Supergravity workshop, Penn State, Sep 7, 2018

UV structure of gravity loop integrands

Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP), UC Davis

in collaboration with Enrico Herrmann (SLAC), 1808.10446

What is the scattering amplitude?

What is the scattering amplitude?

3

2

16

7

4

5

in planar N=4 SYM theory

What is the scattering amplitude?

3

2

16

7

4

5

What about gravity?

What is the scattering amplitude?

Evidence GR is very special

✤ Some pieces of evidence:

✤ In this talk I will show another evidence there is something special in gravity loop integrands

✤ The biggest mysteries are there for trees, not today

BCJ relations for gravity and YMEnhanced cancelationsLarge-z behavior of treesRemarkable Hodge’s MHV formula in 4DUnique gauge invariance

(Bern, Carrasco, Johansson 2007, 2010)

(Bern, Davies, Dennen 2014)

(Cachazo, Svrcek; Bedford, Brandhuber, Spence, Travaglini 2005) (Arkani-Hamed, Kaplan 2008)

(Hodges 2012)

(Arkani-Hamed, Rodina, Trnka 2016)

Lessons from planar N=4 SYM

✤ Identify properties which fix the amplitude uniquely

✤ They must constrain IR and UV

Uniqueness of planar loop integrand

UV

IR

Reformulate theseconstrains as inequalities which define geometry

Homogeneous conditions

3

2

16

7

4

5

Amplituhedron

Y = C · Z

(Arkani-Hamed, Trnka 2013)

Strategy for N=8 SUGRA

✤ Find IR and UV constraints for N=8 amplitudes

✤ We will work with the full non-planar loop integrand, no IBPs, no integration

✤ Must be homogeneous conditions

UV

IR

next step

next stepDoes it fix the

amplitude?Is there geometric picture

for gravity?

Summary of results

✤ Goal: Find special UV properties (cancelations) of gravity loop integrands on cuts in N=8 SUGRA

✤ Result: Cancelations between diagrams happen only in D=4 and it is not restricted only to N=8

⇠ 1

z8 ⇠ 1

z8�L

Loop integrands and cuts

Loop amplitude

✤ In this talk I will talk about loop amplitudes in D=4

✤ We can rewrite it as:

A =X

FD

ZIj d4`1 . . . d4`L

A =X

k

ck

ZIk d4`1 . . . `L

Basis integralsKinematical coefficients

Match the amplitude on cuts, do not neglect

terms which integrate to zero

I ⇠ I +@

@`µI

No total derivative issues

Planar integrand

✤ Planar (large N) limit: we can define global variables

✤ Switch integral and the sum:

x2 x2

x3x3

x4 x4

x1

x1

y1y1

y2

y2

k1 = (x1 � x2)

`1 = (x3 � y1)etc

Dual variables

A =X

k

ck

ZIk d4`1 . . . `L =

ZI d4`1 . . . d

4`L

Loop integrand

Momentum twistors(Hodges 2009)

Planar integrand

✤ Loop integrand is a rational function of momenta

Get the final amplitude: still want to integrate

A1�loop ⇠ Li2, log, ⇣2

AL�loop ⇠ ?polylogs

elliptic polylogsbeyond

AL�loop =

Zd4`1 . . . d

4`L I

Study the integrand insteadsimpler (rational) functionmany variables (loop momenta)properties of the amplitude non-trivially encoded in the integrand

✤ No planarity - no labels, no unique integrand

✤ No planar limit of gravity amplitudes

✤ Some attempts to solve the labeling issue

What is ?

Problem with labels

1 1 1

222 3

3

3 4

44

`

Sum over all labelsLinearized propagators

Nothing completely satisfactorywe have to stick with diagrams

✤ For both planar and non-planar we can perform: the cuts are given by the product of trees

✤ Generalized unitarity

Cuts of the integrand

`2 = (`+Q)2 = 0

other suggestion appeared in the context of Q-cuts [30] and ambitwistor strings [31, 32]

where the non-planar integrand was written using terms with linearized propagators.

For the case above the integrand would be written as a sum of 24 terms of the form,

I =X

�

1

`2(` · p1)(` · p12)(` · p4), (1.9)

where pij = (pi + pj) and � labels the 4! = 24 permutations of external legs. While

this gives a unique prescription there is a problem with spurious poles as I does not

vanish on the residue (` · p1) = 0. However, the representation (1.9) reproduces the

known answer after integration. While both proposals seem promising and the ultimate

solution to finding good loop coordinates for non-planar loop integrands might involve

some of the ideas involved there if we demand that the integrand is absent of spurious

poles or over-counting of singular regions then no such function exists.

1.2 Cuts of integrands

The problem of the non-planar integrand disappears if we consider unitarity cuts of

the integrand by putting some of the propagators on shell. In particular, if we cut

su�ciently many propagators, the cut defines natural coordinates and makes the cut

integrand Icut well-defined. The most extreme example is the maximal cut when all

propagators in a corresponding Feynman integral are set on-shell

, . (1.10)

In this particular case, the integral has 4L propagators so that the maximal cut localizes

all degrees of freedom and constitutes a leading singularity [33]. The on-shell conditions

localize all internal degrees of freedom, so that the residue of the loop-integrand is

a rational function of external kinematics. In a diagrammatic representation of the

amplitude where we only introduce Feynman integrals with at most 4L propagators,

such a maximal cut isolates a single term, and its coe�cient is directly given by this

on-shell function. Note that sometimes it is wise to go beyond diagrams with 4L

propagators to expose special features of a given theory, see e.g. the representations

of N = 4 sYM amplitudes in [34–36]. In a recently developed prescriptive approach

– 6 –

A ! AtreeL

1

`2(`+Q)2Atree

R

we can factorize everythingfurther to 3pt amplitudes

`+Q

`

✤ For both planar and non-planar we can perform: the cuts are given by the product of trees

✤ Generalized unitarity

Cuts of the integrandother suggestion appeared in the context of Q-cuts [30] and ambitwistor strings [31, 32]

where the non-planar integrand was written using terms with linearized propagators.

For the case above the integrand would be written as a sum of 24 terms of the form,

I =X

�

1

`2(` · p1)(` · p12)(` · p4), (1.9)

where pij = (pi + pj) and � labels the 4! = 24 permutations of external legs. While

this gives a unique prescription there is a problem with spurious poles as I does not

vanish on the residue (` · p1) = 0. However, the representation (1.9) reproduces the

known answer after integration. While both proposals seem promising and the ultimate

solution to finding good loop coordinates for non-planar loop integrands might involve

some of the ideas involved there if we demand that the integrand is absent of spurious

poles or over-counting of singular regions then no such function exists.

1.2 Cuts of integrands

The problem of the non-planar integrand disappears if we consider unitarity cuts of

the integrand by putting some of the propagators on shell. In particular, if we cut

su�ciently many propagators, the cut defines natural coordinates and makes the cut

integrand Icut well-defined. The most extreme example is the maximal cut when all

propagators in a corresponding Feynman integral are set on-shell

, . (1.10)

In this particular case, the integral has 4L propagators so that the maximal cut localizes

all degrees of freedom and constitutes a leading singularity [33]. The on-shell conditions

localize all internal degrees of freedom, so that the residue of the loop-integrand is

a rational function of external kinematics. In a diagrammatic representation of the

amplitude where we only introduce Feynman integrals with at most 4L propagators,

such a maximal cut isolates a single term, and its coe�cient is directly given by this

on-shell function. Note that sometimes it is wise to go beyond diagrams with 4L

propagators to expose special features of a given theory, see e.g. the representations

of N = 4 sYM amplitudes in [34–36]. In a recently developed prescriptive approach

– 6 –

we can factorize everythingfurther to 3pt amplitudes

�1 ⇠ �2 ⇠ �3 e�1 ⇠ e�2 ⇠ e�3

`2 = (`+Q)2 = 0

A ! AtreeL

1

`2(`+Q)2Atree

R`

`+Q

✤ For both planar and non-planar we can perform: the cuts are given by the product of trees

✤ Generalized unitarity

Cuts of the integrandother suggestion appeared in the context of Q-cuts [30] and ambitwistor strings [31, 32]

where the non-planar integrand was written using terms with linearized propagators.

