The Lion in

Stanford Institute for Theoretical Physics

�JZ(J)

+ …

=?

17 March 2014

“PT Data = NP Data”

See “Decoding Perturbation theory using Resurgence”, and refs. therein Cherman, Dorigoni, Unsal: 1403.1277

Conjecture: QFT observables are resurgent functions

Resurgence and Transseries in Quantum, Gauge and

String Theories

30 June - 4 July, CERN

predicti

on

set of numbers

theory

prediction

(natural calculations)

description

(difficult calculations)

reformulation

(analytic expressions)

predicti

on

set of numbers

theory

prediction

(natural calculations)

description

Idea: sometimes have to manipulate theory into a form that’s usable for prediction / description

(difficult calculations)

reformulation

(analytic expressions)

organizational principles => physical principles

Shockwaves from Eikonal Limit

A(B)

n

= 2⇡�(2q · (p� p

0))A(A)

n

(3.25)

as desired. Going through the exact same procedure as the gravitational case, we then find

that:

A

(a)

µ

(k) = �igu

µ

D

(a)

1

�(z � t)

Zd

2k?(2⇡)2

e

�ik?·r? 1

k2?

(3.26)

If we insert this vector field into the Yang-Mills equation:

D

µ

F

(a)

µ⌫

= j

(a)

⌫

(3.27)

we find that the corresponding current is:

j

(a)

µ

= �igu

µ

D

(a)

1

�(z � t)�(x)�(y) (3.28)

Note this is exactly the source that gives rise to a QED shockwave, with the identification

of igD(a)

1

being the electromagnetic charge.

As we did for the gravitational case we can also instead show the equivalence using a

gauge transformation:

A

0µ

= A

µ

+ @

µ

⌦ (3.29)

with:

⌦ = igD

(a)

1

⇥(t� z)

Zd

2k?(2⇡)2

e

�ik?·r? 1

k2?

(3.30)

we have:

A

0

= A

z

= 0;

A? = � igD

(a)

1

4⇡⇥(t� z)r ln(r2?) (3.31)

which is exactly the QED result with the identification of igD(a)

1

being the electromagnetic

charge. So we have seen that deriving a shockwave in the eikonal limit causes it to take an

abelian form, even for the case of nonabelian gauge theories such as QCD.

3.3 Relationship Between The Two Shockwaves

It is interesting to note that the double copy relation similar to that which we showed in

section 2.3 can also be seen in the two shockwaves (3.10) and (3.26). For the full scattering

amplitude we looked at the gravitational quantity (�i)

n

(/2)

2nMn

. The analogous quantity to

look at for the gravitational shockwave is:

1

h

µ⌫

= �q

µ

q

⌫

�(z � t)

2q0

Zd

2k?(2⇡)2

e

�ik?·r? 1

k2?

(3.32)

– 16 –

since we have accounted for factors corresponding to graviton couplings and numerator

factors of the propagators that reside in “rest”. Note that we have identified �(z�t)

2q0as

being a propagator as this piece was derived solely from non-numerator contributions.

On the gauge theory side, the quantity we looked at for the full scattering amplitude

was (�i)

n�1

g

2n An

. The analogous quantity to look at for the shockwave is:

1

g

A

µ

= iq

µ

D

(a)

1

�(z � t)

2q0

Zd

2k?(2⇡)2

e

�ik?·r? 1

k2?

(3.33)

where we have accounted for a factor of g that corresponds to gluon couplings that reside

in “rest”. We have also rescaled the color factor to D

(a)

1

= �1

2

D

(a)

1

(instead of scaling

by � ip2

) since there are two factors of D(a)

1

that need to be accounted for in the n = 1

case that the single external gauge field corresponds to (one factor in the gauge field and

another factor residing in the attachment onto “rest”).

We then easily see that if we make the replacement D(a)

1

! iq

µ

in the gauge shockwave

we clearly recover the gravitational result. In concluding this section we would like to point

out that this double copy relation is only obvious in our choice of gauge and is greatly

obscured in any other gauge.

4 Discussion

We have shown that there exists a double copy relation between eikonalized gravity and

gauge theory amplitudes, as long as we consider only the lowest order contribution to

the color factors corresponding to completely uncrossed ladder diagrams. This restric-

tion is necessary as the inclusion of other contributions to the amplitude will result in

collinearly divergences which are known to cancel in gravitational amplitudes. This is a

feature particular to the eikonal approximation, as in this limit the numerator factors have

no dependence on the loop momenta which does not allow one to change the integrals

found in gauge theories once the double copy conjecture is applied.

