l
e
cu
t
s
u
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e
d
t
o
s
p
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d
u
p
t
h
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ca
l
cu
-
lat
ion
.
(
a)
is
a
t
w
o
p
ar
t
ic
le
c
u
t
,
(
b
)
a
b
o
x
c
u
t
an
d
(
c
)
is
a
sa
m
p
l
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a
p
p
l
i
c
a
t
io
n
o
f
B
C
J
a
m
p
li
t
u
d
e
r
e
la
t
i
o
n
s.
T
h
e
e
x
p
o
se
d
lin
e
s
ar
e
all
c
u
t
.
m
e
n
s
io
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s
a
n
d
c
a
n
b
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u
s
e
d
t
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o
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t
a
in
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c
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s
,
f
r
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m
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h
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V
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c
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h
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t r
a
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t e
d
.
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e
s
t
a
r
t
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a
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s
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t
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d
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f
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t
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t
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d
.
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e
n
o
t
e
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h
a
t
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t
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n
s
f
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n
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u
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t
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c
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s
f
o
r
m
a
t
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n
s
[9
,
1
0
,
2
5
].
T
h
e
p
a
r
a
m
e
t
e
r
s
o
f
t
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e
a
n
s
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A
t re
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(1
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A
t re
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(2
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· · ·
A
t re
e
(
m
)
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h
e
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t
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d
it
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n
s
.
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e
s
t
a
r
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f
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o
m
t
h
e
m
a
x
im
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t
s
(
M
C
s
)
w
h
e
r
e
a
ll
1
6
in
t
e
r
n
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p
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t
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f
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(
N
M
C
s
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,
w
it
h
1
5
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t
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t
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M
C
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o
r
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e

v
e
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lo
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r
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in
t
N
=
4
s
Y
M
a
m
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lit
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c
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s
s
t
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a
t
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s
a
t
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,
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t
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s
f
r
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m
c
a
n
-
2
(410)
(404)
(335)
(184)
(284)
1
2
4
3
10
18
20
13
(1)
1
2
4
3
(370)
(12)
4
5
1
2
3
FI
G
.
1
:
S
a
m
p
l
e
g
ra
p
h
s
f
o
r
t
h
e

v
e
-
l
o
o
p
f
o
u
r-
p
o
i
n
t
N
= 4
sY
M
a
m
p
l
i
t
u
d
e
.
T
h
e
g
r
a
p
h
l
a
b
e
l
s
c
o
r
r
e
sp
o
n
d
t
o
t
h
e
o
n
e
s
us
e
d
i
n
t
he
a
nc
i
l
l
a
r
y
fil
e
[
2
3
]
.
t
h
r
e
e
-
lo
o
p
c
o
u
n
t
e
r
t
e
r
m
[2
0
];
t
h
e
c
o
e
f
f
i
c
ie
n
t
o
f
t
h
is
c
o
u
n
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t
e
r
t
e
r
m
h
a
s
r
e
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e
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t
ly
b
e
e
n
e
x
p
lic
it
ly
s
h
o
w
n
t
o
v
a
n
is
h
[
1
2
].
(
S
e
e
r
e
f
.
[2
1
]
f
o
r
a
s
t
r
in
g
-
b
a
s
e
d
a
r
g
u
m
e
n
t
.)
T
h
is
e
x
h
ib
it
s
b
e
t
t
e
r
b
e
h
a
v
io
r
t
h
a
n
im
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lie
d
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k
n
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m
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d
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e
w
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a
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t
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s
s
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b
y
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n
it
a
r
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y
a
r
g
u
m
e
n
t
s
[2
2
].
I
n
p
a
r
t
ic
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la
r
,
it
e
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p
h
a
s
iz
e
s
t
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e
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p
o
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a
n
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ly
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h
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c
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t
h
e
a
m
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lit
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d
e
s
w
h
e
t
h
e
r
e
q
.
(
1
)
h
o
ld
s
f
o
r
N
=
8
s
u
p
e
r
g
r
a
v
it
y
a
t
L
= 5
.
O
u
r
c
o
n
s
t
r
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c
t
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o
f
t
h
e

v
e
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lo
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p
f
o
u
r
-
p
o
in
t
a
m
p
lit
u
d
e
of
N
=
4
s
Y
M
t
h
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o
r
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g
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n
iz
e
s
it
in
t
h
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f
o
r
m
,
A
(5
)
4
=
i g
12
st
A
t re
e
4
S
4
416
i
= 1
9
j
= 5
d
D
l
j
(2
π
)
D
1
S
i
C
i
N
i
20
m
= 5
l
2
m
,
(2
)
w
h
e
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t
h
e
s
e
c
o
n
d
s
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m
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4
1
6
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c
t
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o
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o
m
o
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p
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r
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ly
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u
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r
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n
t
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s
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o
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a
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r
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h
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in

g
.
1
.
T
h
e

r
s
t
s
u
m
r
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s
o
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r
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ll
2
4
p
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4
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h
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y
m
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e
t
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re
m
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x
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d
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e
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a
b
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c
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c
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r
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s
(
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t
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r
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s
w
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a
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s
)
in
t
o
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r
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s
w
it
h
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n
ly
c
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r
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,
b
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u
lt
ip
ly
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b
y
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p
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at
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s
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e
d
e
n
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t
e
r
n
al
m
om
e
n
t
a
b
y
k
i
fo
r
i
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,...
,
4
a
n
d
t
h
e

v
e
in
d
e
p
e
n
d
e
n
t
lo
o
p
m
o
m
e
n
t
a
b
y
l
j
fo
r
j
= 5
,...,
9
.
T
h
e
r
e
m
a
in
in
g
l
j
a
r
e
lin
e
a
r
c
o
m
b
i-
n
a
t
io
n
s
o
f
t
h
e
s
e
.
T
h
e
c
o
lo
r
f
a
c
t
o
r
s
C
i
o
f
a
ll
g
r
a
p
h
s
a
r
e
o
b
t
a
in
e
d
b
y
d
r
e
s
s
in
g
e
v
e
r
y
t
h
r
e
e
-
v
e
r
t
e
x
in
t
h
e
g
r
a
p
h
w
it
h
a f
a
c
t
o
r
o
f
˜
f
ab
c
= T
r
(
[
T
a
,T
b
]
T
c
),
w
h
e
r
e
t h
e
g
a
u
g
e
g
r
o
u
p
ge
n
e
r
at
or
s
T
a
a
r
e
n
o
r
m
a
liz
e
d
a
s
T
r
(
T
a
T
b
) =
δ
ab
.T
h
e
g
a
u
g
e
c
o
u
p
lin
g
is
g
a
n
d
t
h
e
c
r
o
s
s
in
g
s
y
m
m
e
t
r
ic
p
r
e
f
a
c
-
t o
r
st
A
t re
e
4
is
in
t
e
r
m
s
o
f
t
h
e
c
o
lo
r
-
o
r
d
e
r
e
d
D
-
d
im
e
n
s
io
n
a
l
t
r
e
e
a
m
p
lit
u
d
e
A
t re
e
4

A
t re
e
4
(1
,
2
,
3
,
4)
an
d
s
= (
k
1
+
k
2
)
2
an
d
t
= (
k
2
+
k
3
)
2
.
To
c
o
n
s
t
r
u
c
t
t
h
e
n
u
m
e
r
a
t
o
r
s
N
i
,w
e
u
s
e
t
h
e
m
e
t
h
o
d
o
f
m
a
x
im
a
l
c
u
t
s
[8
],
b
a
s
e
d
o
n
t
h
e
u
n
it
a
r
it
y
m
e
t
h
o
d
[
2
4
].
A
p
-
p
lic
a
t
io
n
o
f
t
h
is
m
e
t
h
o
d
a
n
d
v
a
r
io
u
s
s
t
r
a
t
e
g
ie
s
f
o
r
g
r
e
a
t
ly
s
t
r
e
a
m
lin
in
g
t
h
e
c
o
n
s
t
r
u
c
t
io
n
o
f
t
h
e
n
u
m
e
r
a
t
o
r
s
h
a
s
b
e
e
n
d
e
s
c
r
ib
e
d
in
c
o
n
s
id
e
r
a
b
le
d
e
t
a
il
in
r
e
f
.
[6
],
s
o
h
e
r
e
w
e
g
iv
e
o
n
ly
a
b
r
ie
f
s
u
m
m
a
r
y
.
T
h
e
m
e
t
h
o
d
w
o
r
k
s
in
D
d
i-
N
3
MC
NM
C
MC
N
2
MC
FI
G
.
2
:
S
a
m
p
l
e
N
k
-m
a
x
i
m
a
l
c
u
t
s
f
o
r
k
= 0
,
1
,
2
,
3.
T
h
e
e
x
-
po
s
e
d
l
i
n
e
s
a
r
e
a
l
l
c
u
t
.
(b
)
(a
)
=
s
57
s
36
s
67
s
38
(c
)
3
5
6
7
1
2
4
8
3
5
6
7
1
2
4
8
FI
G
.
3
:
E
x
a
m
p
l
e
s
o
f
s
i
m
p
l
e
cu
t
s
u
s
e
d
t
o
s
p
e
e
d
u
p
t
h
e
ca
l
cu
-
lat
ion
.
(
a)
is
a
t
w
o
p
ar
t
ic
le
c
u
t
,
(
b
)
a
b
o
x
c
u
t
an
d
(
c
)
is
a
sa
m
p
l
e
a
p
p
l
i
c
a
t
io
n
o
f
B
C
J
a
m
p
li
t
u
d
e
r
e
la
t
i
o
n
s.
T
h
e
e
x
p
o
se
d
lin
e
s
ar
e
all
c
u
t
.
m
e
n
s
io
n
s
a
n
d
c
a
n
b
e
u
s
e
d
t
o
o
b
t
a
in
lo
c
a
l
e
x
p
r
e
s
s
io
n
s
,
f
r
o
m
w
h
ic
h
U
V
d
iv
e
r
g
e
n
c
e
s
c
a
n
b
e
s
t
r
a
ig
h
t
f
o
r
w
a
r
d
ly
e
x
-
t r
a
c
t e
d
.
W
e
s
t
a
r
t
w
it
h
a
n
a
n
s
a
t
z
f
o
r
t
h
e
d
ia
g
r
a
m
n
u
m
e
r
a
t
o
r
s
c
o
n
t
a
in
in
g
f
r
e
e
p
a
r
a
m
e
t
e
r
s
t
o
b
e
d
e
t
e
r
m
in
e
d
b
y
m
a
t
c
h
in
g
a
g
a
in
s
t
g
e
n
e
r
a
liz
e
d
u
n
it
a
r
it
y
c
u
t
s
.
O
u
r
a
n
s
a
t
z
is
a
p
o
ly
-
n
o
m
ia
l
o
f
d
e
g
r
e
e
f
o
u
r
in
t
h
e
k
in
e
m
a
t
ic
in
v
a
r
ia
n
t
s
,
s
u
b
j
e
c
t
t
o
t
h
e
p
o
w
e
r
-
c
o
u
n
t
in
g
c
o
n
s
t
r
a
in
t
t
h
a
t
n
o
t
e
r
m
h
a
s
m
o
r
e
t
h
a
n
s
ix
p
o
w
e
r
s
o
f
lo
o
p
m
o
m
e
n
t
u
m
.
W
e
a
ls
o
d
e
m
a
n
d
t
h
a
t
e
a
c
h
n
u
m
e
r
a
t
o
r
r
e
s
p
e
c
t
s
t
h
e
a
u
t
o
m
o
r
p
h
is
m
s
y
m
m
e
-
t
r
ie
s
o
f
t
h
e
g
r
a
p
h
.
O
n
c
e
a
s
o
lu
t
io
n
is
f
o
u
n
d
s
a
t
is
f
y
in
g
a
c
o
m
p
le
t
e
s
e
t
o
f
c
u
t
c
o
n
d
it
io
n
s
,
w
e
h
a
v
e
t
h
e
in
t
e
g
r
a
n
d
.
I
f
a
n
in
c
o
n
s
is
t
e
n
c
y
is
e
n
c
o
u
n
t
e
r
e
d
,
t
h
e
a
n
s
a
t
z
m
u
s
t
b
e
e
n
la
r
g
e
d
.
W
e
n
o
t
e
t
h
a
t
t
h
e
s
o
lu
t
io
n
s
f
o
r
n
u
m
e
r
a
t
o
r
s
a
r
e
n
o
t
u
n
iq
u
e
a
n
d
d
i
f
f
e
r
e
n
t
c
h
o
ic
e
s
c
a
n
b
e
m
a
p
p
e
d
in
t
o
e
a
c
h
o
t
h
e
r
b
y
g
e
n
e
r
a
liz
e
d
g
a
u
g
e
t
r
a
n
s
f
o
r
m
a
t
io
n
s
[9
,
1
0
,
2
5
].
T
h
e
p
a
r
a
m
e
t
e
r
s
o
f
t
h
e
a
n
s
a
t
z
a
r
e
d
e
t
e
r
m
in
e
d
f
r
o
m
g
e
n
-
e
r
a
liz
e
d
u
n
it
a
r
it
y
c
u
t
s
t
h
a
t
d
e
c
o
m
p
o
s
e
a
lo
o
p
in
t
e
g
r
a
n
d
in
t
o
p
r
o
d
u
c
t
s
o
f
o
n
-
s
h
e
ll
t
r
e
e
a
m
p
lit
u
d
e
s
s
u
m
m
e
d
o
v
e
r
a
ll
in
t
e
r
m
e
d
ia
t
e
s
t
a
t
e
s
,
st
a
t
e
s
A
t re
e
(1
)
A
t re
e
(2
)
···
A
t re
e
(
m
)
.T
h
e
s
e
c
u
t
s
a
r
e
o
r
g
a
n
iz
e
d
a
c
c
o
r
d
in
g
t
o
t
h
e
n
u
m
b
e
r
o
f
c
u
t
p
r
o
p
-
a
g
a
t
o
r
s
t
h
a
t
a
r
e
r
e
p
la
c
e
d
w
it
h
o
n
-
s
h
e
ll
c
o
n
d
it
io
n
s
.
W
e
s
t
a
r
t
f
r
o
m
t
h
e
m
a
x
im
a
l
c
u
t
s
(
M
C
s
)
w
h
e
r
e
a
ll
1
6
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
c
u
t
.
A
f
t
e
r
o
b
t
a
in
in
g
t
h
e
M
C
s
,
w
e
t
h
e
n
c
o
n
-
s
t
r
u
c
t
s
a
ll
n
e
x
t
-
t
o
-
m
a
x
im
a
l
c
u
t
s
(
N
M
C
s
)
,
w
it
h
1
5
c
u
t
p
r
o
p
a
g
a
t
o
r
s
.
W
e
c
o
n
t
in
u
e
t
h
is
p
r
o
c
e
s
s
,
s
y
s
t
e
m
a
t
ic
a
lly
c
o
n
s
t
r
u
c
t
in
g
a
n
a
ly
t
ic
e
x
p
r
e
s
s
io
n
s
f
o
r
(
n
e
x
t
-
t
o
)
k
-
m
a
x
im
a
l
cu
t
s
(
N
k
M
C
s
)
w
it
h
f
e
w
e
r
a
n
d
f
e
w
e
r
im
p
o
s
e
d
c
u
t
c
o
n
d
i-
t
io
n
s
.
F
o
r
t
h
e