For the case above the integrand would be written as a sum of 24 terms of the form,

I =X

�

1

`2(` · p1)(` · p12)(` · p4), (1.9)

where pij = (pi + pj) and � labels the 4! = 24 permutations of external legs. While

this gives a unique prescription there is a problem with spurious poles as I does not

vanish on the residue (` · p1) = 0. However, the representation (1.9) reproduces the

known answer after integration. While both proposals seem promising and the ultimate

solution to finding good loop coordinates for non-planar loop integrands might involve

some of the ideas involved there if we demand that the integrand is absent of spurious

poles or over-counting of singular regions then no such function exists.

1.2 Cuts of integrands

The problem of the non-planar integrand disappears if we consider unitarity cuts of

the integrand by putting some of the propagators on shell. In particular, if we cut

su�ciently many propagators, the cut defines natural coordinates and makes the cut

integrand Icut well-defined. The most extreme example is the maximal cut when all

propagators in a corresponding Feynman integral are set on-shell

, . (1.10)

In this particular case, the integral has 4L propagators so that the maximal cut localizes

all degrees of freedom and constitutes a leading singularity [33]. The on-shell conditions

localize all internal degrees of freedom, so that the residue of the loop-integrand is

a rational function of external kinematics. In a diagrammatic representation of the

amplitude where we only introduce Feynman integrals with at most 4L propagators,

such a maximal cut isolates a single term, and its coe�cient is directly given by this

on-shell function. Note that sometimes it is wise to go beyond diagrams with 4L

propagators to expose special features of a given theory, see e.g. the representations

of N = 4 sYM amplitudes in [34–36]. In a recently developed prescriptive approach

– 6 –

we can factorize everythingfurther to 3pt amplitudes

on-shell diagrams

`2 = (`+Q)2 = 0

A ! AtreeL

1

`2(`+Q)2Atree

R

`+Q

`

Properties of on-shell diagrams(Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka 2012)

✤ Building matrix with positive minors

✤ Positive Grassmannian

Same diagrams in mathematics

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

C =

✓1 ↵1 0 �↵4

0 ↵2 1 ↵3

◆

Active area of research in algebraic geometry and combinatoricsConnection to cluster algebras, KP equations,…

↵k > 0

Surprising connection

✤ Building matrix with positive minors

✤ For N=4 SYM the value of the diagram is equal to

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

C =

✓1 ↵1 0 �↵4

0 ↵2 1 ↵3

◆

⌦ =d↵1

↵1

d↵2

↵2. . .

d↵n

↵n�(C · Z)

(Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, JT 2012)

Surprising connection

✤ Building matrix with positive minors

✤ For N<4 SYM the value of the diagram is equal to

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

C =

✓1 ↵1 0 �↵4

0 ↵2 1 ↵3

◆

⌦ =d↵1

↵1

d↵2

↵2. . .

d↵n

↵n· J (↵)�(C · Z)

(Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, JT 2012)

Surprising connection

✤ Building matrix with positive minors

✤ For N=8 SUGRA the value of the diagram is equal to

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

C =

✓1 ↵1 0 �↵4

0 ↵2 1 ↵3

◆

⌦ =d↵1

↵31

d↵2

↵32

. . .d↵m

↵3m

Y

v

�v · �(C · Z)

(Herrmann, JT 2016)

Surprising connection

✤ Building matrix with positive minors

✤ For general QFT the value of the diagram is equal to

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

4.6 Coordinate Transformations Induced by Moves and Reduction

Let us now examine how the identification of diagrams via merge-operations, square-

moves, and bubble-deletion is reflected in the coordinates—the edge- or face-variables

—used to parameterize cells C 2 G(k, n). As usual, the simplest of these is the

merge/un-merge operation which trivially leaves any set of coordinates unchanged.

For example, in terms of the face variables, it is easy to see that

(4.62)

The square-move is more interesting. It is obvious that squares with opposite coloring

both give us a generic configuration in G(2, 4), but (as we will soon see), the square-

move acts rather non-trivially on coordinates used to parameterize a cell,

(4.63)

Let us start by determining the precise way the face-variables fi and f 0i of square-

move related diagrams are related to one another. To do this, we will provide perfect

orientations (decorated with edge variables) for both graphs, allowing us to com-

pare the resulting boundary-measurement matrices in each case. Because these two

boundary measurement matrices must represent the same point in G(2, 4), we will

be able to explicitly determine how all the various coordinate charts are related—

including the relationship between the variables fi and f 0i . Our work will be consid-

erably simplified if we remove the GL(1)-redundancies from each vertex, leaving us

with a non-redundant set of edge-variables. Of course, any choice of perfect orienta-

tions for the graphs, and any fixing of the GL(1)-redundancies would su�ce for our

purposes; but for the sake of concreteness, let us consider the following:

✓1 ↵1 0 ↵4

0 ↵2 1 ↵3

◆ ✓1 �2�3�4� 0 �4�

0 �2� 1 �1�2�4�

◆

(4.64)

– 41 –

C =

✓1 ↵1 0 �↵4

0 ↵2 1 ↵3

◆

⌦ = F (↵) �(C · Z)

✤ In a sense defines a theory (as Lagrangian does)F (↵)

Amplitude from recursion relations

✤ In any theory: on-shell diagrams = cuts of the amplitude

✤ In planar N=4 SYM theory we have recursion relations

✤ Even if we do not have recursion the properties of on-shell diagrams are related to properties of the amplitude

We learn about properties of the amplitude

= +X

L,R

Amplituhedron

✤ Pieces in the recursion glue together

THE 3D INDEX OF AN IDEAL TRIANGULATION AND ANGLE STRUCTURES 7

that recover the complete hyperbolic structure. A case-by-case analysis shows that this ex-ample admits an index structure, thus the index IT exists. This example appears in [HRS,Example 7.7]. We thank H. Segerman for a detailed analysis of this example.

2.4. On the topological invariance of the index. Physics predicts that when defined,the 3D index IT depends only on the underlying 3-manifold M . Recall that [HRS] provethat every hyperbolic 3-manifold M that satisfies

(2.9) H1(M,Z/2) → H1(M, ∂M,Z/2) is the zero map

(eg. a hyperbolic link complement) admits an ideal triangulation with a strict angle struc-ture, and conversely if M has an ideal triangulation with a strict angle structure, then M isirreducible, atoroidal and every boundary component of M is a torus [LT08].

A simple way to construct a topological invariant using the index, would be a map

M "→ {IT | T ∈ SM}

where M is a cusped hyperbolic 3-manifold with at least one cusp and SM is the set of idealtriangulations of M that support an index structure. The latter is a nonempty (generallyinfinite) set by [HRS], assuming that M satisfies (2.9). If we want a finite set, we can usethe subset SEP

M of ideal triangulations T of M which are a refinement of the Epstein-Pennercell-decomposition of M . Again, [HRS] implies that SEP

M is nonempty assuming (2.9). Butreally, we would prefer a single 3D index for a cusped manifold M , rather than a finitecollection of 3D indices.

It is known that every two combinatorial ideal triangulations of a 3-manifold are relatedby a sequence of 2-3 moves [Mat87, Mat07, Pie88]. Thus, topological invariance of the 3Dindex follows from invariance under 2-3 moves.

Consider two ideal triangulations T and !T with N and N+1 tetrahedra related by a 2−3move shown in Figure 1.

Figure 1. A 2–3 move: a bipyramid split into N tetrahedra for T and N + 1 tetrahedra for!T .

Proposition 2.13. If !T admits a strict angle structure structure, so does T and I!T = IT .

For the next proposition, a special index structure on T is given in Definition 6.2.

