niels emil jannik bjerrum-bohr
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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
Gronningen 3-5 Dec 2009 6Playing with Gravity
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
Gronningen 3-5 Dec 2009 8Playing with Gravity
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]
Gronningen 3-5 Dec 2009 9Playing with Gravity
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
Gronningen 3-5 Dec 2009 10Playing with Gravity
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
Gronningen 3-5 Dec 2009 21Playing with Gravity
No-triangle hypothesisString based formalism natural basis of integrals is
Constraint from SUSY
Amplitude takes the form
Gronningen 3-5 Dec 2009 22Playing with Gravity
No-triangle hypothesisNow if we look at integrals
Typical expressions
Use+ integration by parts
Generalisation from 5 pts..
Gronningen 3-5 Dec 2009 23Playing with Gravity
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)
Gronningen 3-5 Dec 2009 25Playing with Gravity
No-triangle hypothesisGeneric gravity theories:
Prediction N=4 SUGRA
Prediction pure gravity
N 3 theories constructable from cuts
Gronningen 3-5 Dec 2009 26Playing with Gravity
>
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
Gronningen 3-5 Dec 2009 31Playing with Gravity
• 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
Gronningen 3-5 Dec 2009 32Playing with Gravity
• 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
Playing with Gravity 33
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
Gronningen 3-5 Dec 2009 36Playing with Gravity
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
Gronningen 3-5 Dec 2009 37Playing with Gravity
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
Playing with Gravity 40
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
Gronningen 3-5 Dec 2009 41Playing with Gravity
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..
Gronningen 3-5 Dec 2009 45Playing with Gravity
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