v
e
-
lo
o
p
f
o
u
r
-
p
o
in
t
N
=
4
s
Y
M
a
m
p
lit
u
d
e
t
h
is
p
r
o
c
e
s
s
t
e
r
m
in
a
t
e
s
a
t
k
=
3
,
s
in
c
e
t
h
e
p
o
w
e
r
c
o
u
n
t
-
in
g
o
f
t
h
e
t
h
e
o
r
y
p
r
e
v
e
n
t
s
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
f
r
o
m
c
a
n
-

416 integral topologies:




Used maximal cut method


Bern, Carrasco, HJ,
Kosower





Maximal cuts: 410


Next
-
to
-
MC: 2473


N
2
MC: 7917


N
3
MC: 15156


Unitarity cuts done in
D

dimensions...integrated UV div. in
D
=26/5

H. Johansson, Frascati 2013

N

=4 SYM 5
-
loop UV divergence

Non
-
Planar UV divergence in
D
=26/5:

Double traces and single
-
trace NNLC finite in
D
=26/5,

only single
-
trace LC and NLC are divergent

Vanish!

4
s
e
t
o
f
h
o
m
o
g
e
n
e
o
u
s
c
o
n
s
is
t
e
n
c
y
e
q
u
a
t
io
n
s
.
T
h
e
f
a
c
t
t
h
a
t
n
o
p
o
s
it
iv
e
d
e

n
it
e
in
t
e
g
r
a
l
is
s
e
t
t
o
z
e
r
o
b
y
t
h
is
s
y
s
t
e
m
is
a
s
t
r
o
n
g
c
h
e
c
k
o
n
t
h
e
c
a
lc
u
la
t
io
n
.
T
h
e
s
e
c
o
n
s
is
t
e
n
c
y
r
e
la
t
io
n
s
e
lim
in
a
t
e
m
o
s
t
o
f
t
h
e
v
a
c
u
u
m
d
ia
g
r
a
m
s
.
T
w
o
e
x
a
m
p
le
s
a
r
e
,
V
(j
)
=
24
5
V
(a
)

2
V
(d
)
,
V
(b
)
= 2
V
(c
)
+ 3
5
V
(i
)
+
365
6
V
(d
)

4175
162
V
(e
)

1045
18
V
(f
)

9865
81
V
(g
)
+
305
3
V
(h
)
,
w
h
e
r
e
t
h
e
la
b
e
ls
c
o
r
r
e
s
p
o
n
d
t
o
t
h
e
o
n
e
s
in

g
.
4
.
A
f
t
e
r
u
s
in
g
t
h
e
c
o
n
s
is
t
e
n
c
y
r
e
la
t
io
n
s
,
t
h
e
le
a
d
in
g
U
V
d
iv
e
r
g
e
n
c
e
is
r
e
m
a
r
k
a
b
ly
s
im
p
le
a
n
d
g
iv
e
n
b
y
o
n
ly
t
h
r
e
e
v
a
c
u
u
m
in
t
e
g
r
a
ls
.
F
o
r
SU
(
N
c
)
,
it
is
A
(5
)
4
di v
=

144
5
g
12
st
A
t re
e
4
N
3
c
N
2
c
V
(a
)
+ 1
2
(
V
(a
)
+ 2
V
(b
)
+
V
(c
)
)
×
(
t
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
+
s
˜
f
a
2
a
3
b
˜
f
ba
4
a
1
)
.
(3
)
W
it
h
t
h
e
c
h
o
s
e
n
n
o
r
m
a
liz
a
t
io
n
,
t
h
e
W
ic
k
r
o
t
a
t
e
d
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
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[ar
X
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:1008.3327
[h
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Z
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Be
r
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J
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J
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M.
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ar
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as
c
o,
L
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ix
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(
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X
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]].
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.
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r
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,
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J
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ar
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c
o,
H
.
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A
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X
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Z
.
Be
r
n
,
J
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J
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M.
C
ar
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as
c
o
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oh
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on
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s
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v
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a
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v
:
0
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3
9
9
3
[
h
e
p
-
p
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]
]
.
[10]
Z
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Be
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n
,
J
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J
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M.
C
ar
r
as
c
o
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d
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t
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[
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[11]
Z
.
Be
r
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C
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c
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V
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on
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oh
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on
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s
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c
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t
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q
u
a
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.
T
h
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f
a
c
t
t
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a
t
n
o
p
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it
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d
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n
it
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r
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l
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s
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h
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t
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la
t
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t
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a
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u
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ia
g
r
a
m
s
.
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w
o
e
x
a
m
p
le
s
a
r
e
,
V
(j
)
=
24
5
V
(a
)

2
V
(d
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,
V
(b
)
= 2
V
(c
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+ 3
5
V
(i
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+
365
6
V
(d
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4175
162
V
(e
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1045
18
V
(f
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9865
81
V
(g
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+
305
3
V
(h
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,
w
h
e
r
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t
h
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la
b
e
ls
c
o
r
r
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s
p
o
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d
t
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t
h
e
o
n
e
s
in

g
.
4
.
A
f
t
e
r
u
s
in
g
t
h
e
c
o
n
s
is
t
e
n
c
y
r
e
la
t
io
n
s
,
t
h
e
le
a
d
in
g
U
V
d
iv
e
r
g
e
n
c
e
is
r
e
m
a
r
k
a
b
ly
s
im
p
le
a
n
d
g
iv
e
n
b
y
o
n
ly
t
h
r
e
e
v
a
c
u
u
m
in
t
e
g
r
a
ls
.
F
o
r
SU
(
N
c
)
,
it
is
A
(5
)
4
di
v
=

144
5
g
12
st
A
t re
e
4
N
3
c
N
2
c
V
(a
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+ 1
2
(
V
(a
)
+ 2
V
(b
)
+
V
(c
)
)
×
(
t
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
+
s
˜
f
a
2
a
3
b
˜
f
ba
4
a
1
)
.
(3
)
W
it
h
t
h
e
c
h
o
s
e
n
n
o
r
m
a
liz
a
t
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n
,
t
h
e
W
ic
k
r
o
t
a
t
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v
a
c
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u
m
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r
a
ls
in
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q
.
(
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a
r
e
a
ll
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o
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e

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it
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,
p
r
o
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a
t
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d
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c
a
n
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lla
t
io
n
s
r
e
m
a
in
a
t
L
=
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in
t
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r
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l
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im
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o
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le
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d
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o
r
s
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b
le
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d
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g
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c
o
lo
r
c
o
n
t
r
ib
u
t
io
n
s
.
U
s
in
g
FI
E
S
T
A
[
2
8
]
w
e
h
a
v
e
n
u
m
e
r
ic
a
lly
e
v
a
lu
a
t
e
d
t
h
e
in
t
e
g
r
a
ls
g
iv
in
g
,
V
(a
)
=
0
.
331
K
,V
(b
)
=
0
.
310
K
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(c
)
=
0
.
291
K
,
w
h
e
r
e
t
h
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d
im
e
n
s
io
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a
l
r
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iz
a
t
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p
a
r
a
m
e
t
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is