Y = C · ZLogarithmic volume form

⌦(Y, Zi)

Tree-level + loop integrand

(Arkani-Hamed, JT 2013)

Properties of on-shell diagrams

✤ In N=4 SYM (both planar and non-planar)

IR conditions: logarithmic singularities

A ⇠ dx

xx = 0near any pole

dx dy

xy(x+ y)x=0��! dy

y2more than just single poles:

with scalar numerator

many diagrams which arenaively okay wouldhave double poles

Properties of on-shell diagrams

UV conditions: no poles at infinitythere is no residue for ` ! 1

more than just UV finitenesstriangle has a pole at infinity

d4`

`2(`+ k1)2(`� k2)2`=↵�1

e�2������! d↵

↵

✤ In N=4 SYM (both planar and non-planar)

✤ Planar: baked in the geometry

✤ Non-planar: can not prove it is true for the amplitude

Non-planar N=4 SYM conjecture

✤ Implement both properties term-by-term

✤ Fixing coefficients: homogenous constraints

A =X

i

ai · Ci · Ii

2

I2(`) ⌘d4`

`2(`+ p2 + p3)2; I3(`) ⌘

d4` (p1 + p2)2

`2(`+ p2)2(`� p1)2;

I4(`) ⌘d4` (p1 + p2)2(p2 + p3)2

`2(`+ p2)2(`+ p2 + p3)2(`� p1)2. (2)

While the bubble integration measure is not logarithmic,it is known (see e.g. [8]) that the box can be written indlog-form, I4(↵)=dlog(↵1) ^ · · · ^ dlog(↵4), via:

↵1⌘`2/(` `⇤)2, ↵3⌘(`+p2+p3)2/(` `⇤)2,↵2⌘(`+p2)2/(` `⇤)2, ↵4⌘(` p1)2/(` `⇤)2,

(3)

where `⇤ ⌘ h23ih31i�1

e�2 is one of the quad-cuts of the box.Similarly, the triangle can also be written in dlog-form,I3(↵)=dlog(↵1) ^ · · · ^ dlog(↵4), via:

↵1⌘`2, ↵2⌘(`+p2)2, ↵3⌘(` p1)

2, ↵4⌘(` · `⇤), (4)

where `⇤⌘�1e�2.

Notice that while both the triangle and box integralsare logarithmic, only the box is free of a pole at ` 7!1.And while both integrals are UV-finite (unlike the bub-ble), poles at infinity could possibly signal bad UV be-havior. Although the absence of poles at infinity maynot be strictly necessary for finiteness, the amplitudesfor both N = 4 SYM and N =8 SUGRA are remarkablyfree of such poles through at least two-loops.

There are many reasons to expect that loop amplitudeswhich are logarithmic have uniform (maximal) transcen-dentality; and integrands free of any poles at infinity arealmost certainly UV-finite. This makes it natural to toask whether these properties can be seen term-by-termat the level of the integrand.

LOGARITHMIC FORM OF THE TWO-LOOPFOUR-POINT AMPLITUDE IN N =4 AND N =8

Our experience with planar N = 4 SYM suggests thatthe natural representation of the integrand which makeslogarithmic singularities manifest in terms of on-shell di-agrams, which are not in general manifestly local term-by-term. However at low loop-order, it has also beenpossible to see logarithmic singularities explicitly in par-ticularly nice local expansions [18, 19]. Since we don’t yethave an on-shell reformulation of ‘the’ integrand beyondthe planar limit (which may or may not be clearly definedfor non-planar amplitudes) we will content ourselves herewith an investigation of the singularity structure startingwith known local expansions of two-loop amplitudes.

The four-point, two-loop amplitude in N =4 SYM andN =8 SUGRA has been known for some time, [20]. It isusually given in terms of two integrand topologies—oneplanar, one non-planar—and can be written as follows:

A2-loop4,N =

KN4

X

�2S4

Z hC(P )

�,NI(P )� +C(NP )

�,N I(NP )�

i�4|2N

��·q

�(5)

where � is a permutation of the external legs and�4|2N (�·q) encodes super-momentum conservation with

q⌘(e�, e⌘); the factors KN are the permutation-invariants,

K4 ⌘ [3 4][4 1]

h1 2ih2 3i and K8 ⌘✓

[3 4][4 1]

h1 2ih2 3i

◆2

; (6)

the integration measures I(P )� , I(NP )

� correspond to,

(7)

and

I(NP )1,2,3,4 ⌘ (p1 + p2)

2 ⇥ (8)

for � = {1, 2, 3, 4}; and the coe�cients C(P ),(NP ){1,2,3,4},N are

the color-factors constructed out of structure constantsfabc’s according to the diagrams above for N =4, and areboth equal to (p1 + p2)2 for N =8.While the representation (5) is correct, it obscures

the fact that the amplitudes are ultimately logarithmic,maximally transcendental, and free of any poles atinfinity. This is because the non-planar integral’s

measure, I(NP )� , is not itself logarithmic. We will show

this explicitly below by successively taking residues untila double-pole is encountered; but it is also evidencedby the fact that its evaluation (using e.g. dimensionalregularization) is not of uniform transcendentality,[21]. These unpleasantries are of course cancelled incombination, but we would like to find an alternaterepresentation of (5) which makes this fact manifestterm-by-term. Before providing such a representation,let us first show that the planar double-box integrandcan be put into dlog-form, and then describe how thenon-planar integrands can be modified in a way whichmakes them manifestly logarithmic.

The Planar Double-Box Integral I(P )�

In order to write I(P )1,2,3,4 in dlog-form, we should

first normalize it to have unit leading singularities.This is accomplished by rescaling it according to:eI(P )1,2,3,4⌘s t I(P )

1,2,3,4, where s⌘(p1+p2)2 and t⌘(p2+p3)2

are the usual Mandelstam invariants. Now that it isproperly normalized, we can introduce an ephemeralextra propagator by multiplying the integrand by

(`1+p3)2/(`1+p3)2, and notice that eI(P )1,2,3,4 becomes the

product of two boxes—motivating the following change

2

I2(`) ⌘d4`

`2(`+ p2 + p3)2; I3(`) ⌘

d4` (p1 + p2)2

`2(`+ p2)2(`� p1)2;

I4(`) ⌘d4` (p1 + p2)2(p2 + p3)2

`2(`+ p2)2(`+ p2 + p3)2(`� p1)2. (2)

While the bubble integration measure is not logarithmic,it is known (see e.g. [8]) that the box can be written indlog-form, I4(↵)=dlog(↵1) ^ · · · ^ dlog(↵4), via:

↵1⌘`2/(` `⇤)2, ↵3⌘(`+p2+p3)2/(` `⇤)2,↵2⌘(`+p2)2/(` `⇤)2, ↵4⌘(` p1)2/(` `⇤)2,

(3)

where `⇤ ⌘ h23ih31i�1

e�2 is one of the quad-cuts of the box.Similarly, the triangle can also be written in dlog-form,I3(↵)=dlog(↵1) ^ · · · ^ dlog(↵4), via:

↵1⌘`2, ↵2⌘(`+p2)2, ↵3⌘(` p1)

2, ↵4⌘(` · `⇤), (4)

where `⇤⌘�1e�2.

Notice that while both the triangle and box integralsare logarithmic, only the box is free of a pole at ` 7!1.And while both integrals are UV-finite (unlike the bub-ble), poles at infinity could possibly signal bad UV be-havior. Although the absence of poles at infinity maynot be strictly necessary for finiteness, the amplitudesfor both N = 4 SYM and N =8 SUGRA are remarkablyfree of such poles through at least two-loops.

There are many reasons to expect that loop amplitudeswhich are logarithmic have uniform (maximal) transcen-dentality; and integrands free of any poles at infinity arealmost certainly UV-finite. This makes it natural to toask whether these properties can be seen term-by-termat the level of the integrand.

LOGARITHMIC FORM OF THE TWO-LOOPFOUR-POINT AMPLITUDE IN N =4 AND N =8

Our experience with planar N = 4 SYM suggests thatthe natural representation of the integrand which makeslogarithmic singularities manifest in terms of on-shell di-agrams, which are not in general manifestly local term-by-term. However at low loop-order, it has also beenpossible to see logarithmic singularities explicitly in par-ticularly nice local expansions [18, 19]. Since we don’t yethave an on-shell reformulation of ‘the’ integrand beyondthe planar limit (which may or may not be clearly definedfor non-planar amplitudes) we will content ourselves herewith an investigation of the singularity structure startingwith known local expansions of two-loop amplitudes.

