Niels Emil Jannik Bjerrum-Bohr

Structure of Amplitudes in Gravity

III

Symmetries of Loop and Tree amplitudes, No-Triangle Property, Gravity amplitudes

from String Theory

Playing with Gravity - 24th Nordic Meeting

Gronningen 2009Niels Emil Jannik Bjerrum-BohrNiels Emil Jannik Bjerrum-Bohr

Niels Bohr International AcademyNiels Bohr Institute

Outline

Outline Lecture III• We have considered how to compute tree

and loop amplitudes in gravity• We have seen how new efficient methods

clearly simplifies computations• In this lecture we would like to consider

the new insights that we get into gravity amplitudes from this

• Especially we want to focus on new symmetries and what this might tell us on the high energy limit of gravity

Gronningen 3-5 Dec 2009 3Playing with Gravity

Generic loop

amplitudes

5

Supersymmetric decomposition in YM

• Super-symmetry imposes a simplicity of the expressions for loop amplitudes.– For N=4 YM only scalar boxes appear.– For N=1 YM scalar boxes, triangles and bubbles

appear.

• One-loop amplitudes are built up from a linear combination of terms (Bern, Dixon, Dunbar, Kosower).

General 1-loop amplitudes

Vertices carry factors of loop momentum

n-pt amplitudep = 2n for gravityp=n for Yang-MillsPropagators


Mn = ¹ 2²Z dD `

(2¼)D

Q 2nj (q(2n;j )

¹ j¹̀ j ) + Q 2n¡ 1

j (q(2n¡ 1;j )¹ j

¹̀ j ) + ¢¢¢+ K2̀1 ¢¢¢̀ 2n

RdD ` 2(`¢k1)`2(`¡ k1)2 (¢¢¢) = RdD ` 1

(`¡ k1)2 (¢¢¢) ¡ RdD ` 1`2 (¢¢¢)

@p

For a SUSY theorywith N chargesthere will be N lessderivatives `

Playing with Gravity 7Gronningen 3-5 Dec 2009

(Passarino-Veltman) reduction

Collapse of a propagator

N = 1: @n¡ 1` !X

iQi

1(` ¡ ¦ i )2

N = 4: @n¡ 4` !X

iQi

4Y

k

1(`¡ ¦ ki )2

@2n¡ 8` !X

iQi

8¡ nY

k

1(` ¡ ¦ ki )2

N = 8(SUGRA) :

General 1-loop amplitudes

n=4: boxesn=5: trianglesn=6: bubbles…

5pt cut revisited


Lets consider 5pt 1-loop amplitude in N=8 Supergravity (singlet cut)Cut = RdLIPSM4(1¡ ;2¡ ;`+

1 ;`+2 )M5( ¡̀

1 ; ¡̀2 ;3+;4+;5+) =

h12i7 [12]h1 1̀i h1 2̀i h2 1̀i h2 2̀i h̀ 1 2̀i2£

h̀ 1 2̀i7 (h4 1̀i h̀ 2 3i [34] [̀ 1 2̀]¡ h34i h̀ 1 2̀i [4 1̀] [̀ 2 3])h34i h35i h45i h̀ 1 3i h̀ 1 4i h̀ 1 5i h̀ 2 3i h̀ 2 4i h̀ 2 5i

» s12 £ M5(1¡ ;2¡ ;3+;4+;5+) £ tr(1; l1; l2;2)hl1 1i hl2 2i [l1 1] [l2 2]

» s12 £ h12i6 [23] [45]h14i h15i h23i h34i h35i h45i £ tr(3; l1; l2;1)

hl1 3i hl2 1i [l1 3] [l2 1]

» s45 £ h12i7 [34] [12]h13i h15i h23i h25i h34i h45i2 £ tr(5; l1; l2;3)

hl1 5i hl2 3i [l1 5] [l2 3]