(2
6
/
5

D
)
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2,
K
= 1
/
(4
π
)
13
a
n
d
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m
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ic
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in
t
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g
r
a
t
io
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u
n
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e
r
t
a
in
t
ie
s
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r
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lo
w
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h
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d
is
p
la
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e
d
d
ig
it
s
.
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t
is
in
t
e
r
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e
s
t
in
g
t
h
a
t
t
h
e
r
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t
io
b
e
t
w
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t
h
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b
le
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d
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g
a
n
d
le
a
d
in
g
c
o
n
t
r
ib
u
t
io
n
s
4
5
.
0
/N
2
c
is
r
a
t
h
e
r
c
lo
s
e
t
o
t
h
e
t
h
r
e
e
-
a
n
d
f
o
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r
-
lo
o
p
r
a
t
io
s
,
4
3
.
3
/N
2
c
an
d
44
.
4
/N
2
c
[6
].
A
s
t
r
ik
in
g
f
e
a
t
u
r
e
o
f
t
h
e
r
e
s
u
lt
(
3
)
is
t
h
a
t
t
h
e
d
iv
e
r
g
e
n
c
e
d
o
e
s
n
o
t
c
o
n
t
a
in
t
e
r
m
s
b
e
y
o
n
d
O
(1
/N
2
c
)
s
u
p
p
r
e
s
s
io
n
,
n
o
r
d
o
e
s
it
c
o
n
t
a
in
d
o
u
b
le
-
t
r
a
c
e
c
o
n
t
r
ib
u
t
io
n
s
w
h
e
n
c
o
n
v
e
r
t
e
d
t
o
a
n
SU
(
N
c
)
c
o
lo
r
-
t
r
a
c
e
r
e
p
r
e
s
e
n
t
a
t
io
n
,
in
lin
e
w
it
h
e
x
p
e
c
t
a
-
t
io
n
s
f
r
o
m
lo
w
e
r
lo
o
p
s
[6
].
T
h
e
s
e
c
o
n
d
o
f
t
h
e
s
e
f
e
a
t
u
r
e
s
h
a
s
a
lr
e
a
d
y
b
e
e
n
d
is
c
u
s
s
e
d
in
r
e
f
s
.
[
6
,
2
9
].
F
u
r
t
h
e
r
m
o
r
e
,
t
h
e
t
h
r
e
e
in
t
e
g
r
a
ls
a
n
d
t
h
e
ir
r
e
la
t
iv
e
c
o
e
f
f
i
c
ie
n
t
s
h
a
v
e
a
r
e
m
a
r
k
a
b
le
s
im
ila
r
it
y
w
it
h
t
h
e
c
o
r
r
e
s
p
o
n
d
in
g
o
n
e
s
a
t
f
o
u
r
lo
o
p
s
,
a
s
s
e
e
n
b
y
c
o
m
p
a
r
in
g
t
o
e
q
.
(
5
.3
3
)
o
f
r
e
f
.
[6
].
A
t
lo
w
e
r
lo
o
p
s
,
e
x
a
c
t
ly
t
h
e
s
a
m
e
c
o
m
b
in
a
t
io
n
o
f
in
t
e
g
r
a
ls
a
p
p
e
a
r
in
g
in
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
c
o
n
t
r
ib
u
t
io
n
s
t
o
t
h
e
N
=
4
s
Y
M
d
iv
e
r
g
e
n
c
e
s
a
p
p
e
a
r
in
t
h
e
c
o
r
r
e
s
p
o
n
d
in
g
on
e
s
of
N
=
8
s
u
p
e
r
g
r
a
v
it
y
[6
].
A
n
a
t
u
r
a
l
c
o
n
j
e
c
t
u
r
e
is
t
h
a
t
t
h
e
s
a
m
e
h
o
ld
s
a
t

v
e
lo
o
p
s
,
s
o
t
h
a
t
t
h
e
t
w
o
t
h
e
o
r
ie
s
s
h
a
r
e
t
h
e
s
a
m
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
= 2
6
/
5.
I
n
s
u
m
m
a
r
y
,
t
h
e

v
e
-
lo
o
p
a
m
p
lit
u
d
e
w
e
h
a
v
e
c
o
n
-
s
t
r
u
c
t
e
d
h
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r
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o
f
f
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r
s
d
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t
a
ile
d
in
f
o
r
m
a
t
io
n
o
n
t
h
e
s
t
r
u
c
-
t
u
r
e
o
f
t
h
e
n
o
n
p
la
n
a
r
s
e
c
t
o
r
o
f
N
= 4
s
Y
M
t
h
e
o
r
y
.
A
s
a

r
s
t
a
p
p
lic
a
t
io
n
,
w
e
h
a
v
e
s
h
o
w
n
t
h
a
t
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im
p
le
p
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in
t
h
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d
im
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n
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io
n
w
h
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r
e
t
h
e
y

r
s
t
a
p
p
e
a
r
c
o
n
t
in
u
e
t
o
h
o
ld
t
h
r
o
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g
h

v
e
lo
o
p
s
;
t
h
is
h
in
t
s
t
h
a
t
t
h
e
d
i-
v
e
r
g
e
n
c
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s
a
r
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c
o
n
t
r
o
lle
d
b
y
a
d
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e
p
s
t
r
u
c
t
u
r
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o
f
t
h
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t
h
e
o
r
y
.
O
u
r
c
o
n
s
t
r
u
c
t
io
n
o
f
t
h
e

v
e
-
lo
o
p
f
o
u
r
-
p
o
in
t
a
m
p
lit
u
d
e
is
a
n
e
x
c
e
lle
n
t
s
t
a
r
t
in
g
p
o
in
t
t
o
t
r
y
t
o

n
d
a
r
e
p
r
e
s
e
n
t
a
t
io
n
e
x
h
ib
it
in
g
t
h
e
d
u
a
lit
y
b
e
t
w
e
e
n
c
o
lo
r
a
n
d
k
in
e
m
a
t
ic
s
.
W
e
e
x
p
e
c
t
t
h
a
t
t
h
e
r
e
s
u
lt
s
p
r
e
s
e
n
t
e
d
h
e
r
e
w
ill
b
e
c
r
u
c
ia
l
in
-
p
u
t
f
o
r
o
b
t
a
in
in
g
c
o
r
r
e
s
p
o
n
d
in
g
s
u
p
e
r
g
r
a
v
it
y
a
m
p
lit
u
d
e
s
a
n
d
f
o
r
s
t
u
d
y
in
g
t
h
e
ir
U
V
b
e
h
a
v
io
r
.
W
e
t
h
a
n
k
S
.
D
a
v
ie
s
,
T
.
L
.
D
e
n
n
e
n
,
S
.
F
e
r
r
a
r
a
,
Y
.-
t
.
H
u
a
n
g
,
R
.
K
a
llo
s
h
,
D
.
A
.
K
o
s
o
w
e
r
,
A
.
T
s
e
y
t
lin
,
V
.
A
.
S
m
ir
n
o
v
a
n
d
K
.
S
t
e
lle
f
o
r
h
e
lp
f
u
l
d
is
c
u
s
s
io
n
s
.
W
e
e
s
p
e
c
ia
lly
t
h
a
n
k
L
.J
.
D
ix
o
n
f
o
r
c
o
lla
b
o
r
a
t
io
n
o
n
r
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la
t
e
d
t
o
p
ic
s
a
n
d
f
o
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m
a
n
y
h
e
lp
f
u
l
d
is
c
u
s
s
io
n
s
a
n
d
e
n
c
o
u
r
a
g
e
-
m
e
n
t
.
T
h
is
r
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s
e
a
r
c
h
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s
s
u
p
p
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.
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o,
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.
B
401
,
273
(
1997)
[ar
X
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:
h
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p
-
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/9702424
]
.
[3]
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.
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.
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.
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].
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.
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,
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85
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2012)
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X
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:1106.
4711
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]]
.
[6]
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[11]
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s
.
Bern, Carrasco, HJ,
Roiban


H. Johansson, Frascati 2013

Testing
D
=5 intercept

Plot of critical dimensions of
N

= 8 SUGRA

and
N

= 4 SYM

Known bound for
N

= 4

Bern, Dixon, Dunbar,
Rozowsky
,
Perelstein
; Howe,
Stelle

current trend for
N

= 8

If
N

= 8 div. at
L
=7

calculations:

L
= 7

lowest loop order for possible
D

= 4

divergence

Beisert
,
Elvang
, Freedman,
Kiermaier
, Morales,
Stieberger
;

Björnsson
, Green,
Bossard
, Howe,
Stelle
,
Vanhove

Kallosh
,
Ramond
,
Lindström
,
Berkovits
,
Grisaru
,
Siegel, Russo, and more….


1
-
2 loops:
Green, Schwarz, Brink; Marcus and
Sagnotti


3
-
5 loops:
Bern, Carrasco, Dixon, HJ,
Kosower
,
Roiban

6 loops:
Bern, Carrasco, Dixon, Douglas, HJ, von
Hippel

H. Johansson, Frascati 2013

6
-
Loop Planar
D
=5 SYM

(1
)
(2
)
(3
)
(4
)
(5
)
(6
)
(7
)
(8
)
(9
)
(1
0
)
(1
5
)
(1
3
)
(1
1
)
(1
4
)
(1
2
)
(2
0
)
(1
8
)
(1
6
)
(1
7
)
(1
9
)
(2
5
)
(2
3
)
(2
1
)
(2
4
)
(2
2
)
(3
0
)
(2
8
)
(2
6
)
(2
9
)
(2
7
)
(3
5
)
(3
3
)
(3
1
)
(3
2
)
(3
4
)
FI
G.
1
:
Gr
a
p
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s
1
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3
5
f
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s
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s
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nf
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f
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s
w
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t
a
n
d
a
r
d
[3
2
]
d
u
a
l
v
a
r
ia
b
le
s
x
i

x
j
=
x
i j
,
w
it
h
x
41
=
k
1
,x
12
=
k
2
,x
23
=
k
3
,x
34
=
k
4
,
(
2
.5
)
wh
e
r
e
k
i
a
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e
t
h
e
e
x
t
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r
n
a
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m
o
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t
a
.
A
s
d
is
c
u
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e
d
in
d
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t
a
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in
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f
.
[1
2
],
a
p
r
a
c
t
ic
a
l
w
a
y
o
f
e
x
p
r
e
s
s
in
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in
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r
n
a
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t
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d
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p
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c
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h
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d
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c
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c
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c
u
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g
s
.
A
t
s
ix
lo
o
p
s
,
w
e
c
o
n
s
id
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r
a
s
e
v
e
n
-
p
a
r
t
ic
le
c
u
t
in
t
h
e
s
= (
k
1
+
k
2
)
2
c
h
a
n
n
e
l.
T
h
e
s
e
v
e
n
c
u
t
le
g
s
c
a
r
r
y
m
o
m
e
n
t
a
p
5
,p
6
,...
,p
11
.
T
h
e
s
ix
d
u
a
l
lo
o
p
m
o
m
e
n
t
a
x
5
,x
6
,...
,x
10
ar
e
t
h
e
n
d
e

n
e
d
b
y
,
x
45
=
p
5
,x
56
=
p
6
,x
67
=
p
7
,x
78
=
p
8
,x
89
=
p
9
,x
9
,
10
=
p
10
.
(
2
.6
)
T
h
e
k
e
y
d
u
a
l
c
o
n
f
o
r
m
a
l
p
r
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p
e
r
t
ie
s
f
o
llo
w
f
r
o
m
t
h
e
b
e
h
a
v
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r
o
f
t
h
e
in
t
e
g
r
a
n
d
u
n
d
e
r
d
u
a
l
6
(3
6
)
(3
7
)
(3
8
)
(3
9
)
(4
0
)
(4
1
)
(4
2
)
(4
4
)
(4
5
)
(4
6
)
(4
7
)
(4
3
)
(4
8
)
(4
9
)
(5
0
)
(5
1
)
(5
2
)
(5
3
)
(5
4
)
(5
5
)
(5
6
)
(5
7
)
(5
8
)
(5
9
)
(6
0
)
(6
1
)
(6
2
)
(6
3
)
(6
4
)
(6
5
)
(6
6
)
(6
7
)
(6
8
)
FI
G.
2
:
Gr
a
p
h
s
3
6
t h
r
o
u
g
h
6
8
f
o
r
t h
e
p
l
a
n
a
r
s
i
x
-l
o
o
p
f
o
u
r
-p
o
i
nt
a
m
p
l
i
t
u
d
e
.
c
o
o
r
d
in
a
t
e
in
v
e
r
s
io
n
,
w
h
ic
h
m
a
p
s
x
µ
i

x
µ
i
x
2
i
,x
2
i j

x
2
i j
x
2
i
x
2
j
.
(
2
.7
)
I
n
f
o
u
r
d
im
e
n
s
io
n
s
,
d
u
a
l
c
o
n
f
o
r
m
a
l
in
v
a
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c
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q
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a
t
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c
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in
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r
a
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d
s
c
a
le
s
a
s
[3
2
]
I
i