The four-point, two-loop amplitude in N =4 SYM andN =8 SUGRA has been known for some time, [20]. It isusually given in terms of two integrand topologies—oneplanar, one non-planar—and can be written as follows:

A2-loop4,N =

KN4

X

�2S4

Z hC(P )

�,NI(P )� +C(NP )

�,N I(NP )�

i�4|2N

��·q

�(5)

where � is a permutation of the external legs and�4|2N (�·q) encodes super-momentum conservation with

q⌘(e�, e⌘); the factors KN are the permutation-invariants,

K4 ⌘ [3 4][4 1]

h1 2ih2 3i and K8 ⌘✓

[3 4][4 1]

h1 2ih2 3i

◆2

; (6)

the integration measures I(P )� , I(NP )

� correspond to,

I(P )1,2,3,4 ⌘ (p1 + p2)

2 ⇥ (7)

and

I(NP )1,2,3,4 (8)

for � = {1, 2, 3, 4}; and the coe�cients C(P ),(NP ){1,2,3,4},N are

the color-factors constructed out of structure constantsfabc’s according to the diagrams above for N =4, and areboth equal to (p1 + p2)2 for N =8.While the representation (5) is correct, it obscures

the fact that the amplitudes are ultimately logarithmic,maximally transcendental, and free of any poles atinfinity. This is because the non-planar integral’s

measure, I(NP )� , is not itself logarithmic. We will show

this explicitly below by successively taking residues untila double-pole is encountered; but it is also evidencedby the fact that its evaluation (using e.g. dimensionalregularization) is not of uniform transcendentality,[21]. These unpleasantries are of course cancelled incombination, but we would like to find an alternaterepresentation of (5) which makes this fact manifestterm-by-term. Before providing such a representation,let us first show that the planar double-box integrandcan be put into dlog-form, and then describe how thenon-planar integrands can be modified in a way whichmakes them manifestly logarithmic.

The Planar Double-Box Integral I(P )�

In order to write I(P )1,2,3,4 in dlog-form, we should

first normalize it to have unit leading singularities.This is accomplished by rescaling it according to:eI(P )1,2,3,4⌘s t I(P )

1,2,3,4, where s⌘(p1+p2)2 and t⌘(p2+p3)2

are the usual Mandelstam invariants. Now that it isproperly normalized, we can introduce an ephemeralextra propagator by multiplying the integrand by

(`1+p3)2/(`1+p3)2, and notice that eI(P )1,2,3,4 becomes the

product of two boxes—motivating the following change

Logarithmic singularitesNo poles at infinity

(Arkani-Hamed, Bourjaily, Cachazo, JT 2014) (Bern, Herrmann, Litsey, Stankowicz, JT 2014, 2015)

CutA = 0 ` = ↵p1e.g.(`+Q)2 = 0

Suggests geometric interpretation

hidden symmetry?

✤ Let us check if N=8 SUGRA can have either of these properties

No-go for N=8 SUGRA

z ! 1⇠ dz

zfor

pole at infinity

✤ Let us check if N=8 SUGRA can have either of these properties

No-go for N=8 SUGRA

z ! 1for

multiple pole at infinity

⇠ dz

z4�L

Rules of the game are certainly different

Gravity in IR(Herrmann, JT 2016)

✤ On-shell diagrams and cuts are all about singularities in the IR

✤ For N=4 SYM: this is full story, knowing on-shell diagrams is enough to fix the amplitude — no UV region

✤ N=8 SUGRA: on-shell diagrams capture IR — we can learn about IR properties of gravity loop integrands

From on-shell diagrams to amplitude in IR

`21 = `22 = `23 = `24 = 0

= +X

L,R

Collinearity conditions

✤ On-shell diagrams for gravity: surprising behavior

✤ This is slightly surprising because typical diagram would give

h`1`2i = h`1`3i = h`2`3i = 0

Residue on the cut ⇠ [`1`2]

h`1`2i⇥R

on the support `21 = `22 = 0 1

`23=

1

(`1 + `2)2! 1

h`1`2i[`1`2]Special case:

collinear limit ⇠ [12]

h12i ⇥R The main statementis more general

(Herrmann, JT 2016)

✤ In IR the amplitudes behaves very mildly

✤ They come from different regions but the strongest divergence comes from (soft)-collinear

Connection to singularities in IR

AL�loopYM ⇠ 1

✏2LAL�loop

GR ⇠ 1

✏L

(Herrmann, JT 2016)

3.1 IR of gravity from cuts

Due to the higher derivative nature of the gravity action the infrared divergences of

gravity at loop level are very mild. For Yang-Mills scattering amplitudes, the leading

IR-divergences of an L-loop amplitude calculated in dimensional regularization starts

with the leading 1/✏2L-term in the ✏-Laurent expansion. In contrast, the leading term

in gravity is only

M(L)

⇠1

✏L. (3.1)

The mild IR behavior of integrated gravity amplitudes can be nicely understood

from properties of the on-shell functions at integrand level already. In order to draw

this connection, we have to elaborate on the particular regions of loop-momentum

integration where infrared divergencies can in principle arise. The first possibility for

IR-divergencies comes from collinear regions where the internal loop momentum is

proportional to one of the external momenta, e.g. ` = ↵p1. At the level of on-shell

functions, this region is associated to cuts isolating a single massless external leg.

Res I`2=0=[`1]

= (3.2)

First, we put `2 = 0 on shell where the loop momentum factorizes into a product

of spinor-helicity variables ` = �`e�`. Due to this factorization of `, if the following

propagator-momentum di↵ers from ` by a massless external momentum, say p1 = �1e�1,

then (`� p1)2 also factorizes

(`� p1)2 = h`1i[`1] . (3.3)

In order to approach the collinear region we have to set both factors to zero, h`1i =

[`1] = 0 which localizes ` = ↵p1. Note that we are still cutting only two propagators

`2, (`�p1)2 but we impose three constraints. This residue can be thought of as cutting

two propagators and a Jacobian. The relation between the residue of I on this cut

and the IR divergence of the one-loop amplitude M(1) is as follow: if I has a non-zero

residue on a collinear cut ` = ↵pk, and there is an additional pole corresponding to a

soft-collinear singularity, ↵ = 0 or ↵ = 1, for which either ` = 0 or ` � pk = 0, the

combined IR-divergence of the amplitude is 1

✏2 . If the residue on ` = ↵pk is non-zero

but there are no further poles for ↵ = 0 or ↵ = 1 there is only a collinear divergence 1

✏ .

– 26 –

1

`2(`� p1)2`2=0���! 1

h`1i[`1]

0 0

` = ↵p1

✤ In IR the amplitudes behaves very mildly

✤ They come from different regions but the strongest divergence comes from (soft)-collinear

Connection to singularities in IR

AL�loopYM ⇠ 1

✏2LAL�loop

GR ⇠ 1

✏L

(Herrmann, JT 2016)

3.1 IR of gravity from cuts

Due to the higher derivative nature of the gravity action the infrared divergences of

gravity at loop level are very mild. For Yang-Mills scattering amplitudes, the leading

IR-divergences of an L-loop amplitude calculated in dimensional regularization starts

with the leading 1/✏2L-term in the ✏-Laurent expansion. In contrast, the leading term

in gravity is only

M(L)

⇠1

✏L. (3.1)

The mild IR behavior of integrated gravity amplitudes can be nicely understood

from properties of the on-shell functions at integrand level already. In order to draw

this connection, we have to elaborate on the particular regions of loop-momentum

integration where infrared divergencies can in principle arise. The first possibility for

IR-divergencies comes from collinear regions where the internal loop momentum is

proportional to one of the external momenta, e.g. ` = ↵p1. At the level of on-shell

functions, this region is associated to cuts isolating a single massless external leg.