» s12 £ h12i6 [23] [45]h14i h15i h23i h34i h35i h45i £ tr(3; l2; l1;1)

hl2 3i hl1 1i [l2 3] [l1 1]


5pt cut revisited

tr(1; l2;2; l2) = ¡ h2jP j2](l1 ¡ k1)2 + h1jP j1](l2 + k2)2 ++(l1 ¡ k1)2(l2 + k2)2

tr(1; l2;3; l2) = h1jP j1]h3jP j3]¡ P 2s13¡ h3jP j3](l1 ¡ k1)2 + h1jP j1](l2 + k3)2

+(l1 ¡ k1)2(l2 + k3)2

tr(5; l2;3; l2) = h5jP j5]h3jP j3]¡ P 2s35¡ h3jP j3](l1 ¡ k5)2 + h5jP j5](l2 + k3)2

+(l1 ¡ k5)2(l2 + k3)2

tr(k1; l1; l2;k2) = ¡ tr(k1; l1;k2; l2) + sk1 l1 sk2 l2

Using that

We have


5pt cut revisited

• Surprice?

• Power counting seems to be seriously off?

• 5pt non-singlet shows similar behaviour…

• Part of a pattern..

No-TriangleProperty

12

No-Triangle Hypothesis

History True for 4pt n-point MHV 6pt NMHV (IR) 6pt Proof 7pt evidence n-pt proof

(Bern,Dixon,Perelstein,Rozowsky)(Bern, NEJBB, Dunbar,Ita)

(Green,Schwarz,Brink)

Consequence: N=8 supergravity same one-loop

structure as N=4 SYM

(NEJBB, Dunbar,Ita, Perkins, Risager; Bern, Carrasco, Forde, Ita, Johansson)

Direct evaluation of cuts (NEJBB, Vanhove; Arkani-Hamed,

Cachazo, Kaplan)Gronningen 3-5 Dec 2009 Playing with Gravity

13

No-Triangle Hypothesis: Cuts by cut…Attack different parts of amplitudes 1) .. 2) .. 3) ..

(1) Look at soft divergences (IR) 1m and 2m triangles

(2) Explicit unitary cuts bubble and 3m triangles

(3) Factorisation rational terms.

(NEJBB, Dunbar,Ita, Perkins, Risager; Arkani-Hamed, Cachazo, Kaplan; Badger, NEJBB, Vanhove)

Check that boxes gives the correct IR divergences In double cuts:would scale like

In double cuts:would scale like and

Scaling properties of (massive) cuts.

Gronningen 3-5 Dec 2009 Playing with Gravity

!

!

!

1z

z0 1z

14

No-Triangle HypothesisGravity IR loop relation :

Compact result for SYM tree amplitudes (Bern, Dixon and Kosower; Roiban Spradlin and Volovich)

Check that boxes gives the correct IR divergences

No one mass and two mass triangles (no statement about three mass

triangles

x C(1m) = 0 x C(2m) = 0

Checked until 7pt!

15

No-Triangle Hypothesis

Three mass triangles x C(3m) = 0

16

No-Triangle Hypothesisx C(bubble) = 0

Evaluate double cutsDirectly using various methods,Identify singularities.(e.g. Buchbinder, Britto,Cachazo Feng,Mastrolia)

Playing with Gravity 17

• Scaling behaviour of shifts

3-5 Dec 2009

Supergravity amplitudes

M tree¡(¡ 1̀)¡ ; i;¢¢¢;j ;( 2̀)¡ ¢£ M tree

¡(¡ 2̀)+; j + 1;¢¢¢;i ¡ 1;( 1̀)+¢

=X

i2C0

ci( 1̀ ¡ K i ;4)2( 2̀ ¡ K i ;2)2 +

X

j 2 D 0

dj( 1̀ ¡ K j ;3)2 + ek0 + D( 1̀; 2̀)

Let us consider this equation under the shift of the two-cut legs¸`1 ¡ ! ¸`1 + z¸`2 ;~̧̀