An interesting consequence of this double copy relation between the eikonalized am-

plitudes is that the corresponding shockwave solutions have a double copy relation as

well. While both the gauge and gravity shockwaves had been calculated previously using

completely classical methods, our method directly shows the relationship between eikonal

amplitudes and shockwaves. Furthermore, the double copy relation between the two shock-

waves had not previously been seen. This is because this double copy relation is only

apparent in a particular choice of gauge, which was naturally selected by our method of

calculation.

One aspect of this analysis we would like to comment on is that we did not need to

consider gravity coupled to a dilaton or an anti-symmetric tensor in order to arrive at a

double copy relation in this kinematic regime at leading eikonal order. Such a coupling is

necessary for a double copy relation in the unrestricted kinematic region as discussed in

[1, 2]. However, it is to be expected that we would not need to consider these couplings

as in the soft regime scalars decouple due to power counting arguments. This reasoning

is also why ghosts do not need to be considered in the soft limit [23]. Arguments for the

– 17 –

Saotome and Akhoury, 1210.8111

(cf. also Alex’s talk!)

(inspired by / using perturbative structure)

(exists a gauge transform to Aichelberg-Sexl Metric)

+ … =Duff, PRD Vol 7 #8, 15 April 1973

Due for an update! What gauge solutions double-copy to BHs?

Schwarzchild Solution

What structure do we see in perturbative scattering?

“YM Data” = “Grav Data”

(gen.) gauge freedom makes life easier:

use freedom to impose constraints!

Motto:

In what sense constrained?

•Minimal information in.

•Relations propagate this information to a full solution.

What’s going on?

Consider an Amplitude

Original solution ofthree-loop four-point

N=4 sYM and N=8 sugra

Bern, JJMC, Dixon, Kosower, Johansson, Roiban ’07

Original solution ofthree-loop four-point

N=4 sYM and N=8 sugra

Bern, JJMC, Dixon, Kosower, Johansson, Roiban ’07

Original solution ofthree-loop four-point

N=4 sYM and N=8 sugra

Bern, JJMC, Dixon, Kosower, Johansson, Roiban ’07

Original solution ofthree-loop four-point

N=4 sYM and N=8 sugra

Bern, JJMC, Dixon, Kosower, Johansson, Roiban ’07

Bern, JJMC, Johansson, ‘10

Bern, JJMC, Johansson, ‘10

Bern, JJMC, Johansson, ‘10

Atreem = g(m`2)X

G2cubic

„c(G)n(G)D(G)

«

Color factors and numerator factors

satisfy similar lie algebra properties

Color-Kinematic Duality!

Bern, JJMC, Johansson (‘08)

Generic D-dimensional YM theories have a novel structure at tree-level

Jacobi

Vertex Antisymmetry

= +

= -

Atreem = g(m`2)X

G2cubic

„c(G)n(G)D(G)

«

Color-kinematic duality Bern, JJMC, Johansson (’08,’10)

Atreem = g(m`2)X

G2cubic

„c(G)n(G)D(G)

«

gravity lesson from on-shell 3 vertices: = x

color factors just sitting there obeying antisymmetry and Jacobi relations.

Color-kinematic duality Bern, JJMC, Johansson (’08,’10)

Atreem = g(m`2)X

G2cubic

„c(G)n(G)D(G)

«

gravity lesson from on-shell 3 vertices: = x

= Gravity amplitudein a related theory

Gravity!!

�iM treen =

X

G2cubic

n(G)n(G)D(G)

Double-copy property

Can be proven via recursion relations

= Gravity amplitudein a related theory

Gravity!! Bern, JJMC, Johansson (’08, ’10)

�iM treen =

X

G2cubic

n(G)n(G)D(G)

Double-copy property

Can be proven via recursion relations Bern, Dennen, Huang, Kiermaier (’10)

Antisymmetry + Kinematic Jacobi, makes manifest a (n-3)! basis (BCJ relations)

BCJ ’08

leads to proofs of all sorts of things (KLT etc), but the best thing, is a linear relation that can be inverted at tree-level to find Jacobi

satisfying numerators.

A(1,2, _, ..., _)

A(1,2,3, _, ... , _)

Relations between Color-Ordered Amplitudes

Antisymmetry => (n-2)! basis (Kleiss Kuijf relations)

= -

= +

In general for n-point:

Imposing Jacobi: (n-2)! master numerators only (n-3)! need be non-vanishing

leaves (n-3)!(n-3) free parameters

4 6 8 10 12 141

100

104

106

108

number of points

Hn-3L!Hn-

3Lfreeparameters

Post-Jacobi Freedom

(2n-5)!! cubic graphs

105 cubic graphs at 6 pt

24 master graphs at 6 pt

24 master graphs at 6 pt

Could it have something to do with loops?

