Gravity as the Square of Gauge Theory
Michael Kiermaier
Princeton University
Great Lakes Strings Conference, April 29, 2011
a
a
based mainly on arXiv:1004.0693 with Zvi Bern, Tristan Dennen, Yu-tin Huang
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
1 / 25
How to Square Gauge Theory?
KLT relations
4-point:
M4 (1, 2, 3, 4) = −is12 A4 (1, 2, 3, 4)Ã4 (1, 2, 4, 3) .
5-point:
M5 (1, 2, 3, 4, 5) =is12 s34 A5 (1, 2, 3, 4, 5)Ã5 (2, 1, 4, 3, 5)
+ is13 s24 A5 (1, 3, 2, 4, 5)Ã5 (3, 1, 4, 2, 5) .
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
2 / 25
How to Square Gauge Theory?
KLT relations
4-point:
M4 (1, 2, 3, 4) = −is12 A4 (1, 2, 3, 4)Ã4 (1, 2, 4, 3) .
5-point:
M5 (1, 2, 3, 4, 5) =is12 s34 A5 (1, 2, 3, 4, 5)Ã5 (2, 1, 4, 3, 5)
+ is13 s24 A5 (1, 3, 2, 4, 5)Ã5 (3, 1, 4, 2, 5) .
n-point?
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
2 / 25
How to Square Gauge Theory?
KLT relations, n-point
h
X
Mn (1, 2, . . . , n) = i(−)n+1 An (1, 2, . . . , n)
f (i1 , . . . , ij )f¯(l1 , . . . , lj )
perms
i
Ãn (i1 , . . . , ij , 1, n − 1, l1 , . . . , lj , n)
+ P(2, . . . , n − 2) .
with
{i1 , . . . , ij } ∈ P(2, . . . , bn/2c) ,
and
f (i1 , . . . , ij ) = s1,ij
{l1 , . . . , lj 0 } ∈ P(bn/2c + 1, . . . , n − 2) .
j−1
Y
m=1
s1,im +
j
X
!
g (im , ik )
,
k=m+1
where g (i, j) = sij for i > j and g (i, j) = 0 otherwise.
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
3 / 25
How to Square Gauge Theory?
Problem
KLT relations in this form express the unordered gravity amplitude in
terms of color-ordered gauge theory amplitudes!
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
4 / 25
How to Square Gauge Theory?
Problem
KLT relations in this form express the unordered gravity amplitude in
terms of color-ordered gauge theory amplitudes!
Possible Solutions
Use “ordered” gravity amplitudes
⇒ Drummond, Spradlin, Volovich, Wen [arXiv:0901.2363]
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
4 / 25
How to Square Gauge Theory?
Problem
KLT relations in this form express the unordered gravity amplitude in
terms of color-ordered gauge theory amplitudes!
Possible Solutions
Use “ordered” gravity amplitudes
⇒ Drummond, Spradlin, Volovich, Wen [arXiv:0901.2363]
Use full unordered gauge-theory amplitude
⇒ Bern, Carrasco, Johansson [arXiv:0805.3993]
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
4 / 25
How to Square Gauge Theory?
BCJ duality
The gauge theory amplitude can be written as
X ni ci
Q
,
An =
sαi
diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
5 / 25
How to Square Gauge Theory?
BCJ duality
The gauge theory amplitude can be written as
X ni ci
Q
,
An =
sαi
diags. i
diagrams i only contain cubic vertices:

Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
 ...
Great Lakes
April 29, 2011
5 / 25
How to Square Gauge Theory?
BCJ duality
The gauge theory amplitude can be written as
X ni ci
Q
,
An =
sαi
diags. i
diagrams i only contain cubic vertices:
 ...

numerators ni satisfy Jacobi-like relations (“BCJ duality”):
ci + cj + ck = 0
s
Michael Kiermaier (Princeton University)
⇒
ni + nj + nk = 0 .
t
Gravity as the Square of Gauge Theory
u
Great Lakes
April 29, 2011
5 / 25
How to Square Gauge Theory?
conjectured BCJ squaring relations
Gauge theory amplitudes
X ni c i
Q
An =
,
s αi
diags. i
X
Ãn =
diags. i
ñ c
Qi i
s αi
with numerators satisfying Jacobi-like relations:
ci + cj + ck = 0
Michael Kiermaier (Princeton University)
⇒
ni + nj + nk = 0 ,
Gravity as the Square of Gauge Theory
ñi + ñj + ñk = 0 .