4
j
= 1
x
2
j
10
l
= 5
(
x
2
l
)
4
I
i
.
(
2
.8
)
T
h
e
in
t
e
g
r
a
n
d
s
o
f
p
la
n
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r
M
S
Y
M
in
D
d
im
e
n
s
io
n
s
h
a
v
e
b
e
e
n
s
h
o
w
n
t
o
t
r
a
n
s
f
o
r
m
in
e
x
a
c
t
ly
t
h
e
s
a
m
e
f
a
s
h
io
n
t
o
a
ll
lo
o
p
o
r
d
e
r
s
,
a
t
le
a
s
t
f
o
r
D

6
[3
7
,
3
8
].
T
h
is
p
r
o
p
e
r
t
y
is
s
u
f
f
i
c
ie
n
t
f
o
r
o
u
r
p
u
r
p
o
s
e
s
,
s
in
c
e
w
e
a
r
e
m
a
in
ly
in
t
e
r
e
s
t
e
d
in
t
h
e
in
t
e
g
r
a
n
d
in
D
= 5
.
Th
e
(
L
+
1
)
-
p
a
r
t
ic
le
c
u
t
s
c
a
n
a
ls
o
b
e
u
s
e
d
t
o
g
e
n
e
r
a
t
e
t
h
e
c
o
m
p
le
t
e
lis
t
o
f
g
ra
p
h
s
n
e
e
d
e
d
a
t
s
ix
lo
o
p
s
.
O
n
e
c
o
n
s
id
e
r
s
a
ll
p
o
s
s
ib
le
s
e
w
in
g
s
o
f
t
w
o
t
r
e
e
-
le
v
e
l
c
u
b
ic
g
r
a
p
h
s
t
h
a
t
a
p
p
e
a
r
in
t
h
e
s
e
c
u
t
s
[1
2
].
(
W
e
m
o
d
if
y
t
h
e
p
r
o
c
e
d
u
r
e
s
lig
h
t
ly
c
o
m
p
a
r
e
d
t
o
r
e
f
.
[1
2
]
b
y
in
c
lu
d
in
g
o
n
ly
d
ia
g
r
a
m
s
w
it
h
c
u
b
ic
v
e
r
t
ic
e
s
.)
I
n
p
r
in
c
ip
le
t
h
e
r
e
a
r
e
d
u
a
l
c
o
n
f
o
r
m
a
l
g
r
a
p
h
s
w
it
h
f
o
u
r
-
o
r
h
ig
h
e
r
-
p
o
in
t
v
e
r
t
ic
e
s
t
h
a
t
a
r
e
n
o
t
g
e
n
e
r
a
t
e
d
b
y
t
h
e
p
r
o
d
u
c
t
o
f
t
re
e
g
ra
p
h
s
o
f
t h
e
(
L
+ 1
)
-
p
a
r
t
ic
le
c
u
t
s
;
h
o
w
e
v
e
r
,
a
ll
s
u
c
h
p
o
t
e
n
t
ia
l
c
o
n
t
r
ib
u
t
io
n
s
,
in
c
lu
d
in
g
t
h
o
s
e
n
o
t
d
e
t
e
c
t
a
b
le
in
t
h
e
(
L
+
1
)
-
p
a
r
t
ic
le
c
u
t
s
,
c
a
n
b
e
a
s
s
ig
n
e
d
t
o
g
r
a
p
h
s
w
it
h
o
n
ly
c
u
b
ic
v
e
r
t
ic
e
s
b
y
m
u
lt
ip
ly
in
g
a
n
d
7

68 planar diagrams


Given by dual conformal invariance (up to integer 0,1,
-
1,2,
-
2...
prefactors
)


Independently constructed by:


Eden,
Heslop
,
Korchemsky
,
Sokatchev
;






Bourjaily
,
DiRe
,
Shaikh
,
Spradlin
,
Volovich


Bern, Carrasco, Dixon,

Douglas, HJ, von
Hippel


6
-
Loop Planar
D
=5 SYM

The Parking Spot Escalation

our secret collaborator…

6
-
Loop
D
=5 SYM divergence


Using integration by parts identities, div. simplifies to 3 integrals



Numerical
integration


modified version of
FIESTA





1000
node cluster at Stony Brook


Result: divergence is nonzero.




What cancels this divergence ?
Solitons
/KK modes ?
Douglas; Lambert et al.


Bern, Carrasco, Dixon, Douglas, HJ, von
Hippel


(
l
1
+
l
3
)
2
2
×
(a
)
l
3
l
1
(
l
1
+
l
3
)
2
×
(
l
2
+
l
4
)
2
×
(b
)
l
3
l
1
l
4
l
2
(
l
1
+
l
3
)
2
×
(d
)
l
1
l
3
(
l
1
+
l
3
)
2
×
(
l
2
+
l
4
)
2
×
l
3
l
1
l
4
l
2
(c
)
(f
)
(
l
1
+
l
3
)
2
×
(e
)
l
3
l
1
FI
G.
5
:
T
h
e
s
i
x
d
i
s
t i
n
c
t
v
a
c
u
u
m
d
i
a
g
r
a
m
s
t h
a
t
a
p
p
e
a
r
i
n
e
q
s
.
(
3.
3
)
an
d
(
3.
4
)
.
E
ac
h
d
ot
i
n
d
i
c
at
e
s
t h
a
t
t h
e
c
o
r
r
e
s
p
o
n
d
i
n
g
p
r
o
p
a
g
a
t o
r
s
h
o
u
l
d
b
e
s
q
u
a
r
e
d
(d
o
u
b
l
ed
)
i
n
t
h
e
i
n
t
eg
r
a
n
d
.
T
h
e

v
e

t
en
s
o
r

in
t
e
g
r
a
ls
h
a
v
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
ic
a
t
e
d
b
y
t
h
e
p
re
f
a
c
t
o
rs
.
T
h
e
n
u
m
e
ra
t
o
r
f
a
c
t
o
rs
a
re
bui
l
t
f
r
o
m
m
o
m
e
n
t
um
i
n
v
a
r
i
a
n
t
s
i
n
v
o
l
v
i
ng
a
s
ubs
e
t
o
f
t
he
l
o
o
p m
o
m
e
n
t
a
,
l
a
b
e
l
e
d
b
y
l
1
,l
2
,l
3
,l
4
.
wh
e
r
e
A
i,
x
an
d
B
i,
x
a
r
e
r
a
t
io
n
a
l
n
u
m
b
e
r
s
d
e
t
e
r
m
in
e
d
b
y
t
h
e
e
x
p
a
n
s
io
n
.
(
W
e
w
ill
n
o
t
lis
t
t
h
e
s
e
c
o
e
f
f
i
c
ie
n
t
s
s
e
p
a
r
a
t
e
ly
f
o
r
e
a
c
h
d
ia
g
r
a
m
.)
A
f
t
e
r
t
h
e
a
b
o
v
e
v
a
c
u
u
m
in
t
e
g
r
a
n
d
s
V
(
x
)
ar
e
in
t
e
g
r
a
t
e
d
o
v
e
r
t
h
e
s
ix
lo
o
p
m
o
m
e
n
t
a
p
5
,p
6
,...
p
10
in
D
= 5

2
,
w
it
h
t
h
e
m
e
a
s
u
r
e
10
l
= 5
d
5

2
p
l
(2
π
)
5
,
(
3
.2
)
w
e
o
b
t
a
in
s
ix
v
a
c
u
u
m
in
t
e
g
r
a
ls
,
V
(a
)
,V
(b
)
,...
,V
(f
)
,
s
h
o
w
n
in

g
.
5
.
T
h
e
s
e
in
t
e
g
r
a
ls
h
a
v
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
w
h
ic
h
a
r
e
in
d
ic
a
t
e
d
t
o
t
h
e
le
f
t
o
f
e
a
c
h
g
r
a
p
h
,
a
n
d
e
it
h
e
r
o
n
e
o
r
t
w
o
d
o
u
b
le
d
p
r
o
p
a
g
a
t
o
r
,
w
h
o
s
e
lo
c
a
t
io
n
is
in
d
ic
a
t
e
d
b
y
a
d
o
t
.
T
h
e
in
t
e
g
r
a
ls
V
(
x
)
c
o
n
t
a
in
n
o
s
u
b
d
iv
e
r
g
e
n
c
e
s
;
e
a
c
h
in
t
e
g
r
a
l
h
a
s
a
s
in
g
le
o
v
e
r
a
ll
U
V
d
iv
e
r
g
e
n
c
e
in
D
=
5
w
h
e
n
a
ll
s
ix
lo
o
p
m
o
m
e
n
t
a
b
e
c
o
m
e
la
r
g
e
.
H
e
n
c
e
t
h
e
in
t
e
g
r
a
ls
h
a
v
e
o
n
ly
s
im
p
le
p
o
le
s
in
.
C
o
lle
c
t
in
g
t
h
e
c
o
n
t
r
ib
u
t
io
n
s
f
r
o
m
t
h
e
6
8
d
is
t
in
c
t
in
t
e
g
r
a
ls
in
t
h
e
s
ix
-
lo
o
p
a
m
p
lit
u
d
e
e
q
.
(
2
.2
)
,
w
e
o
b
t
a
in
t
h
e
f
o
llo
w
in
g
U
V
d
iv
e
r
g
e
n
c
e
A
(6
)
4
D= 5
,
d
i
v
.
= 6
st
u
A
t r
e
e
4
(1
,
2
,
3
,
4)
(
V
(a
)
+
V
(b
)
+ 2
V
(c
)
+ 4
V
(d
)
+ 2
V
(e
)

2
V
(f
)
)
.
(
3
.3
)
10
Divergence to all loop
o
rders ?

Bern, Carrasco, Dixon, Douglas, HJ, von
Hippel


Intriguing pattern of UV divergence in critical dimension of maximal susy YM

2
3
4
5
6
0.5
1.0
5.0
10.0
50.0
Ρ
L
2
3
4
5
6
0.985
0.990
0.995
1.000
1.005
1.010
1.015
2
3
4
5
6
L
loops
Ρ
L
f
i
t
Ρ
L
FI
G.
1
2
:
T
h
e
d
o
t s
i
n
d
i
c
a
t e
d
t h
e
n
u
m
e
r
i
c
a
l
c
o
e
f
f
i
c
i
e
n
t s
i
n
e
q
.
(
5.
6
)
of
t
h
e
U
V
d
i
v
e
r
ge
n
c
e
s
i
n
t
h
e
cr
i
t
i
ca
l
d
i
m
en
s
i
o
n
s
.
T
h
e
s
o
l
i
d
(
b
l
u
e)
l
i
n
e
i
s
t
h
e
r
es
u
l
t
o
f

t t i
n
g
t h
e
p
a
r
a
m
e
t r
i
c
f
o
r
m
i
n
e
q
.
(5
.
7
)
t o
t h
e
d
i
s
p
l
a
y
e
d
r
e
s
u
l
t s
f
o
r
L
= 2
,
3
,
4
,
5
,
6.
T
h
e
d
as
h
e
d
(
p
u
r
p
l
e
)
l
i
n
e
i
s
a

t
t
o
t
h
e
p
ar
am
e
t
r
i
c
f
or
m
i
n
eq
.
(
5
.
1
0
)
.
T
h
e
l
o
w
er
p
a
n
el
s
h
o
w
s
t
h
a
t
t
h
e
r
el
a
t
i
v
e
er
r
o
r
b
et
we
e
n
t
h
e
p
o
i
n
t
s
a
n
d
t
h
e

t
i
n
e
q
.
(
5
.
7
)
is
w
it
h
in
1
%
.
p
r
e
v
io
u
s
ly
[4
,
9
,
1
0
,
1
3
].
C
o
r
r
e
c
t
in
g
a
c
o
u
p
le
o
f
o
v
e
r
a
ll
s
ig
n
s
,
t
h
e
y
a
re
ρ
2
=
π
20
,
ρ
3
=
1
3
,
ρ
4
= 6
512
5
Γ
(
3
4
)
4