Res I`2=0=[`1]

= (3.2)

First, we put `2 = 0 on shell where the loop momentum factorizes into a product

of spinor-helicity variables ` = �`e�`. Due to this factorization of `, if the following

propagator-momentum di↵ers from ` by a massless external momentum, say p1 = �1e�1,

then (`� p1)2 also factorizes

(`� p1)2 = h`1i[`1] . (3.3)

In order to approach the collinear region we have to set both factors to zero, h`1i =

[`1] = 0 which localizes ` = ↵p1. Note that we are still cutting only two propagators

`2, (`�p1)2 but we impose three constraints. This residue can be thought of as cutting

two propagators and a Jacobian. The relation between the residue of I on this cut

and the IR divergence of the one-loop amplitude M(1) is as follow: if I has a non-zero

residue on a collinear cut ` = ↵pk, and there is an additional pole corresponding to a

soft-collinear singularity, ↵ = 0 or ↵ = 1, for which either ` = 0 or ` � pk = 0, the

combined IR-divergence of the amplitude is 1

✏2 . If the residue on ` = ↵pk is non-zero

but there are no further poles for ↵ = 0 or ↵ = 1 there is only a collinear divergence 1

✏ .

– 26 –

1

`2(`� p1)2`2=0���! 1

h`1i[`1]

0 0

` = ↵p1

We know thatgravity integrand ⇠ [`1]

h`1i[`1]��! 0 vanishes

✤ Requires cancelation between diagrams even at 1-loop

✤ The behavior on the cut is stronger then just collinear vanishing, it puts one power of in the numerator

IR cancelations

Integrand cut

Integral 1

✏21

✏21

✏21

✏=+ +

` = ↵p1C1(↵) C2(↵) C3(↵) 0+ + =

Then the amplitude can be written as

A =

Zd4y1 d

4y2 . . . d4yL I , (1.5)

where the integrand I is now uniquely defined, and one does not have to refer to the sum

of Feynman integrals (1.1) anymore. This allowed to find BCFW recursion relations for

the loop integrand in planar N = 4 sYM [28] which were then reformulated in terms

of on-shell diagrams [25]. Naively even with good global coordinates the integrand is

still not uniquely defined because we can add terms proportional to total derivatives

I ⇠ I +@

@`eI . (1.6)

This is true if we are interested in amplitudes, A, directly (as the total derivatives

integrate to zero). However, if we want to obtain the integrand I in the context of

generalized unitarity as the function which satisfies all field theory cuts then no total

derivatives can be added as they would spoil matching the cuts. In other words, any

function which is a total derivative would change the value of the cuts which are already

matched by I or introduce unphysical poles.

In contrast, for non-planar theories the above set of unique labels in terms of dual

face variables is not available and we are forced to think about the integrand in the

context of (1.1) as sum of individual Feynman integrals. Let us demonstrate this for

the one-loop four-point amplitude in N = 8 supergravity first calculated by Brink,

Green and Schwarz [29] as low energy limit of string amplitudes. The amplitude can

be written in terms of three scalar box integrals,

�iM(1)

4= stu M

(0)

4

hIbox4

(s, t) + Ibox4

(u, t) + Ibox4

(s, u)i, (1.7)

where M(0)

4is the tree-level amplitude and the individual Feynman integrals

Ibox4

(s, t) = , Ibox4

(u, t) = , Ibox4

(s, u) = . (1.8)

are defined with unit numerators. In (1.7), the usual sij-dependent box-normalization

is included in the totally crossing-symmetric stu M(0)

4prefactor.

The question here is again how to choose the loop variables ` in individual diagrams.

One natural instruction is to sum over all choices of labeling an edge by `. While this

gives a unique function there is some intrinsic over-counting in this prescription. The

– 5 –[. . . ]

(Herrmann, JT 2016)

Gravity in UV: first encounter

✤ Pole at infinity can not be checked for the non-planar integrand — no global loop variables

✤ On maximal cut: poles at infinity -> diagram numerator

Pole at infinity on cuts

` ! 1

⇠ dz

z4�L

n = (`1 · `2)2L�6

only one diagram contributes - fixes

the numerator

1

2 3

4

cut, cut, cut, cut,…

Full amplitudeAll integrals contributeCan not check if there are poles at infinity

Maximal cutOne integral contributesThere are (higher) poles at infinity

We want to do this but cannot due to the lack of variables

Pole at infinity on cuts

1

2 3

4

cut, cut, cut, cut,…

Full amplitudeAll integrals contributeCan not check if there are poles at infinity

Maximal cutOne integral contributesThere are (higher) poles at infinity

Stop half-way in the cut structure: allow for

cancelations between diagrams

Pole at infinity on cuts

Non-trivial behavior at infinity

✤ We perform a cut where more diagrams contribute

✤ Send loop momenta to infinity:

✤ Any cancelation on any cut would be interesting

AL�loop4 = + + + . . .

z ! 1`k ! 1 by sending

zn = zm1 zm2 zm3 + . . .+ +

n < max(m1,m2, . . . )

✤ The minimal cut which defines good variables

✤ The residue is the sum of products trees:

✤ The set of all channels hits all diagrams

Multi-unitarity cut

`2k = 0X

k

`k = p1 + p2

multi-unitarity cut

The ` ! 1 scaling of the numerator can subsequently be related to the UV

behavior of the Feynman integral. When integrating (2.7) we can extract the leading

UV divergence from sending all loop momenta to infinity,

I =

Zd`

`15�2L= divergent for L � 7 . (2.8)

Following this line of reasoning, for maximal cuts there is a direct relation between

the degree of the pole at infinity, the loop-momentum dependence of the numerator

of the corresponding integral and the degree of the UV divergence. Up to this point

everything seems very predictable and unsurprising. One might expect that once poles

at infinity are present in maximal cuts, they also appear for lower cuts as well. Fur-

thermore, without relying on any surprises, the naive expectation is that all Feynman

integrals that appear in the expansion of the amplitude have the same (or lower) degree

poles at infinity as the cut integrand. However, we already saw at the end of section 1

that this is not true even at one-loop, where we found cancelations between box in-

tegrals on the triple cut (1.18). As we will show below, the unexpected behavior also

appears at higher loops.

2.2 Multi-particle unitarity cut

As we have seen in the previous subsection, no surprising features were found on max-

imal cuts mainly because they isolate individual Feynman integrals in the expansion

and there is no room for cancelations. We learned in the one-loop example (1.18) that

surprises appear when multiple diagrams contribute on a given lower cut. To this end,

we now consider the opposite to maximal cuts: we only cut a minimal number of prop-

agators which still gives us unique loop labels and allows us to approach infinity. The

particular cut of our interest is the multi-unitarity cut

. (2.9)

We will probe cancelations of the large loop momentum behavior between di↵erent

Feynman integrals that contribute to (2.9). The presence of such cancelations and

better behavior of the amplitude in comparison to individual integrals then points to

some novel mechanism or symmetry we have not yet unraveled in the context of gravity

amplitudes.

– 13 –

ResA =X

AtreeL Atree

R

✤ Unitarity cut at one-loop

✤ In this simple case we can rewrite the result

First example: one-loopThe absolute large z scaling powers of (2.13) are not really important. What mat-

ters is the relative di↵erence between the on-shell function and the individual integral.

In this simple one-loop example, there is an easy analytic proof of this enhanced scaling

behavior of the complete cut in comparison to individual diagrams. All box integrals

have the same crossing-symmetric prefactor 8 =⇣stA4,YM

tree (1234)⌘2

= stuM(0)

4, so that

the cut is given in a local expansion as the sum of four boxes,

= 8

"+ + +

#(2.14)

⇠1

(`1 · 2)(`1 · 3)+

1

(`1 · 1)(`1 · 3)+

1

(`1 · 2)(`1 · 4)+

1

(`1 · 1)(`1 · 4)

=((`1 · 1)+(`1 · 2)) ((`1 · 3)+(`1 · 4))

(`1 · 1)(`1 · 2)(`1 · 3)(`1 · 4)=

s212

(`1 · 1)(`1 · 2)(`1 · 3)(`1 · 4)

which directly shows the improved behavior of the full cut in comparison to individual

local diagrams. For N = 8 supergravity, the form of the Feynman integral expansion

as well as the structure of the super-amplitudes makes it clear that this is the same

result for any helicity configuration of the external states.