Great Lakes
April 29, 2011
6 / 25
How to Square Gauge Theory?
conjectured BCJ squaring relations
Gauge theory amplitudes
X ni c i
Q
An =
,
s αi
X
Ãn =
diags. i
diags. i
ñ c
Qi i
s αi
with numerators satisfying Jacobi-like relations:
ci + cj + ck = 0
⇒
ni + nj + nk = 0 ,
ñi + ñj + ñk = 0 .
Gravity amplitude:
−iMn =
X
diags. i
Michael Kiermaier (Princeton University)
n ñ
Qi i .
s αi
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
6 / 25
How to Square Gauge Theory?
conjectured BCJ squaring relations
Gauge theory amplitudes
X ni c i
Q
An =
,
s αi
X
Ãn =
diags. i
diags. i
ñ c
Qi i
s αi
with numerators satisfying Jacobi-like relations:
ci + cj + ck = 0
⇒
ni + nj + nk = 0 ,
ñi + ñj + ñk = 0 .
Gravity amplitude:
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Why do these squaring relations hold?
What are the implications?
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
6 / 25
Important related work
Stringy approach/generalizations of BCJ
Bjerrum-Bohr, Damgaard, Vanhove [0907.1425]
Stieberger, Mafra, Schlotterer [0907.2211, 1104.5224]
Tye, Zhang [1003.1732]
Bjerrum-Bohr, Damgaard, Sondergaard, Vanhove
[1003.2396,1003.2403]
Bern, Dennen [1103.0312]
Applications of Squaring Relations at loop level
Bern, Carrasco, Johansson [1004.0476]
Vanhove [1004.1392]
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
7 / 25
1
Generalized Gauged Invariance
2
Field Theory Derivation of the Squaring Relations
3
The Squaring Relations from a Lagrangian Viewpoint
4
BCJ at loop level
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
8 / 25
1
Generalized Gauged Invariance
2
Field Theory Derivation of the Squaring Relations
3
The Squaring Relations from a Lagrangian Viewpoint
4
BCJ at loop level
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
9 / 25
Generalized Gauge Invariance
Generalized Gauge Transformations
Gauge theory amplitude
An =
X
diags. i
n c
Qi i
sαi
is invariant under
ni
with
→
ni + ∆ i
X ∆i ci
Q
= 0.
s αi
diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
10 / 25
Generalized Gauge Invariance
Generalized Gauge Transformations
Gauge theory amplitude
An =
X
diags. i
n c
Qi i
sαi
is invariant under
ni
with
→
ni + ∆ i
X ∆i ci
Q
= 0.
s αi
diags. i
∆i “move around” contact terms, can be local or non-local
Preserves Jacobi-like relations if
∆i + ∆ j + ∆ k = 0 .
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
10 / 25
Generalized Gauge Invariance
The Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
n ñ
Qi i .
s αi
April 29, 2011
11 / 25
Generalized Gauge Invariance
The Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Generalized Gauge Transformation of the Squaring Relations
ni
→
ni + ∆ i
with
X ∆i ci
Q
= 0,
s αi
∆i + ∆ j + ∆ k = 0
diags. i
Gravity amplitude transforms as
−iMn
→
−iMn +
X ∆i ñi
Q
.
s αi
diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
11 / 25
Generalized Gauge Invariance
The Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Consistency Identity
If ∆i , ñi satisfy Jacobi-like relations:
X ∆i c i
Q
=0
s αi
diags. i
Michael Kiermaier (Princeton University)
⇒
X ∆i ñi
Q
= 0.
s αi
diags. i
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
12 / 25
Generalized Gauge Invariance
The Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Consistency Identity
If ∆i , ñi satisfy Jacobi-like relations:
X ∆i c i
Q
=0
s αi
diags. i
⇒
X ∆i ñi
Q
= 0.
s αi
diags. i
Origin: ci are color factors any gauge group
⇒ identity only relies on algebraic properties of ci
⇒ must work for ci → ñi
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
12 / 25
Generalized Gauge Invariance
The Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Consistency Identity
If ∆i , ñi satisfy Jacobi-like relations:
X ∆i c i
Q
=0
s αi
diags. i
⇒
X ∆i ñi
Q
= 0.
s αi
diags. i
Origin: ci are color factors any gauge group
⇒ identity only relies on algebraic properties of ci
⇒ must work for ci → ñi
⇒ ∆i actually do not need to satisfy Jacobi-like relations!