2048
105
Γ
(
3
4
)
3
Γ
(
1
2
)
Γ
(
1
4
)
1
.
553
,
ρ
5
9
.
537
,
(
5
.6
)
w
h
e
r
e
t
h
e
e
x
p
r
e
s
s
io
n
s
t
h
r
o
u
g
h
L
= 4
a
r
e
e
x
a
c
t
.
T
h
e
L
=
5
e
x
p
r
e
s
s
io
n
is
a
p
p
r
o
x
im
a
t
e
,
b
u
t
it
is
a
c
c
u
r
a
t
e
t
o
t
h
e
d
ig
it
s
g
iv
e
n
.
T
h
e
lin
e
a
r
b
e
h
a
v
io
r
b
e
y
o
n
d
L
=
2
in
t
h
e
u
p
p
e
r
p
a
n
e
l
o
f

g
.
1
2
m
a
k
e
c
le
a
r
t
h
a
t
t
h
e
c
o
e
f
f
i
c
ie
n
t
s
o
f
t
h
e
d
iv
e
r
g
e
n
c
e
s
h
a
v
e
a
n
a
p
p
r
o
x
im
a
t
e
ly
e
x
p
o
n
e
n
t
ia
l
b
e
h
a
v
io
r
.
T
h
is
o
b
s
e
r
v
a
t
io
n
m
o
t
iv
a
t
e
s
a
s
im
p
le
A
n
s
a
t
z
f
o
r
t
h
e
a
p
p
r
o
x
im
a
t
e
f
o
r
m
o
f
t
h
e
d
iv
e
r
g
e
n
c
e
s
a
t
a
n
y
lo
o
p
o
r
d
e
r
L

2,
ρ
L
b
1
c
L
+
a
1
/L
1
.
(
5
.7
)
T
h
e
s
o
lid
c
u
r
v
e
in
t
h
e
u
p
p
e
r
p
a
n
e
l
o
f

g
.
1
2
is
b
a
s
e
d
o
n
e
q
.
(
5
.7
)
w
it
h
t
he
pa
r
a
m
e
t
e
r
s
a
1
= 3
.
99
,b
1
= 1
.
74
×
10

5
,c
1
= 9
.
77
.
(
5
.8
)
24

Accurate to <1%



Why do the UV divergences


follow this approximate curve?



Is it asymptotically exact ?



H. Johansson, Frascati 2013

5pt
N
=8 SUGRA calculations

H. Johansson, Frascati 2013

C
-
K
amplitudes at 1

loop

2

4

1

3

Green, Schwarz,

Brink (1982)

D
uality
-
satisfying loop amplitudes:

2

4

1

3

N=4 SYM:

All
-
plus QCD:

N=4 SYM and

All
-
plus QCD:

1106.4711 [hep
-
th]

Carrasco, HJ

SYM UV div in
D
=8:

Carrasco, HJ

1106.4711

[
hep
-
th
]

SU(8) violating SG

UV div in
D
=8:


SG UV div in
D
=8:

1
-
loop 5
-
pts UV divergences


counterterms:

H. Johansson, Frascati 2013

2
-
loop 5
-
pts
N
=4 SYM and
N
=8 SG

The 2
-
loop 5
-
point
amplitude with
Color
-
Kin. duality

N
= 8 SG obtained
from numerator
double copies

Carrasco, HJ


1106.4711

[
hep
-
th
]

2
-
loop 5
-
pts UV divergences

SYM UV div in
D
=7:

SG UV div in
D
=7:

Carrasco, HJ

1106.4711

[
hep
-
th
]

SU(8) violating SG UV div in
D
=8:

H. Johansson, Frascati 2013

3
-
loop 5
-
point SYM
and
N
=8 SG

(

ladder
-
like


diagrams)

Carrasco, HJ

(to appear)

Non
-
planar
D
-
dimensional amplitude
with
manifest
color
-
kinematics duality

(
N
= 8 SG obtained from
squaring the numerators)

H. Johansson, Frascati 2013

3
-
loop 5
-
point SYM
and
N
=8 SG

some

Mercedes
-
like


diagrams


Carrasco, HJ

(to appear)

H. Johansson, Frascati 2013

3
-
loop 5
-
point SYM
and
N
=8 SG

Carrasco, HJ

(to appear)


in total 42 diagrams.

For SYM
the UV divergent diagrams (in
D
=6) are very simple:

(for SG the UV div. comes from the other diagrams as well)

H. Johansson, Frascati 2013

3
-
loop 5
-
pts UV divergences

N

=8 SG 5
-
loop Status

Working on reorganizing 5
-
loop
N
=4 SYM


Bern
, Carrasco, HJ,
Roiban



(in progress)

2
(410)
(404)
(335)
(184)
(284)
1
2
4
3
10
18
20
13
(1)
1
2
4
3
(370)
(12)
4
5
1
2
3
FI
G
.
1
:
S
a
m
p
l
e
g
ra
p
h
s
f
o
r
t
h
e

v
e
-
l
o
o
p
f
o
u
r-
p
o
i
n
t
N
= 4
sY
M
a
m
p
l
i
t
u
d
e
.
T
h
e
g
r
a
p
h
l
a
b
e
l
s
c
o
r
r
e
sp
o
n
d
t
o
t
h
e
o
n
e
s
us
e
d
i
n
t
he
a
nc
i
l
l
a
r
y
fil
e
[
2
3
]
.
t
h
r
e
e
-
lo
o
p
c
o
u
n
t
e
r
t
e
r
m
[2
0
];
t
h
e
c
o
e
f
f
i
c
ie
n
t
o
f
t
h
is
c
o
u
n
-
t
e
r
t
e
r
m
h
a
s
r
e
c
e
n
t
ly
b
e
e
n
e
x
p
lic
it
ly
s
h
o
w
n
t
o
v
a
n
is
h
[
1
2
].
(
S
e
e
r
e
f
.
[2
1
]
f
o
r
a
s
t
r
in
g
-
b
a
s
e
d
a
r
g
u
m
e
n
t
.)
T
h
is
e
x
h
ib
it
s
b
e
t
t
e
r
b
e
h
a
v
io
r
t
h
a
n
im
p
lie
d
b
y
k
n
o
w
n
s
y
m
m
e
t
r
y
c
o
n
-
s
id
e
r
a
t
io
n
s
a
n
d
is
in
lin
e
w
it
h
c
a
n
c
e
lla
t
io
n
s
s
u
g
g
e
s
t
e
d
b
y
u
n
it
a
r
it
y
a
r
g
u
m
e
n
t
s
[2
2
].
I
n
p
a
r
t
ic
u
la
r
,
it
e
m
p
h
a
s
iz
e
s
t
h
e
im
p
o
r
t
a
n
c
e
o
f
d
ir
e
c
t
ly
c
h
e
c
k
in
g
t
h
e
a
m
p
lit
u
d
e
s
w
h
e
t
h
e
r
e
q
.
(
1
)
h
o
ld
s
f
o
r
N
=
8
s
u
p
e
r
g
r
a
v
it
y
a
t
L
= 5
.
O
u
r
c
o
n
s
t
r
u
c
t
io
n
o
f
t
h
e

v
e
-
lo
o
p
f
o
u
r
-
p
o
in
t
a
m
p
lit
u
d
e
of
N
=
4
s
Y
M
t
h
e
o
r
y
o
r
g
a
n
iz
e
s
it
in
t
h
e
f
o
r
m
,
A
(5
)
4
=
i g
12
st
A
t re
e
4
S
4
416
i
= 1
9
j
= 5
d
D
l
j
(2
π
)
D
1
S
i
C
i
N
i
20
m
= 5
l
2
m
,
(2
)
w
h
e
r
e
t
h
e
s
e
c
o
n
d
s
u
m
r
u
n
s
o
v
e
r
a
s
e
t
o
f
4
1
6
d
is
t
in
c
t
(
n
o
n
-
is
o
m
o
r
p
h
ic
)
g
r
a
p
h
s
w
it
h
o
n
ly
c
u
b
ic
(
t
r
iv
a
le
n
t
)
v
e
r
-
t
ic
e
s
.
S
o
m
e
s
a
m
p
le
g
r
a
p
h
s
a
r
e
s
h
o
w
n
in

g
.
1
.
T
h
e

r
s
t
s
u
m
r
u
n
s
o
v
e
r
a
ll
2
4
p
e
r
m
u
t
a
t
io
n
s
o
f
e
x
t
e
r
n
a
l
le
g
la
b
e
ls
in
d
ic
a
t
e
d
b
y
S
4
.T
h
e
s
y
m
m
e
t
r
y
f
a
c
t
o
r
s
S
i
re
m
o
v
e
o
v
e
r
c
o
u
n
t
s
,
in
c
lu
d
in
g
t
h
o
s
e
a
r
is
in
g
f
r
o
m
in
t
e
r
n
a
l
a
u
t
o
-
m
o
r
p
h
is
m
s
y
m
m
e
t
r
ie
s
w
it
h
e
x
t
e
r
n
a
l
le
g
s

x
e
d
.
H
e
r
e
w
e
a
b
s
o
r
b
a
ll
c
o
n
t
a
c
t
t
e
r
m
s
(
i.e
.
t
e
r
m
s
w
it
h
f
e
w
e
r
t
h
a
n
t
h
e
m
a
x
im
u
m
n
u
m
b
e
r
o
f
p
r
o
p
a
g
a
t
o
r
s
)
in
t
o
g
r
a
p
h
s
w
it
h
o
n
ly
c
u
b
ic
v
e
r
t
ic
e
s
,
b
y
m
u
lt
ip
ly
in
g
a
n
d
d
iv
id
in
g
b
y
a
p
p
r
o
p
r
i-
at
e
p
r
op
agat
or
s
.
W
e
d
e
n
ot
e
e
x
t
e
r
n
al
m
om
e
n
t
a
b
y
k
i
fo
r
i
= 1
,...
,
4
a
n
d
t
h
e

v
e
in
d
e
p
e
n
d
e
n
t
lo
o
p
m
o
m
e
n
t
a
b
y
l
j
fo
r
j
= 5
,...,
9
.
T
h
e
r
e
m
a
in
in
g
l
j
a
r
e
lin
e
a
r
c
o
m
b
i-
n
a
t
io
n
s
o
f
t
h
e
s
e
.
T
h
e
c
o
lo
r
f
a
c
t
o
r
s
C
i
o
f
a
ll
g
r
a
p
h
s
a
r
e
o
b
t
a
in
e
d
b
y
d
r
e
s
s
in
g
e
v
e
r
y
t
h
r
e
e
-
v
e
r
t
e
x
in
t
h
e
g
r
a
p
h
w
it
h
a f
a
c
t
o
r
o
f
˜
f
ab
c
= T
r
(
[
T
a
,T
b
]
T
c
),
w
h
e
r
e
t h
e
g
a
u
g
e
g
r
o
u
p
ge
n
e
r
at
or
s
T
a
a
r
e
n
o
r
m
a
liz
e
d
a
s
T
r
(
T
a
T
b
) =
δ
ab
.T
h
e
g
a
u
g
e
c
o
u
p
lin
g
is
g
a
n
d
t
h
e
c
r
o
s
s
in
g
s
y
m
m
e
t
r
ic
p
r
e
f
a
c
-
t o
r
st
A
t re
e
4
is
in
t
e
r
m
s
o
f
t
h
e
c
o
lo
r
-
o
r
d
e
r
e
d
D
-
d
im
e
n
s
io
n
a
l
t
r
e
e
a
m
p
lit
u
d
e
A
t re
e
4

