uv structure - pennsylvania state...
TRANSCRIPT
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