twistors and perturbative gravity

Twistors and Perturbative Gravity

Emil Bjerrum-Bohr

UK Theory Institute 20/12/05UK Theory Institute 20/12/05

Steve Bidder

Harald Ita

Warren Perkins +Zvi Bern (UCLA) and Kasper Risager (NBI)

Dave Dunbar, Swansea University

D Dunbar UK Inst 05 2/46

Plan• Recently a duality between Yang-Mills and twistor string theory has inspired a variety of new techniques in perturbative Yang-Mills theories. First part of talk will review these • Look at Gravity Amplitudes -which, if any, features apply to gravity

• Application: Loop Amplitudes N=4 Yang –Mills N=8 Supergravity

• Consequences and Conclusions


Duality with String Theory

Witten (2003) proposed a Weak-Weak duality between

• A) Yang-Mills theory ( N=4 )• B) Topological String Theory with twistor target

space

-Since this is a `weak-weak` duality perturbative S-matrix of two theories should be identical order

by order-True for tree level scattering

Rioban, Spradlin,Volovich


Featutures of Duality Topological String Theory with twistor target space CP3

-open string instantons correspond to Yang-Mills states

-theory has conformal symmetry, N=4 SYM

-closed string states correspond to N=4 superconformal gravity

- N < 4 ??

Berkovits+Witten, Berkovits


Is the duality useful?Theory A :Theory A :

hard, hard, interestinginteresting

Theory B: Theory B: easyeasy

Perturbative Perturbative Gauge Theories,Gauge Theories,hard, interestinghard, interesting

TopologicalTopologicalString TheoryString Theory::

harder, uninteresting harder, uninteresting

-duality may be useful -duality may be useful indirectlyindirectly


Twistor Definitions

• Consider a massless particle with momenta

• We can realise as

• So we can express

where are two component Weyl spinors


•This decomposition is not unique but

We can also turn polarisation vector into fermionic objects, ``Spinor Helicity`` formalism Xu, Zhang,Chang 87

-Amplitude now a function of spinor variables


Transform to Twistor Space

Twistor Space is a complex projective (CP3) space

n-point amplitude is defined on (CP3)n

new coordinates-note we make a choice which to transform

Penrose+


Twistor Structure• Conjecture (Witten) : amplitudes have non-zero

support on curves in twistor space • support should be a curve of degree (number of –ve helicities)+(loops) -1

Carrying out the transform is problematic, instead we can test structure by acting with differential operators


We test collinearity and coplanarity by acting with differential operators Fijk and Kijkl

-action of F is obtained using fact that points Zi collinear if

Allows us to test without determining


Collinearity of MHV amplitudes• We organise gluon scattering amplitudes

according to the number of negative helicities• Amplitude with no or one negative helicities

vanish[ for supersymmetric theories to all order; for non-supersymmetric true for tree amplitudes]• Amplitudes with exactly two negative helicities

are refered to as `MHV` amplitudes

Parke-Taylor, Berends-Giele

(amplitudes are color-ordered)


Collinearity of MHV amplitudes• MHV amplitudes only depend upon

• So, for Yang-Mills, FijkAn=0 trivially

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

Penrose transform yields a function after integration .


MHV amplitudes have suppport on line only

Curve of degree 1 (= 0+2-1)


NMHV amplitudes in twistor space

• amplitudes with three –ve helicity known as NMHV amplitudes

• remarkably NMHV amplitudes have coplanar support in

twistor space• prove this not directly but by showing

- time to look at techniques motivated by duality


Techniques:I MHV-vertex construction

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

• Works for photons

-No known derivation from a Lagrangian (but…… Khoze, Mason, Mansfield)

Ozeren+Stirling

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

Wu,Zhu; Su,Wu; Georgiou Khoze

Cachazo Svrcek Witten, Nair

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


+

_

___

_+

+

+ +

+

+

__

_

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

-1

A MHV diagram


eg for NMHV amplitudes

3-1-

k+

2-

k+1+

2(n-3) diagrams

+

Topology determined by number of –ve helicity gluons

- +q


Coplanarity-byproduct of MHV vertices

Two intersecting lines in twistor space define the plane

-NMHV amplitudes is sum of two MHV vertices

Curve is a degenerate curve of degree 2


Techniques:2 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 q(z)