A
t re
e
4
(1
,
2
,
3
,
4)
an
d
s
= (
k
1
+
k
2
)
2
an
d
t
= (
k
2
+
k
3
)
2
.
To
c
o
n
s
t
r
u
c
t
t
h
e
n
u
m
e
r
a
t
o
r
s
N
i
,w
e
u
s
e
t
h
e
m
e
t
h
o
d
o
f
m
a
x
im
a
l
c
u
t
s
[8
],
b
a
s
e
d
o
n
t
h
e
u
n
it
a
r
it
y
m
e
t
h
o
d
[
2
4
].
A
p
-
p
lic
a
t
io
n
o
f
t
h
is
m
e
t
h
o
d
a
n
d
v
a
r
io
u
s
s
t
r
a
t
e
g
ie
s
f
o
r
g
r
e
a
t
ly
s
t
r
e
a
m
lin
in
g
t
h
e
c
o
n
s
t
r
u
c
t
io
n
o
f
t
h
e
n
u
m
e
r
a
t
o
r
s
h
a
s
b
e
e
n
d
e
s
c
r
ib
e
d
in
c
o
n
s
id
e
r
a
b
le
d
e
t
a
il
in
r
e
f
.
[6
],
s
o
h
e
r
e
w
e
g
iv
e
o
n
ly
a
b
r
ie
f
s
u
m
m
a
r
y
.
T
h
e
m
e
t
h
o
d
w
o
r
k
s
in
D
d
i-
N
3
MC
NM
C
MC
N
2
MC
FI
G
.
2
:
S
a
m
p
l
e
N
k
-m
a
x
i
m
a
l
c
u
t
s
f
o
r
k
= 0
,
1
,
2
,
3.
T
h
e
e
x
-
po
s
e
d
l
i
n
e
s
a
r
e
a
l
l
c
u
t
.
(b
)
(a
)
=
s
57
s
36
s
67
s
38
(c
)
3
5
6
7
1
2
4
8
3
5
6
7
1
2
4
8
FI
G
.
3
:
E
x
a
m
p
l
e
s
o
f
s
i
m
p
l
e
cu
t
s
u
s
e
d
t
o
s
p
e
e
d
u
p
t
h
e
ca
l
cu
-
lat
ion
.
(
a)
is
a
t
w
o
p
ar
t
ic
le
c
u
t
,
(
b
)
a
b
o
x
c
u
t
an
d
(
c
)
is
a
sa
m
p
l
e
a
p
p
l
i
c
a
t
io
n
o
f
B
C
J
a
m
p
li
t
u
d
e
r
e
la
t
i
o
n
s.
T
h
e
e
x
p
o
se
d
lin
e
s
ar
e
all
c
u
t
.
m
e
n
s
io
n
s
a
n
d
c
a
n
b
e
u
s
e
d
t
o
o
b
t
a
in
lo
c
a
l
e
x
p
r
e
s
s
io
n
s
,
f
r
o
m
w
h
ic
h
U
V
d
iv
e
r
g
e
n
c
e
s
c
a
n
b
e
s
t
r
a
ig
h
t
f
o
r
w
a
r
d
ly
e
x
-
t r
a
c
t e
d
.
W
e
s
t
a
r
t
w
it
h
a
n
a
n
s
a
t
z
f
o
r
t
h
e
d
ia
g
r
a
m
n
u
m
e
r
a
t
o
r
s
c
o
n
t
a
in
in
g
f
r
e
e
p
a
r
a
m
e
t
e
r
s
t
o
b
e
d
e
t
e
r
m
in
e
d
b
y
m
a
t
c
h
in
g
a
g
a
in
s
t
g
e
n
e
r
a
liz
e
d
u
n
it
a
r
it
y
c
u
t
s
.
O
u
r
a
n
s
a
t
z
is
a
p
o
ly
-
n
o
m
ia
l
o
f
d
e
g
r
e
e
f
o
u
r
in
t
h
e
k
in
e
m
a
t
ic
in
v
a
r
ia
n
t
s
,
s
u
b
j
e
c
t
t
o
t
h
e
p
o
w
e
r
-
c
o
u
n
t
in
g
c
o
n
s
t
r
a
in
t
t
h
a
t
n
o
t
e
r
m
h
a
s
m
o
r
e
t
h
a
n
s
ix
p
o
w
e
r
s
o
f
lo
o
p
m
o
m
e
n
t
u
m
.
W
e
a
ls
o
d
e
m
a
n
d
t
h
a
t
e
a
c
h
n
u
m
e
r
a
t
o
r
r
e
s
p
e
c
t
s
t
h
e
a
u
t
o
m
o
r
p
h
is
m
s
y
m
m
e
-
t
r
ie
s
o
f
t
h
e
g
r
a
p
h
.
O
n
c
e
a
s
o
lu
t
io
n
is
f
o
u
n
d
s
a
t
is
f
y
in
g
a
c
o
m
p
le
t
e
s
e
t
o
f
c
u
t
c
o
n
d
it
io
n
s
,
w
e
h
a
v
e
t
h
e
in
t
e
g
r
a
n
d
.
I
f
a
n
in
c
o
n
s
is
t
e
n
c
y
is
e
n
c
o
u
n
t
e
r
e
d
,
t
h
e
a
n
s
a
t
z
m
u
s
t
b
e
e
n
la
r
g
e
d
.
W
e
n
o
t
e
t
h
a
t
t
h
e
s
o
lu
t
io
n
s
f
o
r
n
u
m
e
r
a
t
o
r
s
a
r
e
n
o
t
u
n
iq
u
e
a
n
d
d
i
f
f
e
r
e
n
t
c
h
o
ic
e
s
c
a
n
b
e
m
a
p
p
e
d
in
t
o
e
a
c
h
o
t
h
e
r
b
y
g
e
n
e
r
a
liz
e
d
g
a
u
g
e
t
r
a
n
s
f
o
r
m
a
t
io
n
s
[9
,
1
0
,
2
5
].
T
h
e
p
a
r
a
m
e
t
e
r
s
o
f
t
h
e
a
n
s
a
t
z
a
r
e
d
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t
e
r
m
in
e
d
f
r
o
m
g
e
n
-
e
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a
liz
e
d
u
n
it
a
r
it
y
c
u
t
s
t
h
a
t
d
e
c
o
m
p
o
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e
a
lo
o
p
in
t
e
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r
a
n
d
in
t
o
p
r
o
d
u
c
t
s
o
f
o
n
-
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ll
t
r
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e
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m
p
lit
u
d
e
s
s
u
m
m
e
d
o
v
e
r
a
ll
in
t
e
r
m
e
d
ia
t
e
s
t
a
t
e
s
,
st
a
t
e
s
A
t re
e
(1
)
A
t re
e
(2
)
· · ·
A
t re
e
(
m
)
.T
h
e
s
e
c
u
t
s
a
r
e
o
r
g
a
n
iz
e
d
a
c
c
o
r
d
in
g
t
o
t
h
e
n
u
m
b
e
r
o
f
c
u
t
p
r
o
p
-
a
g
a
t
o
r
s
t
h
a
t
a
r
e
r
e
p
la
c
e
d
w
it
h
o
n
-
s
h
e
ll
c
o
n
d
it
io
n
s
.
W
e
s
t
a
r
t
f
r
o
m
t
h
e
m
a
x
im
a
l
c
u
t
s
(
M
C
s
)
w
h
e
r
e
a
ll
1
6
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
c
u
t
.
A
f
t
e
r
o
b
t
a
in
in
g
t
h
e
M
C
s
,
w
e
t
h
e
n
c
o
n
-
s
t
r
u
c
t
s
a
ll
n
e
x
t
-
t
o
-
m
a
x
im
a
l
c
u
t
s
(
N
M
C
s
)
,
w
it
h
1
5
c
u
t
p
r
o
p
a
g
a
t
o
r
s
.
W
e
c
o
n
t
in
u
e
t
h
is
p
r
o
c
e
s
s
,
s
y
s
t
e
m
a
t
ic
a
lly
c
o
n
s
t
r
u
c
t
in
g
a
n
a
ly
t
ic
e
x
p
r
e
s
s
io
n
s
f
o
r
(
n
e
x
t
-
t
o
)
k
-
m
a
x
im
a
l
cu
t
s
(
N
k
M
C
s
)
w
it
h
f
e
w
e
r
a
n
d
f
e
w
e
r
im
p
o
s
e
d
c
u
t
c
o
n
d
i-
t
io
n
s
.
F
o
r
t
h
e

v
e
-
lo
o
p
f
o
u
r
-
p
o
in
t
N
=
4
s
Y
M
a
m
p
lit
u
d
e
t
h
is
p
r
o
c
e
s
s
t
e
r
m
in
a
t
e
s
a
t
k
=
3
,
s
in
c
e
t
h
e
p
o
w
e
r
c
o
u
n
t
-
in
g
o
f
t
h
e
t
h
e
o
r
y
p
r
e
v
e
n
t
s
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
f
r
o
m
c
a
n
-

416 + 336 = 752 integral topologies


BCJ: 2500 functional Jacobi
eqns

Relaxing
ansatz
:




Non
-
manifest crossing symmetry


Allow for non
-
local numerators


Gauge variant numerators


Relax power counting


Allow for triangle diagrams




Once we have integrand, integration will take ~ 1 day:


No
subdivergences

in
D
=24/5


No IR divergences since D>4, and absence of bubbles


No more difficult than IBP:s for
N
=4
SYM <
1
day


If needed,
n
umerical integration ~ few days

SYM
SG


?