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
12 / 25
1
Generalized Gauged Invariance
2
Field Theory Derivation of the Squaring Relations
3
The Squaring Relations from a Lagrangian Viewpoint
4
BCJ at loop level
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
13 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
n ñ
Qi i .
s αi
April 29, 2011
14 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Strategy
Squaring relations trivial at 3-point:
−i M3 = A3 × Ã3 .
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
14 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Strategy
Squaring relations trivial at 3-point:
−i M3 = A3 × Ã3 .
Proceed inductively, using on-shell recursion relations
X
X
i
i
ÂL ÂR ,
M̂L M̂R ,
An =
Mn =
sα
sα
α
α
A L
Michael Kiermaier (Princeton University)
s
A R
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
14 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Assumptions
A local choice of ni exists such that
X ni ci
Q
,
ni + nj + nk = 0 .
An =
sαi
diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
15 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Assumptions
A local choice of ni exists such that
X ni ci
Q
,
ni + nj + nk = 0 .
An =
sαi
diags. i
Complex on-shell deformations of momenta
pa → p̂a (z) = pa + zqa ,
pa · qa = qa2 = 0
exist such that
M̂n (z) → 0 ,
Ân (z) → 0 ,
Èn (z) → 0
as
z → ∞.
(BCFW particularly suitable)
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
15 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Outline of derivation
Express gravity and gauge-theory amplitudes using recursion relation
Use lower-point squaring relations for subamplitudes
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
16 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Outline of derivation
Express gravity and gauge-theory amplitudes using recursion relation
Use lower-point squaring relations for subamplitudes
Crucial manipulation of gravity amplitude:
"
#
X i X
˜ α n̂i (zα)
˜α
∆αi ∆
n̂i (zα)ñˆi (zα) ∆αi ñˆi (zα)+ ∆
i
i
Q
Q
−
+Q
.
Mn =
s
ŝ
(z
)
ŝ
(z
)
ŝ
(z
)
α
α
α
α
α
α
α
i
i
i
α
α-diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
16 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Outline of derivation
Express gravity and gauge-theory amplitudes using recursion relation
Use lower-point squaring relations for subamplitudes
Crucial manipulation of gravity amplitude:
"
#
X i X
˜ α n̂i (zα)
˜α
∆αi ∆
n̂i (zα)ñˆi (zα) ∆αi ñˆi (zα)+ ∆
i
i
Q
Q
−
+Q
.
Mn =
s
ŝ
(z
)
ŝ
(z
)
ŝ
(z
)
α
α
α
α
α
α
α
i
i
i
α
α-diags. i
∆αi ,
˜ α are generalized gauge transformations:
∆
i
X
∆α c
Q i i
= 0
ŝαi (zα )
α-diags. i
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
16 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Outline of derivation
Express gravity and gauge-theory amplitudes using recursion relation
Use lower-point squaring relations for subamplitudes
Crucial manipulation of gravity amplitude:
"
#
X i X
˜ α n̂i (zα)
˜α
∆αi ∆
n̂i (zα)ñˆi (zα) ∆αi ñˆi (zα)+ ∆
i
i
Q
Q
−
+Q
.
Mn =
s
ŝ
(z
)
ŝ
(z
)
ŝ
(z
)
α
α
α
α
α
α
α
i
i
i
α
α-diags. i
∆αi ,
˜ α are generalized gauge transformations:
∆
i
X
X
∆α c
Q i i
= 0
⇒
ŝαi (zα )
α-diags. i
Michael Kiermaier (Princeton University)
α-diags. i
Gravity as the Square of Gauge Theory
∆α ñˆ (z )
Qi i α = 0
ŝαi (zα )
Great Lakes
April 29, 2011
16 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Outline of derivation
Express gravity and gauge-theory amplitudes using recursion relation
Use lower-point squaring relations for subamplitudes
Crucial manipulation of gravity amplitude:
Mn =
∆αi ,
X i
sα
α
X
α-diags. i
n̂i (zα)ñˆi (zα)
Q
ŝαi (zα )
.