What’s this freedom?

“One should always generalize.” - C. Jacobi

Could it have something to do with loops?

What’s this freedom?

(`i)Lgn`2+2L

Aloop=X

G2cubic

Z LY

l=1

dDpl(2ı)D

1

S(G)n(G)c(G)D(G)

What’s the correct loop-level CK generalization?

C-K holds for integrands!

= +d c

a b

1

2

3

n

d

a b

c

1

2

3

n

1

2

3

n

d

a b

c

4

n− 1

4 4

n− 1 n− 1

CONJECTURE: can find for any pure YM theory

(`i)Lgn`2+2L

Aloop=X

G2cubic

Z LY

l=1

dDpl(2ı)D

1

S(G)n(G)c(G)D(G)

LOOP LEVEL DOUBLE COPY

(`i)Lgn`2+2L

Aloop=X

G2cubic

Z LY

l=1

dDpl(2ı)D

1

S(G)n(G)c(G)D(G)

(`i)L+1(»=2)n`2+2L

Mloop

=

X

G2cubic

Z LY

l=1

dDpl(2ı)D

1

S(G)n(G)~n(G)D(G)

LOOP LEVEL DOUBLE COPY

TREMENDOUS CONSTRAINT AT LOOP-LEVEL

MINIMAL INFORMATION => FULL SOLUTION

Only need maximal cut information of (e) graphto build full amplitude!

BCJ (2010)

u = (k1 + k3)2t = (k1 + k4)2s = (k1 + k2)2 fii;j = 2ki ´ lj

Only need maximal cut information of (e) graphto build full amplitude!

BCJ (2010)

u = (k1 + k3)2t = (k1 + k4)2s = (k1 + k2)2 fii;j = 2ki ´ lj

Note:

BOTH N=4 sYM and N=8 sugra

manifestly have same overall powercounting!

u = (k1 + k3)2t = (k1 + k4)2s = (k1 + k2)2 fii;j = 2ki ´ lj

BCJ (2010)

JJMC, Johansson (2011)

Venerable form satisfies duality (no freedom)

Five point 1-loop N=4 SYM & N=8 SUGRA

JJMC, Johansson (2011)

Venerable form satisfies duality (no freedom)

Five point 1-loop N=4 SYM & N=8 SUGRA

JJMC, Johansson (2011)

Venerable form satisfies duality (no freedom)

Five point 1-loop N=4 SYM & N=8 SUGRA

Five point 2-loop N=4 SYM & N=8 SUGRAJJMC, Johansson (2011)

Five point 2-loop N=4 SYM & N=8 SUGRAJJMC, Johansson (2011)

Five point 2-loop N=4 SYM & N=8 SUGRAJJMC, Johansson (2011)

JJMC, Johansson (to appear)Five point 3-loop N=4 SYM & N=8 SUGRA

JJMC, Johansson (to appear)Five point 3-loop N=4 SYM & N=8 SUGRA

JJMC, Johansson (to appear)Five point 3-loop N=4 SYM & N=8 SUGRA

Full four loop N=4 SYM & N=8 SUGRA Bern, JJMC, Dixon, Johansson, Roiban (2012)

Full four loop N=4 SYM & N=8 SUGRA Bern, JJMC, Dixon, Johansson, Roiban (2012)

Full four loop N=4 SYM & N=8 SUGRA Bern, JJMC, Dixon, Johansson, Roiban (2012)

Bern, JJMC, Dixon, Johansson, Roiban

4-loops Maximal SUSY

So$the$leading$UV$pole$in$d=11/2$is$

M(4)

4

���pole

= �238

⇣�

2

⌘10

stu(s2 + t2 + u2)2 M tree

4

( + 2 + )

�256 +2025

8117propagator$integrals;$same$as$in$sYM$

127$and$137propagator$integrals$

As$for$comparison$with$the$single7trace$subleading$color$sYM$$

A(4)

4

���SU(Nc)

pole

= �6 g10KN2

c

⇣N2

c + 12 ( + 2 + )⌘

⇥⇣s (Tr

1324

+ Tr1423

) + t (Tr1243

+ Tr1342

) + u (Tr1234

+ Tr1432

)⌘

It$seems$unlikely$that$rela9on$is$a$coincidence;$its$origin$and$$implica9ons$however$are$not$clear;$may$con9nue$at$higher$loops$