Two-loops

The next-to-simplest case is the three-particle cut of two-loop amplitudes. This example

is simple enough to keep track of all terms but we already have to choose a particular

way how to send the loop momenta to infinity. In our four-dimensional cut analysis,

we make such a choice by performing a collective shift on all cut legs. As we will

explain below, this appears to be the most uniform choice possible. We point out that

this particular limit of approaching infinity is special to D = 4 where we have spinor-

helicity variables at our disposal. This multi-particle cut has been analyzed before [42]

and was revisited in [12] in an attempt to understand the enhanced cancellations in

half-maximal supergravity in D = 5. The outcome of their analysis was that the

improved UV behavior of the amplitude in comparison to individual integrals can not

be seen at the integrand level. The authors of [12] checked that for some limit `i ! 1

there is no improvement in the large loop-momentum behavior after summing over all

terms, compared to the behavior of a single cut integral. For the particular amplitude

that was studied, the non-existence of an integrand level cancellation was reduced to

the statement that once a certain loop-momentum-dependent, permutation-invariant

prefactor is extracted, the remaining sum of diagrams is precisely the same one that

– 15 –

The absolute large z scaling powers of (2.13) are not really important. What mat-

ters is the relative di↵erence between the on-shell function and the individual integral.

In this simple one-loop example, there is an easy analytic proof of this enhanced scaling

behavior of the complete cut in comparison to individual diagrams. All box integrals

have the same crossing-symmetric prefactor 8 =⇣stA4,YM

tree (1234)⌘2

= stuM(0)

4, so that

the cut is given in a local expansion as the sum of four boxes,

= 8

"+ + +

#(2.14)

⇠1

(`1 · 2)(`1 · 3)+

1

(`1 · 1)(`1 · 3)+

1

(`1 · 2)(`1 · 4)+

1

(`1 · 1)(`1 · 4)

=((`1 · 1)+(`1 · 2)) ((`1 · 3)+(`1 · 4))

(`1 · 1)(`1 · 2)(`1 · 3)(`1 · 4)=

s212

(`1 · 1)(`1 · 2)(`1 · 3)(`1 · 4)

which directly shows the improved behavior of the full cut in comparison to individual

local diagrams. For N = 8 supergravity, the form of the Feynman integral expansion

as well as the structure of the super-amplitudes makes it clear that this is the same

result for any helicity configuration of the external states.

Two-loops

The next-to-simplest case is the three-particle cut of two-loop amplitudes. This example

is simple enough to keep track of all terms but we already have to choose a particular

way how to send the loop momenta to infinity. In our four-dimensional cut analysis,

we make such a choice by performing a collective shift on all cut legs. As we will

explain below, this appears to be the most uniform choice possible. We point out that

this particular limit of approaching infinity is special to D = 4 where we have spinor-

helicity variables at our disposal. This multi-particle cut has been analyzed before [42]

and was revisited in [12] in an attempt to understand the enhanced cancellations in

half-maximal supergravity in D = 5. The outcome of their analysis was that the

improved UV behavior of the amplitude in comparison to individual integrals can not

be seen at the integrand level. The authors of [12] checked that for some limit `i ! 1

there is no improvement in the large loop-momentum behavior after summing over all

terms, compared to the behavior of a single cut integral. For the particular amplitude

that was studied, the non-existence of an integrand level cancellation was reduced to

the statement that once a certain loop-momentum-dependent, permutation-invariant

prefactor is extracted, the remaining sum of diagrams is precisely the same one that

– 15 –

`1 ! 1cancelation

in D dimensions

Two-loops

✤ Next case is 2-loops: studied for half maximal SUGRA where enhanced cancelations of UV divergences happen in D=5

Is there a cancelation at the level of integrand?

`k ! t`kt ! 1

No cancelations!

Same kinematical statement applies for N=8 SUGRA

Re-scale cut momenta

(Bern, Enciso, Parra-Martinez, Zeng 2017)

✤ Using the no-triangle property of one-loop amplitudes

Old observation

of the cuts. By definition the behavior of maximal cuts is as bad as that for individual

diagrams. However, the logic here is as follows. If we see any improved behavior even

for descendant cuts of the original (L+1)-particle unitarity cut discussed previously, it

signals that the original (L+1)-cut itself, as well as the full uncut amplitude, if we are

able to find good variables, have an improved UV behavior at infinity too (perhaps even

further improved in comparison to the descendent cut due to additional cancellations).

There is an old example of a similar attempt to study the UV-structure of gravity

scattering amplitudes in terms of multi-unitarity cuts by cutting an L-loop amplitude

into a one-loop piece ⇥ a tree-level amplitude [10, 43] and potential cancellations were

related to the ”no-triangle property” of gravity at one-loop,

, (2.20)

which is to be compared to the scaling behavior of either the maximal cuts or iter-

ated two-particle cuts. The original argument for cancellations given in [10, 43] starts

from the observation, that the iterated two-particle cuts demand high powers of loop-

momentum ` in the numerator of the associated local integrals,

) N ⇠ stuM(0)

4(1234)

⇥t(`+ p1)

2⇤2(L�2)

. (2.21)

When writing the L-loop amplitude as a sum of local integrals many di↵erent diagrams

contribute on this cut including

⇠

Zd4` [(`+ p1)2]

2(L�2)

[`2]L+2, (2.22)

– 18 –

(Bern, Dixon, Roiban 2006) (Bern, Carrasco, Forde, Ita, Johannson 2007)

can be expandedin terms of boxes

of the cuts. By definition the behavior of maximal cuts is as bad as that for individual

diagrams. However, the logic here is as follows. If we see any improved behavior even

for descendant cuts of the original (L+1)-particle unitarity cut discussed previously, it

signals that the original (L+1)-cut itself, as well as the full uncut amplitude, if we are

able to find good variables, have an improved UV behavior at infinity too (perhaps even

further improved in comparison to the descendent cut due to additional cancellations).

There is an old example of a similar attempt to study the UV-structure of gravity

scattering amplitudes in terms of multi-unitarity cuts by cutting an L-loop amplitude

into a one-loop piece ⇥ a tree-level amplitude [10, 43] and potential cancellations were

related to the ”no-triangle property” of gravity at one-loop,

, (2.20)

which is to be compared to the scaling behavior of either the maximal cuts or iter-

ated two-particle cuts. The original argument for cancellations given in [10, 43] starts

from the observation, that the iterated two-particle cuts demand high powers of loop-

momentum ` in the numerator of the associated local integrals,

) N ⇠ stuM(0)

4(1234)

⇥t(`+ p1)

2⇤2(L�2)

. (2.21)

When writing the L-loop amplitude as a sum of local integrals many di↵erent diagrams

contribute on this cut including

⇠

Zd4` [(`+ p1)2]

2(L�2)

[`2]L+2, (2.22)

– 18 –

⇠ d4`

(`2)4

n = [(`+ k1)2]2L�4

⇠ d4`

(`2)6�LCan we make the scaling manifest?

✤ Using the no-triangle property of one-loop amplitudes

Old observation

of the cuts. By definition the behavior of maximal cuts is as bad as that for individual

diagrams. However, the logic here is as follows. If we see any improved behavior even

for descendant cuts of the original (L+1)-particle unitarity cut discussed previously, it

signals that the original (L+1)-cut itself, as well as the full uncut amplitude, if we are

able to find good variables, have an improved UV behavior at infinity too (perhaps even

further improved in comparison to the descendent cut due to additional cancellations).