˜ α are generalized gauge transformations:
∆
i
X
X
∆α c
Q i i
= 0
⇒
ŝαi (zα )
α-diags. i
Michael Kiermaier (Princeton University)
α-diags. i
Gravity as the Square of Gauge Theory
∆α ñˆ (z )
Qi i α = 0
ŝαi (zα )
Great Lakes
April 29, 2011
16 / 25
Deriving the Squaring Relations
ni +nj +nk = 0 ,
ñi +ñj +ñk = 0
⇒
−iMn =
X
diags. i
n ñ
Qi i .
s αi
Outline of derivation
Express gravity and gauge-theory amplitudes using recursion relation
Use lower-point squaring relations for subamplitudes
Crucial manipulation of gravity amplitude:
Mn =
∆αi ,
X i
sα
α
X
α-diags. i
n̂i (zα)ñˆi (zα)
Q
ŝαi (zα )
.
˜ α are generalized gauge transformations:
∆
i
X
X
∆α c
Q i i
= 0
⇒
ŝαi (zα )
α-diags. i
α-diags. i
∆α ñˆ (z )
Qi i α = 0
ŝαi (zα )
⇒ consistency identity implies squaring relations!
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
16 / 25
1
Generalized Gauged Invariance
2
Field Theory Derivation of the Squaring Relations
3
The Squaring Relations from a Lagrangian Viewpoint
4
BCJ at loop level
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
17 / 25
The Squaring Relations from a Lagrangian Viewpoint
Motivation
Amplitudes computed from the ordinary YM Lagrangian
do not satisfy Jacobi-like relations!
Can Jacobi-like relations
ni + nj + nk = 0
arise from a Lagrangian?
In what sense is
Lgravity = (Lgauge )2
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
?
Great Lakes
April 29, 2011
18 / 25
The Squaring Relations from a Lagrangian Viewpoint
Ordinary LYM does not lead to BCJ-compatible amplitudes
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
19 / 25
The Squaring Relations from a Lagrangian Viewpoint
Ordinary LYM does not lead to BCJ-compatible amplitudes
Strategy
Expand gauge theory Lagrangian as
L = LYM + L5 + L6 + . . .
Determine Ln , n ≥ 5 to make Jacobi-like relations manifest
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
19 / 25
The Squaring Relations from a Lagrangian Viewpoint
Ordinary LYM does not lead to BCJ-compatible amplitudes
Strategy
Expand gauge theory Lagrangian as
L = LYM + L5 + L6 + . . .
Determine Ln , n ≥ 5 to make Jacobi-like relations manifest
Ln , n ≥ 5 must not alter amplitudes!
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
19 / 25
The Squaring Relations from a Lagrangian Viewpoint
Ordinary LYM does not lead to BCJ-compatible amplitudes
Strategy
Expand gauge theory Lagrangian as
L = LYM + L5 + L6 + . . .
Determine Ln , n ≥ 5 to make Jacobi-like relations manifest
Ln , n ≥ 5 must not alter amplitudes!
Use auxiliary fields to turn Lagrangian cubic
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
19 / 25
The Squaring Relations from a Lagrangian Viewpoint
Ordinary LYM does not lead to BCJ-compatible amplitudes
Strategy
Expand gauge theory Lagrangian as
L = LYM + L5 + L6 + . . .
Determine Ln , n ≥ 5 to make Jacobi-like relations manifest
Ln , n ≥ 5 must not alter amplitudes!
Use auxiliary fields to turn Lagrangian cubic
Square cubic interactions in momentum space ⇒ Lgravity
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
19 / 25
The Squaring Relations from a Lagrangian Viewpoint
5-point
No covariant, local L5 can ensure Jacobi
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
20 / 25
The Squaring Relations from a Lagrangian Viewpoint
5-point
No covariant, local L5 can ensure Jacobi
Instead:
L5 = − 21 g 3 (f a1 a2 b f ba3 c + f a2 a3 b f ba1 c + f a3 a1 b f ba2 c )f ca4 a5
1
× ∂[µ Aaν]1 Aaρ2 Aa3 µ (Aa4 ν Aa5 ρ ) .
non-local and vanishing by Jacobi-identity
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
20 / 25
The Squaring Relations from a Lagrangian Viewpoint
5-point
No covariant, local L5 can ensure Jacobi
Instead:
L5 = − 21 g 3 (f a1 a2 b f ba3 c + f a2 a3 b f ba1 c + f a3 a1 b f ba2 c )f ca4 a5
1
× ∂[µ Aaν]1 Aaρ2 Aa3 µ (Aa4 ν Aa5 ρ ) .