2 ¡ ! ~̧̀2 ¡ z ~̧̀

1

Playing with Gravity

Scaling behaviour

18

Yang-Mills

Gravity

QED

(hi,hj) : (+,+), (-,-), (+,-) (hi,hj) : (-,+)

(hi,hj) : (+,+), (-,-), (+,-) (hi,hj) : (-,+)

(hi,hj) : (+,-) (hi,hj) : (-,+)

(n-pt graviton amplitudes)

(n-pt 2 photon amplitudes)

(n-pt gluon amplitudes)Amazingly good behaviour

3-5 Dec 2009

» 1z

» z3

» 1z2

» z6

» z3¡ n

» z5¡ n

No-Triangle HypothesisN=4 SUSY Yang-Mills

N=8 SUGRA

QED (andsQED)

No-triangle property: YES Expected from power-counting and z-scaling properties

No-triangle property: YES NOT expected from naïve power-counting (consistent with string based rules)

No-triangle property: from 8ptNOT as expected from naive power-counting (consistent with string based rules)

String based formalism

No-triangle hypothesisGeneric loop amplitude (gravity / QED)

Passarino-VeltmanNaïve counting!!

(NEJBB, Vanhove)

Tensor integrals derivatives in Qn


No-triangle hypothesisString based formalism natural basis of integrals is

Constraint from SUSY

Amplitude takes the form


No-triangle hypothesisNow if we look at integrals

Typical expressions

Use+ integration by parts

Generalisation from 5 pts..


24

No-triangle hypothesisN=8 Maximal Supergravity(r = 2 (n – 4), s = 0)

(r = 2 (n – 4) - s, s >0)

Higher dimensional contributions – vanish by amplitude gauge invariance

Proof of No-triangle hypothesis

(NEJBB, Vanhove)

Gronningen 3-5 Dec 2009 Playing with Gravity

No-triangle hypothesisQED

(r = n, s = 0)

Higher dimensional contributions – vanish by amplitude gauge invariance

(NEJBB, Vanhove)

(from n = 8)


No-triangle hypothesisGeneric gravity theories:

Prediction N=4 SUGRA

Prediction pure gravity

N 3 theories constructable from cuts


>

No-triangle at multi-

loops

No-triangle for multi-loops

Two-particle cut might miss certain cancellations

Three/N-particle cutIterated two-particle cut

No-triangle hypothesis 1-loop

Consequences for powercounting arguments above one-loop..

Possible to obtain YM bound??D = 6/L + 4 for gravity???

Explicitly possible to

see extra cancellations!

(Bern, Dixon, Perelstein, Rozowsky; Bern, Dixon, Roiban)Gronningen 3-5 Dec

2009 28Playing with Gravity

29

No-triangle for multiloops

(Bern,Rozowsky,Yan)(Bern,Dixon,Dunbar, Perelstein,Rozowsky)Explicit at two loops :

‘No-triangle hypothesis’ holds at two-loops 4pt

(Bern, Carrasco, Dixon, Johansson, Kosower, Roiban)

…and even higher loops.

Still general principle for simplicity lacking…

Finiteness of N=8 SUGRA?

Finiteness Question


• For finiteness of N=8 supergravity we need a strong symmetry to remove the possible UV divergences that can be encountered at n-loop order.

• We know that SUSY limits the possibilities for UV divergences in supergravity considerably

• 4-loop computation explicit shows that particular divergences which could be present are in fact not

• Still however such divergences are not in conflict with SUSY – they can be adapted within formalism

• There will be a make or break point around 7-9 loops however…(this is far beyond present capabilities)

Finiteness Question


• The no-triangle property is not related to SUSY it is a symmetry of the amplitude which is also present in pure gravity

• Combined with SUSY we get a temendous simplification of the N=8 one-loop amplitudes– This is related to scaling behaviour at tree-level