So$the$leading$UV$pole$in$d=11/2$is$

M(4)

4

���pole

= �238

⇣�

2

⌘10

stu(s2 + t2 + u2)2 M tree

4

( + 2 + )

�256 +2025

8117propagator$integrals;$same$as$in$sYM$

127$and$137propagator$integrals$

As$for$comparison$with$the$single7trace$subleading$color$sYM$$

A(4)

4

���SU(Nc)

pole

= �6 g10KN2

c

⇣N2

c + 12 ( + 2 + )⌘

⇥⇣s (Tr

1324

+ Tr1423

) + t (Tr1243

+ Tr1342

) + u (Tr1234

+ Tr1432

)⌘

It$seems$unlikely$that$rela9on$is$a$coincidence;$its$origin$and$$implica9ons$however$are$not$clear;$may$con9nue$at$higher$loops$

In the new manifest representation, we have the power to identify remarkable structure between YM and Gravity

BCDJR

An interesting development at 4-loops!

D=11/2

D=11/2 5-loops? Need the integrand first!

5-loops N=4 sYM:~900 cubic graphs with no bubbles, and no triangles

Jacobi’s fix to a set of 3 planar masters!

can impose a consistent power-counting on a local ansatz

can impose all symmetries

(or 2 if you allow non-planar masters)

but maximal cuts break almost immediately!

Just like 3-loops before including:

Generic multiloop methods and application to N = 4 super-Yang-Mills 32

2

4(c)1

35 6 7

Jc

5 6 7

Jb 4(b)

32

1

5 6 7

Ja(a)

32

1 4

7

Ji

1 (i) 4

32

5

6

2

Jh

(h) 41

3

5

7 6

6

7 Jg

1 4(g)

2 3

5

3

65

7Jl

(l)1

2 3

4

6

7

5

Jk

(k)

2

1

3

4

5

7

6

Jj

(j)1

2

4

7 Jn

1 4(n)

2 3

5 6Jm 4(m)

32

1

6 75

(e) 41

2 3

5 6

7 7

6

Jf

(f)1

2 3

4

5

5 6 7

Jd

3

(d)

2

41

Figure 18. Three-loop four-point cubic graphs considered in the main text. Theexternal momenta is outgoing and the shaded (red) edges mark the application ofkinematic Jacobi relations used in (55). Note that only graphs (a)–(l) contribute tothe N = 4 sYM amplitude where the duality between color and kinematics is mademanifest.

5.3. Three-loop example

In this section we reexamine the four-point three-loop N = 4 sYM amplitude using

the duality between color and kinematics [28]. This amplitude was originally given in

[26, 27] in terms of nine cubic diagrams. For this exercise we start by considering a

larger set of 25 graphs, which are related to any of the original nine diagrams by a

single application of a kinematic Jacobi relation. However, eleven of these diagramscontain triangle subgraphs, which the no-triangle property of N = 4 sYM [1] suggests

will not contribute. After removing those with one-loop triangle subgraphs we have the

14 graphs depicted in figure 18. We will see that this set of diagrams is sufficiently large

to admit a manifest representation of the duality.

Now we will introduce the kinematic Jacobi relations that the numerators of each

diagram must satisfy. Each numerator depends on three independent external momenta

max cuts => local representation entirely inconsistent with manifest same critical dimension for N=8 SG as N=4 SYM

only UV happy if cancellations between local diagrams

Consider every edge cut--on shell (max cut)

Max cut Grav = (Max cut YM)^2

Yet, relaxing cut conditions => cancellations between diagrams.Bern, Dixon, Roiban ’06

Relaxing ansatz:

Asymmetric representations (maximize avail gauge freedom from 14 pt tree!)

Allow for non-local numerators Relax power countingAllow for add’l graphs (e.g non-planar triangles)Generalize prescription to handle unusual graphs

The sea of possibilities to explore is vast at 5 loops

The sea of possibilities to explore is vast at 5 loops

Maximally asymmetric (loop-level) representations allow coherent path to non-local representations

4 6 8 10 12 141

100

104

106

108

number of points

Hn-3L!Hn-

3Lfreeparameters

Post-Jacobi Freedom

Can learn lessons from Planar Loop-level Recursion

This has all been a very combinatorial/discrete/algorithmic discussion.

Open question: Does there exist a geometric interpretation?

cf. Arkani-Hamed,Trnka (for planar N=4

sYM)

shoals of understanding

when in doubt, calculate

ASK THE RIGHT QUESTIONS

beauty that trivializes calculations is very special

shoals of understanding