non-local and vanishing by Jacobi-identity
can add one “self-BCJ” term:
∆L5 ∝ g 3 f a1 a2 b f ba3 c f ca4 a5 ∂(µ Aaν)1 Aaρ2 Aa3 µ + ∂(µ Aaν)2 Aaρ3 Aa1 µ
1
+ ∂(µ Aaν)3 Aaρ1 Aa2 µ
(Aa4 ν Aa5 ρ ) .
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
20 / 25
The Squaring Relations from a Lagrangian Viewpoint
5-point
No covariant, local L5 can ensure Jacobi
Instead:
L5 = − 21 g 3 (f a1 a2 b f ba3 c + f a2 a3 b f ba1 c + f a3 a1 b f ba2 c )f ca4 a5
1
× ∂[µ Aaν]1 Aaρ2 Aa3 µ (Aa4 ν Aa5 ρ ) .
non-local and vanishing by Jacobi-identity
can add one “self-BCJ” term:
∆L5 ∝ g 3 f a1 a2 b f ba3 c f ca4 a5 ∂(µ Aaν)1 Aaρ2 Aa3 µ + ∂(µ Aaν)2 Aaρ3 Aa1 µ
1
+ ∂(µ Aaν)3 Aaρ1 Aa2 µ
(Aa4 ν Aa5 ρ ) .
introducing auxiliary fields ⇒ local and cubic
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
20 / 25
The Squaring Relations from a Lagrangian Viewpoint
6-point
L5 not sufficient to ensure Jacobi at 6-point
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
21 / 25
The Squaring Relations from a Lagrangian Viewpoint
6-point
L5 not sufficient to ensure Jacobi at 6-point
L6 contains terms of the form
1
1
(∂Aa1 Aa2 Aa3 ) (Aa4 Aa5 )∂Aa6,
1 a1 a2
1
(A A )∂Aa3 (∂Aa4 Aa5 )Aa6, . . .
L6 vanishes by Jacobi-identity
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
21 / 25
The Squaring Relations from a Lagrangian Viewpoint
6-point
L5 not sufficient to ensure Jacobi at 6-point
L6 contains terms of the form
1
1
(∂Aa1 Aa2 Aa3 ) (Aa4 Aa5 )∂Aa6,
1 a1 a2
1
(A A )∂Aa3 (∂Aa4 Aa5 )Aa6, . . .
L6 vanishes by Jacobi-identity
30 different “self-BCJ” terms
⇒ BCJ seems easy to satisfy at tree-level
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
21 / 25
The Squaring Relations from a Lagrangian Viewpoint
6-point
L5 not sufficient to ensure Jacobi at 6-point
L6 contains terms of the form
1
1
(∂Aa1 Aa2 Aa3 ) (Aa4 Aa5 )∂Aa6,
1 a1 a2
1
(A A )∂Aa3 (∂Aa4 Aa5 )Aa6, . . .
L6 vanishes by Jacobi-identity
30 different “self-BCJ” terms
⇒ BCJ seems easy to satisfy at tree-level
n-point: General structure
need to add new “vanishing” terms Ln for all n
full local cubic Lagrangian ⇒ infinitely many auxiliary fields
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
21 / 25
The Squaring Relations from a Lagrangian Viewpoint
6-point
L5 not sufficient to ensure Jacobi at 6-point
L6 contains terms of the form
1
1
(∂Aa1 Aa2 Aa3 ) (Aa4 Aa5 )∂Aa6,
1 a1 a2
1
(A A )∂Aa3 (∂Aa4 Aa5 )Aa6, . . .
L6 vanishes by Jacobi-identity
30 different “self-BCJ” terms
⇒ BCJ seems easy to satisfy at tree-level
n-point: General structure
need to add new “vanishing” terms Ln for all n
full local cubic Lagrangian ⇒ infinitely many auxiliary fields
Non-polynomial structure not surprising: gives covariant Lgravity !