3
loops,
D
=6:


4 loops,
D
=11/2
:


SYM SG

SYM SG

H. Johansson, Frascati 2013

Fortune
-
telling from pattern

5 loops,
D
=26/5
:


V
1
V
2
V
8
FI
G.
1
7
:
T
h
e
b
a
s
i
c
f
o
u
r
-l
o
o
p
v
a
c
u
u
m
i
n
t e
g
r
a
l
s
V
1
,
V
2
an
d
V
8
,t
o
w
h
i
c
h
a
l
l o
t
h
e
r
s
c
a
n
b
e
r
e
d
u
c
e
d
.
r
e
v
e
a
ls
t
h
a
t
t
h
e
le
a
d
in
g
U
V
b
e
h
a
v
io
r
c
o
m
e
s
s
o
le
ly
f
r
o
m
in
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
.
T
h
e
s
e
s
ix
in
t
e
g
r
a
ls
h
a
v
e
1
1
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
,
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
e
p
e
n
d
e
n
t
o
f
t
h
e
lo
o
p
m
o
m
e
n
t
u
m
.
T
h
e
r
e
f
o
r
e
t
h
e
y
d
iv
e
r
g
e

r
s
t
in
D
c
= 1
1
/
2
,
w
h
ic
h
m
a
t
c
h
e
s
t
h
e
e
x
p
e
c
t
e
d
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
c
= 4
+
6
/L
w
it
h
L
= 4
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
1
P
I
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
h
a
v
e
1
3
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
T
h
e
ir
n
u
-
m
e
r
a
t
o
r
s
w
o
u
ld
h
a
v
e
t
o
b
e
q
u
a
r
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
f
o
r
t
h
e
m
t
o
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
w
e
n
o
t
e
f
r
o
m
e
q
.
(
3
.1
4
)
t
h
a
t
t
h
e
m
a
s
t
e
r
n
u
m
e
r
a
t
o
r
s
N
18
an
d
N
28
a
r
e
q
u
a
d
r
a
t
ic
in
t h
e
τ
i j
,
a
n
d
h
e
n
c
e
m
e
r
e
ly
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
T
h
e
J
a
c
o
b
i
r
e
la
t
io
ns
(
A4
)
a
nd
(
A
5
)
p
r
e
s
e
r
v
e
t
h
is
q
u
a
d
r
a
t
ic
b
e
h
a
v
io
r
f
o
r
a
ll
n
u
m
e
r
a
t
o
r
s
.
T
h
e
r
e
f
o
r
e
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
a
r
e

n
it
e
in
D
c
= 1
1
/
2
.
T
h
e
1
P
R
in
t
e
g
r
a
ls
I
53
t h
ro
u
g
h
I
79
h
a
v
e
1
2
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
I
f
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
w
e
r
e
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
,
t
h
e
n
t
h
e
y
w
o
u
ld
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
it
is
e
a
s
y
t
o
s
e
e
f
r
o
m
e
q
s
.
(
A
5
)
a
n
d
(
B
1
)
t
h
a
t
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
a
r
e
a
ll
lin
e
a
r
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
I
n
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
r
e
d
u
c
e
e
a
s
ily
t
o
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
t
h
e
lim
it
t
h
a
t
t
h
e
e
x
t
e
r
n
a
l
m
o
m
e
n
t
a
v
a
n
is
h
.
T
h
e
p
la
n
a
r
in
t
e
g
r
a
ls
I
80
an
d
I
83
r
e
d
u
c
e
t
o
t
h
e
v
a
c
u
u
m
in
t
e
g
r
a
l
V
1
d
e
p
ic
t
e
d
in

g
.
1
7
.
W
h
ile
in
t
e
g
r
a
ls
I
81
an
d
I
84
a
r
e
n
o
n
p
la
n
a
r
a
s
f
o
u
r
-
p
o
in
t
g
r
a
p
h
s
,
in
t
h
e
v
a
c
u
u
m
lim
it
t
h
e
y
r
e
d
u
c
e
t
o
t
h
e
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
2
.
F
in
a
lly
,
in
t
e
g
r
a
ls
I
82
an
d
I
85
re
d
u
c
e
t o
t
h
e
n
o
n
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
8
.
A
s
w
a
s
t
h
e
c
a
s
e
a
t
t
h
r
e
e
lo
o
p
s
,
t
h
e
c
o
lo
r
f
a
c
t
o
r
s
f
o
r
t
h
e
le
a
d
in
g
U
V
g
r
a
p
h
s
a
r
e
r
e
la
t
e
d
b
y
c
o
lo
r
J
a
c
o
b
i
id
e
n
t
it
ie
s
.
I
n
t
h
is
c
a
s
e
w
e
c
a
n
s
u
b
t
r
a
c
t
,
f
o
r
e
x
a
m
p
le
,
C
81
fr
o
m
C
80
,a
n
d
u
s
e
a
J
a
c
o
b
i
id
e
n
t
it
y
o
p
e
r
a
t
in
g
o
n
t
h
e
b
o
x
a
t
t
h
e
t
o
p
c
e
n
t
e
r
o
f
t
h
e
g
r
a
p
h
s
in

g
.
1
1
.
T
h
e
n
w
e
r
e
d
u
c
e
t
h
e
r
e
s
u
lt
in
g
t
r
ia
n
g
le
s
u
b
g
r
a
p
h
s
it
e
r
a
t
iv
e
ly
,
t
o

n
d
C
80
,
83

2
N
4
c
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
=
C
81
=
C
82
=
C
84
=
C
85
.
(
4
.1
3
)
A
g
a
in
,
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
p
a
r
t
s
o
f
a
ll
c
o
n
t
r
ib
u
t
in
g
c
o
lo
r
f
a
c
t
o
r
s
a
r
e
e
q
u
a
l.
Fr
o
m
e
q
.
(
A
5
)
,
t
h
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
o
b
e
y
N
80
=
N
81
=
N
82
an
d
N
83
=
N
84
=
N
85
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
t
h
e
c
o
m
b
in
a
t
o
r
ia
l
f
a
c
t
o
r
o
f
I
81
in
e
q
.
(
3
.3
)
is
t
w
ic
e
a
s
la
r
g
e
a
s
t
h
o
s
e
fo
r
I
80
an
d
I
82
,
a
n
d
s
im
ila
r
ly
f
o
r
I
84
w
it
h
r
e
s
p
e
c
t
t
o
I
83
an
d
I
85
.
T
a
k
in
g
in
t
o
a
c
c
o
u
n
t
b
o
t
h
c
o
m
b
in
a
t
o
r
ia
l
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
,
t
h
e
c
o
n
t
r
ib
u
t
io
n
o
f
I
83
is

9
8
t
im
e
s
t
h
a
t
o
f
I
80
,a
n
d
s
im
ila
r
ly
f
o
r
t
h
e
o
t
h
e
r
t
w
o
p
a
ir
s
o
f
g
r
a
p
h
s
.
C
o
m
b
in
in
g
a
ll
t
e
r
m
s
a
n
d
s
w
it
c
h
in
g
t
o
t
h
e
c
o
lo
r
-
t
r
a
c
e
b
a
s
is
,
w
e

n
d
t
h
a
t
t
h
e
U
V
d
iv
e
r
g
e
n
c
e
in
t
h
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
D
= 1
1
/
2
is
g
iv
e
n
by
,
A
(4
)
4
(1
,
2
,
3
,
4)
SU
(
N
c
)
po
l
e
=

6
g
10
K
N
2
c
N
2
c
V
1
+ 1
2
(
V
1
+ 2
V
2
+
V
8
)
(
4
.1
4
)
×
s
( T
r
1324
+ T
r
1423
) +
t
( T
r
1243
+ T
r
1342
) +
u
( T
r
1234
+ T
r
1432
)
,
33
V
1
V
2
V
8
FI
G.
1
7
:
T
h
e
b
a
s
i
c
f
o
u
r
-l
o
o
p
v
a
c
u
u
m
i
n
t e
g
r
a
l
s
V
1
,
V
2
an
d
V
8
,t
o
w
h
i
c
h
a
l
l o
t
h
e
r
s
c
a
n
b
e
r
e
d
u
c
e
d
.
r
e
v
e
a
ls
t
h
a
t
t
h
e
le
a
d
in
g
U
V
b
e
h
a
v
io
r
c
o
m
e
s
s
o
le
ly
f
r
o
m
in
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
.
T
h
e
s
e
s
ix
in
t
e
g
r
a
ls
h
a
v
e
1
1
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
,
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
e
p
e
n
d
e
n
t
o
f
t
h
e
lo
o
p
m
o
m
e
n
t
u
m
.
T
h
e
r
e
f
o
r
e
t
h
e
y
d
iv
e
r
g
e

r
s
t
in
D
c
= 1
1
/
2
,
w
h
ic
h
m
a
t
c
h
e
s
t
h
e
e
x
p
e
c
t
e
d
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
c
= 4
+
6
/L
w
it
h
L
= 4
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
1
P
I
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
h
a
v
e
1
3
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
T
h
e
ir
n
u
-
m
e
r
a
t
o
r
s
w
o
u
ld
h
a
v
e
t
o
b
e
q
u
a
r
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
f
o
r
t
h
e
m
t
o
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
w
e
n
o
t
e
f
r
o
m
e
q
.
(
3
.1
4
)
t
h
a
t
t
h
e
m
a
s
t
e
r
n
u
m
e
r
a
t
o
r
s
N
18
an
d
N
28
a
r
e
q
u
a
d
r
a
t
ic
in
t h
e
τ
i j
,
a
n
d
h
e
n
c
e
m
e
r
e
ly
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
T
h
e
J
a
c
o
b
i
r
e
la
t
io
ns
(
A4
)
a
nd
(
A
5
)
p
r
e
s
e
r
v
e
t
h
is
q
u
a
d
r
a
t
ic
b
e
h
a
v
io
r
f
o
r
a
ll
n
u
m
e
r
a
t
o
r
s
.
T
h
e
r
e
f
o
r
e
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
a
r
e

n
it
e
in
D
c
= 1
1
/
2
.
T
h
e
1
P
R
in
t
e
g
r
a
ls
I
53
t h
ro
u
g
h
I
79
h
a
v
e
1
2
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
I
f
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
w
e
r
e
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
,
t
h
e
n
t
h
e
y
w
o
u
ld
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
it
is
e
a
s
y
t
o
s
e
e
f
r
o
m
e
q
s
.
(
A
5
)
a
n
d
(
B
1
)
t
h
a
t
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
a
r
e
a
ll
lin
e
a
r
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
I
n
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
r
e
d
u
c
e
e
a
s
ily
t
o
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
t
h
e
lim
it
t
h
a
t
t
h
e
e
x
t
e
r
n
a
l
m
o
m
e
n
t
a
v
a
n
is
h
.
T
h
e
p
la
n
a
r
in
t
e
g
r
a
ls
I
80
an
d
I
83
r
e
d
u
c
e
t
o
t
h
e
v
a
c
u
u
m
in
t
e
g
r
a
l
V
1
d
e
p
ic
t
e
d
in

g
.
1
7
.
W
h
ile
in
t
e
g
r
a
ls
I
81
an
d
I
84
a
r
e
n
o
n
p
la
n
a
r
a
s
f
o
u
r
-
p
o
in
t
g
r
a
p
h
s
,
in
t
h
e
v
a
c
u
u
m
lim
it
t
h
e
y
r
e
d
u
c
e
t
o
t
h
e
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
2
.
F
in
a
lly
,
in
t
e
g
r
a
ls
I
82
an
d
I
85
re
d
u
c
e
t o
t
h
e
n
o
n
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
8
.
A
s
w
a
s
t
h
e
c
a
s
e
a
t
t
h
r
e
e
lo
o
p
s
,
t
h
e
c
o
lo
r
f
a
c
t
o
r
s
f
o
r
t
h
e
le
a
d
in
g
U
V
g
r
a
p
h
s
a
r
e
r
e
la
t
e
d
b
y
c
o
lo
r
J
a
c
o
b
i
id
e
n
t
it
ie
s
.
I
n
t
h
is
c
a
s
e
w
e
c
a
n
s
u
b
t
r
a
c
t
,
f
o
r
e
x
a
m
p
le
,
C
81
fr
o
m
C
80
,a
n
d
u
s
e
a
J
a
c
o
b
i
id
e
n
t
it
y
o
p
e
r
a
t
in
g
o
n
t
h
e
b
o
x
a
t
t
h
e
t
o
p
c
e
n
t
e
r
o
f
t
h
e
g
r
a
p
h
s
in

g
.
1
1
.
T
h
e
n
w
e
r
e
d
u
c
e
t
h
e
r
e
s
u
lt
in
g
t
r
ia
n
g
le
s
u
b
g
r
a
p
h
s
it
e
r
a
t
iv
e
ly
,
t
o

n
d
C
80
,
83

2
N
4
c
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
=
C
81
=
C
82
=
C
84
=
C
85
.
(
4
.1
3
)
A
g
a
in
,
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
p
a
r
t
s
o
f
a
ll
c
o
n
t
r
ib
u
t
in
g
c
o
lo
r
f
a
c
t
o
r
s
a
r
e
e
q
u
a
l.
Fr
o
m
e
q
.
(
A
5
)
,
t
h
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
o
b
e
y
N
80
=
N
81
=
N
82
an
d
N
83
=
N
84
=
N
85
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
t
h
e
c
o
m
b
in
a
t
o
r
ia
l
f
a
c
t
o
r
o
f
I
81
in
e
q
.
(
3
.3
)
is
t
w
ic
e
a
s
la
r
g
e
a
s
t
h
o
s
e
fo
r
I
80
an
d
I
82
,
a
n
d
s
im
ila
r
ly
f
o
r
I
84
w
it
h
r
e
s
p
e
c
t
t
o
I
83
an
d
I
85
.
T
a
k
in
g
in
t
o
a
c
c
o
u
n
t
b
o
t
h
c
o
m
b
in
a
t
o
r
ia
l
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
,
t
h
e
c
o
n
t
r
ib
u
t
io
n
o
f
I
83
is