There is an old example of a similar attempt to study the UV-structure of gravity

scattering amplitudes in terms of multi-unitarity cuts by cutting an L-loop amplitude

into a one-loop piece ⇥ a tree-level amplitude [10, 43] and potential cancellations were

related to the ”no-triangle property” of gravity at one-loop,

, (2.20)

which is to be compared to the scaling behavior of either the maximal cuts or iter-

ated two-particle cuts. The original argument for cancellations given in [10, 43] starts

from the observation, that the iterated two-particle cuts demand high powers of loop-

momentum ` in the numerator of the associated local integrals,

) N ⇠ stuM(0)

4(1234)

⇥t(`+ p1)

2⇤2(L�2)

. (2.21)

When writing the L-loop amplitude as a sum of local integrals many di↵erent diagrams

contribute on this cut including

⇠

Zd4` [(`+ p1)2]

2(L�2)

[`2]L+2, (2.22)

– 18 –

(Bern, Dixon, Roiban 2006) (Bern, Carrasco, Forde, Ita, Johannson 2007)

can be expandedin terms of boxes

⇠ d4`

(`2)4The scaling in otherloops obscured: irreducibleproblem pushed from one

loop to another

which have high poles at infinity and violate the ”no-triangle hypophysis”. Based on

this observation, it was argued that certain cancellations between diagrams had to

occur. Note that (` · p1) = (`+ p1)2 on the cut `2 = 0 so either choice would lead to the

same behavior. The di↵erence between both numerator choices is their continuation

o↵-shell. In this particular case, the worst behaved UV-terms can be pushed into

contact terms in contrast to our later example (2.29). The one-loop amplitude in

(2.20) has a box expansion and should therefore scale like d4`[`2]�4 compared to the

scaling of the higher-loop analog to the tennis-court integral of d4`[`2]L�6. We see

that, starting at L = 3, there is a mismatch between the tennis-court integral and the

one-loop box expansion. If one attempts to re-express the amplitude on the cut (2.20)

using a di↵erent set of Feynman integrals which manifest the box-type behavior of the

uncut loop, one encounters some trouble. Even though the power-counting for the

loop involving ` has been made manifest, this comes at a cost of spoiling the box-like

UV-behavior on the other side of the diagram which can now involve triangles,

. (2.23)

There seems to be some irreducible problem which can be pushed back and forth

between di↵erent loops. With a particular choice of representation, we can make the

UV-behavior of an individual loop momentum manifest, but not all at the same time.

Here, the problem is that we are approaching infinity via independent limits. The

ultimate check if a given sum of Feynman integrals that contribute on a cut can show

a global improvement of the UV-scaling is to send all loop momenta to infinity at

the same time. Similar arguments apply to another cut where the bottom tree-level

gravity amplitude in (2.20) is replaced by another uncut loop. On this cut, the iterated

two-particle analysis implies that some of the Feynman integrals need even higher

tensor-power numerators with a worse UV-behavior.

As mentioned before, at higher loops it becomes prohibitively complicated to de-

termine the numerator structures of the Feynman integrals completely. Instead of

summing all integrals which contribute on a given cut, we are going to focus directly

on the on-shell function. It is the product of tree-level amplitudes which encodes the

physical information independent of any particular integral expansion. We focus here

on a cut that allows us to probe the simultaneous large loop-momentum scaling in all

– 19 –

⇠ d4`

(`2)4

Need to send all loop momenta to infinity

(Herrmann, JT 2018)

Gravity in UV: cancelations(Herrmann, JT 2018)

Revisit two-loops

✤ Consider the N=8 SUGRA amplitude in D=4

`k = �ke�k

`k ! `k + zck�ke⇣

Generalized chiral shift

X

k

ck�k = 0

On-shell preservedMomentum conservation

z ! 1Send all loop momenta to infinity

`k = zck�ke⇣ ! 1

Loop momenta go to infinity in particular chiral direction

✤ In N=4 SYM no cancelation happens

Revisit two-loops

⇠ 1

z4

✤ In N=4 SYM no cancelation happens

✤ In N=8 SUGRA:

Revisit two-loops

⇠ 1

z4

✤ In N=4 SYM no cancelation happens

✤ In N=8 SUGRA: cancelation

Revisit two-loops

⇠ 1

z4

Same happensalso for

N<8 SUGRA

Higher loops

✤ We can push the same check at higher loops

✤ We checked it at 3-loops: again cancelation happened

✤ At higher loops: need to control tree-level amplitudes for higher multiplicity and all helicities

The ` ! 1 scaling of the numerator can subsequently be related to the UV

behavior of the Feynman integral. When integrating (2.7) we can extract the leading

UV divergence from sending all loop momenta to infinity,

I =

Zd`

`15�2L= divergent for L � 7 . (2.8)

Following this line of reasoning, for maximal cuts there is a direct relation between

the degree of the pole at infinity, the loop-momentum dependence of the numerator

of the corresponding integral and the degree of the UV divergence. Up to this point

everything seems very predictable and unsurprising. One might expect that once poles

at infinity are present in maximal cuts, they also appear for lower cuts as well. Fur-

thermore, without relying on any surprises, the naive expectation is that all Feynman

integrals that appear in the expansion of the amplitude have the same (or lower) degree

poles at infinity as the cut integrand. However, we already saw at the end of section 1

that this is not true even at one-loop, where we found cancelations between box in-

tegrals on the triple cut (1.18). As we will show below, the unexpected behavior also

appears at higher loops.

2.2 Multi-particle unitarity cut

As we have seen in the previous subsection, no surprising features were found on max-

imal cuts mainly because they isolate individual Feynman integrals in the expansion

and there is no room for cancelations. We learned in the one-loop example (1.18) that

surprises appear when multiple diagrams contribute on a given lower cut. To this end,

we now consider the opposite to maximal cuts: we only cut a minimal number of prop-

agators which still gives us unique loop labels and allows us to approach infinity. The

particular cut of our interest is the multi-unitarity cut

. (2.9)

We will probe cancelations of the large loop momentum behavior between di↵erent

Feynman integrals that contribute to (2.9). The presence of such cancelations and

better behavior of the amplitude in comparison to individual integrals then points to

some novel mechanism or symmetry we have not yet unraveled in the context of gravity

amplitudes.

– 13 –

`k = �ke�k

`k ! `k + zck�ke⇣

z ! 1

1

2 3

4

All-loop cuts

single diagramno cancelations

all diagramsnot well defined

The ` ! 1 scaling of the numerator can subsequently be related to the UV

behavior of the Feynman integral. When integrating (2.7) we can extract the leading

UV divergence from sending all loop momenta to infinity,

I =

Zd`

`15�2L= divergent for L � 7 . (2.8)

Following this line of reasoning, for maximal cuts there is a direct relation between

the degree of the pole at infinity, the loop-momentum dependence of the numerator

of the corresponding integral and the degree of the UV divergence. Up to this point

everything seems very predictable and unsurprising. One might expect that once poles

at infinity are present in maximal cuts, they also appear for lower cuts as well. Fur-

thermore, without relying on any surprises, the naive expectation is that all Feynman

integrals that appear in the expansion of the amplitude have the same (or lower) degree

poles at infinity as the cut integrand. However, we already saw at the end of section 1

that this is not true even at one-loop, where we found cancelations between box in-

tegrals on the triple cut (1.18). As we will show below, the unexpected behavior also

appears at higher loops.

2.2 Multi-particle unitarity cut

As we have seen in the previous subsection, no surprising features were found on max-

imal cuts mainly because they isolate individual Feynman integrals in the expansion

and there is no room for cancelations. We learned in the one-loop example (1.18) that

surprises appear when multiple diagrams contribute on a given lower cut. To this end,

we now consider the opposite to maximal cuts: we only cut a minimal number of prop-

agators which still gives us unique loop labels and allows us to approach infinity. The

particular cut of our interest is the multi-unitarity cut

. (2.9)

We will probe cancelations of the large loop momentum behavior between di↵erent

Feynman integrals that contribute to (2.9). The presence of such cancelations and

better behavior of the amplitude in comparison to individual integrals then points to

some novel mechanism or symmetry we have not yet unraveled in the context of gravity

amplitudes.

