uv structure of n=8 supergravity

UV structure of N=8 Supergravity

Emil Bjerrum-Bohr, IAS

Windows on Quantum Gravity 18Windows on Quantum Gravity 18thth June June 0808

Harald Ita, , UCLAUCLA

Warren Perkins

Dave Dunbar, Swansea University

Kasper Risager, NBI

Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager, ``The no-triangle hypothesis for N = 8 supergravity,'‘ JHEP 0612 (2006) 072 , hep-th/0610043.

D Dunbar Windows on Quantum Gravity

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Objective• Is N=8 Supergravity a self-consistent Quantum Field theory?

• Does the theory have ultra-violet singularities or is it a ``finite’’ field theory.


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N=8 Maximal Supergravity?

• Field theory with N=8 supersymmetry

• One graviton, eight gravitinos………….70 scalars

• Maximal supersymmetry consistent with spins <=2

• Field theory which couples gravity to all sorts of particles

• Endless symmetries…

• …really complicated Lagrangian

• Descendant of N=1 in D=11

g 2 1

3/2 8A 1 28 ½ 56 0 70(35

)

Cremmer,Julia, Scherk(ungauged)


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Superstring Theory

2) Look at supergravity embedded within string theory

N=8 Supergravity

1) Approach problem within the theory

Dual Theory

3) Find a dual theory which is solvable

Green, Russo, Van Hove, Berkovitz, Chalmers

Abou-Zeid, Hull, Mason

``Finite for 8 loops but not beyond’’


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…..Perturbative Quantum Gravity


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Quantum Problems: Renormalisability-calculate scattering amplitudes using Feynman vertices etc

- Only works if g is dimensionless

= +

Then we remove singularities by renormalising


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Gravity• Gravity cannot be renormalised (in D=4)• Infinities must be removed by adding terms to

lagrangian not present initially.

• If we have to continually add terms then theory looses predictive power

• Can avoid this if theory has no UV divergences (finite)

(eg N=4 SYM in D=4 and String Theory)

-can we find a finite field theory of gravity???


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• Feynman diagram approach to perturbative quantum gravity is not terrible useful

Using traditional techniques even the four-point tree amplitude for four graviton scattering is very difficult

Sannan,86

Supergravity calculations where we must calculate using all particles in multiplet are even more difficult…..


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however……N=8 supergravity has a lot of particles but it has enormous symmetry amongst them

Although computations are very difficult end results which must express this symmetry can be rather simple

New techniques which use symmetry to generate scattering amplitudes are particularly useful for supergravity

-return to S-matrix theory?


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-try to derive behaviour from N=4 SYM

• N=4 SYM is a finite field theory

• Try to exploit links to this for N=8 supergravity

• In string theory, closed string ~ open string x open string

So… N=8 ~ ( N=4) x (N=4)

Mandelstam


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Kawai-Lewellen-Tye Relations

-derived from string theory relations

-become complicated with increasing number of legs

-involves momenta prefactors

-applies to N=8/N=4 (and consequently pure YM/gravity)

Kawai,Lewellen Tye, 86


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Loop Calculations in N=8 Supergravity

• Desperately complicated using Feynman diagrams• Pre strings revolution of 1984 people believed theory was

finite. [Only candidate for quantum gravity…]• Post 1984 people believed theory was non-renormalisable

and only appeared as a low energy effective theory [ of string theory]

• In D=4 ``expect infinities’’ at 3-loops. [At this time nodefinite calculation of any infinity in D=4 in any supergravity

theory]

• In D > 4 they appear earlier ( …..s dDp » s |p|D-1d|p| )


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One-Loop Amplitudes• Calculated by Green Schwarz and Brink using

string theory

I4 (s,t) is scalar box integral

Remarkably similar to the N=4 Yang-Mills results

(colour ordered/leading in colour part/planar)

2

1

3

4


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Two-Loop Supergravity, all D form,

Bern,Dixon,Dunbar,Perelstein,Rozowsky

-N=8 amplitudes very close to N=4

(planar part) Bern, Rozowsky, Yan


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Proof: use unitarity methodsBern, Dixon, Dunbar, Kosower 94,95..-reconstruct amplitude using its unitary cuts:

Eg for 4pt two-loop amplitude


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For N=4 SYM/ N=8 SUGRA Key Identity

T estingT estertesttester

propagators-pair of propagaters is exactly the cut in a scalar box integral

1

2 3

4l1

l2


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-equivalent identity for N=8

-derivable using KLT relations


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Using Identity also works for 2-loop

3

4

-this (plus other work) gives two-loop result

-consider

-which is


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l

l l

l-three loop cuts, YM

s2 s [ l1 . 4 ]

1

2

Loop momentum caught in integral

-gives ansatz for multiloop terms


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Using Identity for multiloop

4, N=8

-beyond 2 loops N=4 SYM and N=8 SUGRA have different functions


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UV behaviour of diagramsWorst behaved integral has integrand

Infinite if…..

Or Finite if


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UV pattern of Pattern 98,07

D=11

0 #/ "

D=10

0(!) #/"

D=9 0 #/"D=8 #/" #’/"2+#”/

"D=7 0 #/"D=6 0 0D=5 0 0 0D=4 0 0 0 0

L=1 L=2 L=3 L=4 L=5 L=6

N=4 Yang-Mills

Honest calculation/ conjecture (BDDPR)

Based upon 4pt amplitudes

N=8 Sugra

UV pattern of Pattern 98


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Caveats :Caveats: 1) not all functions touched

2) assume no cancellations between diagrams

-gets N=4 SYM correct Howe, Stelle


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New Results: driven by progress in QCD

• More loops

• More legs

• Formal Proofs • Start with more legs…


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General Decomposition of One- loop n-point Amplitude

Vertices involve loop momentum propagators

p

degree p in l

p=n : Yang-Millsp=2n Gravity


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Passarino-Veltman reduction

Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator


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Passarino-Veltman reduction

•process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) In going from 4 -> 3 scalar boxes are generated•similarly 3 -> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. •so in general, for massless particles

Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator


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N=4 SUSY Yang-Mills• In N=4 Susy there are cancellations between the

states of different spin circulating in the loop.• Leading four powers of loop momentum cancel (in

well chosen gauges..)