-results in recursive on-shell relation

(three-point amplitudes must be included)

1 2

( cf Berends-Giele off-shell recursive technique )

q

Amplitude has poles Amplitude is poles


MHV vs BCF recursion• Difference MHV asymmetric between helicity sign BCF chooses two special legs For NMHV : MHV expresses as a product of two

MHV : BCF uses (n-1)-pt NMHV • Similarities-• both rely upon analytic structure • both for trees but… Loops: MHV: Bedford, Brandhuber,Spence, Travaglini Recursive: Bern,Dixon Kosower; Bern, Bjerrum-Bohr, Dunbar, Ita, Perkins


Gravity-Strategy1) Try to understand twistor structure2) Develop formalisms

- a priori we might expect Einstein gravity to contain no knowledge of twistor structure since duality contains conformal gravity


…..Perturbative Quantum Gravity…first some review


• Feynman diagram approach to perturbative 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

-involves momenta prefactors

-MHV amplitudes calculated using this

Kawai,Lewellen Tye, 86

Berends,Giele, Kuijf


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


Gravity MHV amplitudes

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

• Consequently

• In fact…..

• Upon transforming Mn has a derivative of function support


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


MHV construction 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 q2(ri) 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+

+- +


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

• Loop amplitudes are sensitive to the entire theory• For loops we must be specific about which theory

we are studying


Tale of two theories, N=4 SYM vs N=8 Supergravity

N=4 SYM is maximally supersymmetric gauge theory (spin · 1 )N=8 Supergravity is maximal theory with gauged supersymmetry (spin · 2 )-both appear in low energy limit of superstring theory

-S-matrix of both theories is constrained by a rich set of symmetries-N=4 key in Weak-Weak duality-in D=4 YM has dimensionless coupling constant wheras gravity has a dimensionful coupling constant

-both theories are extremelly important models: toy or otherwise

Cremmer, Julia, Scherk


General Decomposition of One- loop n-point Amplitude

Vertices involve loop momentumpropagators

p

degree p in l

p=n : Yang-Millsp=2n Gravity


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


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- 4pt…. Green, Schwarz, Brink- MHV,6pt 7pt,gluinos Bern, Dixon, Del Duca Dunbar,

Kosower Britto, Cachazo, Feng; Roiban Spradlin

Volovich Bidder, Perkins, Risager


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

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


Box Coefficients-Twistor Structure

• Box coefficients has coplanar support for NMHV 1-loop

• amplitudes

-true for both N=4 and QCD!!!


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

Bjerrum-Bohr, Dunbar,Ita

Green,Schwarz,Brink

consequences?• One-Loop amplitudes N=8 SUGRA look just like N=4

SYM


Beyond one-loopsTwo-Loop Result obtained by reconstructing amplitude from cuts


Two-Loop SYM/ Supergravity

Bern,Rozowsky,Yan

Bern,Dixon,Dunbar,Perelstein,Rozowsky (BDDPR)

-N=8 amplitudes very close to N=4

IPs,t planar double box integral


Beyond 2-loops: UV pattern (98)

D=11

0 #/

D=10

0(!) #/

D=9 0 #/D=8 #/ #’/

+#”/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)

N=8 Sugra

Based upon 4pt amplitudes


Pattern obtained by cuttingBeyond 2 loop , loop momenta get ``caught’’ within the integral functions

Generally, the resultant polynomial for maximal supergravity of 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

BUT…………..


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? relations to work of Green and Van Hove


• 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. Four point amplitudes very similar

• Is N=8 SUGRA perturbatively finite?????


Conclusions• Perturbation theory is interesting and still contains

many surprises• Recent “discoveries” are interesting and useful• Studying on-shell amplitudes can give information not

obvious in the Lagrangian

• Gravity calculations amenable to many of the new twistor inspired techniques

-both recursion and MHV– vertex formulations exist -perturbative expansion of N=8 seems to be

surprisingly simple. This may have consequences for the UV behaviour

• Consequences for the duality?

twistors and perturbative gravity

Documents

twistor string theory

yang mills n

twistor space support

gravity amplitudes

loop amplitudes n

yangmills statestheory

twistor spacetwistor

twistor spaceamplitudes