twistors and pertubative gravity including work (2005) with z bern, s bidder, e bjerrum-bohr, h ita,...

Twistors and Pertubative Gravity

including work (2005) with Z Bern, S Bidder, E Bjerrum-Bohr,

H Ita, W Perkins, K Risager

From Twistors to Amplitudes From Twistors to Amplitudes 2005 2005

Dave Dunbar QMUL, Nov 05 2/36

Summary

• Review of Perturbative Gravity KLT approach• Recursive approach• MHV vertex approach •Loops • N=8, 1-loop comparison with gravity• beyond one-loop•Conclusions


Perturbative Quantum Gravity


• Feynman diagram approach to quantum gravity is extremely complicated

• Gravity = (Yang-Mills)2

• Feynman diagrams for Yang-Mills = horrible mess

• How do we deal with (horrible mess)2

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

Sannan,86


Kawai-Lewellen-Tye Relations

-pre-twistors one of few useful techniques

-derived from string theory relations

-become complicated with increasing number of legs

-contains unneccessary info

-MHV amplitudes calculated using this

KLT,86

Berends,Giele, Kuijf


Double-Poles

• Naively, products of Yang-Mills amplitudes would contain double poles

• A(1,2,3,4,5)xA(2,1,3,4,5)

• Cancelled by momentum prefactors s34 s12

• Factorisation structure not manifest• Crossing Symmetric although not manifest


Twistor Structure Of Gravity Amplitudes • Look for Twistor inspired formalism• Not obvious such formalism exist (conformal gravity..)

• Can we examine twistor structure by action of differential operators?


Collinearity of MHV amplitudes

• For Yang-Mills FijkAn=0 trivially

• This implies MHV amplitudes have collinear support when transforming to a function in twistor space

• Independence upon implies has a function


Gravity MHV amplitudes

• For Gravity Mn is polynomial in with degree (2n-6), eg

• Consequently

• In fact…..

• Upon transforming M has a derivative of function support


MHV amplitudes have suppport on line only

-For Yang-Mills there is function

-For Gravity it is a derivative of a function


CoplanarityNMHV amplitudes in Yang-Mills have coplanar support

For Gravity we have verified

n=5 by Giombi, Ricci, Robles-Llana Trancanelli

n=6,7,8 Bern, Bjerrum-Bohr,Dunbar


Coplanarity-MHV vertices

Two intersecting lines in twistor space define the plane

-Points on one MHV vertex


Recursion Relations • Return of the analytic S-matrix!• Shift amplitude so it is a complex function of z

Amplitude becomes an analytic function of z, A(z)

Full amplitude can be reconstructed from analytic properties

Britto,Cachazo,Feng (and Witten)

Within the amplitude momenta containing only one of the pair are z-dependant P(z)


Recursion for Gravity

• Gravity, seems to satisfy the conditions to use recursion relations

• Allows (re)calculation of MHV gravity tree amps

• Expression for six-point NMHV tree

Bedford, Brandhuber, Spence, Travaglini

Cachazo,Svrcek

Bedford, Brandhuber, Spence, Travaglini

Cachazo,Svrcek


MHV-vertex construction

• Works for gluon scattering tree amplitudes• Works for (massless) quarks• Works for Higgs and W’s

• Works for photons• Works for gravity……. Bjerrum-Bohr,DCD,Ita,Perkins, Risager

Ozeren+Stirling

Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia

Wu,Zhu; Su,Wu; Georgiou Khoze

Cachazo Svrcek Witten++

• Promotes MHV amplitude to fundamental object by off-shell continuation


+

_

___

_

+

+

+ +

+

+

_

_

_

-three point vertices allowed-number of vertices = (number of -)

-1


-problem for gravity• Need the correct off-shell continuation• Proved to be difficult• Resolution involves continuing the of the negative helicity legs

• The ri are chosen so that a) momentum is conserved b) multi-particle poles P(z) are on-shell

-this fixes them uniquely

Shift is the same as that used by Risager to derive MHV rules using analytic structure


Eg NMHV amplitudes

3-1-

k+

2-

k+1+

+- +


applying momentum conservation gives

-this a combination of three BCF shifts

-demanding P(z)2=0 gives the condition on z

-which fixes z and so determines prescription


• Makes MHV apparent as a analytic shift• Has interpretation as contact terms since

• and the P2 can cancel pole between MHV vertices • Construction ``expands’’ contact terms in a

consistent manner


Loop Amplitudes

• Loop amplitudes perhaps the most interesting aspect of gravity calculations

• UV structure always interesting

• Chance to prove/disprove our prejudices

• Studying Amplitudes may uncover symmetries not obvious in Lagrangian


Supersymmetric DecompositionSupersymmetric decomposition important

for

QCD amplitudes

-this can be inverted


Decomposition of Graviton One-Loop Scattering Amplitude

Known for Four-Point only

-N=8 Green Schwarz & Brink ’ ! 0 limit of string theory, 1985

-N=0 Grisaru & Zak, 1980

-remainder Dunbar & Norridge, 1996

-focus upon N=8 for rest of talk


General Decomposition of One- loop n-point Amplitude

Vertices involve loop momentumpropagators

p

degree p in l

p=n : Yang-Mills

p=2n Gravity


Passarino-Veltman reduction

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

k

l

l-k


•-process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators)

•-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


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 (BDDK)

• Determining rational ci determines amplitude- 4pt…. Green, Schwarz, Brink

- MHV,6pt 7pt,gluinos Bern, Dixon, Del Duca Dunbar, Kosower

Britto, Cachazo, Feng; Roiban Spradlin Volovich

Bidder, Perkins, Risager

• UV finiteness of one-loop amplitudes trivial


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

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


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)

• Beyond 4-point amplitude contains triangles..bubbles

• Beyond 6-point amplitude is not cut-constructible


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

Evidence?true for 4pt

n-point MHV

6pt NMHV

-factorisation suggests this is true for all one-loop amplitudes

Bern,Dixon,Perelstein,Rozowsky

Bern, Bjerrum-Bohr, Dunbar,Ita

Green,Schwarz,Brink


consequences?• One-Loop amplitudes look just like N=4 SYM• UV finiteness obvious• …..as it is from field theory analysis• ..but no so for N<8

Dunbar,Julia,Seminara,Trigiante, 00


Two-Loop SYM/ Supergravity

Bern,Rozowsky,Yan

Bern,Dixon,Dunbar,Perelstein,Rozowsky

-N=8 amplitudes very close to N=4

IP planar double box integral


Beyond 2-loops: UV pattern

D=11

0 #/

D=10

0(!) #/

D=9 0 #/

D=8 #/ #’/2+#”/

D=7 0 #/

D=6 0 0

D=5 0 0 0

D=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)

N=8 Sugra


• Does ``no-triangle hypothesis imply perturbative expansion of N=8 SUGRA more similar to that of N=4 SYM than power counting/ field theory arguments suggest????

• If factorisation is the key then perhaps yes.


Conclusions

• Gravity calculations amenable to many of the new techniques

• Both recursion and MHV– vertex formulations exist

• Perturbative expansion of N=8 seems to be surprisingly simple. This may have consequences

• Consequences for the duality?

twistors and pertubative gravity including work (2005) with z bern, s bidder, e bjerrum-bohr, h ita,...

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