9
8
t
im
e
s
t
h
a
t
o
f
I
80
,a
n
d
s
im
ila
r
ly
f
o
r
t
h
e
o
t
h
e
r
t
w
o
p
a
ir
s
o
f
g
r
a
p
h
s
.
C
o
m
b
in
in
g
a
ll
t
e
r
m
s
a
n
d
s
w
it
c
h
in
g
t
o
t
h
e
c
o
lo
r
-
t
r
a
c
e
b
a
s
is
,
w
e

n
d
t
h
a
t
t
h
e
U
V
d
iv
e
r
g
e
n
c
e
in
t
h
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
D
= 1
1
/
2
is
g
iv
e
n
by
,
A
(4
)
4
(1
,
2
,
3
,
4)
SU
(
N
c
)
po
l
e
=

6
g
10
K
N
2
c
N
2
c
V
1
+ 1
2
(
V
1
+ 2
V
2
+
V
8
)
(
4
.1
4
)
×
s
( T
r
1324
+ T
r
1423
) +
t
( T
r
1243
+ T
r
1342
) +
u
( T
r
1234
+ T
r
1432
)
,
33
V
1
V
2
V
8
FI
G.
1
7
:
T
h
e
b
a
s
i
c
f
o
u
r
-l
o
o
p
v
a
c
u
u
m
i
n
t e
g
r
a
l
s
V
1
,
V
2
an
d
V
8
,t
o
w
h
i
c
h
a
l
l o
t
h
e
r
s
c
a
n
b
e
r
e
d
u
c
e
d
.
r
e
v
e
a
ls
t
h
a
t
t
h
e
le
a
d
in
g
U
V
b
e
h
a
v
io
r
c
o
m
e
s
s
o
le
ly
f
r
o
m
in
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
.
T
h
e
s
e
s
ix
in
t
e
g
r
a
ls
h
a
v
e
1
1
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
,
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
t
h
a
t
a
r
e
in
d
e
p
e
n
d
e
n
t
o
f
t
h
e
lo
o
p
m
o
m
e
n
t
u
m
.
T
h
e
r
e
f
o
r
e
t
h
e
y
d
iv
e
r
g
e

r
s
t
in
D
c
= 1
1
/
2
,
w
h
ic
h
m
a
t
c
h
e
s
t
h
e
e
x
p
e
c
t
e
d
c
r
it
ic
a
l
d
im
e
n
s
io
n
,
D
c
= 4
+
6
/L
w
it
h
L
= 4
.
I
n
c
o
n
t
r
a
s
t
,
t
h
e
1
P
I
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
h
a
v
e
1
3
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
T
h
e
ir
n
u
-
m
e
r
a
t
o
r
s
w
o
u
ld
h
a
v
e
t
o
b
e
q
u
a
r
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
f
o
r
t
h
e
m
t
o
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
w
e
n
o
t
e
f
r
o
m
e
q
.
(
3
.1
4
)
t
h
a
t
t
h
e
m
a
s
t
e
r
n
u
m
e
r
a
t
o
r
s
N
18
an
d
N
28
a
r
e
q
u
a
d
r
a
t
ic
in
t h
e
τ
i j
,
a
n
d
h
e
n
c
e
m
e
r
e
ly
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
T
h
e
J
a
c
o
b
i
r
e
la
t
io
ns
(
A4
)
a
nd
(
A
5
)
p
r
e
s
e
r
v
e
t
h
is
q
u
a
d
r
a
t
ic
b
e
h
a
v
io
r
f
o
r
a
ll
n
u
m
e
r
a
t
o
r
s
.
T
h
e
r
e
f
o
r
e
in
t
e
g
r
a
ls
I
1
t h
ro
u
g
h
I
52
a
r
e

n
it
e
in
D
c
= 1
1
/
2
.
T
h
e
1
P
R
in
t
e
g
r
a
ls
I
53
t h
ro
u
g
h
I
79
h
a
v
e
1
2
in
t
e
r
n
a
l
p
r
o
p
a
g
a
t
o
r
s
.
I
f
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
w
e
r
e
q
u
a
d
r
a
t
ic
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
,
t
h
e
n
t
h
e
y
w
o
u
ld
d
iv
e
r
g
e
in
D
= 1
1
/
2.
H
o
w
e
v
e
r
,
it
is
e
a
s
y
t
o
s
e
e
f
r
o
m
e
q
s
.
(
A
5
)
a
n
d
(
B
1
)
t
h
a
t
t
h
e
ir
n
u
m
e
r
a
t
o
r
s
a
r
e
a
ll
lin
e
a
r
in
t
h
e
lo
o
p
m
o
m
e
n
t
a
.
I
n
t
e
g
r
a
ls
I
80
t h
ro
u
g
h
I
85
r
e
d
u
c
e
e
a
s
ily
t
o
v
a
c
u
u
m
in
t
e
g
r
a
ls
in
t
h
e
lim
it
t
h
a
t
t
h
e
e
x
t
e
r
n
a
l
m
o
m
e
n
t
a
v
a
n
is
h
.
T
h
e
p
la
n
a
r
in
t
e
g
r
a
ls
I
80
an
d
I
83
r
e
d
u
c
e
t
o
t
h
e
v
a
c
u
u
m
in
t
e
g
r
a
l
V
1
d
e
p
ic
t
e
d
in

g
.
1
7
.
W
h
ile
in
t
e
g
r
a
ls
I
81
an
d
I
84
a
r
e
n
o
n
p
la
n
a
r
a
s
f
o
u
r
-
p
o
in
t
g
r
a
p
h
s
,
in
t
h
e
v
a
c
u
u
m
lim
it
t
h
e
y
r
e
d
u
c
e
t
o
t
h
e
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
2
.
F
in
a
lly
,
in
t
e
g
r
a
ls
I
82
an
d
I
85
re
d
u
c
e
t o
t
h
e
n
o
n
p
la
n
a
r
v
a
c
u
u
m
in
t
e
g
r
a
l
V
8
.
A
s
w
a
s
t
h
e
c
a
s
e
a
t
t
h
r
e
e
lo
o
p
s
,
t
h
e
c
o
lo
r
f
a
c
t
o
r
s
f
o
r
t
h
e
le
a
d
in
g
U
V
g
r
a
p
h
s
a
r
e
r
e
la
t
e
d
b
y
c
o
lo
r
J
a
c
o
b
i
id
e
n
t
it
ie
s
.
I
n
t
h
is
c
a
s
e
w
e
c
a
n
s
u
b
t
r
a
c
t
,
f
o
r
e
x
a
m
p
le
,
C
81
fr
o
m
C
80
,a
n
d
u
s
e
a
J
a
c
o
b
i
id
e
n
t
it
y
o
p
e
r
a
t
in
g
o
n
t
h
e
b
o
x
a
t
t
h
e
t
o
p
c
e
n
t
e
r
o
f
t
h
e
g
r
a
p
h
s
in

g
.
1
1
.
T
h
e
n
w
e
r
e
d
u
c
e
t
h
e
r
e
s
u
lt
in
g
t
r
ia
n
g
le
s
u
b
g
r
a
p
h
s
it
e
r
a
t
iv
e
ly
,
t
o

n
d
C
80
,
83

2
N
4
c
˜
f
a
1
a
2
b
˜
f
ba
3
a
4
=
C
81
=
C
82
=
C
84
=
C
85
.
(
4
.1
3
)
A
g
a
in
,
t
h
e
s
u
b
le
a
d
in
g
-
c
o
lo
r
p
a
r
t
s
o
f
a
ll
c
o
n
t
r
ib
u
t
in
g
c
o
lo
r
f
a
c
t
o
r
s
a
r
e
e
q
u
a
l.
Fr
o
m
e
q
.
(
A
5
)
,
t
h
e
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
o
b
e
y
N
80
=
N
81
=
N
82
an
d
N
83
=
N
84
=
N
85
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
t
h
e
c
o
m
b
in
a
t
o
r
ia
l
f
a
c
t
o
r
o
f
I
81
in
e
q
.
(
3
.3
)
is
t
w
ic
e
a
s
la
r
g
e
a
s
t
h
o
s
e
fo
r
I
80
an
d
I
82
,
a
n
d
s
im
ila
r
ly
f
o
r
I
84
w
it
h
r
e
s
p
e
c
t
t
o
I
83
an
d
I
85
.
T
a
k
in
g
in
t
o
a
c
c
o
u
n
t
b
o
t
h
c
o
m
b
in
a
t
o
r
ia
l
a
n
d
n
u
m
e
r
a
t
o
r
f
a
c
t
o
r
s
,
t
h
e
c
o
n
t
r
ib
u
t
io
n
o
f
I
83
is

9
8
t
im
e
s
t
h
a
t
o
f
I
80
,a
n
d
s
im
ila
r
ly
f
o
r
t
h
e
o
t
h
e
r
t
w
o
p
a
ir
s
o
f
g
r
a
p
h
s
.
C
o
m
b
in
in
g
a
ll
t
e
r
m
s
a
n
d
s
w
it
c
h
in
g
t
o
t
h
e
c
o
lo
r
-
t
r
a
c
e
b
a
s
is
,
w
e

n
d
t
h
a
t
t
h
e
U
V
d
iv
e
r
g
e
n
c
e
in
t
h
e
c
r
it
ic
a
l
d
im
e
n
s
io
n
D
= 1
1
/
2
is
g
iv
e
n
by
,
A
(4
)
4
(1
,
2
,
3
,
4)
SU
(
N
c
)
po
l
e
=

6
g
10
K
N
2
c
N
2
c
V
1
+ 1
2
(
V
1
+ 2
V
2
+
V
8
)
(
4
.1
4
)
×
s
( T
r
1324
+ T
r
1423
) +
t
( T
r
1243
+ T
r
1342
) +
u
( T
r
1234
+ T
r
1432
)
,
33
r
elated to diagrams in the

quartic
Casimir


Summary

Explicit calculations in
N
= 8

SUGRA
up to four loops show that the
power counting exactly follows that of
N
= 4

SYM
--

a finite theory


5 loop calculation in
D
=24/5 probes the potential 7
-
loop
D
=4
counterterm

--

will provide critical input to the
N
= 8

question !


D
=5
SYM have a 6
-
loop UV divergence, showing that the standard
perturbative

expansion misses some of the (2,0) theory contributions.


Color
-
Kinematics
duality
a
llows for gravity calculations for
multiloop

multipoint amplitudes
--

greatly facilitating UV analysis in gravity.


Numbers in UV divergences of
N

=8 SUGRA and

N

=4 SYM
coincide, suggesting a deeper connection between the theories


Stay tuned for the 5
-
loop SUGRA result…