• N=4 lie in a subspace of the allowed amplitudes (Bern,Dixon,Dunbar,Kosower, 94??)

• Determining rational ci determines amplitude- Tremendous progress in last few years

Green, Schwarz, Brink, Bern, Dixon, Del Duca, Dunbar, Kosower

Britto, Cachazo, Feng; Roiban Spradlin Volovich Bjerrum-Bohr, Ita, Bidder, Perkins, Risager,

Brandhuber,Spence, Travaglini


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Basis in N=4 Theory‘‘easy’ two-mass easy’ two-mass boxbox

‘‘hard’ two-mass hard’ two-mass boxbox


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N=8 Supergravity • Loop polynomial of n-point amplitude of degree 2n.

• Leading eight-powers of loop momentum cancel (in well chosen gauges..) leaving (2n-8) or (2r-8)

• Beyond 4-point amplitude contains triangles and bubbles but

only after reduction

• Expect triangles n > 4 , bubbles n >5 , rational n > 6

r


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No-Triangle Hypothesis-against this expectation, it might be the case that…….

Evidence?true for 4pt 5+6pt-point MHV General feature 6+7pt pt NMHV

Bern,Dixon,Perelstein,Rozowsky

Bern, Bjerrum-Bohr, Dunbar

Green,Schwarz,Brink (no surprise)

• One-Loop amplitudes N=8 SUGRA look ``just like’’ N=4 SYM

Bjerrum-Bohr, Dunbar, Ita,Perkins Risager


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Evidence???

• Attack different parts by different methods

• Soft Divergences -one and two mass triangles

• Unitary Cuts –bubbles and three mass triangles

• Factorisation –rational terms


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Soft-DivergencesOne-loop graviton amplitude has soft divergences

The divergences occur in both boxes and triangles (with at least one massless leg

For no-triangle hypothesis to work the boxes alone must completely produce the expected soft divergence.

(closely connected to BCFW recursion)


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Soft-Divergences-II

= =0[ ] ][C C

-form one-loop amplitude from boxes-check the soft singularities are correct

-if so we can deduce one-mass and two-mass triangles are absent

-this has been done for 5pt, 6pt and 7pt


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Triple Cuts

[ ]C =0

-only boxes and a three-mass triangle contribute to this cut

-if boxes reproduce C3 exactly (numerically)

-tested for 6pt +7pt (new to NMHV, not IR)


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-assuming no-triangle is correct..

in loop momentum

n+4 powers cancel -8 powers by SUSY, (n-4) by ????

-look for where cancelation occurs


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Large z shift on cuts

-use trick to look at the two-particle cuts

-normally s dLIPS doesn’t probe UV limit

-use analytic continuation to look at UV limit


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Use Spinor Form of Amplitudes (Twistor)

• Consider a massless particle with momenta

• We can realise as

• So we can express

where are two component Weyl spinors


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-probe UV by shifting cut legs (BCF)

-keeps legs onshell, effectively momentum becomes complex

-useful because the behaviour of tree amplitudes under this shift is known

Analytic structure of tree amplitudes under this shift has led

to “on-shell recursion” Britto,Cachazo,Feng


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Look at large z behaviour


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-use behaviour of treesValid for MHV and NMHV

+

+

-

-

x +

-

-

+

s

ss- Consistent with boxes

Bedford Brandhuber Spence Travaglini

Cachazo Svercek, BDIPR, Benincasa Boucher-Veronneau Cachazo


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-supersymmetry

-any gravity amplitude

-much of cancelation already present in gravity theories


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Does no-triangle have implication beyond one-loop?

-cancellation is stronger than expected-cancellation is NOT diagram by diagram (unlike YM) -cancellation is unexplained….In general, for higher loops we expect,

M must satisfy a wide range of factorisation/unitary conditions –are integral functions with sub-triangles disallowed?


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Implications beyond one-loop, e.g.Beyond 2 loop , loop momenta get ``caught’’ within the integral functionsGenerally, the resultant polynomial for maximal supergravity is the square of that for maximal super yang-mills

eg in this case YM :P(li)=(l1+l2)2

SUGRA :P(li)=((l1+l2)2)2

I[ P(li)]

l1 l2

However…..


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on the three particle cut..

For Yang-Mills, we expect the loop to yield a linear pentagon integralFor Gravity, we thus expect a quadratic pentagon

However, a quadratic pentagon would give triangles which are not present in an on-shell amplitude -indication of better behaviour in entire

amplitude


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Three Loops Result

SYM: K3D-18

Sugra: K3D-16

Finite for D=4,5 , Infinite D=6-actual for Sugra

-again N=8 Sugra looks like N=4 SYM

Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07


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Large Shifts on Multiparticle Cuts

L+1 particle cut in L loop amplitude (sample)

-work in progress, Dunbar, Ita, Bjerrum-Bohr, Perkins


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Conclusions/Consequences

- Lots of recent progress in perturbation theory based upon analytic and physical properties.

-the finiteness or otherwise of N=8 Supergravity is still unresolved although all explicit results favour finiteness

-does it mean anything? Possible to quantise gravity with only finite degrees of freedom.

-is N=8 supergravity the only finite field theory containing gravity? ….seems unlikely….N=6/gauged….


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Rockall versus Hawai

uv structure of n=8 supergravity

Documents

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