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
21 / 25
The Squaring Relations from a Lagrangian Viewpoint
6-point
L5 not sufficient to ensure Jacobi at 6-point
L6 contains terms of the form
1
1
(∂Aa1 Aa2 Aa3 ) (Aa4 Aa5 )∂Aa6,
1 a1 a2
1
(A A )∂Aa3 (∂Aa4 Aa5 )Aa6, . . .
L6 vanishes by Jacobi-identity
30 different “self-BCJ” terms
⇒ BCJ seems easy to satisfy at tree-level
n-point: General structure
need to add new “vanishing” terms Ln for all n
full local cubic Lagrangian ⇒ infinitely many auxiliary fields
Non-polynomial structure not surprising: gives covariant Lgravity !
To find general Ln : systematic approach? symmetry principle?
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
21 / 25
1
Generalized Gauged Invariance
2
Field Theory Derivation of the Squaring Relations
3
The Squaring Relations from a Lagrangian Viewpoint
4
BCJ at loop level
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
22 / 25
The Squaring Relations at Loop Level
KLT at loop level
KLT used in unitarity cuts for tree subamplitudes:
Only applicable on the cut, and different for each cut
Of practical importance, but no loop-level KLT relation
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
23 / 25
The Squaring Relations at Loop Level
Squaring relations at loop level
see also: Bern, Carrasco, Johansson [arXiv:1004.0476]
through the unitarity method, tree derivation generalizes to loop level
large z behavior ↔ cut-constructability
⇒ No issue if cuts are carried out in D dimensions
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
24 / 25
The Squaring Relations at Loop Level
Squaring relations at loop level
see also: Bern, Carrasco, Johansson [arXiv:1004.0476]
through the unitarity method, tree derivation generalizes to loop level
large z behavior ↔ cut-constructability
⇒ No issue if cuts are carried out in D dimensions
assumption: numerators arranged to satisfy Jacobi-like relations:
L
X Z Y
d D la ni (l1 , . . . , lL ) ci
Q
(−i)L AL-loop
=
, ni + nj + nk = 0 .
n
sαi (l1 , . . . , lL )
(2π)D
diags. i
Michael Kiermaier (Princeton University)
a=1
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
24 / 25
The Squaring Relations at Loop Level
Squaring relations at loop level
see also: Bern, Carrasco, Johansson [arXiv:1004.0476]
through the unitarity method, tree derivation generalizes to loop level
large z behavior ↔ cut-constructability
⇒ No issue if cuts are carried out in D dimensions
assumption: numerators arranged to satisfy Jacobi-like relations:
L
X Z Y
d D la ni (l1 , . . . , lL ) ci
Q
(−i)L AL-loop
=
, ni + nj + nk = 0 .
n
sαi (l1 , . . . , lL )
(2π)D
diags. i
a=1
Then:
L+1
(−i)
ML-loop
n
L
X Z Y
d D la ni (l1 , . . . , lL ) ñi (l1 , . . . , lL )
Q
=
,
sαi (l1 , . . . , lL )
(2π)D
diags. i
a=1
holds for arbitrary loop momenta (with internal lines off-shell)
A universal relation, not a different one for each cut
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
24 / 25
Summary and Outlook
Summary of results
Origin of Squaring Relations understood from a QFT perspective
Squaring implemented at a Lagrangian level
Better understanding of “ gravity=(gauge)2 ” (for trees and loops)
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
25 / 25
Summary and Outlook
Summary of results
Origin of Squaring Relations understood from a QFT perspective
Squaring implemented at a Lagrangian level
Better understanding of “ gravity=(gauge)2 ” (for trees and loops)
Various useful new expressions for gauge and gravity amplitudes
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
25 / 25
Summary and Outlook
Summary of results
Origin of Squaring Relations understood from a QFT perspective
Squaring implemented at a Lagrangian level
Better understanding of “ gravity=(gauge)2 ” (for trees and loops)
Various useful new expressions for gauge and gravity amplitudes
Open problems
Simple, explicit expression for local, Jacobi-satisfying numerators
For recent progress: see Stieberger, Mafra, Schlotterer [1104.5224]
Better understanding of BCJ at loop level
Can we see BCJ in the Grassmannian for planar N = 4 SYM?
(reconcile manifest locality with manifest planarity)
Implications for the UV properties of N = 8 supergravity
Non-perturbative analog of BCJ?
Michael Kiermaier (Princeton University)
Gravity as the Square of Gauge Theory
Great Lakes
April 29, 2011
25 / 25