– 13 –

many diagramscancelationsno all-L check

1

2 3

4

Cutting more

The ` ! 1 scaling of the numerator can subsequently be related to the UV

behavior of the Feynman integral. When integrating (2.7) we can extract the leading

UV divergence from sending all loop momenta to infinity,

I =

Zd`

`15�2L= divergent for L � 7 . (2.8)

Following this line of reasoning, for maximal cuts there is a direct relation between

the degree of the pole at infinity, the loop-momentum dependence of the numerator

of the corresponding integral and the degree of the UV divergence. Up to this point

everything seems very predictable and unsurprising. One might expect that once poles

at infinity are present in maximal cuts, they also appear for lower cuts as well. Fur-

thermore, without relying on any surprises, the naive expectation is that all Feynman

integrals that appear in the expansion of the amplitude have the same (or lower) degree

poles at infinity as the cut integrand. However, we already saw at the end of section 1

that this is not true even at one-loop, where we found cancelations between box in-

tegrals on the triple cut (1.18). As we will show below, the unexpected behavior also

appears at higher loops.

2.2 Multi-particle unitarity cut

As we have seen in the previous subsection, no surprising features were found on max-

imal cuts mainly because they isolate individual Feynman integrals in the expansion

and there is no room for cancelations. We learned in the one-loop example (1.18) that

surprises appear when multiple diagrams contribute on a given lower cut. To this end,

we now consider the opposite to maximal cuts: we only cut a minimal number of prop-

agators which still gives us unique loop labels and allows us to approach infinity. The

particular cut of our interest is the multi-unitarity cut

. (2.9)

We will probe cancelations of the large loop momentum behavior between di↵erent

Feynman integrals that contribute to (2.9). The presence of such cancelations and

better behavior of the amplitude in comparison to individual integrals then points to

some novel mechanism or symmetry we have not yet unraveled in the context of gravity

amplitudes.

– 13 –

cutting more propagatorsfewer diagrams would contributeless/none cancelations expected

✤ Choose cuts which we can calculate to all loops✤ Compare to diagrams which numerators known to all loops✤ If cancelations here: expect cancelations in unitarity cuts too

All-loop cut I

✤ The cut chosen such that n-pt amplitude is MHV

✤ Parametrization of the cut

The`!

1scalingof

thenu

merator

can

subsequ

ently

berelated

totheUV

behaviorof

theFeynman

integral.W

hen

integrating(2.7)wecanextractthelead

ing

UV

divergence

from

sendingallloop

mom

enta

toinfinity,

I=

Zd`

`15�2L=

divergent

forL�

7.

(2.8)

Followingthislineof

reason

ing,

formax

imal

cuts

thereisadirectrelation

between

thedegreeof

thepoleat

infinity,

theloop

-mom

entum

dep

endence

ofthenu

merator

ofthecorrespon

dingintegral

andthedegreeof

theUV

divergence.Upto

this

point

everythingseem

svery

predictable

andunsurprising.

Onemight

expectthat

once

poles

atinfinityarepresent

inmax

imal

cuts,they

also

appearforlower

cuts

aswell.

Fur-

thermore,

withou

trelyingon

anysurprises,thenaive

expectation

isthat

allFeynman

integralsthat

appearin

theexpan

sion

oftheam

plitudehavethesame(orlower)degree

poles

atinfinityas

thecutintegran

d.How

ever,wealread

ysaw

attheendof

section1

that

this

isnot

trueeven

aton

e-loop

,wherewefoundcancelation

sbetweenbox

in-

tegralson

thetriple

cut(1.18).Aswewillshow

below

,theunexpectedbehavioralso

appears

athigher

loop

s.

2.2

Multi-particle

unitaritycut

Aswehaveseen

inthepreviou

ssubsection,nosurprisingfeatureswerefoundon

max

-

imal

cuts

mainly

becau

sethey

isolateindividual

Feynman

integralsin

theexpan

sion

andthereisnoroom

forcancelation

s.Welearned

intheon

e-loop

exam

ple

(1.18)

that

surprisesap

pearwhen

multiple

diagram

scontribute

onagivenlower

cut.

Tothisend,

wenow

consider

theop

positeto

max

imal

cuts:weon

lycutaminim

alnu

mber

ofprop-

agatorswhichstillgivesusuniqueloop

labelsan

dallowsusto

approachinfinity.

The

particularcutof

ourinterest

isthemulti-unitaritycut .

(2.9)

Wewillprobecancelation

sof

thelargeloop

mom

entum

behaviorbetweendi↵erent

Feynman

integralsthat

contribute

to(2.9).

Thepresence

ofsuch

cancelation

san

d

betterbehaviorof

theam

plitudein

comparison

toindividual

integralsthen

pointsto

somenovelmechan

ism

orsymmetry

wehavenot

yetunraveledin

thecontextof

gravity

amplitudes.

–13

–

Compact expression by Hodges (Hodges 2012)

`i = �xie�2

ri =

0

@iX

j=1

�xi � �2

1

Ae�2

i = 1, . . . , L� 1

more complicatedfor i=L,L+1

shift�xi ! �xi + ↵⇠

send ↵ ! 1

All-loop cut II

✤ Cancelation happens

✤ Explicitly checked up to L=4 but very likely continues

✤ Could not identify the diagram here known to all loops

✤ Consider another cut which hits Zvi’s favorite diagram which diverges at L=7 and beyond

All-loop cut II

✤ Consider another cut which hits Zvi’s favorite diagram which diverges at L=7 and beyond

All-loop cut II

✤ Consider another cut which hits Zvi’s favorite diagram which diverges at L=7 and beyond

All-loop cut II

Allow for cancelations

between permutations

of legs

✤ Consider another cut which hits Zvi’s favorite diagram which diverges at L=7 and beyond

✤ Parametrize cut in a similar way and send

All-loop cut II

n = (`1 · `2)2L�6

↵ ! 1

irreduciblenumerator

✤ Cut of the amplitude and the diagram to all loops

All-loop cut II

✤ Cut of the amplitude and the diagram to all loops

Cancelation!

All-loop cut II

Final remarks

✤ Cutting more: less cancelations — expect strongest for multi-unitarity cut or full non-planar integrand!

✤ Same cancelations for N<8 including pure GR, just the orders in different due to susy

✤ Generic gravity cancelations at “chiral” infinity in D=4

✤ Explanation? Hidden property or symmetry? Relation to UV of amplitude (e.g. controlling the divergence)?

↵

Fixing the amplitude

✤ Homogeneous constraints — UV, IR and absence of unphysical singularities to fix the amplitude in N=8

✤ Very different story than in N=4 SYM

✤ N=8 SUGRA is very different: nothing can be made manifest term by term —- gravity does not like diagrams!

(Edison, Herrmann, Langer, Parra-Martinez, JT, in progress)

A =X

k

ck

ZIk IR and UV

conditions manifest

absence of unphysical singularities fixed the coefficients

Gravity amplitudes

Naively, gravity amplitudes are much more complicated than gauge theory amplitudes

GR <<<<< YM2

12

Three Vertices

About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess.

Three-graviton vertex:

Three-gluon vertex:

Standard Feynman diagram approach.

KLT formulas proves this is not correct way to look at gravity 12

Three Vertices

About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess.

Three-graviton vertex:

Three-gluon vertex:

Standard Feynman diagram approach.

KLT formulas proves this is not correct way to look at gravity

Gravity amplitudes

KLT and recently BCJ discovered that in fact the amplitudes are remarkably related

GR = YM2

10

A New Way to Think About Gravity Kawai-Lewellen-Tye string relations in low-energy limit:

gravity gauge theory color ordered

Gravity is obtainable from gauge theory. Standard QFT offers no hint why this is possible.

Pattern gives explicit all-leg form

KLT (1985)

Gravity amplitudes

In this talk I tried to convince you there are additional special properties in gravity amplitudes

GR � YM2

Gravity amplitudes

GR >>>>>>>> YM2

?

I strongly believe there is a beautiful story, much richer and more exciting than for the gauge theory

Gravity amplitudes

I strongly believe there is a beautiful story, much richer and more exciting than for the gauge theory

GR >>>>>>>> YM2

My suspicion is that it is all hidden in the trees

(as e.g. Amplituhedron was)

Gravity amplitudes

Thank you for your attention