• Origin is however still not understood..• To understand results at multi-loop level no-

triangle must be a key element– Clues from string theory: Unorderness of amplitudes

(and gauge invariance) – KEY: to get a better fundamental description of gravity


Summery of cookbook• We use cut techniques for gravity• Problems: cuts with many legs get

more and more cumbersome – Problem but can be dealt with using

more numerical techniques• Solution (maybe)

– Recursive inspired techniques – String based techniques

Gronningen 3-5 Dec 2009

What can be new developments

• Recent years seen automated computations for QCD and Yang-Mills– Much of this should be simple to

adapted to Gravity• Recursion techniques for gravity (also at

loop level) is something one thing one could consider..

• Automated numerical cut techniques to fix the whole amplitude including rational parts (i.e. Blackhat programs etc) (Berger et al)

• Multi-loop need better tools esp integral basis..Gronningen 3-5 Dec

2009 34Playing with Gravity

Monodromyrelations

Monodromy relations for Yang-Mills amplitudes

Monodromy related

Real part :

Imaginary part :(Kleiss – Kuijf) relations

New relations (Bern, Carrasco, Johansson)

(n-3)! functions in basis


See S¿ndergaard'sand Boel's talks

Monodromy and KLT

Double poles

x x xx. .

12

3

M

...+ +=1

2

1 M 12

3

s12 s1M s123

(1)

(2)(4)

(4)

(s124)

4pt

Cyclicity and flip


Monodromy and KLT

Completely Left-Right symmetric formula Fantastic simplicity comparing to Lagrangian complexity…. N-3! basis functions

5pt

N ptGronningen 3-5 Dec 2009 38Playing with Gravity

Summery


Summery• Good news

– Today we can do many more computations than 10 years ago

– This opens a window to further push limits for our understanding of gravity

– We have seen how to do tree and loops with great efficiency

– Need better understanding and techniques still

multi-loop level This is important for finiteness

questionGronningen 3-5 Dec 2009

Observations Gravity amplitudes: Simpler than expected

• Lagrangian hides simplicity• Amplitudes satisfy KLT squaring relation

KLT can be made more symmetric due to monodromy

• Amplitude has simplicity due to unorderedness/diffeomorphism invariance.

• Lead to no-triangle property• Simplicity already present in trees..

• Amplitude has many properties inherited fromYang-Mills : e.g. twistor space structure


Conclusions

43

Conclusions• The calculation of gravity amplitudes benefit hugely

from the use of new techniques.• More perturbative calculations of loop amplitudes

from unitarity will be helpful to understand the symmetry that we see…

• Importance of supersymmetry for cancellations not completely understood.– Will theories with less supersymmetry have similar surprising

cancellations?? N=6 (string theory says: YES)

– KLT seems to play an important roleGravity = (Yang Mills) x (Yang Mills’)

– ‘No-triangle cancellations’ needs to be understood at 1-loop • Calculations beyond 4pt could be important : 5pt 2-loop maybe?

44

Conclusions• The perturbative expansion of N=8 seems to be surprisingly

simple and very similar to N=4 at one-loop. At three loop no worse UV-divergences than N=4!

• This may have important consequences .. • Hints from String theory?? Explaination ???

• Perturbative finite / Non-perturbative completion???

(Abou-Zeid, Hull and Mason)

(Berkovits)(Green, Russo, Vanhove)

(Schnitzer)

Twistor-string theory for gravity??

Mass-less modes with non-perturbative origin??(Green, Ooguri, Schwarz)

Conclusions Clear no-triangle property at one-loop leads to constrains for amplitude at higher loops. Enough for finiteness… open question still Important to understand in full details :

• KLT squaring relation for gravity• Diffeomorphism invariance and unorderedness

of gravityKEY: We need better way to express this better

in order to understand symmetryPossible twistor space construction of gravity (Arkani-Hamed, Cachazo, Cheung, Kaplan)

Development of new and even better techniques for computations important..


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